<%BANNER%>

Spatiotemporal Filtering Methodology for Single-Trial Event-Related Potential Component Estimation

University of Florida Institutional Repository
Permanent Link: http://ufdc.ufl.edu/UFE0022519/00001

Material Information

Title: Spatiotemporal Filtering Methodology for Single-Trial Event-Related Potential Component Estimation
Physical Description: 1 online resource (135 p.)
Language: english
Creator: Li, Ruijiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: eeg, estimation, event, potential, single, spatiotemporal, trial
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Event-related potential (ERP) is an important technique for the study of human cognitive function. In analyzing ERP, the fundamental problem is to extract the waveform specifically related to the brain?s response to the stimulus from electroencephalograph (EEG) measurements that also contain the spontaneous EEG, which may be contaminated by artifacts. A major difficulty for this problem is the low (typically negative) signal-to-noise ratio (SNR) in EEG data. The most widely used tool analyzing ERP has been to average EEG measurements over an ensemble of trials. Ensemble averaging is optimal in the least square sense provided that the ERP is a deterministic signal. However, over four decades of research have shown that the nature of ERP is a stochastic process. In particular, the latencies and the amplitudes of the ERP components can have random variation between repetitions of the stimulus. Under these circumstances, estimation of the ERP on a single-trial basis is desirable. Traditional single-trial estimation methods only consider the time course in a single channel of the EEG. With the advent of dense electrode EEG, a number of spatiotemporal filtering methods have been proposed for the single-trial estimation of ERP using multiple channels. In this work, we introduce a new spatiotemporal filtering method for the problem of single-trial ERP component estimation. The method relies on modeling of the ERP component local descriptors (latency and amplitude) and thus is tailored to extract faint signals in EEG. The model allows for both amplitude and latency variability in the actual ERP component. The extracted ERP component is constrained through a spatial filter to have minimal distance (with respect to some metric) in the temporal domain from a template ERP component. The spatial filter may be interpreted as a noise canceler in the spatial domain. Study with simulated data shows the effectiveness of the proposed method to signal to noise ratios down to -10 dB. The method is also tested in real ERP data from cognitive experiments where the ERP are known to change, and corroborate experimentally the expected behavior.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ruijiang Li.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Principe, Jose C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022519:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022519/00001

Material Information

Title: Spatiotemporal Filtering Methodology for Single-Trial Event-Related Potential Component Estimation
Physical Description: 1 online resource (135 p.)
Language: english
Creator: Li, Ruijiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: eeg, estimation, event, potential, single, spatiotemporal, trial
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Event-related potential (ERP) is an important technique for the study of human cognitive function. In analyzing ERP, the fundamental problem is to extract the waveform specifically related to the brain?s response to the stimulus from electroencephalograph (EEG) measurements that also contain the spontaneous EEG, which may be contaminated by artifacts. A major difficulty for this problem is the low (typically negative) signal-to-noise ratio (SNR) in EEG data. The most widely used tool analyzing ERP has been to average EEG measurements over an ensemble of trials. Ensemble averaging is optimal in the least square sense provided that the ERP is a deterministic signal. However, over four decades of research have shown that the nature of ERP is a stochastic process. In particular, the latencies and the amplitudes of the ERP components can have random variation between repetitions of the stimulus. Under these circumstances, estimation of the ERP on a single-trial basis is desirable. Traditional single-trial estimation methods only consider the time course in a single channel of the EEG. With the advent of dense electrode EEG, a number of spatiotemporal filtering methods have been proposed for the single-trial estimation of ERP using multiple channels. In this work, we introduce a new spatiotemporal filtering method for the problem of single-trial ERP component estimation. The method relies on modeling of the ERP component local descriptors (latency and amplitude) and thus is tailored to extract faint signals in EEG. The model allows for both amplitude and latency variability in the actual ERP component. The extracted ERP component is constrained through a spatial filter to have minimal distance (with respect to some metric) in the temporal domain from a template ERP component. The spatial filter may be interpreted as a noise canceler in the spatial domain. Study with simulated data shows the effectiveness of the proposed method to signal to noise ratios down to -10 dB. The method is also tested in real ERP data from cognitive experiments where the ERP are known to change, and corroborate experimentally the expected behavior.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ruijiang Li.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Principe, Jose C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022519:00001


This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101114_AAAAGM INGEST_TIME 2010-11-14T18:22:15Z PACKAGE UFE0022519_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 17760 DFID F20101114_AACHLK ORIGIN DEPOSITOR PATH li_r_Page_055.QC.jpg GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
9f43b145fc07665655d52408deed530b
SHA-1
a70fc70c9627ba8589f54fd01114804ecd46e745
27087 F20101114_AACHKV li_r_Page_032.QC.jpg
5fdffaf8380184ce66854283da1bd54e
4c90725a078967fd50a61a72dd4eae19e9059253
42890 F20101114_AACGHT li_r_Page_047.pro
c44c130aadad9c9298b185f6e80fed08
b2367e8dadf346cc243ef0677bd98f5832b61add
9735 F20101114_AACGII li_r_Page_077.QC.jpg
4278b6abef5bb4eb7daade67845b8731
4766633dfab6e8ef5d347260022be2d0e72d31f5
5045 F20101114_AACHLL li_r_Page_055thm.jpg
ecaee1c33c26bdb1e43d54dd846c5618
5b3714a30a29aae6a191c41a33432a4217481ac3
22500 F20101114_AACHKW li_r_Page_033.QC.jpg
348c8fd724236dd6c0cc8a9c1224de10
8e7e8a0e35aa962b6bb589838e2d462d0a203474
75976 F20101114_AACGHU li_r_Page_048.jp2
9f422903c22e525f69f52ab06e4d8fd6
41e6fcdcb871d25c1b40fcceec5e8bc3860a1ac7
50916 F20101114_AACGIJ li_r_Page_023.pro
fcfc1f4b4dc6ab8588630a0ad963fb20
100fd4fa9a81d9a7243d950531d790fdcf04a08b
11732 F20101114_AACHMA li_r_Page_084.QC.jpg
5933a6a1afaef1e96c699f8f9d29ada5
222ed3b3d904982d7a6b943d2042b97f0631e544
13066 F20101114_AACHLM li_r_Page_056.QC.jpg
d06c0f3951f92e89cf96142a61735756
c99092efb2a094112017939a7b7412b0ff93fd0f
7015 F20101114_AACHKX li_r_Page_035thm.jpg
b0ec897cc9dfabedcdabc205db0689a1
13011f01981638ad117db9e27050d629a79a7237
100780 F20101114_AACGHV li_r_Page_021.jp2
2d0ef49936882450174432ceea7cdbfa
879e71e08004e644663b8acbd3d8118de0b6c6a1
1053954 F20101114_AACGIK li_r_Page_130.tif
e58c47431459766bbdd7835133c3b2bb
81d28189e37e3ed8e19ab5f598c9c6cd8f5c5a47
3443 F20101114_AACHMB li_r_Page_087thm.jpg
4a69d6ac1def201f6bda9e6d98e8860e
de8ccf1d5d6ae35aa2b7cf011081a9a353a8d22c
26121 F20101114_AACHKY li_r_Page_036.QC.jpg
1b45102061cb90bd0dd188cf0514a5ee
fb38920fec1565a2456b5e3eb4825e30d8dff08b
6949 F20101114_AACGHW li_r_Page_133thm.jpg
dd02363d5f7b5de8604357afc03b18f8
383b042e4d747c600b4d3d6c66742bbc4458a680
10986 F20101114_AACHMC li_r_Page_088.QC.jpg
1c9511601e8164092905032c1b69402f
dd391868349c7e2e8503b8510dea80d2a4cf48f9
6822 F20101114_AACHLN li_r_Page_060thm.jpg
0897eab59a0797d43bbeabf237d89ba2
fa8918bf32bc0626f601aff4b70d56bb9301e4ee
6947 F20101114_AACHKZ li_r_Page_037thm.jpg
aeec25a94823c36cb3911f5ba407c7b4
bec37ff3192007082cc0f46fdd8ec6eaaece21de
942 F20101114_AACGHX li_r_Page_011.txt
39a8514559f9344797a3b3f6f268fe57
d22c01306303cd92ecef61be183ce8338a491776
2159 F20101114_AACGJA li_r_Page_092.txt
8aa84d4822e96559ffd7ee5ef5a09720
c168eb0302744fd5e8e11d94feaac437501fc680
76172 F20101114_AACGIL li_r_Page_013.jpg
2036e535ff21f6b2c48869f69d8d5711
5e92205b5202ac834803f88b176ce40575ddd8b7
3119 F20101114_AACHMD li_r_Page_088thm.jpg
279f1cdaa4d975864803264884601c65
d0a976edd0185e51d383d43135d8e9d67a828b07
5214 F20101114_AACHLO li_r_Page_061thm.jpg
cf002f80443e457b92ffd8a40defe311
ee34823ef2e8e913e51998169a24155416950091
21444 F20101114_AACGHY li_r_Page_019.QC.jpg
c9e0f4783d7d66be534cbbeeffd5cbd0
ed533d16eeffd6fa6809218ebf7608f0422a8213
3811 F20101114_AACGJB li_r_Page_005.txt
5f2c85abe42e355ea0fc80a19a577448
270db42e87a7ffb50fd4286891747d7d0c012adc
873 F20101114_AACGIM li_r_Page_104.txt
59c204446ad386b09a6dd7b6ba75653c
3dc37415a891c6b3a2d74ffec4ca5c681ea9504e
7059 F20101114_AACHME li_r_Page_092thm.jpg
4efec9d79c457640bab1840cc87e2f56
1f2e6fb7fabba3a466a409ea99ec797aeb5297d2
21527 F20101114_AACHLP li_r_Page_066.QC.jpg
cf97acda720f57c341be3340724fd097
6e1a82ca3a067a11ba42c5b1b2fa60b63d8a2f7a
16276 F20101114_AACGHZ li_r_Page_074.jpg
df7a5a08de7b6a79f1d83f6e0cbd73fa
6c41871d285b8065ebacbea2cbab46688bdd8bbd
3712 F20101114_AACGJC li_r_Page_102thm.jpg
eead9c6a49a6117f021fa3d6cd135fde
2b45f9461afd508ca03f80015166ed27efbf5f8e
6959 F20101114_AACGIN li_r_Page_123thm.jpg
18713a93671d14821e385beff3a12aa5
96ac33d80eaafe108eea26068bf895bd31810a78
24993 F20101114_AACHMF li_r_Page_093.QC.jpg
6255a4facc2aedbc18dec419e51c7567
3ea04849b838989122f04e937f5d64512196d666
6939 F20101114_AACHLQ li_r_Page_068thm.jpg
d901b91d36c55d4188da7ea1efd34096
534f956eb0c5d399f32069d09ce26f813e2f353a
134034 F20101114_AACGJD li_r_Page_129.jp2
83e3bf1e7b9fc84e11fc74123a0fcb58
c6a415833799c2d19da71e3becf1833bdb4c7b7e
17528 F20101114_AACGIO li_r_Page_043.QC.jpg
14b522861170d2a1ed1460005b826255
12f34eb7be0ae27f81d717a7a85ee76ac6b7f6e6
6985 F20101114_AACHMG li_r_Page_095thm.jpg
94959e18d044cf0329bf653538e568fd
9d69f794ed293686d0aad4026e1397f0806e4367
19689 F20101114_AACHLR li_r_Page_072.QC.jpg
f799dbbe3f5287bd0ac421b529076233
149e1a0c09ee5379eefe49d754d00acf786a91da
5486 F20101114_AACGJE li_r_Page_081.pro
f6b405683bbdbcfb75686ba5dc094a12
0cad279f0ce1c2a6cabee0461c33e303b60f68db
4024 F20101114_AACGIP li_r_Page_054thm.jpg
21886fcaf25dc5057cf43a83d76c742d
cbf8f5712c8e716da31e929a895e98de3e5be800
25235 F20101114_AACHMH li_r_Page_096.QC.jpg
1d71774b54817297e3c9a4a08599e73f
5e941c69d842345adf848ede3fa1ca3fa7468e24
5582 F20101114_AACHLS li_r_Page_074.QC.jpg
1f48fdd4ee929edfa9a07d6da541f84e
7e398295bb74dbfed2023d3fb79065000a189fe0
85343 F20101114_AACGJF li_r_Page_132.jpg
e297deca576929b6b22afb0589516139
bd295384fb0103e9ddd7a3056b0d8ab556a6137a
34085 F20101114_AACGIQ li_r_Page_089.jpg
c6f2975392ca4580414e61a236466087
3571fdf32e5c7b5c2750bfd2d0ca5c434ae0900e
6735 F20101114_AACHMI li_r_Page_098thm.jpg
1f478116dcb8002fe7503124694c1b96
70808c0c5762504f7529a4319dd6e329414f6fa6
17016 F20101114_AACHLT li_r_Page_075.QC.jpg
c3fb24ae9b2f9196b0db2b3121d22a4c
039a291879716a71d98f649abe3a994ea271a597
4959 F20101114_AACGJG li_r_Page_075thm.jpg
e77cfbd74f7c92766c4eebfae88ef020
3280997097624c5e56e2a89780574dedc5feea88
23485 F20101114_AACGIR li_r_Page_099.QC.jpg
a48fa9808a54e8ddea297673d455efa1
249c90712b30fee3251b88a6cb561e2481f5552b
6393 F20101114_AACHMJ li_r_Page_099thm.jpg
5eb6a8cd1755e72580e9453f583e11c9
772d5d720ed152f942b927a93b6f3e03a3493555
4153 F20101114_AACHLU li_r_Page_076thm.jpg
d4cf14130d7727e44261cadec6ece257
ae8a0189773d533b5b7a6657322dea7b18c6143a
23762 F20101114_AACGJH li_r_Page_006.QC.jpg
fa72bf47688b16f51102e6b3379a0cec
335f3a96d6e7a879786c0aaa7de2e3b9e823fd46
109148 F20101114_AACGIS li_r_Page_090.jp2
0aaddde57593b4baa55f0aeff348b970
2bb714f6b69478fbd419b99f2dab152528090e9d
24747 F20101114_AACHMK li_r_Page_100.QC.jpg
252d5bbbb75dc62e9c429a409727a377
f2f3b4b32b4ff0bbfb8b6773a120d414e128392f
2404 F20101114_AACHLV li_r_Page_078thm.jpg
a63582d1a451de84d1a1a4e6688e4f67
d94bd7e265d9da34d11d5c1302714bdc0d7ecfa4
61282 F20101114_AACGJI li_r_Page_118.jp2
303235d178d9a83608571ea886dcc50a
987a79bec4c25bca15d8572d20cfc3d9f23229f9
74306 F20101114_AACGIT li_r_Page_055.jp2
6458f1d7ca9505988c544a1cb5c4951a
a218709fd304251cab1349e850684df35ac5cf42
10815 F20101114_AACHML li_r_Page_104.QC.jpg
d168f3b4c4059d1d4bca8fe766c0a1dc
b066e3f6a95b4caed6048015954f714d4a27a9be
10701 F20101114_AACHLW li_r_Page_079.QC.jpg
d7dd188c68da53d917684117c13a0f53
b45478ca86b53af848bc6bdcbf8e40050713a0e2
52594 F20101114_AACGJJ li_r_Page_061.jpg
d209ac6310fa404e619b4f462080dca5
b444f88ad4cfa3008dccbce453a6eb7ba4807819
1110 F20101114_AACGIU li_r_Page_089.txt
3bf2e9c2d2e60ed49a305d124835ded6
f804a486c9cb046e518df9b3953a72991d52237b
16508 F20101114_AACHNA li_r_Page_134.QC.jpg
b607c938955a3fcf12bce0c7795d29bb
482cdaf34f175f7f5a18e76054d47e043de46412
11255 F20101114_AACHMM li_r_Page_106.QC.jpg
8fe0719ebfc6b0c6f0718c81a6c9b406
5e9f37683e283032268be949264389ed6ab16906
6639 F20101114_AACHLX li_r_Page_080.QC.jpg
8c351fd888895b2a5d00ef9cebfb7017
73fd78c5658d2513b51fe4e4560033697b282179
117098 F20101114_AACGJK li_r_Page_060.jp2
1836dfe0402dd533aabfc68aff0c2d25
929684f66b0696a3d77a801ebf3ab77b3ccc00b7
2516 F20101114_AACGIV li_r_Page_126.txt
d3e5bc39645216427180207096332483
6f8b805c0ed744eabff02a1f3a08ed8433f1c25b
24963 F20101114_AACHMN li_r_Page_112.QC.jpg
c239fb9dfbdb2b81865e25e318e0d3c5
4cf3cc95e046710f531b9dd5327d4726c1cf680b
2260 F20101114_AACHLY li_r_Page_080thm.jpg
6081ae9eb7b3b1a005b2285e91007e68
a313da70a9c6c0d0dd8715fc779c96aa537ee137
2278 F20101114_AACGJL li_r_Page_001thm.jpg
e6031dd44bf96bbf9f53e2fe07ecfda9
1ce09e4bbd01531c7fec3428141b21430731d049
19386 F20101114_AACGIW li_r_Page_104.pro
e565cba0bbe5320d35ef043f7b26a9fc
cce45fcd6111c93a9643c4c298de590be611a1c0
3490 F20101114_AACHLZ li_r_Page_083thm.jpg
9df2279bb7626484fbd5a7db02c07942
8bc83b80d267c22e3c9bb889f4e2f69c944744cb
1051920 F20101114_AACGKA li_r_Page_032.jp2
680cb4cbc7166bf5e44c91bfba3c6ec1
a98cc0d1a58dd58da408db1c3b3f3d5d61ab2b96
1947 F20101114_AACGIX li_r_Page_074thm.jpg
a28df0ef1cad9c26c8e403bae6940daf
9ace243a93cbef65b286dc6f8cbc32037c39dd70
4439 F20101114_AACHMO li_r_Page_120thm.jpg
5756c5324a4b435c9aff3bd1ddd273a7
921f7853aecc793265b5fadc0c63e924969684cb
25108 F20101114_AACGKB li_r_Page_020.QC.jpg
11e68d096105e18eacfd1ded749297fb
e5da51f2a250cda024b3d34a107aa569bca2e22a
36767 F20101114_AACGJM li_r_Page_073.pro
8de2f2e4c3d7ae70c12e9de162d9b308
4ff04009922e8371db6bf2e9705fd51ea37bb62f
F20101114_AACGIY li_r_Page_119.tif
83b60d134f0de4146e507508358016e1
bacab167163f43c70dd949188956ef77651d0a58
2177 F20101114_AACHMP li_r_Page_121thm.jpg
1550f3a9e90d0586a4f77de737c70c3a
b1f13db9cbe6981ab666cb77db07114e773c37a1
32284 F20101114_AACGKC li_r_Page_109.jpg
1a8ea11809e1f0d46e8b37fa8ec0dd48
0f65ded30ad76e2fb7f9169458261148d21aaaab
1982 F20101114_AACGJN li_r_Page_066.txt
06555ad792786c6fdbf9f7330dfb11c2
8e628c461048cb4a7f5edcdfd59da247e0a2f04f
2141 F20101114_AACGIZ li_r_Page_053.txt
403b5abd282e95e9ec7689651ef355cc
383eb67f9fd9cde6d80fd4ae0841f6ef06ec4d48
23892 F20101114_AACHMQ li_r_Page_123.QC.jpg
4295123a3fec66ed7a2b0517697242af
8bb240a760b9b713db2b5d994656cc9600ce7884
115406 F20101114_AACGKD li_r_Page_013.jp2
a32afa2da7e03886fc606e5699f1cd29
e8cd0c44a2f7357a100e4a9d9c4f4126db8f38a7
132396 F20101114_AACGJO li_r_Page_124.jp2
2e8ec2f7d6d4b116304d09f6e9ce5dc3
1e780822bcf45a35e7a570a764ab86c11d336811
6762 F20101114_AACHMR li_r_Page_124thm.jpg
f9989d4804038089bd6fd882f92e4a1c
2c232bbefbb1504726546cc57883feabe803f7ad
F20101114_AACGKE li_r_Page_094.tif
5416d4a0e9704d6277aaefe87c867c6c
d15dd930224740314a0d38c3436c40d49879666c
144514 F20101114_AACGJP li_r_Page_130.jp2
c44b59d7846697b933a5145ec0b92278
f2cd2078c3a6fb276ad6869a3131f3b4bb9a03c9
25907 F20101114_AACHMS li_r_Page_126.QC.jpg
3119ec5d005b1d06feac4f947868d153
e549a8e8952efede33b385b93367b7fe6c54d656
562 F20101114_AACGKF li_r_Page_105.txt
667ffaf09b4a09bf479c3170061522c6
5363fbe66334d833c5c7fcca56f320e981246b46
25271604 F20101114_AACGJQ li_r_Page_058.tif
b8ed077fad1c9f584e730d75299157a9
c8de7fdaed50785b681c16000a43f4b9b2d255c4
7024 F20101114_AACHMT li_r_Page_126thm.jpg
006d7a12cf35c77b63d472ed3050b944
848041d62f7979b0f8b647a52e45bd49c7f6a581
70039 F20101114_AACGKG li_r_Page_065.jpg
d1f56ef45abf10a5ad04d11ae46624f3
d283fa9ba01acec7749f71ae04d9fdd3c2f525e6
34608 F20101114_AACGJR li_r_Page_048.pro
4498343e8ec30765bb6184b3407d82d5
030cf1b1cdc28fbd955a28429f8cdb8c6f070b85
22388 F20101114_AACHMU li_r_Page_127.QC.jpg
e0be1b89fef8077f07096ba7848b98b4
0877a2186fdac4543cb8ddbe0960378e34093e28
52781 F20101114_AACGKH li_r_Page_112.pro
620352b4d6c1125d87794088246e5be7
dc45ef41db0268fe1928c71c4391d7fa30bc5281
61919 F20101114_AACGJS li_r_Page_054.jp2
5602e99860235465701a49c80fba09d0
dbf6c0c0d16224d7aef0cd93bf511c9109ee6f67
26042 F20101114_AACHMV li_r_Page_128.QC.jpg
2f0ce4b7fa244ccb6b13c285b87175e7
ba627bf1fa930ba3a701d856aa6a603fc6efbe34
31437 F20101114_AACGKI li_r_Page_030.pro
01cdc555ba860a541837a10b056141eb
b210b70bf59f037e4457c2389a4bd7701e55edba
24848 F20101114_AACGJT li_r_Page_095.QC.jpg
01beaddda943af384a50be708206dedc
a3ca5bbe33457b9933cde3f8b1553c326aeee3eb
7155 F20101114_AACHMW li_r_Page_128thm.jpg
b7d2aa79e37b6937567dfab7b5ac3609
8c7b21e2925f81dd2a188f215c220d077a211bb3
24619 F20101114_AACGKJ li_r_Page_038.QC.jpg
8abe6bf721100dd9b1a96751db70d8d0
8e0366a1c81e7ace1827fe1112116a1dce9aaf33
51195 F20101114_AACGJU li_r_Page_048.jpg
f81c5de1971368855f18dfc3867223bd
4f1aa3562f045fbdb8776b56927c3e49befe0da5
7105 F20101114_AACHMX li_r_Page_129thm.jpg
93207d4047c22c4a9d7c6bfe0014ec3c
4efed497b7e2a7d080f37211666aa73b483a86c0
91542 F20101114_AACGKK li_r_Page_069.jp2
72b474a8ca424d10be1d79cd65ec50a7
9fa84003bd6b20f3fec630e5a17999874c28835b
44165 F20101114_AACGJV li_r_Page_064.pro
0ea40c53d8f1db6fade50ae405823597
92b5fdafe02dee4e651daf4ffa06eefb45ccfc53
27850 F20101114_AACHMY li_r_Page_130.QC.jpg
02f994f4b8bab7cdc5c870d49b0c2bd4
dca7143274fc53cf80e71a13809155236f344178
106953 F20101114_AACGKL li_r_Page_099.jp2
12308d8de7f640307d76758f9d7c0058
9b446cd043967287618d0e220a6c861bccab7ef8
6713 F20101114_AACGJW li_r_Page_033thm.jpg
f4225a866f792fe12da42b8361fdbc9f
301285f30e9e5e528c90ad1ced01608b69eab980
24679 F20101114_AACHMZ li_r_Page_133.QC.jpg
28f5bb856d687b7d9bf25c43c06ac768
7d7eaf761bb0ce8d4eae09c1f29ec13bd3d56a6a
94840 F20101114_AACGKM li_r_Page_006.pro
82df5bda467b4c43ce27182e7988498a
6be9488b575c548a21d7bf829d4c6985f6b97e1b
7212 F20101114_AACGJX li_r_Page_097thm.jpg
1805e8d9652498d02291fb62308a8f65
a02627d2840cda6a80f984af7da55204d62dc301
2332 F20101114_AACGLA li_r_Page_131.txt
fc952fe7f9efce4c59b1bb5699da4455
b6527dc29d3c47189337791a7a87a6e0cc1bf490
27752 F20101114_AACGJY li_r_Page_004.pro
20121cb40864b8fc043e15e0ca6be980
04d59ac1ea56ec4dff6be64ecf114e26178b827f
51282 F20101114_AACGLB li_r_Page_056.jp2
c0ad4e296ec0083936a415ca76e66dbb
1d42c4ebffc250f4d65c8abb89cf9beba3c3ecb0
F20101114_AACGKN li_r_Page_055.tif
4619ccd675a425121188853a27c51af1
abe946117a85c9a59940372706531baec4ef44e8
81959 F20101114_AACGJZ li_r_Page_123.jpg
490f483b3da0f60c84468d3759d429bd
9d583ee57d0a602edd18c120200555dd16fc1902
14166 F20101114_AACGLC li_r_Page_117.pro
9e812576c5dcba27e032d9cb1d752b86
7f10e213d8072927ba0a380fc74c21b1a02f7e0a
2225 F20101114_AACGLD li_r_Page_095.txt
d72e45763d3ca426c63f657b474005ae
8efd8d3fb9badd5121d5e9af98cd0840ddd0a08c
87204 F20101114_AACGKO li_r_Page_062.jp2
2145c42ea8456fded862716ac2f8ce30
4f4a63b28354a173940a5c781c99d66cb6e69724
53028 F20101114_AACGLE li_r_Page_101.jp2
edb7e23fb2f1c9c8e5b03e78242fb2c1
f04225d8fb90675eea6228c687c5313e4d969303
F20101114_AACGKP li_r_Page_031.tif
1069ba7b3d9a59c171f8a1ed0a07b896
494090fcd6befedc201adfc7c7338f4608003bf3
19226 F20101114_AACGLF li_r_Page_081.jpg
6cccfdf7a4aabe8e4c5a4ddf12c1ef77
0f943abc7c63d7b1cce806f8c8ccbf4ff841536f
F20101114_AACGKQ li_r_Page_013.tif
9321193f0f7207d2fb196118084c84de
41dcb561d7a14b275a048d045bfac53015bdf853
58678 F20101114_AACGLG li_r_Page_133.pro
68aa7dc063a74ba1670f74d5afa425ab
560ff843989c274ebbe590919f595dd5378e3c28
42738 F20101114_AACGKR li_r_Page_008.pro
9dddfb570409e42c7bb21444e8b01cab
9e402eed83d86a2f9c68ee489fb34aec08ff7204
1914 F20101114_AACGLH li_r_Page_029.txt
9b8f1cf304b801858b2c8b01da9bac35
5eb26fef665b5331c00e45863e4c2c4f859cdd84
F20101114_AACGKS li_r_Page_008.tif
46705e859d405f0ee57408669ebd02c1
ac46285b7d4d2a8209e90edeb7c82c6b48996949
1183 F20101114_AACGLI li_r_Page_082.txt
45b069bc4e0e68f5d3eb764fbe6f9d4c
ad79bbf15c5c3caa7a7b5c5141ecb413389d346e
2029 F20101114_AACGKT li_r_Page_127.txt
e875cb8353813c2f9e1b908fb99d7293
9db1282314b78cb079cc87d3f52dd05baeb0d9f3
1152 F20101114_AACGLJ li_r_Page_004.txt
bb9d260ea8e020b2b8ac747fb789054d
048eb974a61368c89d0ce827fef5d745ab362316
54869 F20101114_AACGKU li_r_Page_092.pro
40e3566deb5eb336733d5d8dcc8a2a90
016fb4c6a490367a73687a1f6727d631d9304317
2053 F20101114_AACGLK li_r_Page_022.txt
166e080deda920a5bfd3cb17e9906571
a8c6a295ade2bd8fd88400afdfb8ac84fee6b409
66376 F20101114_AACGKV li_r_Page_012.jpg
d6fe397b87aa508a35e7a723ae1a0d1d
a537ebf4a6831b12174cd01e0b2dba3b678736f2
6893 F20101114_AACGLL li_r_Page_034thm.jpg
fcf0b825fbe2b400b9338c60aca3556e
09055c522ba743ae3899bbe93a3e9042114fa5f2
115959 F20101114_AACGKW li_r_Page_096.jp2
79476e9f062c60e4a367db10f320e10f
c43700842c1349efe5bfce89a851ff28fa998746
1808 F20101114_AACGMA li_r_Page_072.txt
527c34dc61d86d07bd3960a0620854ff
f30ef10146e9685bc6e37f443654cd16be96b5eb
F20101114_AACGLM li_r_Page_032.tif
39480de21f43aa4c53ad68ffe95c6b8e
c2afffe834daa5e363b0bf4baa2eb71eb40b82a3
50620 F20101114_AACGKX li_r_Page_022.pro
09922518589a17a615cfb1aa89349f5c
d68b9a6b24500cdaefe8437a24d6458dae54139d
15329 F20101114_AACGMB li_r_Page_078.pro
41e8891831de3a5ed6d9f7a970f47533
a7281765f18f297c5848a59280c24f762aef22a0
77427 F20101114_AACGLN li_r_Page_053.jpg
3afd6b099238fadf670c5b817f4b0729
ca20ecbc37dc88db498be22ecad481647507b426
F20101114_AACGKY li_r_Page_060.tif
a50a13c6fdcbe76128906569bddb9c78
c8f6b451444dc3f2d8614f57c5826a89e13fb4eb
76263 F20101114_AACGMC li_r_Page_093.jpg
278e9e2f13549ee6cb16310ae1624b8b
efd0e863bd2c8cc3cdb9fc1358504a316415b32b
14565 F20101114_AACGKZ li_r_Page_054.QC.jpg
748b2d0ff522bc90e88d5eea9516ba5b
274ac86909e73c464c167d229f8704d2e7db4e47
622396 F20101114_AACGMD li_r_Page_087.jp2
1eb7bf588a0d29882be68ae7cfbefaad
146533393b45ded1427a18773cb3aeaf79ae99c4
6683 F20101114_AACGLO li_r_Page_121.QC.jpg
2b684c44e71ff88399e8826352bc8aec
23e7006910e69f75c928acecdf334a5f24c10312
F20101114_AACGME li_r_Page_120.tif
5d57063dfc585b18870aa1bb458d7cd1
d0db631680b1cdcad5f30e017d149cb4199dd18b
F20101114_AACGLP li_r_Page_101.tif
bf0668daecf2513c58e2d45b1d262232
5fa265891a6be9fbe64a355e7f8f7f36c29d32a3
23683 F20101114_AACGMF li_r_Page_016.QC.jpg
0d8acfcf4ae73c1c77eff0c9173e8955
b47a22027fb2aab456fc6aa788f5e30cd69fbfec
5887 F20101114_AACGLQ li_r_Page_047thm.jpg
befbcd89b05196e40b2bff017ee0c741
158476ef608d32a0c3420cc18811c8b5d19c0e08
5016 F20101114_AACGMG li_r_Page_048thm.jpg
d7a46ecdab23216bd795771617a5e2d7
3491864e253e2be2da06f4ae4fe6ef6984ebc38d
2057 F20101114_AACGLR li_r_Page_016.txt
071bf2986041d9324a352ca9497f7127
0b95d6b6c728b27175f6b1028e7b61cc828c546b
96738 F20101114_AACGMH li_r_Page_026.jp2
9e4c78650794bcbf614993928cf17675
e8d92ae74a00618060c038f6a9a1f0007d87ed66
F20101114_AACGLS li_r_Page_050.tif
24caf85228d559d9a76ab6f5eb201eb8
d6e9b6b4fc0b090e9f119f3640aa5dc1ebfa2c73
6027 F20101114_AACGMI li_r_Page_005thm.jpg
ed86b20cd907b58b5dcd3281822794e2
a45f74d70ba8e7667cbe07938290729b8b5a23d0
117769 F20101114_AACGLT li_r_Page_092.jp2
ef9f80e18c9aebceb8a5025c02ceb3f5
62dedfc5556e3ad1898a0591f6f5b477e9c53c2a
108691 F20101114_AACGMJ li_r_Page_049.jp2
bfef4757ce50e489782845dfd4c1aea9
608f5c415325f0c868ec69bc478b1e6cd848c9d0
499 F20101114_AACGLU li_r_Page_058.txt
0a46dc950f592802f52d6073a48219ca
1442f94717ef3a8c91c21362290522563bee100b
26285 F20101114_AACGMK li_r_Page_078.jpg
3f67d214625cd84fb63f37d499d07641
ec70c9330e96b2c8049698b0f6165c8208db7d1d
20007 F20101114_AACGLV li_r_Page_085.pro
ccac35584f871b42a93e7e9cc33584db
d752ff7abd02edca65f4a7bc23bff9474e4d848c
59855 F20101114_AACGML li_r_Page_069.jpg
75efd7ae703f1297ecfd24ed2d877bb9
d47f8d65f9ac62ef0423015b470e21f3bec96bab
9828 F20101114_AACGLW li_r_Page_039.QC.jpg
f9391f5d36c16bf7408d0341015fb939
7e6a9f037574ada796fa48b0f42120366ae5c57a
26304 F20101114_AACGMM li_r_Page_056.pro
e77d8f6c6e86d2aca7abdeb68e116844
dc1cd2cecf39c3cd6b6edf2f87c07da05671b26f
2491 F20101114_AACGLX li_r_Page_129.txt
a70ec7b198fbb5bd82cb74bfdb309f64
27794ef85c4d73680c38920219bb564ec9aacee3
2806 F20101114_AACGNA li_r_Page_009.txt
29ae4cf7f4c98ffed6b7465d2ba5b051
30ec7ec9263dbd10fcf5712044b96d36a8be01d9
15254 F20101114_AACGMN li_r_Page_030.QC.jpg
4805981d30619663a059cbb3788357d5
c88f97e31921cd64e2ba127ac82094f70bb9eec8
F20101114_AACGLY li_r_Page_034.tif
7924997fa960c77e4d4a427aa293fb7e
7269eeac23a455d37882f32ccd5da87b8503ea33
18708 F20101114_AACGNB li_r_Page_041.QC.jpg
5df7605467d9500d6683bc5f4b7161e5
153566433217b58f143da6dea1aa2900728b8fe8
F20101114_AACGMO li_r_Page_083.tif
9fc9710c923b5c588fce8dba9795b940
aea59fc93a961ea934bb21d28eb771e7fb1f6995
51525 F20101114_AACGLZ li_r_Page_015.pro
dc05a2efaa5a87a12212eab948b39dcd
6136ac4a179b74ed8df362aed5fb3ce4061bf64f
1951 F20101114_AACGNC li_r_Page_099.txt
1300dab7d3e2dba8bb423adc33471047
5951ccbd48d256e4eeb8fc10cc97c86e285a1cdf
4471 F20101114_AACGND li_r_Page_118thm.jpg
9807afe585bf6fe0f75717dfa8e0ed09
fc35713ac4d116db60bd5bc8ec2f4ef5729d0ce2
73942 F20101114_AACGMP li_r_Page_028.jpg
4766b0146497fee6c9d6ff50202c9987
f53c7931641796b90c9ce9e280ebbf91a581e79b
109966 F20101114_AACGNE li_r_Page_016.jp2
6d64e20a716f994b840a8186146a6b74
c8ebd73eb914b0d39d2cdd8709e52af6d651fb29
F20101114_AACGMQ li_r_Page_053.tif
d14c4078868af8c993055cd73d833ecc
1d23efc50ad81832d744a96d13ba5143472d98c0
F20101114_AACGNF li_r_Page_067.tif
b4ed4400e1d4f42eafcbac898bb18f10
e812bac6b28bb2e3e450bde9ee493f7e9872f165
479 F20101114_AACGMR li_r_Page_001.txt
dc2e778c78e54aed311f34c6e85e0651
0a0f86423e5196e56b4f9cbd85e2f984af1f1d94
19612 F20101114_AACGNG li_r_Page_062.QC.jpg
5b257b71e7a2855e2aee47d0a2aef0a7
9e63238dc52416a98e813027457a8aecb8946ee1
23194 F20101114_AACGMS li_r_Page_017.QC.jpg
71746eee71293acbb7e7701210eafdc9
218db73be2265dd0c1d4982bf9a7e4c77a577503
27410 F20101114_AACGNH li_r_Page_054.pro
4cb5c6f0d5b82ae63fc5f458ff52558b
37d90b4129d19994f6d73e74b6682b1eb51d15e4
103250 F20101114_AACGMT li_r_Page_067.jp2
6641cbecf93f3d97ae956ffd0a923fd8
2f9db61e45a1e0647062309f33ab3c82ecdfdf02
115529 F20101114_AACGNI li_r_Page_093.jp2
76b6f89fedf7600abf8674cbd1d01b74
99ae3af2b7ccf9ebadf01d9ab6d298752047bb5b
74616 F20101114_AACGMU li_r_Page_112.jpg
960ac621f10f3efa508195a2cf0cb2c2
45b516e8734846626678f20f01a6190e81052782
49922 F20101114_AACGNJ li_r_Page_014.pro
24dd4ac5f9e25ef412daf87038b7b240
32ef75a38eb90171082b3c112bfcaef102bde796
6205 F20101114_AACGMV li_r_Page_064thm.jpg
080a887087a337d45de3738147cd3762
496188702779a3b6ea4b0633ddf674e049dc8230
64811 F20101114_AACGNK li_r_Page_125.pro
ed28500a896f70f92b76b89727a4bfa5
1338dacb484e88ed690fcd12dd125c3170a21ea0
3145 F20101114_AACGMW li_r_Page_117thm.jpg
eda423ffd8e1207e397918ecec54520a
b7184575426339bb5aa4202dea5aff49fb0048d7
1180 F20101114_AACGNL li_r_Page_108.txt
23adc39baf93ccd3418d532557dd10b8
1979ca6f5f3901a3237db625d5a72a0edd56d4d7
78785 F20101114_AACGMX li_r_Page_036.jpg
d0393e0d60827b87e6c057fda822b7e4
fe0d42f16fa0b3e60cecda578707f303ba18255a
25338 F20101114_AACGOA li_r_Page_037.QC.jpg
2c74c47450fecce1f1347a6fd3403c0a
c437e7d127de9b9261aa176a45d466102bf7017d
54997 F20101114_AACGNM li_r_Page_111.pro
4f2a9c1349dada73bed0f1b0dd367748
09c628acaae5fd94fe1b3f6268866ecc6c8ed43e
51114 F20101114_AACGMY li_r_Page_028.pro
57b15a9505b3b208fde41b6360ba1966
c6c3ac56a649f5700407946cd58ccaa5c8a45a07
6523 F20101114_AACGOB li_r_Page_046thm.jpg
938eb8834f4d57fe8a323fa094b58d4c
724cf1cd76afd85621b242987ab118e0e31d2761
6432 F20101114_AACGNN li_r_Page_067thm.jpg
37d131111d88653b4b65f51ee80e2262
858504987400f7e67febaacabcb6a86648f2351a
81571 F20101114_AACGMZ li_r_Page_043.jp2
17532812374e94dba9b5fb7c45877773
16207501a23c940a0938ee462815f4584231d920
60410 F20101114_AACGOC li_r_Page_062.jpg
8becb2fe15b5d73b7f17a583f63c9920
b15ec00fbfaae687cd073d7ad397759078539e36
2123 F20101114_AACGNO li_r_Page_096.txt
d13d6c6e02ba329b14aaec3a1c9d05b8
c21e80c830a2fbc912f580460167e63e0ec390b3
F20101114_AACGOD li_r_Page_112.tif
ac4b0f1d76c2cee31bb2f6b2dd581609
9ebc256e13b4a8f7479e97f7e04963e71c8b0b88
23668 F20101114_AACGNP li_r_Page_014.QC.jpg
b99538db8fca805b870fc60bb1e76fde
04277d17fbe42b59e64193599279f78607d8eac0
2447 F20101114_AACGOE li_r_Page_128.txt
aa9a9d434398fbc180b75560bacb26d4
30a5f4708f9c50607290b93a3559ae0e44672ed9
7022 F20101114_AACGOF li_r_Page_036thm.jpg
6679284820e8806eab22b1900ed45aae
dc176af91f6af0b85c7f3689d471084f30846d0e
2118 F20101114_AACGNQ li_r_Page_098.txt
bcf7470355629373bdf497184be36e36
be429c849a561ffb1cb2d70dcb35ec1d89f3163a
6350 F20101114_AACGOG li_r_Page_065thm.jpg
0b9ca6e7bcc6501d4d9ca05db96d1305
fe5e9baac017a50c49b79fb2f28878cb5c871554
22578 F20101114_AACGNR li_r_Page_067.QC.jpg
6bac7034d27667c30ec9a414323f10de
a9aef40c5256593fe9686692a4404d939481a970
16379 F20101114_AACGOH li_r_Page_070.QC.jpg
33e3d1da97dea9ce8fb980ba6452a80d
4af31f7e4a705af8d0dfca27a17492a4ca6e2a15
1053 F20101114_AACGNS li_r_Page_086.txt
679f82846ae1aff8825d4824da104433
142cf9452e6220e61062dd12e6d3845b8f5be3e3
60839 F20101114_AACGOI li_r_Page_027.jpg
5d0f84c14182ad23e319e388483a8b4e
0114e4f7d32975b78f1142bc6d882c737dfc14a7
570630 F20101114_AACGNT li_r_Page_086.jp2
89e6a85518e1f53b1dafb3b033a06e13
9d87edcba3eea001a7c0487d39e49f5dd29e2da5
6298 F20101114_AACGOJ li_r_Page_081.QC.jpg
82c2477f02b2d4f5f55d2d5a195173a2
72346959e727905d10547677ad009eab2dd2716a
7986 F20101114_AACGNU li_r_Page_121.pro
52d9e2402062ebf8bea9802034b16c40
332092cd03af0cc33f8c488061f005b981a3e4ed
F20101114_AACGOK li_r_Page_080.tif
df56947967f046f5cfd6f098f5be4dfd
877699219f6d5281172b82b51ae32727bac8f7d4
54464 F20101114_AACGNV li_r_Page_115.jpg
d5c47337bf7171b8c6c5206e6ef034f9
9852c404b02653741e694315882a3c7b362f2554
2869 F20101114_AACGOL li_r_Page_119thm.jpg
e697d08bfe79a85d49e11e5a99b327c1
acb061d308f70b1a3ff75a868a221f0780ea22c5
115710 F20101114_AACGNW li_r_Page_034.jp2
41c27ed226825cc7167e673e2de37e4f
65729f39563efe386f1e522a8d1f8784829367c0
F20101114_AACGPA li_r_Page_075.tif
959064287448790ea8386066d110ca46
32fea94dd6be6b1822bfec7041b9bebc6759c043
F20101114_AACGOM li_r_Page_087.tif
911684d16b0bf4c7e2f92abe63baecc1
c04f0463916309306c30f9e99fb85bdd6a7fcb2d
F20101114_AACGNX li_r_Page_091.tif
d739291b05e011e3ce58f10561535b93
6efaa31552f46fe38bb1bb7b5fe57075e8331f35
29596 F20101114_AACGPB li_r_Page_103.jpg
1f58edbc84350f5cdc5ac2dadd3a74eb
fb40d537540dcbf725dfc433be304c810ebef1b6
54855 F20101114_AACGON li_r_Page_063.pro
6659af35e66f4c38455e4f1138b19984
feec26fdfea0448f91483b3d4043da1a9bf5d565
32005 F20101114_AACGNY li_r_Page_106.jpg
8621a7c2b3015717f8f9d606926fe3ee
f7eb45ad95b29cc0d1dd82423bbf6b17c819e092
6466 F20101114_AACGPC li_r_Page_057.pro
8f9dcc238a0511fb3c81bcf2ced458cf
034de101fa2eff693000af47dc7fbce959924d6c
67203 F20101114_AACGOO li_r_Page_030.jp2
dd238d24afbab949b6337a46e9a0592a
622c8e78581d0da2db9db3061de9aacaa64f4de6
3405 F20101114_AACGNZ li_r_Page_084thm.jpg
69a06457d3cb04b066194373ac44dc32
42f6cc47670af6f6bec61a0eff6e4e503d68fd78
27165 F20101114_AACGPD li_r_Page_135.jpg
de51f1562863e1e71e044c31db8a654b
d35ac8f5d2837b98df7ec642fc81c85554acf162
82287 F20101114_AACGOP li_r_Page_131.jpg
f650147bc6b2cf1ed2f4b793385cd0f5
2470db1881407f966fb3eeee91da13b499c459c1
6813 F20101114_AACGPE li_r_Page_132thm.jpg
9043fe0efd4c476d818a4783128d8305
0858c267ca8f493dfdc3983c0f29a5d1650a9ee2
12518 F20101114_AACGOQ li_r_Page_102.QC.jpg
60b5add18ac85baa1d2b021ac1e761de
8a536c0d8416a7251f095757643c568a037799fd
25410 F20101114_AACGPF li_r_Page_124.QC.jpg
212113c9b225d70b4686495c576de16e
ec7fb50377595b4c240f910acff479bfa481d112
88442 F20101114_AACGPG li_r_Page_128.jpg
63bb5bd2b53478c1b6ad96e74f5d4ae0
8c538ff504ecb1e315921821e3dd43f48f9cce82
1920 F20101114_AACGOR li_r_Page_047.txt
14457d905794d0c8f7a330bf1ee21b12
d71a6a8308b00f109ab7900594983f5fdad5b68a
77155 F20101114_AACGPH li_r_Page_052.jpg
4bd0796f6c7612f1667902adbe7ad1a1
7b846b9a917ef48dcf4d20ea4461a47d539d38a5
255 F20101114_AACGOS li_r_Page_074.txt
55b6a7811a6a2b184d8225c54248b3d9
a41ef0de6eb4becef77aa9be6772a0fbc036f518
F20101114_AACGPI li_r_Page_051.tif
6afd62f6418892e27a03e65f1f29721c
e84df99b63e45130d5d451aea80f904fab9c1747
25938 F20101114_AACGOT li_r_Page_092.QC.jpg
f02d89e72dcb900e2321cac1ad53ddb7
36ddca8929420fbd730ec396b8cc34d5e7c0b9d9
48373 F20101114_AACGPJ li_r_Page_065.pro
011a32c05939bd3d07128650b05f295b
2e54a4fc23a0ff1766a07f9c474a796569d12ece
6053 F20101114_AACGOU li_r_Page_069thm.jpg
1dbe8a3e97be5b5d909fd582cb7c5654
11f9eec5d49fb0deb0296e82e7da91c71b5b5c3c
F20101114_AACGPK li_r_Page_105.tif
d5db9a13fcc40140d560d27c15eeebe7
7b3b4c113e63654523c888a512e490fc1e41dd70
53672 F20101114_AACGOV li_r_Page_038.pro
b3f1dbfe6b2ada3e6bab26c7a76d34ea
7ca45994ddc38eb65b9a613dd580dfe60d46db5d
2024 F20101114_AACGPL li_r_Page_049.txt
364db632246a8ab63cff2502fa186c01
10895e222e5c66caec1ea26063243bd565bc521f
77484 F20101114_AACGOW li_r_Page_092.jpg
310d0ac1cd17fd141b568274074a3cc9
e33687d79c52dd4e83e5bf4149d62379d8716b66
53845 F20101114_AACGOX li_r_Page_068.pro
6dbcce027c5dd94912a83225154d2791
a041ac5b0e37494a5f3476f5da0ead4cdcc5a592
58123 F20101114_AACGQA li_r_Page_055.jpg
dd971362b5ddc0626dad68d99d7f51fa
cbdb934395bb39dfb6d2c197e0b1476e26984a11
25647 F20101114_AACGPM li_r_Page_035.QC.jpg
1b498de3f80149e0d444a15aebe24454
8a40beb71c88d2337855d0585aadeb582bb46224
1763 F20101114_AACGOY li_r_Page_070.txt
3b8c688df399a594eef8ee981ade1862
b5d37d6f472d948988cca3c988bf67c9f9f0d58a
40230 F20101114_AACGQB li_r_Page_062.pro
36f350fb80d1f87fa2dc25caf542339d
47420f975958409f80618873d18ea080ca0d7235
54038 F20101114_AACGPN li_r_Page_095.pro
e6b6053e954ee0582b82ff4ad3669bad
3d24687e8963f27c33a1c191cf78f117812a56a0
17377 F20101114_AACGOZ li_r_Page_044.QC.jpg
06cca3d24857df5b112e29e0bd98dae2
153c7e352edd8e700e59b6d7ae98b59c1acbd75a
93811 F20101114_AACGQC li_r_Page_006.jpg
4ffa31a8c0bfa1b4124fa29c0075e272
0664650c0e36d81c759efbe30e19d34670d88f2e
70335 F20101114_AACGPO li_r_Page_099.jpg
8979d05fbd7cac8ff501ffa1f0f343c3
b0e45f97d793132578545d35a928bee0530b8d77
2178 F20101114_AACGQD li_r_Page_024.txt
373e5d2241efbaa5b1d2282131b14d3e
8c1bb3fe465fe91d77dd9f54321b74420e2609df
36922 F20101114_AACGPP li_r_Page_117.jp2
26f8f8d97e0183af6a43275e1f414d86
aeb57de0ab3706a8ebd5f3e114ce86c5a79112f6
24484 F20101114_AACGQE li_r_Page_028.QC.jpg
29cba20d8265979f589f8e6bc44d4bb0
6a6c4dce1fcdbfd199734e63951e10b5ebe92ce7
12846 F20101114_AACGPQ li_r_Page_108.QC.jpg
076b2d5517b284d8fe8e520136994739
3a46fa4368fe7d017c2cfcd984882fbdb71c977e
8927 F20101114_AACGQF li_r_Page_135.QC.jpg
053e7095af6b80ab4b65e9cbd7fe548e
b398dbd6e14597fef18a64d74a81f9c2ddcc9350
16639 F20101114_AACGPR li_r_Page_008.QC.jpg
c97f7e45041853c910a4f2e9face1e9d
d1cf89b56d4710f1fd2eeb26c44b344148eff821
6627 F20101114_AACGQG li_r_Page_100thm.jpg
5d4f477bfb065657b9eb2d9e30aaa4ff
16d5b4118a371c4d9b0a9ce43fd1db929249a1eb
63474 F20101114_AACGQH li_r_Page_076.jp2
603435c508195b272e55b866a0179198
949f996a598b747056fc1910dddd3828f0038495
78498 F20101114_AACGPS li_r_Page_035.jpg
c5f333e0094daeab4b8091f8013d0b43
fdb059328a4bf99effe2466aff918c89c84d0722
55929 F20101114_AACGQI li_r_Page_134.jpg
cc2ecf6552bcc039fd8bd5b1a100d8f3
1929189b12f47f0b77564e7f10fb31c41005aa7e
55791 F20101114_AACGPT li_r_Page_041.jpg
ba4bf1588442a547fd08e7fc65da0632
2c1a24f250e176ad5093a83bbce5a59bd84be6dc
F20101114_AACGQJ li_r_Page_078.txt
e242263fa8b297395a559d3f87e16d84
b5485b483ad07daf638b1009cfd62427d3c8d467
20342 F20101114_AACGPU li_r_Page_080.jpg
ee32a5739fdfda6f775ee4515329dcf2
c65b81a240d99065832241a2d04637cdabe8f8f5
65855 F20101114_AACGQK li_r_Page_018.jpg
c14e37f8a5f20843d51da188ef9bf776
91260ca17d2380ee0f20f54a5ae80e895eb541e3
10835 F20101114_AACGPV li_r_Page_089.QC.jpg
d988ede9d2eb9586c3beb164289b6c32
66b0e178ddad5f7969c1f63d12a5edd1620707fe
1887 F20101114_AACGQL li_r_Page_012.txt
28136b0e880bfb2095feab93fcd0117f
8cc2ef0fae0c97d0de35a1377e076561b3a0dc43
F20101114_AACGPW li_r_Page_100.tif
d0bb8d01d8b99bb05d6bfd33639b0c97
91c390d4a3663195191fd66bc6e9ac6e4d02a1ac
F20101114_AACGRA li_r_Page_099.tif
4f510916d39231c607eb9514bd92a2e5
5bb78809036fe4e345fd7e42f4bf974624fcc69b
6882 F20101114_AACGQM li_r_Page_023thm.jpg
95ccbd040889404e66ce3764abca4267
5b4ee0eed47131df55008793bffa11a9ebfc28e3
384 F20101114_AACGPX li_r_Page_081.txt
b4720e0b8732c6858b960fbb131cc188
be21701ff36d48c9d4696ed6738eb2d0b0b48daa
16839 F20101114_AACGRB li_r_Page_074.jp2
90f9b212311c5c2ffbd64ca262f47162
abc5a84947262507485048c7926d3b2b571c4444
474 F20101114_AACGQN li_r_Page_007.txt
09fe2308b833bbf7e77adae3e6d99d61
2d61cff6bb33ec13fdc5d412c2fc653de63b3535
5758 F20101114_AACGPY li_r_Page_062thm.jpg
3c6aac907437bc7d6fca48b226b4f13c
47668159d056bb857f19fccd738f0b5467d76f51
F20101114_AACGRC li_r_Page_112thm.jpg
058f9db362c40fd2a7e9d5d96e3b2695
a4a423d411faaa698394f849df7918ef7b35eca8
38840 F20101114_AACGQO li_r_Page_043.pro
61b67857f84ea877f37ac6ec45a343ab
93c3427f473aac97475dc8a421a0223d504978e9
1813 F20101114_AACGPZ li_r_Page_061.txt
69bd88f0a3565f6633731a024cb96c9a
064eb452a62990437cc00e773198e174fa8962ca
6531 F20101114_AACGRD li_r_Page_012thm.jpg
eec270bd616a13f1a37e4f19c7681e22
1b83657ed850da2fcc12a53c4eec88c55b86409c
1966 F20101114_AACGQP li_r_Page_062.txt
1afdd249f34a5dfb2509ef8496818f23
4bcb0c592b78d2c1bdfb0465d9a1f352dc19f82e
114336 F20101114_AACGRE li_r_Page_100.jp2
6126e7380c64b8de810d5bcb01028dc6
3dd6635e7b8a8b0949ec110504b8b9abb502e3fd
53870 F20101114_AACGQQ li_r_Page_052.pro
6b44b3758056d294137917b2b9809433
fc27bf7597981bfe4edfbeb2c7322b9e249587c8
5890 F20101114_AACGRF li_r_Page_072thm.jpg
22e77563975e63e6db32fd74cb668dc0
151a135222871e3c5ea4d9ed00f4adcff62f6bff
1087 F20101114_AACGQR li_r_Page_054.txt
613a99526fbd85811c801e3daf003e95
e09c36e66d584708fdab66ef595d6cb0a563d14e
F20101114_AACGRG li_r_Page_041.tif
5569edf01746e575fc98ef923b4fb088
e5ac0e9296e9c61473664bcebe6948b712a4a0ab
F20101114_AACGQS li_r_Page_014.tif
10628006521fcf9578ad6ae81eb96186
96a628c9e3c8e2c04edbbc559f177f3bf11698f9
2036 F20101114_AACGRH li_r_Page_045.txt
5a003cbf1f930159f1c4c62c91be0c09
74dcdcc3bbb05469de1e24db988f2c00dd2c5e5e
2884 F20101114_AACGRI li_r_Page_122thm.jpg
834c5eadc2aa43dca76b71da61949b36
046463b656b22858855a835b34d3ce6df5e4258e
2043 F20101114_AACGQT li_r_Page_015.txt
415190446d9d4068eb5f37ab9e0c8bc0
094e96a2b48f5ab14032d20b6dc8c13def28f87d
2556 F20101114_AACGRJ li_r_Page_105thm.jpg
266e2a5b9c5cc1c3d3873eed4f081d06
3e39873bc3cf043d8af1124bc8ed76c8df3c7dd5
25819 F20101114_AACGQU li_r_Page_053.QC.jpg
c9c28d4b47bf2563457bd96575a9fd2d
091c04653d342e9a5a651f20841704a466c173c4
99333 F20101114_AACGRK li_r_Page_019.jp2
6b9bf2ad7103cf56d645bdaff88526e4
726303434e222fffb39161e7390e4f77587b8bf2
1950 F20101114_AACGQV li_r_Page_067.txt
c4e0c8e8be7904d99e0ca21debc2d195
04d4e4132c86e24bea2838487371f0bd62fb0f0c
F20101114_AACGRL li_r_Page_064.tif
5b3f685faf9b1ef171fdc071ead7e922
55fb6b62e642d586501e491089bbf9e455e54c9b
18628 F20101114_AACGQW li_r_Page_079.pro
30b949d347e5ea05a8bfd9535bec1865
06529d2d47cdd700b2a1d0ffa61a06b0bb103692
42111 F20101114_AACGRM li_r_Page_072.pro
d196647ba3c439c3e405766e8afba153
8f294a2e830aa18e9c5e202e91700c30cb18d547
7036 F20101114_AACGQX li_r_Page_024thm.jpg
5106d3716fe7e2de80a65568ae544bfa
d204c3d0c7747343192a8a33365cf502648b9fe0
F20101114_AACGSA li_r_Page_133.tif
32c09cc84c4e95e589f5d294191d47a8
d52d824f5ea933557e4a0f28fee90c9e5072c795
65798 F20101114_AACGRN li_r_Page_026.jpg
97278b2af2660b0d71ab431267bd2c66
a63418e49b09e29f395311c4085f9478eae223a4
37784 F20101114_AACGQY li_r_Page_088.jpg
94902a3df438580d024ebc87fd7f1d3d
704312a1a523a8ffa74aab7f6d1edf518c5164a3
547311 F20101114_AACGSB li_r_Page_079.jp2
1de33af968214149c0f1eb85792e0c54
994a2ca32a95ce8571e271101380b4ffd9478db3
1752 F20101114_AACGRO li_r_Page_115.txt
8f2f041ff156608f4c80bcae39fdd445
8a2466a78949e797723b236183a2392a4b54f714
51718 F20101114_AACGQZ li_r_Page_044.jpg
33145023d1b5fae93b5248dec843f831
459b881e2167a792cee85e1a8ffb44de08c75586
34591 F20101114_AACGSC li_r_Page_077.jp2
804d3631f9ee8541736a24fe5e3cdb82
76bfab770e07c4db8de2e6d2a9395f0992890744
23104 F20101114_AACGRP li_r_Page_065.QC.jpg
b746385e39a3059128fb8bbbbddd2a96
7017e90b128701acd4c93f2f8172d515f5db3407
3492 F20101114_AACGSD li_r_Page_107thm.jpg
71020e9d236f6f01bfc1315613b4355e
db52ae69cbbae0953aa6302d84cb6b501d866d60
3473 F20101114_AACGRQ li_r_Page_085thm.jpg
4eced7e8e494609330c006380eb76439
54fceae8c1d04b61654c03a938df786855e918c4
5734 F20101114_AACGSE li_r_Page_073thm.jpg
ca965373006efd3b0c9649d06d8d46d8
44ad332baa4904c72100e08ea4fc64b86dc7591f
853 F20101114_AACGRR li_r_Page_113.txt
0c7b9aff311a07ff1786de805e62ff8b
fd0e2b33e6f36e4c6119ae852ad1f27b2c183d57
135829 F20101114_AACGSF li_r_Page_128.jp2
72484b600a0689028187b80643c76102
99b7cd237eddfe44a3a3fe548f67dd944a56f311
85020 F20101114_AACGRS li_r_Page_041.jp2
3086fc0347104dc592b83957ad01870e
94ef7e9390c105fa5e929523cb36bbe1cb55f2f4
9738 F20101114_AACGSG li_r_Page_002.jpg
c355da3dfbc9c15379f4313a4070a4ba
afe4d220149bfde7e28b7f07b0a3f1fc40e56604
1785 F20101114_AACGRT li_r_Page_041.txt
2dbed3b3c6f5a6f654181abd677a2821
2fd252dd1edc65d6f8205a1f83cfe793dfc83874
F20101114_AACGSH li_r_Page_036.txt
c6f8d78d7a48fcd43d6920825e904a8d
4e11777fdcbd4e8fba28304cd7addb76506ebabb
F20101114_AACGSI li_r_Page_039.tif
60bdb197efbea0c777cc8557ded9a98f
5fa9e9187c8572ad470c27b458c910ed5f7d5d4b
2131 F20101114_AACGRU li_r_Page_023.txt
b89feb569baa6dcbef725c76822b7303
50de880d37e3a456c64e64d897d727b15764bbaa
74183 F20101114_AACGSJ li_r_Page_095.jpg
efa21f1bbbf70ae87afe4a1c75acb947
6c1ee68d8cd76311b213c1af302a6f0a7566d920
21722 F20101114_AACGRV li_r_Page_040.QC.jpg
d5c3eaf38dca0bd27f5fcdcc44f92f8f
fe39087cf8b39f1a1e47485be4c766b72c3694ac
6596 F20101114_AACGSK li_r_Page_026thm.jpg
653ad2afef893a0dce9cac67adee553c
beec62a19fc2e8100d12fdd83c6ca53aa4292a30
F20101114_AACGRW li_r_Page_005.tif
8478a597172deda62ec8529fdd0b2f5c
28dd2d7cde821fa5901a7d77eeb5edc70856ed94
23922 F20101114_AACGSL li_r_Page_101.pro
b54289845b25a53998a03c5fd3273b43
6d666fa9245b2c4ae313e17a5c24fa2868c891aa
39167 F20101114_AACGRX li_r_Page_039.jp2
2e656f2fe3e48bb3238754fa99dc885e
aaf3281aec9ecf372f40279f6bfa8db362836626
25196 F20101114_AACGTA li_r_Page_013.QC.jpg
34491c4eead540ed6b78d46696b16cfd
bc4d4fbd6a70ec4e4526e51efd066df84926c701
1051984 F20101114_AACGSM li_r_Page_006.jp2
543ad7cda2c35a6b6f2314dcb3840c85
28fa5c018d2cfd685e1950db77ece9db0c561094
7884 F20101114_AACGRY li_r_Page_078.QC.jpg
215e54979dacb4030e4ad3f902ab4b7c
50910740c20a31aae5d2e36c6596a3d692811d96
5996 F20101114_AACGTB li_r_Page_071thm.jpg
fcaa2d6730c55c44620f49e4b86db037
ce3978f91870b3bbe659be5d595bad5c4d8c8ab9
53141 F20101114_AACGSN li_r_Page_011.jp2
54b5adad009f9b10f4e4f3b2e5a3a8f2
6f573a3d9f66415deac14d6821ab28e05b34e68c
5995 F20101114_AACGRZ li_r_Page_045thm.jpg
abe5401001f88102ae29a527ed7365f4
afd2e487bc5673cd01e2e774ee56ca9da857f7ad
7318 F20101114_AACGTC li_r_Page_130thm.jpg
33d6001ffa7ef448b778e1e18d086587
18699c5b08b37692709822195304d7e62eeec2cb
75600 F20101114_AACGSO li_r_Page_044.jp2
ebbe182d2c64853f7ed1a460f7e31e7d
06202fb2bbf25b1a77dedc662273261e36585ac5
12983 F20101114_AACGTD li_r_Page_119.pro
87cb59c851f0d2668244619b12b07e2f
22286efe65fafb1f09d3694bef6b11ffd4cd9c1a
1639 F20101114_AACGSP li_r_Page_071.txt
5f94ad0e5ecb1a3ad92d96b4538fc10e
f10c2b4ac1fd4f7abb2ebcebd0718f590ae59a38
47842 F20101114_AACGTE li_r_Page_067.pro
29b1fd6f052f676b7e5318f0ac27164d
73d224b7f5668578608346651fd5321572c9650f
63665 F20101114_AACGSQ li_r_Page_064.jpg
4dd1edf98d5ed969a56477934ebe5f18
d417d3607625df2570c4ab7e86f667c5c6365965
72122 F20101114_AACGTF li_r_Page_014.jpg
d2012e036074a6aa92ef388778725575
d027aaae96faf5bb94164d90269e0d2306fe0e5e
24443 F20101114_AACGSR li_r_Page_131.QC.jpg
8796b6b1afc84c98af6bb8abaf9143fc
b0e3ac5bb76ceefcfcd31c799f925251ae138909
F20101114_AACGTG li_r_Page_110.tif
7348ec19cd697f0599b7fa4b4767f89c
bb06cf09b6931a9cd7c61316ccc0725c361ebaee
4393 F20101114_AACGSS li_r_Page_004thm.jpg
45b7341b24acea85b946946232fa0479
5a11f21d07fc344245dda95c98c5ab39789bb587
12257 F20101114_AACGTH li_r_Page_082.QC.jpg
15f2595b599221839e8f5ef7398d49b5
96472e506729604492a92b8fb1a3140e830349be
44077 F20101114_AACGST li_r_Page_004.jpg
f3ea0a9e745a55a31906bfcb7ef67f46
b8bd33c980d7488e6e743a41901f108f910dbd62
258968 F20101114_AACGTI li_r_Page_080.jp2
a4884e272ef7e20451dc4cd340b86686
58de3d3ee900bd337fc655f445ff9a0bfaef7afe
880694 F20101114_AACGSU li_r_Page_106.jp2
523ac881b46504b29532a1a9a98fd27a
9113dacc95e2b1bec0cadac13a2926f93fbadc92
F20101114_AACGTJ li_r_Page_095.tif
7f1d01d67794541cefd6dfea54186c0d
a440d581b43fe8eefd34e1d02158df413977df5c
F20101114_AACGTK li_r_Page_066.tif
f1c66a6727ba02490368b5f558a1c6f4
b6edea4ab6714b807d0fa7583bbab923e818bad3
26303 F20101114_AACGSV li_r_Page_024.QC.jpg
9c81ddf627229855050409506685129c
8754c44d91906b14fbfab74220c11f311272f59f
34035 F20101114_AACGTL li_r_Page_079.jpg
a2d112e1e8fe5bdbc2bfd38f8981a873
7c5394dc243b6accefde3130a41516cdd801e85c
6361 F20101114_AACGSW li_r_Page_029thm.jpg
629a3f0a6d8ac62e8178500478ce6d53
91c6d42f6a9d6b44f570bce2ff54c4b2ae00dc13
F20101114_AACGUA li_r_Page_109.tif
cce18a01b011a37467151e00f588840e
033caad5958080de83a25b974731eb46b1a9c2cd
F20101114_AACGTM li_r_Page_069.tif
607f8966b3a53114065e15d647130757
aa131322bdd1e2f14e17811089b1118d8ac199ff
23804 F20101114_AACGSX li_r_Page_082.pro
3c4e24052fec3079531b3fd7bf7474c0
24800cb0177a05a7121f76eb9adba8036401d914
F20101114_AACGUB li_r_Page_111.tif
12e8f997b0308270676360558ecab07a
b2672b1d4f5c8668af590054e1c3f53409ebcbf7
749 F20101114_AACGTN li_r_Page_117.txt
740abc02136812d46c15c9e665e99839
9f3f294029eb227041bcbfeb096c91b279d47335
7144 F20101114_AACGSY li_r_Page_105.QC.jpg
96c5061ad165ac446e0c387956d5a5f3
792544e3365a2a820f9e2911cafe759d7321e51c
4367 F20101114_AACGUC li_r_Page_108thm.jpg
3ca041907a452b08eda69c28965415ad
85cec0fea31a140a788fac4996e5df222489dc70
F20101114_AACGTO li_r_Page_077.tif
fbd8de72d992f19e784bdf0981a5ddaf
00caa5079b66bd6507096b635ea955601161d4b7
56996 F20101114_AACGSZ li_r_Page_045.jpg
24430b9fd5bf07ebed553082be6eba3c
a9ad5f5a1d6ec6f274d015433178b891bc13e067
2348 F20101114_AACGUD li_r_Page_081thm.jpg
de14440cea1de35706adddf351a487a9
99162d0b440249742ff741313fc9d4edeb216530
76602 F20101114_AACGTP li_r_Page_068.jpg
4250411caeb6617bde23e9e3ba732f7b
d8a43325d43fc3866bf28fb39ded7507c41d3319
F20101114_AACGUE li_r_Page_096.tif
ac741ab3cf037013830cdcbb2f3d71a5
1271e7ef4bdc965c8db6ba565511228870ad8882
F20101114_AACGTQ li_r_Page_028.txt
bb1063bb6daf14a171d2de4b16a87832
5b5c3c78d963fd331652b31f3f6e3075a96f09ac
F20101114_AACGUF li_r_Page_045.tif
9f74acac1417e6eec5fdb5c88b7216bb
11e9d770a72b0d857fa58739123cd57abbd382ae
7094 F20101114_AACGTR li_r_Page_094thm.jpg
4db1b53ce18fddc4780cc37778fa549b
a66096e3770d8fc00c8495dc4713f31cc6bb70cb
1350 F20101114_AACHAA li_r_Page_102.txt
e585f291e178c9fd2b2ce9323f4e37b0
465e28432929413c2732683b249f79f3c459f87d
68820 F20101114_AACGUG li_r_Page_067.jpg
7364aca9bf95ba5019bd412ca9a59070
e8247a1269de760c93bcb7c1255aef5ce805f309
61193 F20101114_AACGTS li_r_Page_128.pro
99e6d7d957c7974bce0a916edec3b39d
00102bcacb295d828495c54d940b7af0223d503d
24049 F20101114_AACHAB li_r_Page_023.QC.jpg
049812c0b45f8f95989aa6d27779a9d7
ffa642bb9e9b4951fedad1ec59d30b854ae6b0f7
118039 F20101114_AACGUH li_r_Page_094.jp2
daac077ae4eb8ed518d1b27beddcab3b
9ecc4d765d184686184f4d456f0de2eb488e817c
636490 F20101114_AACGTT li_r_Page_085.jp2
2bb238f7ca06d4fd69a19fd514b8b078
ba52b8e768a2b0fbe289db85fbbbac817385c9cf
7364 F20101114_AACGUI li_r_Page_001.QC.jpg
70c873e8d1953638d01be9128c6bb5bb
72e647a865a34ae815801c461461deae693b50be
54035 F20101114_AACGTU li_r_Page_110.pro
5820a7bc22643b535a951d9d41e8fb2b
093cb99ddaaa8402b896f2926d7aaab2b3a1f289
F20101114_AACHAC li_r_Page_073.tif
e9278ffebf0afca4df4e4cf80a0d99dd
6982d3520521d1d60b2580880e6b2434c8884147
44048 F20101114_AACGTV li_r_Page_027.pro
eab876ffc82d25a0b26022a6dc63a9d5
9306963d1ac5687ff692e9b4864f09d562abf69c
66522 F20101114_AACHAD li_r_Page_010.jpg
a3e9ce988c58995accddc2a1604d8fc6
822ee0be41c83287e4f9454f18a5e74c68a6448a
53830 F20101114_AACGUJ li_r_Page_051.pro
6272520b6d33c262824d325813559121
0b3a2dc5ae86c61bdab9ece9a75e1760ffdba974
24802 F20101114_AACHAE li_r_Page_098.QC.jpg
1fbe74e0534fd51c1652e1a4eabedf48
8d56adf9c4bf8c9f76c5ceba909d84a7e92e5822
46143 F20101114_AACGUK li_r_Page_031.jpg
ac3f4277c82db30d951e25a37a0bd4ba
e2035e26cbd8a91bd1ee50e536e86db50f491dc6
76636 F20101114_AACGTW li_r_Page_051.jpg
acaa77c75f0228f6b85f332297fc0f96
f468140a678e6f809343baa7c3966052b66f0900
25004 F20101114_AACHAF li_r_Page_120.pro
bd46d837dfedabc2fe24d9a4c9023096
a0683f3aec23b6d116e0b56e4594c367019324b6
6493 F20101114_AACGUL li_r_Page_050thm.jpg
1ca62b30aabfb839c4bc4f633fc3b53a
a34ff90991360b471b44d9a34bd9e1327f54564a
34765 F20101114_AACGTX li_r_Page_085.jpg
f88cd186a7a756e73dcf013edf36d06a
6a43a10fbf2b7ad8139fabae6955f4e9e3570155
73852 F20101114_AACHAG li_r_Page_049.jpg
863cdc3d8f4107f7ae32b9997d157b62
7ee6d3ccd8fc331a2bdb70841a5042c7e6f8f1a4
1949 F20101114_AACGVA li_r_Page_017.txt
b723248a9b84660c5cb915eb6c8d7928
27020786ec703c5e76a4b69c4b1ab5e0efbee3d0
F20101114_AACGUM li_r_Page_131.tif
c47e295452a93b22cf8348fc822f02c5
04a3a327ca07aeba6b8035f8b85b4ab82a0d0adf
6809 F20101114_AACGTY li_r_Page_111thm.jpg
cf1e1417469eb07ca6b07637bdb57f7d
ec34ddd3a75a75364c4d9d7bc9e99af051d83c05
38646 F20101114_AACHAH li_r_Page_101.jpg
948c9bd08b78f7578e00293db04c625b
63a816fb76046c61a54a68cb1e83ff1b8445f218
F20101114_AACGVB li_r_Page_042.tif
45fec8c7ea3228d44476e203c833a924
34f39ff3d5578588a00f774f81d911a5669e2ef7
6601 F20101114_AACGUN li_r_Page_127thm.jpg
071077f3e64ea5c4655463c9640ea624
c00a940b506f126bafc32977ac066d056a2bcd78
F20101114_AACGTZ li_r_Page_072.tif
fa4b67127c23e53856b00c5610b3c75d
14e842e51ee8edc6d7d9be5539cba8b618227b7d
50313 F20101114_AACHAI li_r_Page_049.pro
117d002208601115054a90d9529454e2
0893612dc74003205ee713b0d0572d1799c332d1
25088 F20101114_AACGVC li_r_Page_034.QC.jpg
901b3da62f7906c2e8737100a1eddea9
248b2de7081feb892bf9a810ad09b9bb1b48452a
9586 F20101114_AACGUO li_r_Page_117.QC.jpg
78dd2cbebd425d03fd202ebc674bb6ad
db2181b4e4a85efb74c48429238e73a9c8a176ab
F20101114_AACHAJ li_r_Page_116.txt
08c1ac0949305a7f7cbdff65e4270503
9ee8de9b3ae45c85b4ed6497765a743b41be1835
72936 F20101114_AACGVD li_r_Page_016.jpg
fcca42846ad0d7f9ab8588ea08e85abe
edd30cb9f86c0627cd819bc7a4cd8ac9120b555a
55333 F20101114_AACGUP li_r_Page_036.pro
2e25e283f98b54ab657986d8fd1770c5
b5c641c0fa5ae5e2269a330a2affc5682c96c4ec
1051982 F20101114_AACHAK li_r_Page_009.jp2
d383d32359d71a5284b81950e60d0d1a
8323b6222370f4b7dd4d1213b1fc72e6a0194e5f
8841 F20101114_AACGVE li_r_Page_122.QC.jpg
54587e5b63b3eb2d618848bc7c3706a9
e5ccc098a307f53b2d350cff2442c655839e2c1f
55094 F20101114_AACGUQ li_r_Page_097.pro
c1170a6297918ec8c6c6f1c26b7957ca
4c3d28ac45481cfff9e1353a38bf0d83e0d5b93f
5999 F20101114_AACHBA li_r_Page_018thm.jpg
10492f8255ed034b133af3b1ca43471d
a1e13e1a68c96576f9d6903e95b77e0bc4e064fd
6960 F20101114_AACHAL li_r_Page_063thm.jpg
0842995a5fe6e36924be8cc25c5b2a48
b8fe0c712775b1afd6b0d252c94cb3f9e9064747
1219 F20101114_AACGVF li_r_Page_079.txt
3c0e5aad4a692ccd8d8522d234173178
dbbb4cac767f8eb32011b59251c6e6deec9d6aac
60572 F20101114_AACGUR li_r_Page_071.jpg
2988d0f159f9e4e25b632aae77627252
3718ea500d877b7826b98aa2311bb5afe3548b49
F20101114_AACHBB li_r_Page_108.tif
377ec2fb45180dfd068bed1757c23a65
2479f9c946b7967de313859f2bffda944e71dfd8
2052 F20101114_AACHAM li_r_Page_027.txt
6c1cfb100aa7bbfc4b535c105896699c
3cbd19f3680d11dc2920bb4138369806bf4b8347
2647 F20101114_AACGVG li_r_Page_130.txt
f093ab38fc9066ff3bdc54f96891931d
c190a9686f90a57b3ddf0e04589114f23d342cf8
F20101114_AACGUS li_r_Page_016.tif
ba01b5f65c8b9e24242ba8fc700ed0b5
fd75ed4737aef0bab37d343d322b4c5e6cc6065c
2396 F20101114_AACHBC li_r_Page_124.txt
60a8ea5df2346255902753df7dcd2fca
a43b450fe75365e4746cd73754841b4f40cd4398
6560 F20101114_AACHAN li_r_Page_014thm.jpg
2fb2f9e3bc0bc58669852114512d4029
ddd24f27186fd18ab86ec6ed298bb98f5e4c0697
119724 F20101114_AACGVH li_r_Page_024.jp2
665fd82aff4fb95614dc49c4341647fa
1ebbed3014a470321172379a5a50ce853be84934
F20101114_AACGUT li_r_Page_027.tif
0723a1e3c3f9d6c98c09a2733c9bdfc8
de8a78b148b9b02a75d6a82f58c98e496aafc653
118198 F20101114_AACHAO li_r_Page_052.jp2
e098651de7124b6c6c13095fa723bcc4
91b220fc3b8073630622b6bdef2fce142e77926c
F20101114_AACGVI li_r_Page_102.tif
6ba314218eaf95da9614570617910c0c
53cfa3a77ba1daa8eda1d58fe0c4719287f47192
29645 F20101114_AACGUU li_r_Page_039.jpg
4f332bcc4c4050ab3d593d8c36058811
c067d90eb463bf3baac7f751b54bf14bfb94d6c9
40616 F20101114_AACHBD li_r_Page_084.jpg
6d2a82c20aa8007555b73c05ecfda310
1e26cd793e6037d5075b0968dab712f57e00b30e
204 F20101114_AACHAP li_r_Page_003.txt
c6e0f16fb11cfaf6b602919b5f0b769b
6aca0005c028a7d4ce6cd89b9224bdf02775ff17
9780 F20101114_AACGVJ li_r_Page_109.QC.jpg
f3d042d89ae20b71e0d1f8fa625cce83
b1543859ed046a7b8187db069a0edf6a5332c810
1051985 F20101114_AACGUV li_r_Page_005.jp2
3ca5943627d21614c485ed3a38cb2132
38b9e70aaccace601f2d719b42b5aec52c3b88ab
44314 F20101114_AACHBE li_r_Page_118.jpg
9ad32398b2963232ea4efaf7f539854a
ed422891a1545a31c7f5983207c275af58ce5c2c
43073 F20101114_AACHAQ li_r_Page_045.pro
33cce7742dc586778003848ea9693b27
749340e915d8296907267a8f4ac47850ba6fec75
34867 F20101114_AACGVK li_r_Page_070.pro
d6eebffef918118e5e8b05fa5fe40ecf
4dfdfdb649af0a3fc00253cad6ed1d6f99a994bb
112849 F20101114_AACGUW li_r_Page_020.jp2
d41e669f4f2967ef0d2356d022e2cb5c
47490860ef27fd6b86c685caccfa6e31fd51badd
F20101114_AACHBF li_r_Page_024.tif
898cfc9737d9d51b36fab00504ade569
83d0c1b746b99e348a4eb99ec3f38e90bb8bd890
50551 F20101114_AACHAR li_r_Page_127.pro
9525868586e1d2f1e0aa3d86bc7110ab
60391b47d6465c1fa855b5e259fe52dcb10b7cf3
F20101114_AACGVL li_r_Page_106.tif
d9c44eff9bf9abd5e7c8877ae7756167
871f8dd45b5daabda2269b75bbb1634ba55468d0
25853 F20101114_AACGWA li_r_Page_129.QC.jpg
ee38692293364b75e8f34c5238bb53a8
8a4bf563f7cfd219827e456e2a82c603f8f637ef
35957 F20101114_AACHAS li_r_Page_058.jpg
3e5794698c2c46c482f311505f297085
52742ed081999b2be73cb5cb8a69a35948574d02
15217 F20101114_AACGVM li_r_Page_031.QC.jpg
eb298c5bcb16c09fdcdac2635509ac99
3e89d1f9c0a824dbeeca9573c9e18e39189ce414
24213 F20101114_AACGUX li_r_Page_119.jpg
a3bc9548a94cff3c04d2df9923b9ec01
0b48a895fb83cbd97d62a083b575d255cc9ed572
55210 F20101114_AACHBG li_r_Page_024.pro
3aa819f5b1f0cce034d006d71ebf006c
98f1acd4c28017497038fd29379206777167597e
25849 F20101114_AACGWB li_r_Page_097.QC.jpg
483a4ec87b313985b9c2030f29e55a45
a26160a43cfe7c82b7bdda3e1703833a9c25c310
77719 F20101114_AACHAT li_r_Page_110.jpg
ea9dc9ae239e91209a8bc19ac5173bee
c4443b259d5ff6b4fe5dfb943ce718f2f3d4f3d7
98228 F20101114_AACGVN li_r_Page_066.jp2
e17a517000ec87c9208af8258e011d87
f396f823bcee03be05845a1470220b850cb2991d
21076 F20101114_AACGUY li_r_Page_064.QC.jpg
1b8e19eccd982abd6e95698570b40025
e807e8e8ab43e118a6e15927736af9563fa4b51b
540326 F20101114_AACHBH li_r_Page_104.jp2
04b5fea351cec2611f40b650e17deaa3
bc9043dbe7c3d63cfc6a829817c2100eca53e1a8
61807 F20101114_AACGWC li_r_Page_004.jp2
8447307f5ac212d2a8a7e4153767fbea
b154bb3f3837a559a8cd4f2a837851c8235cdee2
54101 F20101114_AACHAU li_r_Page_098.pro
c2401026e119dfc4c8ce4aec5ff637e9
d7cd4a98e05d6d689e407955e4bb6b30476258a4
29489 F20101114_AACGVO li_r_Page_119.jp2
0457c520f48d0fc141d3e524495b8ba2
b11976d0111f48e1fdbf08b7c8316ca8b0969327
55515 F20101114_AACGUZ li_r_Page_008.jpg
6865fc6dc303fe56f3edfa6900bcd15f
79d4c749dd33655b17a09ac897e9998690fa7270
23770 F20101114_AACHBI li_r_Page_001.jpg
c4d44edb67b7eaa057240f38b1c1fbff
472fc06b58ebfda271455f2bc37c158ae290054f
25758 F20101114_AACGWD li_r_Page_060.QC.jpg
29e8e8ca91a7a3b0166f3805ebf483c5
dfe4f906bd3499ed59d0f3f9fe398d6937100eb0
25826 F20101114_AACHAV li_r_Page_068.QC.jpg
ac3317a633811b707cc0d59aa38d3b9d
8574b2dcd060be9d5d7564d684f2e13ba4d11253
11085 F20101114_AACGVP li_r_Page_058.QC.jpg
d6055629ebf5c555a124eaaf070634b8
70393513d4d2c6f91d34ab7c529e15e5ce1d8197
7046 F20101114_AACHBJ li_r_Page_053thm.jpg
1dec276d04c2993072595a89c4b492f1
052ae1040a412448d543b9b4aa222b9fa6d00d59
F20101114_AACGWE li_r_Page_017.tif
306ff0a287e04372e0fc8d965e3da1d3
90d41e8c477cf70ef77fd4c4b7fa24cfd242bd0e
113946 F20101114_AACHAW li_r_Page_127.jp2
d6db1bb1704954e7d3d74bf8a0429c48
b4378800f304eea583c9a4bf2cfcac2a17bf1405
51883 F20101114_AACGVQ li_r_Page_020.pro
cf83dd787f520b398a8a2c9841d9648c
d1f1fd5050def6504f8781a68f520befcca94862
15743 F20101114_AACHBK li_r_Page_114.QC.jpg
d8c58587bb3a1116cef6168efb568a29
2d546d618430e007346fa69ab74c43c94bce8d6e
F20101114_AACGWF li_r_Page_116.tif
e6b9949b1cb4e2c7d533fb77146f3054
e2e0b8de17529b0e9f1726b3c721997badc8f68d
2824 F20101114_AACHAX li_r_Page_057thm.jpg
86592763df629317fc226db55fb5a48b
3e918f10ddf6b0e6a9f00aad5c5eac6f9ede94d5
20942 F20101114_AACGVR li_r_Page_025.QC.jpg
2bef9a5ad29df5d339cbfbb95fabcfd2
170036f36c6683197f2061b603fc9917246a8bb2
348 F20101114_AACHCA li_r_Page_106.txt
bb378f2ba028f061d2b7f556c3b5f176
b7341a7d3abe3b3b1abf2d6521abd418746a8d04
F20101114_AACHBL li_r_Page_077.txt
b946a56b51b57bd7a1ffc9bee7264072
44683c79b18355bd29513c7748113903d463739c
61033 F20101114_AACGWG li_r_Page_047.jpg
04338ac1ea330ad49f4efa592c9a7740
cd8f694aa80acb12c8b7d4726f698b60d5efba3c
104054 F20101114_AACHAY li_r_Page_033.jp2
d78c7a72b9747ea1d3ba4b8113e2d995
a85c6a38d3b62706b914f26f1dad62fdccdeff4f
3057 F20101114_AACGVS li_r_Page_077thm.jpg
00d567e466848f02d5dfad826d35cf19
126b3702614993c2e7bed4cbaf0f57b90e926ccc
54189 F20101114_AACHCB li_r_Page_096.pro
bd33613cfa1fb296b72a250ce1a66721
cbb2b9a764216f5d1d71fb9fa4a6baf87e83fa81
20958 F20101114_AACHBM li_r_Page_010.QC.jpg
badf435179c5a6bfbe5929410af6a035
873fd4e4dfad23af16c5967e412ddd3cc437561c
90427 F20101114_AACGWH li_r_Page_047.jp2
989b4c080ddf322bb94d536828cca570
769105f33909f9f973928bc3ba138048c1d127be
24926 F20101114_AACHAZ li_r_Page_132.QC.jpg
48bda8386ead49f4ae34735585c1d8ec
0aee72673323264a3c4df83e2916efece8ed2c24
1873 F20101114_AACGVT li_r_Page_026.txt
3b13245822d51f4a0bcce6ad8a0577c4
a6457b8d443a49a76014bebe451a86678c93c107
1687 F20101114_AACHCC li_r_Page_030.txt
0885b52aa9d1575b4101167c42e29c82
95e2c2464a6a77c4dc57f819d765b335887f9538
19981 F20101114_AACHBN li_r_Page_121.jp2
daaf3b170ca559c0ae60adf53d37f894
1288600895d5b26c23be2093869fca2c3ab6edb7
94980 F20101114_AACGWI li_r_Page_130.jpg
132eb68e94c5b5b788ac7545c509d133
66f50548551dcc602277f684a2930d5f98413448
F20101114_AACGVU li_r_Page_089.tif
f0015856416d219d0d5f134a3396e88f
78f76b417ba769e83f1cf1dcc4a445dd5276fbf5
85682 F20101114_AACHCD li_r_Page_129.jpg
bfc26d8eea402f0d2cc97c93cac3077a
bea5ba8442bcfd67fadc73ea5adcef7bdcd0405c
111358 F20101114_AACHBO li_r_Page_028.jp2
762d896b68f0131d96e594feba20ce40
886ee514d561f5869b4f1465b75aa2194c795170
23062 F20101114_AACGWJ li_r_Page_116.pro
72c9e725001d738e41a0dfd01c327a14
9a8047c6011ea00b5c5fa3a391947539ad2b9c40
5921 F20101114_AACGVV li_r_Page_025thm.jpg
e706b610d2708f84b04cfc9421435ecc
34e2aaaf786e8906270de355087b986c8e88393a
49332 F20101114_AACHBP li_r_Page_099.pro
fc4f71b721130e7db6941d245ed3a5d1
f51f885b73a0eadec6efc9cbaaa7d3cc733018a9
4609 F20101114_AACGWK li_r_Page_031thm.jpg
9ac795c8d910045bc1f04e5743537329
04dfce45142cff53da798e0ce7dee52fd47596f7
16960 F20101114_AACGVW li_r_Page_039.pro
739d25f1f537403efba4762723e8b96a
c4709cc0272a904bf01b74f03dee6a0081ee7639
3459 F20101114_AACHCE li_r_Page_079thm.jpg
d77b447dfbac1d48f3ebb4805c8d1877
a9bb5879ce6d61d89d76229a29f99febb53c1dc7
24149 F20101114_AACHBQ li_r_Page_022.QC.jpg
5817daf015cbe42af18410b8af5262d3
2e25fe6edacbc0ff091ffc307e79ad2debe51a87
125350 F20101114_AACGWL li_r_Page_131.jp2
d69e4669dc867835787fc3931a50f7f4
17d1fa39d28a792e50e740504f4e8a13b365d6a7
F20101114_AACGVX li_r_Page_081.tif
bc6c4a9168ae9f832f7b4c6f13e310e8
c95c52406cb154d5fb57ea6c859e06f1bd0e2d02
46720 F20101114_AACHCF li_r_Page_114.jpg
2890be9915f1d5ee097923dbe82ee3da
0c093a57478781a73666792c493a48a0a28bdaa3
F20101114_AACHBR li_r_Page_114.tif
658894ae5ebceb9b94239966062de0e6
bccda498efd1f0d46d1435c8924bd6643e06fbd1
1418 F20101114_AACGWM li_r_Page_056.txt
b51b7157a32923d9800aeb6a8df85d26
85a2bd049110d09972afbf2d531906468d24aab6
153253 F20101114_AACHCG UFE0022519_00001.mets FULL
755591cc781f24abd3a165b9b70a3c88
cd92e0bb76bdd45c6242e54c120302490d71994e
56800 F20101114_AACGXA li_r_Page_120.jp2
49e56c2a28f64e8b280e33b9949328ad
5ae5af50848c7288e9fa90a5f9dd339106d81d0a
54550 F20101114_AACHBS li_r_Page_053.pro
808d221fb4204a618c437ba58c8a7c4b
b13e75e69bd3472b46f2e26db54828e85e5835e1
51849 F20101114_AACGWN li_r_Page_046.pro
c6695a8e3c17ebed8d365eef2b2655ed
ea5f62109beeca4ade689ba84a6a5681aaabfd10
98375 F20101114_AACGVY li_r_Page_018.jp2
caa4e1f711e706ac90cdbf20fa48147d
b1d4dc0dd7b212a77d44c4baab6323f6bc9733e9
136301 F20101114_AACGXB li_r_Page_126.jp2
0aae1160ef4309b3a16770dc85c09e17
acd90130aaebe34a5da4db859174a962c0b01b77
5336 F20101114_AACHBT li_r_Page_070thm.jpg
65ec77886c100167c5887cb5461f836d
9d1e6b9b4f89db6eaa75088ef754eaf1544b94c5
20021 F20101114_AACGWO li_r_Page_071.QC.jpg
17f21abaff18c6373ebf68347cca4b44
62b706243e0402412415dceac6e8cbe70c7302a7
6450 F20101114_AACGVZ li_r_Page_021thm.jpg
d5776d8000203f993cb73ca9c692736e
8b8d6b373195a8e976859d45366834b1737063b7
F20101114_AACGXC li_r_Page_052.tif
9df4fbac4995943f81104997d332132f
8271c5cf0e7a207cea0fccd3b88a7930e6575928
219213 F20101114_AACHBU li_r_Page_081.jp2
61354dc5ab046402a75c4804abc6c3f3
677b6646a9f9d5bfcfc7710b9d97286bf9aead7a
3781 F20101114_AACGWP li_r_Page_003.pro
19ed81c69e845f4ab25fcc66e41ae747
19d428c92521c0cc4160c4ab9d6ebd538bf2a804
91063 F20101114_AACHCJ li_r_Page_005.jpg
ef7715613cdad15022975ea17f9dc7d6
fa57a44ba683f7089e0d32a4ac37767533fb2b3c
13476 F20101114_AACGXD li_r_Page_135.pro
47593d8bfc063cd975a4ec696c58bddf
feecda6da392d25c57735485d23d412ecb19802a
104559 F20101114_AACHBV li_r_Page_017.jp2
6b4d33a5e30916d2aebc71695dc52217
d3d1b1b4062067cda7c3adc8ecd600d63ce89284
F20101114_AACGWQ li_r_Page_121.tif
ee006a6b48f1c87fa17e28be6ef8192a
55d6b4b30c09a45a978d9fbc0231e6e27b54ae3c
20206 F20101114_AACHCK li_r_Page_007.jpg
3b0f7b372f7d09ff3dbec4dd52522aba
309bb3cd5007e3d834ae713f7194a091d8018486
1829 F20101114_AACGXE li_r_Page_031.txt
d3b0bae93304c51612057658c05b5f52
a525c65dab4903df839241810e44a76df2644d6d
2664 F20101114_AACHBW li_r_Page_135thm.jpg
bbd4d4fcd7922b4e458f006f395f2317
86c38387c57d4f03a5ed92261fbb6f9a7d0e2738
111867 F20101114_AACGWR li_r_Page_023.jp2
c50a9ff3dadea30fd7beb2f2862fb900
37f45f1476c4f1a97924ae8f5a5c2eb151c9ad26
55075 F20101114_AACHDA li_r_Page_043.jpg
c5c97905411582edec17572bfa7b3e03
3d675626e96bf6df7b9a18a8a7fbc023f7b00425
88007 F20101114_AACHCL li_r_Page_009.jpg
24eeb408429cdada78e9f78d0b190d9d
8ca17187fb9020465d0465463b921c097a8c5263
30716 F20101114_AACGXF li_r_Page_077.jpg
abbf108842e268a3e19c9d4af74d76ea
1e661f878b247a84cce62f9401ea049c620c0522
4267 F20101114_AACHBX li_r_Page_008thm.jpg
7780b79453d7a7788c1bc090e8b0b7f7
1e69c23af377ad0d4aeda1625aa7d2be9f76c22f
F20101114_AACGWS li_r_Page_063.tif
aaf4c9665441024525f3edfc5982c967
585ab555d405f731000281fb8bf28837a2667263
72404 F20101114_AACHDB li_r_Page_046.jpg
8008eb9b3848a837e6f311eb7e7c2d5e
b0a422c78f9f91a5c7b706e9459a7a7dc444addb
38783 F20101114_AACHCM li_r_Page_011.jpg
019210abfdf5e5d8faf5d04a041e2e69
42340568d50a4e77a37184645659b70b9312833f
1112 F20101114_AACGXG li_r_Page_087.txt
b7534fa453a3921b822caa568e10289f
94cb51b4b94572ded6fc6ccb760de43d465790a8
342428 F20101114_AACHBY li_r_Page_078.jp2
c1b82194f8c4c8a12e50c798e547c0bd
c9c2849e1f6d46d4faa7ef5f6ccf91fec3d53e0f
2121 F20101114_AACGWT li_r_Page_100.txt
ce85d303ce85b9ab890498562d298b90
9b0e4178ba69b82d0053cc0389b39a22fe4ff7ff
66227 F20101114_AACHDC li_r_Page_050.jpg
611da5a3d2c5eb1bf820f18f8192bb4c
626f8667364e2668cd34452821fd758f2c6ae148
73897 F20101114_AACHCN li_r_Page_015.jpg
0edb5e0e20b7f1889f82f6289bf4af31
51022903d7870915f71b71b6cf0ad7aec51e4804
F20101114_AACGXH li_r_Page_085.tif
c95f026aaa5b08c5033f0e7c33e2fb59
71f13c76d89ba172e6cfb49b7638624cb978e7b9
85779 F20101114_AACHBZ li_r_Page_124.jpg
e1610ac870a6c917cb57106b6c737e47
636541902b43beab21bc44e974284cc2ef44ece2
38952 F20101114_AACGWU li_r_Page_086.jpg
6f9214a6c8a42e79c01c51db1aafb6fc
a92769cbd95d1a6bb96ffe5dbcec33870b0dbd62
42278 F20101114_AACHDD li_r_Page_056.jpg
8739e212822e0643c98e130cfcca87e6
962e956486342efd1f136c3d34236ee5905fca0a
75621 F20101114_AACHCO li_r_Page_020.jpg
59fdfb9caa25b06d0a622167fc0a06ec
86e5468f0d4a27eb6566edefd048c2f838e8fc4b
35099 F20101114_AACGXI li_r_Page_061.pro
643d59a841a917fd3d81c6ff9f88030d
4ac73cfc53edf1ed2c0713b3ee34d9eb3f6bfc4d
25549 F20101114_AACGWV li_r_Page_063.QC.jpg
3f6c44db5aaf4b269e231cc1c416cc7c
371329933aa2b67ed1adbd96eb9a8a5ca482e0d4
25584 F20101114_AACHDE li_r_Page_057.jpg
134751bb0b02167dbd0a6e8c3d1bb82a
c2a147c4be6e4a78950d517440a9a4a7e9bbd2ad
66784 F20101114_AACHCP li_r_Page_021.jpg
592db769974cc189820e28a6ca5dfd11
be27638636b7f1c2ad3a361f9141eaf063487993
6491 F20101114_AACGXJ li_r_Page_028thm.jpg
11723d8f6871124852377bfedee4f630
df0311badf6b2deda71873da8c49c19ae981a825
1902 F20101114_AACGWW li_r_Page_010.txt
0550c2cdba24896e26a50f2f2c5dd315
e81b57c6b82fd2a71b601e9683f00482a3cf08c1
72397 F20101114_AACHCQ li_r_Page_023.jpg
6c78045dbbdb77ab8537c4a143dc6653
31cf212d2ff608bf92cdcff96b1a8e4774cae5a8
11271 F20101114_AACGXK li_r_Page_086.QC.jpg
52b08cded359becf5ba4d06756114dfe
bc998632723a55a54191b442b1e66d0d5ff4b2e3
15398 F20101114_AACGWX li_r_Page_076.QC.jpg
ddac18b8c57d5e2d525beb6ea204c353
997b838f5129705ea363d1373b05edfb81d57e35
77197 F20101114_AACHDF li_r_Page_060.jpg
5d8cccc3dd6e0b00eb7745eed5f6a333
995a17ead2b5493476ad84fbf4620d51f8ae6b52
79511 F20101114_AACHCR li_r_Page_024.jpg
8f29833a40f25bde9a905a4830382775
65df822f099a31d5556ca825fdd282a29b38767e
7593 F20101114_AACGXL li_r_Page_032thm.jpg
a05d300d05a152f8e7479e7acd0b30d5
10287583349d2155c03b07720bddf83606219092
19438 F20101114_AACGWY li_r_Page_045.QC.jpg
f0a6e8535f1a22efa0fadab32572f61e
c682ea5bf7c263671929de7d0dd993903b3993a4
64079 F20101114_AACHDG li_r_Page_066.jpg
b6417859c7cfd984a2158bb92161434e
8feecb783aa2f9fbab5f606e8e7a610e6e605f28
65938 F20101114_AACGYA li_r_Page_040.jpg
8a9dd61a858d8c011d71c2720776ea80
4b9bb23a8ef50fcc94ee856b3c681f9a29e4446a
61077 F20101114_AACHCS li_r_Page_025.jpg
62f38739126d9bdcbfd41fb6befce856
e15ba21c585cbe36f4f051d665188bdae8c4477c
6967 F20101114_AACGXM li_r_Page_110thm.jpg
8198660424a0dcc4bc18704a6c1619f2
c6d466cdef355e8ca10745d35e3ffd43752e7ca8
48012 F20101114_AACHDH li_r_Page_070.jpg
0228d646ba8da8bfc9212700386f7a01
21124d0ebcdd5342a94afdcf66f5c4c17c117f1d
8317 F20101114_AACGYB li_r_Page_057.QC.jpg
cd5e4a72893b90640954277f9bca48c4
d0cc490e26f49e2064285c472f1a96a512f66542
70578 F20101114_AACHCT li_r_Page_029.jpg
7c2d6828bf706276b4c02780be0cccde
fd2022a1cc18a570ecce88f8905f83a22244cddf
F20101114_AACGXN li_r_Page_061.tif
d7c8d4225ec7632e8bc9b642f1686fd9
2fae0a33f4e667fdb851afbb6c4d12540bbbccf1
1944 F20101114_AACGWZ li_r_Page_073.txt
c0b3f2877e91dd6b82bc3bc8284faefa
47fc3af9f1578336ec59732b98e3c3e11dba2cee
62252 F20101114_AACHDI li_r_Page_072.jpg
a611bb133cc08190d31793120803f7db
14502fc9aca959f9c875ab0c3741aedfde5c6b46
3327 F20101114_AACGYC li_r_Page_104thm.jpg
d3ef92d0f98b6db6407e59d06e37c33a
41d6786b07081824212e9c43733f85eae9c93053
46000 F20101114_AACHCU li_r_Page_030.jpg
1248dfdef850c39991af0cbdb43ee444
781ae2915383239abb206bc05ecafdcd220f63b1
34154 F20101114_AACGXO li_r_Page_087.jpg
0647d545fcf5593ea21ddeeab7b0c481
3cd00c8809b49f2abe75ea4f853ea0d3a330edf9
59051 F20101114_AACHDJ li_r_Page_073.jpg
6a722f6713e5e4b5c8020cb56bd149b8
9f6f916d8a4e4c18069ded4a3ed52bf477e348fb
23859 F20101114_AACGYD li_r_Page_090.QC.jpg
8b1cf6c80d81e26a9b6bc647e97d8428
a41cd7edec59f2abae622dc93738f89da4b04d90
85353 F20101114_AACHCV li_r_Page_032.jpg
b0f4b61c4be9e75d48ffd981ff6e9ad2
4a4f0ca598b2be1ad2a073bed68c9f5334f7e73b
39179 F20101114_AACGXP li_r_Page_055.pro
b65468e34727a1715b3805f33ad34321
a99f4a675ef186c69dec166e5d55b2d4e58d611b
34005 F20101114_AACHDK li_r_Page_083.jpg
d20aaac0de9e5fe6238fb4699b495c1b
e9d6bbe51b17542c2df51a4efe3364486e2b4793
422 F20101114_AACGYE li_r_Page_121.txt
258a368f3057a09495ef2aed738d6532
6c7085438afe4aabcb3d1d2e0436a3202a2cee03
77127 F20101114_AACHCW li_r_Page_034.jpg
332b8a14ce60975e8dea2ba40dc846da
78ca51434e4697b35527d88082e0260d0bfce270
F20101114_AACGXQ li_r_Page_029.tif
a00d6a0b6f0d32fe84e1b22d23ed5ac2
6baa9b353689e2d6f5fa61ead52d2402f7100a31
72915 F20101114_AACHDL li_r_Page_090.jpg
94d0596160389c75a0ac22a6ed6ac094
5b7ba2516c965f6c87014fbeaccd36cfb5c67147
12599 F20101114_AACGYF li_r_Page_101.QC.jpg
caade566c4ce1bcc866c70f63e1ffd07
6d8f3c80701a7e230d0fbdbd318f256ad7def7b2
76788 F20101114_AACHCX li_r_Page_037.jpg
969d58cdd37df8990d5b5093c69e18c3
98134d6f67a9e6f704d9e67009033ccdec8a1401
55558 F20101114_AACGXR li_r_Page_116.jp2
16ee06a12d1af3112154eea68f4a69cd
c75b54812d27b49a8728091ba694fa7f6faf929e
91139 F20101114_AACHEA li_r_Page_025.jp2
d55ab15053734c09d7177793f9febbd0
1a52797fc3edee7a3b79553958278ae11043d3e7
78843 F20101114_AACHDM li_r_Page_097.jpg
b7e36536bf80a297c7baa16b847c0fa1
5b0aaad997fc5cd549c94b7c5256135a5450644c
29104 F20101114_AACGYG li_r_Page_114.pro
315bde1a4733ef7abf0d14c94d62495c
186d4e204f682bc5e1cfc965e26c4b7b65aa4149
75314 F20101114_AACHCY li_r_Page_038.jpg
8ead372c7b6e382b7c17421b4b583120
a270c5d485e161c835a082181370d635c0261fa0
25453 F20101114_AACGXS li_r_Page_111.QC.jpg
53bfea37266eeb359caaa4a64fb58c86
b7b657d28acf0ae50d3909bc35f56840c8e64172
91030 F20101114_AACHEB li_r_Page_027.jp2
e49f38f3752624fcf6a893d8cc843abf
8010f11a7dfafc6a9e2dd0c57eb8214ac1224891
75896 F20101114_AACHDN li_r_Page_098.jpg
522f6615fbead4644bd5b4b893ba1552
d47782b0d2f67ed5583ce09c73043e0387384d26
66437 F20101114_AACGYH li_r_Page_130.pro
eeb64da1a6dcdba4af87756a55e64e5c
470243e4f694742a6cc0f918a3a94ffa87a6a7cb
73566 F20101114_AACHCZ li_r_Page_042.jpg
4329112afbd3aa6026a11b8aa40ca92e
7186e0d0f2743ffecdeb4194c44d34b75aac86b2
F20101114_AACGXT li_r_Page_012.tif
41f8dc67482dee9cca262f95b4a10b10
52e6e4ca593cb1a482ec46402fc64371441c6945
64704 F20101114_AACHEC li_r_Page_031.jp2
2056b76628d9b3443b589bab3b699231
fc237ff4979394c8ec5edcfd035ad2d782675f60
75840 F20101114_AACHDO li_r_Page_100.jpg
90c6171ba2420b38659ab0e06433b884
e5c7a1bda95f9a5507e66e875e771d21deb3e1b2
33270 F20101114_AACGYI li_r_Page_135.jp2
2215723529c6d9a9b23a6091a43d9a5f
19f47c7763bd601aba51a4964d41117b835ac31c
49288 F20101114_AACGXU li_r_Page_033.pro
e994fc8046c6ac4d97e33ed0176ff8df
03532fede222ec60ee582129419a45cdd3b3905e
119664 F20101114_AACHED li_r_Page_035.jp2
2bbab9f13e59b060fba760de12cc397a
6886e362d09284222bc225ef8ade9dd15d900dca
21920 F20101114_AACHDP li_r_Page_105.jpg
6057cffdb03013f3ef76df044ebcd41c
54ac46c6d6047d5b58f26df8df4bda51ba063177
3413 F20101114_AACGYJ li_r_Page_089thm.jpg
7687ea1b80ed8ef9bc1ad76b66c83542
7692f428377b19e266d6f1e97e52316b71627415
F20101114_AACGXV li_r_Page_003.tif
8241fe345d19d89cb9e71a1524ae2219
dee1fbb11444ef6928dda52c98e9b88a60f7074a
118508 F20101114_AACHEE li_r_Page_036.jp2
f5175d8e65e33323aa5fc78a8ede9bae
107c1efd4ebcff5034236937f1316f292d27da29
35304 F20101114_AACHDQ li_r_Page_113.jpg
2dc0401759ef33bd9b015887093d85f9
e0320388b78964b80a8f7c4e67ce7326106a1f34
F20101114_AACGYK li_r_Page_059.tif
cce6acb7788f951f1906b4617ecd0f53
25bfb37a630e485638053922c7eba9ccdc74d460
634043 F20101114_AACGXW li_r_Page_082.jp2
7be29c6241919b821df8a81063ac0919
95523724b4c3136dc8d7a893a91c0d4fe13ab789
114556 F20101114_AACHEF li_r_Page_038.jp2
c968cefa7825cd7b71164d62f130406c
118ca15c3350a49d84b0009ecf4688369a5cf8c7
41691 F20101114_AACHDR li_r_Page_116.jpg
9b3723a31fc2892a92a29c782c4432fc
1e9e7cbdc82cdebd1dd3084ab8b87feb639ad747
40641 F20101114_AACGYL li_r_Page_108.jpg
72760ff4af559f68d6026fcb9262b088
7633d7d7a283f0f39b89c53d34205d8957f67d7a
41054 F20101114_AACGXX li_r_Page_120.jpg
cdff513d04d7e9e58e79d1c8b3b935a1
e36c18b37ae4ef36fbf3ef7194560ceaf99b8d45
115446 F20101114_AACGZA li_r_Page_068.jp2
83d4327feb3f1f1d118067a32ab77577
ee96e77ce2e1f0646ae60e42d45b6cbf37bbb83d
29658 F20101114_AACHDS li_r_Page_117.jpg
bdeca4e5297ab6804c2c9bc72ff2a064
de83cbcf35e6dd217f70eb24b600a6ca2cb4d2cc
F20101114_AACGYM li_r_Page_043.tif
ca0430bc3e6d1ab88c6ad00d9b3414c1
fff2956510902f5b9acce9ff08b91bbdeeeefb2f
45305 F20101114_AACGXY li_r_Page_018.pro
8de5e2bfba3b95a429e484446412aa2d
381db1d7637b6dd1bc62736f40676ad5326c20f7
94826 F20101114_AACHEG li_r_Page_040.jp2
c6cda00e08992a4e0465b078c7c0b661
27a5449bbe7b91c328026da5a7e9e1e0c48091d0
13200 F20101114_AACGZB li_r_Page_003.jpg
328a80b24b5f0d9995046c2eb11f5a60
0d08d04cf86d111c76339b1c3eb18e5b3e4de6e0
25574 F20101114_AACHDT li_r_Page_122.jpg
a5f99b7fd4e0d2eb798abc21963e1e70
b72c91826e6064faad37716927aafcab4af54987
11475 F20101114_AACGYN li_r_Page_107.QC.jpg
016132fe44663782982c58a6884999c0
560c158f56add9cfa045fd73f99a3843d7274133
41149 F20101114_AACGXZ li_r_Page_071.pro
8da4eebb9fee8764cf4e6f3e0157f088
5695098838bb11b6f8c091ba3badd798669e394a
88833 F20101114_AACHEH li_r_Page_045.jp2
19b71f9f6cd627103f48356e96931b08
e2fd60547c1392d3c29f49a23f55e2dae161b306
1942 F20101114_AACGZC li_r_Page_025.txt
3d4cec45aa3f94f9bb6dabba96a85540
bfd21d68c34963bc4388f239d3c16611c513418a
90741 F20101114_AACHDU li_r_Page_125.jpg
7d167dadc96af69b9a8cee954d473bda
bcbdea92b886dde97e8e64349a136c2687582bfc
5334 F20101114_AACGYO li_r_Page_115thm.jpg
c1e5664d3161ff57bd02ee18035936f1
312b093b574fa1b6ff288b2364d5eb8bff545f40
109693 F20101114_AACHEI li_r_Page_046.jp2
f7cd3415ce2dac82e2677d986d1b8bc6
45aed8ee6a7b090faca26913376e4d6273271816
384626 F20101114_AACGZD li_r_Page_007.jp2
25a86dc9e82c54ac19abde3a8b922b0e
d5ba236403de516d5ca82d9dc64184a7ca2b21d5
87341 F20101114_AACHDV li_r_Page_126.jpg
bfa1b79432afdfb6afabbde73868a481
223e9dec1c79c86473794af4f2d14b1a4fbce10e
6824 F20101114_AACGYP li_r_Page_096thm.jpg
baa1b4eb5c0579f3a8e992deae346359
327fc2d4d57b0dc37dff6441333ddfac237bd643
99803 F20101114_AACHEJ li_r_Page_050.jp2
7d872307ca21f111c5a162711147d677
9843a0c45761d2c1732739bdb96eabe4f874ec0b
16813 F20101114_AACGZE li_r_Page_115.QC.jpg
c837d39890fc2cd0540a1905d190654c
e333065f5acedf21d6f7a962631acafdb0a7a152
74129 F20101114_AACHDW li_r_Page_127.jpg
fbc5c01e994c9acf2d5fcd61094d0ac5
a657d6eeff17a742ae2dd9f7bc0d702650c12c6e
25507 F20101114_AACGYQ li_r_Page_052.QC.jpg
b5684357bf5fdad9e934eb1307f277d9
a1d6bf8cd0fb4baf69c9b37869e82611810e3366
116942 F20101114_AACHEK li_r_Page_053.jp2
2bacd1d6b232a2873aabb696e81cd5cd
93dec72ed62a1db89b02d884690a48ff98c67337
F20101114_AACGZF li_r_Page_044.tif
de5c3bc9415b769d7baa5c114eafef50
e443faeecc552e7d59578f38850e3c49cb29ebb8
1051963 F20101114_AACHDX li_r_Page_008.jp2
3cc610b7daff924a8ec5f6f3a062f3c4
652770a031cd24a67ded77674c04211aa417c28b
112949 F20101114_AACGYR li_r_Page_112.jp2
a4bf6955bb8b00e2d811ed125af2ce03
0dafe537b829c292ffb648d6a308a5292f07a44a
1047012 F20101114_AACHFA li_r_Page_108.jp2
b1c1ef1f73ee9b44d0eec760cde8ee9b
99e2a9bb747e2b19a5da46ac3604416aec47c2f9
395355 F20101114_AACHEL li_r_Page_057.jp2
26143fb951ba03fdf929a6eb8ac60d76
857d1cfe2807bc3a41f2fe3bb2bb5af2494aa96d
97570 F20101114_AACHDY li_r_Page_012.jp2
5ee004fefef60d2124de2676744c26be
9f7fb8b50938e05f735a7cd0dd9fc30074274ca9
F20101114_AACGYS li_r_Page_125.tif
25e5158680869e4d54af16713f339e50
eae9afb5406307a997ffa508e1f0c4ce18eff4e6
546578 F20101114_AACHFB li_r_Page_109.jp2
8ee531022ab670c5e321fd5b661cda9d
01a9a4e38404148d22eb7f1157bd8f558f16e6d9
77218 F20101114_AACHEM li_r_Page_061.jp2
ca072f5d5747b82a12b4ed0eb1ee9d88
9605bec63d9b08b4f565d747df2ab8d7e130677f
3192 F20101114_AACGZG li_r_Page_086thm.jpg
48830dd786e38fcd8f48e39b916dbb66
23127bcc0f20522d6a87378f76383444fd33fe22
111264 F20101114_AACHDZ li_r_Page_015.jp2
804edf18f191578593352657321076fe
d125af22b3bbba71adca63e011f755ab3ad38431
70334 F20101114_AACGYT li_r_Page_017.jpg
f020f7cb28941745ca8f7486e9a8af6e
cb6371ab4e9ada24a90209b1f40f115f297d39b9
115703 F20101114_AACHFC li_r_Page_110.jp2
4a7600d4bc73264627cf07c5ec1ac9df
dfcb1c53388d3c85838d85973f0863b59fa7347f
117004 F20101114_AACHEN li_r_Page_063.jp2
7827823623a3d0813876394c96971c2a
1815b13a6ffaed208b5ef07d96268c538ea04669
49022 F20101114_AACGZH li_r_Page_017.pro
6299ca0ee44ef69c28205fb6e3f37f58
5c73a712281c92e4771dd7e1f3e80e6c26da2a4a
F20101114_AACGYU li_r_Page_019.pro
26b9cee1d4eb8cab6fbbb8ea9f1ce270
cab4ace155be2e958f74fcc35f5960378862ede7
117164 F20101114_AACHFD li_r_Page_111.jp2
c213ae2c9537fa0580834dc7fd8f71e5
11f5e1d5fd9c3c2360f45d355b40d4691ef31817
95129 F20101114_AACHEO li_r_Page_064.jp2
12408b22c0529195815750138c51339f
7370f5016bf0b89e752ad5f0fc20b2d6b2639077
20486 F20101114_AACGZI li_r_Page_069.QC.jpg
501fe586f32efce6c5c848dfe5b65dc8
803828907603bb881fc77025621ec1f1ccb06143
3778 F20101114_AACGYV li_r_Page_011thm.jpg
af35bcd0759e277d63646686f4fac357
afa6338dcdc4cf849695cc6e5cd83bf9d80a09df
47892 F20101114_AACHFE li_r_Page_113.jp2
c7e58eeca03e4073c34793c8eeabccc3
3dd948b964c139e20997e356236ea7681a0cf586
105495 F20101114_AACHEP li_r_Page_065.jp2
35bbf5552a57c9e323fa377bdcfb3da1
47e4c87899deab186114f9de8d36506e99b26d3e
61019 F20101114_AACGZJ li_r_Page_132.pro
ebc297612bf2860083e89d6972f2577a
a121391f3ecef0eb5b394d6582d799fa10a34eda
64226 F20101114_AACGYW li_r_Page_114.jp2
746e502aa011aad225fe1289bedcc76f
1589d1050b1cb249cdf171c801148606440dfcc1
78725 F20101114_AACHFF li_r_Page_115.jp2
46deb1294e35c0317a8b61e63f2a1a27
5e22187b01ed79ebf4dcc8df10af23d052e67ffb
89461 F20101114_AACHEQ li_r_Page_071.jp2
31763779b44c5a529adc31f7ef14cfe1
c115a639172bf426beb2c0d5ce2d211d50afa460
11594 F20101114_AACGZK li_r_Page_003.jp2
bd7d253964de8bb8e2c59fe070316a8e
778c29ea60be27c11126b8d0979bd86801a53312
6373 F20101114_AACGYX li_r_Page_059thm.jpg
fb31bc598b96b6725c8bb8ab1355c55f
8e5e056114b0eb1cb09b4075fe338b0f418c7fd8
124173 F20101114_AACHFG li_r_Page_123.jp2
b4de30dae775ffda47a9b6e134e8af7b
da85b97a979f73d8876ee791f1aabb509063a67b
776896 F20101114_AACHER li_r_Page_073.jp2
df41b661553d7b47a07cbfbd3fd06294
ecab922555205712997a54f65f030236338890d5
6782 F20101114_AACGZL li_r_Page_052thm.jpg
4321931cc576f98b5799da0359a86fb6
0b94d846f12cf504e992894c7b6acccbc06dc8fa
78219 F20101114_AACGYY li_r_Page_091.jpg
b2f1cfe3c681878cd1141c22fe7ef95e
7131fc96a4f2d8bb729913ab0c4eee2a5cc1ac69
622622 F20101114_AACHES li_r_Page_083.jp2
6388c682362e0b2ddb6d336a46c413d7
fc090c648d002b33e7c23dbc2372b80eb0e154d0
62258 F20101114_AACGZM li_r_Page_129.pro
b1c5fe85b5c7c9fa2f5fc652dce35255
fc74a57c07969d55a50f12e8464ccbfa9f87142c
63230 F20101114_AACGYZ li_r_Page_126.pro
04d28276d5a1ea43e701c7990c3e76b0
87e28ac9f468b0ce8e65ff6f37518aa4dcb580eb
133542 F20101114_AACHFH li_r_Page_132.jp2
0cdc17f5d4c2c4bb230a2e77498fa7ec
bc2e2e137ba11699d71d538d7542ab0f9a5bc08a
586991 F20101114_AACHET li_r_Page_084.jp2
8b7c62c40dc56a9617b2236736f434d0
dc713a191bf323a67ed2a6af71ca6b740d8c32d1
55290 F20101114_AACGZN li_r_Page_093.pro
362bdd43284aaeef5b3f2caaf05f4f4a
07f5ed840b0fb0cc1888bfa83a226e473f41f185
129136 F20101114_AACHFI li_r_Page_133.jp2
8298859e7d1862a7ff591f3fdfd95a6a
0f8a4f44efd9c00d1304c1c9eab79d97a6d1e3de
630301 F20101114_AACHEU li_r_Page_089.jp2
8bad9c3ba8de25529d2521ba99064538
baf278a9673a5e31356c5d11d0011d44cce6113c
1120 F20101114_AACGZO li_r_Page_107.txt
e177a6f60975b56d0a92d3b960119e55
0811f2a68ed002ce05744f2827f129ff9ba43f72
82895 F20101114_AACHFJ li_r_Page_134.jp2
8642990a9547f0d1cbe4efeb15e8d7dc
76eb85450cd9dba64baf8cb5201c2d21c2ca8279
118203 F20101114_AACHEV li_r_Page_091.jp2
0477bb21e7ea571dd87a0e1053a815bc
6110177852bf65cf0e4f9faf29667a9eec9931d8
575 F20101114_AACGZP li_r_Page_135.txt
c3c9cf2c7085eb3cba0abe45083cd55c
0b290a2bfc7020fc9520faabb84d8b2310dc7749
F20101114_AACHFK li_r_Page_001.tif
e8326b578cb1ff4646639b0b218fc922
a46cbc22ca43acaf01093f3a99a055429dcc7cb4
113799 F20101114_AACHEW li_r_Page_095.jp2
74387c7e099571edab80d3874757465c
4540ddfce3a2820043f542b13c342133d1ee5c7e
3906 F20101114_AACGZQ li_r_Page_103thm.jpg
c611d5c6d2aaf437991c802c294e1df2
6cce1a6159172b387e562ffa9c865c51594c538f
F20101114_AACHGA li_r_Page_040.tif
6853b8ac7b833e43fbcd2dc97f7b67a5
1b4c7a1c10dc2a3b52b089d2fe97ae6e06dd17ab
F20101114_AACHFL li_r_Page_004.tif
a7fe3ee13e162ec05a4c464e91210437
5b67de214a37a3368a6a7699bf44b1cf6cae06d2
118148 F20101114_AACHEX li_r_Page_097.jp2
0a198ee4b900be4f143ed2c466a3e32c
b8a4d87a4fb5a7830d16c3bd2bd2ae77dfe1ffcd
1830 F20101114_AACGZR li_r_Page_021.txt
0371e19868ddd9fdb8d7d75476590066
38dd587865c373d7a692d16a841e30efd2b64a75
F20101114_AACHGB li_r_Page_046.tif
3fcc31ef4216dc0386d621ffa1f8b234
ac0a863deced19d3ad961de7261ce32255aba925
F20101114_AACHFM li_r_Page_006.tif
6d365cd6a88b5b9fb3fcb4b8c075acde
e724957243a92c8c855e7443e2d98660fb3f2b46
50055 F20101114_AACHEY li_r_Page_102.jp2
0569997b27f465c56595ffc2685d1c0b
9c1de36f094b18a434e551090bf6f38ddfdf7d85
69208 F20101114_AACGZS li_r_Page_033.jpg
312b3c363b4a7acfcee71173f84cea04
963ee713c50014a65e8397a9a8670e26db4d20f0
F20101114_AACHGC li_r_Page_048.tif
d094c7ba7d5bd51b2658630e82179ad9
013f639ab08f864ecbd0984d3d3f08fd18ac0dbd
F20101114_AACHFN li_r_Page_010.tif
144e04c19586ccd7b582446eb80c1834
b79c76c9b5b48dc080725fd444aa71ceffeae352
663333 F20101114_AACHEZ li_r_Page_107.jp2
1d3473b4e8ac072ecb874dbfabfcf5ff
46853e770352718bde8ca44433a83cdc58d682de
19068 F20101114_AACGZT li_r_Page_121.jpg
00aa803cbcbb54d73bd11b7f1b839a27
821e8169db477cf5ad17d81c4848442ddf23b259
107698 F20101114_AACGDA li_r_Page_042.jp2
05fedb5fb637fe62526920d1724d1eea
487a122ecc146c4ade2d83781a371f14257cb836
F20101114_AACHGD li_r_Page_049.tif
d03a233575a861dff610d8f67e094fe8
7f6ac496bcacdd3c89067fec403737db0683acc0
F20101114_AACHFO li_r_Page_011.tif
4972219f79779db882f65b462dce6780
d4a749c0f4ea327a744dc0f2863b41014c156083
90659 F20101114_AACGZU li_r_Page_072.jp2
9c607b041bf377de7ebe6828e987918f
a657894070efda63b94078c9cf54f013dd5ca062
21781 F20101114_AACGDB li_r_Page_012.QC.jpg
f02da82b797933059e47c7503d75c069
9f8bfde3415c38e2adf556813aaec0f641f633fb
2025 F20101114_AACGCN li_r_Page_007thm.jpg
b3f026bdf3a553049db5f9856756f483
c9f9074d5fa19a7d9105ea9acdf4936a985b22e5
F20101114_AACHFP li_r_Page_015.tif
b56f6f5e9a2ce23cb87752d5231a677d
7aa79c688a193bff05fcb5b5fe401a5bca25040b
5862 F20101114_AACGZV li_r_Page_080.pro
ef709efd3990cc19012ae44f3502d9b4
20ab6fb3284a0d70a57a14754c95a02e5aa7df7b
41484 F20101114_AACGDC li_r_Page_075.pro
ae4f0d33e3311eb1b162fd70a6823757
9aa5a04d462d0db1c7d8dcd623299fd114c66be7
F20101114_AACHGE li_r_Page_054.tif
c24ad6671f8ca2ce2184e906f45df6d1
1af3c9c12a03f6bc12b4b65cf12275ea62c29353
6750 F20101114_AACGCO li_r_Page_022thm.jpg
79fb5864fa05574af231a413ea86abb2
3bdb479fcbc7fbd706b6c1e00e3bf0b21f611229
F20101114_AACHFQ li_r_Page_018.tif
dc428e28cb70cf605bfd67460143d689
f71ffda341fb2615579db7b107d15f2b3b23b099
1807 F20101114_AACGZW li_r_Page_050.txt
b54f5d786b8a547d3b41cd9900b60a11
91d0ba2f8d04e3e35ad481e62d7e9461380a44da
3200 F20101114_AACGDD li_r_Page_109thm.jpg
d2a6d2d9131809ba3ba2fc2107222911
cf750481e7a6459aa4f6b7d69ca7f3e1121b37f2
F20101114_AACHGF li_r_Page_057.tif
1445e2e7200b680a2bba63672219c596
423cf489c29c00de8e77458d90f9f7825f46b76c
F20101114_AACGCP li_r_Page_028.tif
41a53930304776f4868d6cbc7dc32f23
21de3a0644cc52a8d2cc07bc639d6ec32f5fe174
F20101114_AACHFR li_r_Page_023.tif
a5a12363b9b22256b003aa70e40c6131
bdc425680efbb8403896a0f9f95344424d2fe69d
1148 F20101114_AACGZX li_r_Page_084.txt
83747e2079448be0bc33fed609dff453
30e2e8b2c0ee7598df3594b4f68fdc3a00522521
17321 F20101114_AACGDE li_r_Page_089.pro
e2ffd70b4205185a466a9dfeec0aea6b
88aace574cd150281b1b1207cbc4c7b9b703b639
F20101114_AACHGG li_r_Page_062.tif
c469bbdf4a4da967cc5924159ea113d9
906ad8db3324227317018b6c0b93f405110fd74b
556045 F20101114_AACGCQ li_r_Page_058.jp2
f349210e6ce3df1b6109b3fb15c599fc
b76e809d78c341223da8ba3a6c1fdcc0be5f7351
F20101114_AACHFS li_r_Page_025.tif
ca8caa856c47380416804880213ea46c
22826436165e634e50f8877a1b7859b6cbd5bb9c
8513 F20101114_AACGZY li_r_Page_119.QC.jpg
ba4dc41e2f64072277bcd13ff616c4d3
48c7bf45896b993f6106526f8077b1cbc6e25336
24632 F20101114_AACGDF li_r_Page_009.QC.jpg
ef52a1e1c3fdd8b1fe439c884523feb6
9ef0859008efeb12dc6d5a0e38bf6f7535939031
F20101114_AACHGH li_r_Page_068.tif
f0bd0208811e49d615a63856eb533523
389a5be7cdc930533a94a4a8084550493c83aecd
116545 F20101114_AACGCR li_r_Page_037.jp2
6ca9045d23763eb09ec9e991fcc7d6bd
e4134d13934aa21ca0c628afbc1a2ec89ee4100e
F20101114_AACHFT li_r_Page_026.tif
e279b4aac7bd6be18821aac57dd8471b
d5766be8f758ede6eed8872ddce9d51c8cebc1ed
F20101114_AACGZZ li_r_Page_002.tif
d80d2f851c4039741f3dfea772dab178
dfbc7c75a669460b57d6fce654cb264b6da3455b
4876 F20101114_AACGCS li_r_Page_030thm.jpg
90e041367f0266729eb2479c76289690
8ac57104548a923f8af2917cc4eab453eadf60d4
F20101114_AACHFU li_r_Page_030.tif
721a697b8e0ac3a7230a92b6f700ae01
9cbb8fbf53e234ca8cbcff02935030f2f710be23
4099 F20101114_AACGDG li_r_Page_003.QC.jpg
9d4d4f53a33882499d436a344d8b5078
1cb89a7b0c38dd1eac4371dd99be6a0154198c54
F20101114_AACHGI li_r_Page_070.tif
89dfa3e4c6f71c5af39ccf8f3c6c3c27
c968dbfdae03ebd7579d0ea5a5d98fc1f9076c2c
25591 F20101114_AACGCT li_r_Page_051.QC.jpg
f995d2d98feb77b38ba05c7623356db6
e6901ae2ca87c65735050588bf9e02f3f4eba89d
F20101114_AACHFV li_r_Page_033.tif
d8f4594f66221bf5f8e7d88f17374be3
fcf9ce14bd982cc975895904d624f3182cfb20b3
21960 F20101114_AACGDH li_r_Page_018.QC.jpg
d3adf65e3e60e36add40c010bb6ce927
c691e5123125935f8cfe4469562a275d87161825
F20101114_AACHGJ li_r_Page_074.tif
0b3f2eb6808a607f1ac303a704e89200
fb93e7ac5147e2833c07c45f92bdd218321fcdd4
F20101114_AACGCU li_r_Page_071.tif
ecc049679534a99b2b63e84d9c4eead2
a776c8addc6ffc471242c2b0ac7411090411a875
F20101114_AACHFW li_r_Page_035.tif
f0a7ff8e2b06aadb5bfb8287bff1c937
f402f39ee4abd6a73aaa7b0905e7e5315c0f8a03
1448 F20101114_AACGDI li_r_Page_134.txt
a8ea110cbc5bd4b9fb78d1276b2a8fbd
ebd137821bdac5fa02bd0ad51d655e3113e4c386
F20101114_AACHGK li_r_Page_076.tif
117d3f153c304d26ec2dbca496f629bf
d604bd20dd74185ac49d57813ba3ee133dcf895c
F20101114_AACHHA li_r_Page_132.tif
db204130093bea547eb9ab81bc41a460
da0f04730c4a920a5ce6168e0da22e12997dae91
2168 F20101114_AACGCV li_r_Page_093.txt
3bb96549f70af1cf8b5b260269b080ff
c06bb1cf2b014d5351e058295f7c98c5d45cad83
F20101114_AACHFX li_r_Page_036.tif
bcc8e84f3e312427a77b0cb245abb3dc
05a6572cec83bd00d73bf43a28881917031d3dcd
39502 F20101114_AACGDJ li_r_Page_107.jpg
50967f8bb0da40efb2b16b18f1f8e5f0
b8e69a938b0609576e74bb63a2310184d03592d2
F20101114_AACHGL li_r_Page_078.tif
af5c4c15ccc258442ebe41421d54a71b
a6cdbaef9c682283a23355b807408312210f2e28
8580 F20101114_AACHHB li_r_Page_001.pro
44fa0c22303aae305626739a929feb0a
f483603140e5d298004f95e2b7e26e6f135bf43c
F20101114_AACGCW li_r_Page_083.txt
644aa9fa51ee8b9097e07fa77b392b85
1ca36b783841a441e36ca3c628711b2bcd95f0d4
F20101114_AACHFY li_r_Page_037.tif
f19b72836aaa2c59f427fceba80c25f7
3600240e7ada3dfac4d3fc492e9cf014695c3ae9
16695 F20101114_AACGDK li_r_Page_048.QC.jpg
cc1de17f543bc73bd95488eaedb62a5c
7e1b6ca7d088f6ec862ea307790c975b8ca0c0c3
F20101114_AACHGM li_r_Page_082.tif
057bf86f5931596b551ee701b7232660
aa94815ef31e194ff6402e5109b0522526447421
752 F20101114_AACHHC li_r_Page_002.pro
ec5a1d7026b047a8395e15c02d152a17
e0120388ae70bb3556b1c73b4ca041165f3ba226
72947 F20101114_AACGCX li_r_Page_022.jpg
e66e0fe25a59b3397c2c27190f3c1edf
cd53255422d2714f571f84e70e7b682b400cdd17
F20101114_AACHFZ li_r_Page_038.tif
02420534fca84b32d72fa4e24ed4d535
0c0d5de17a5195015e3e93b78b3f32b744ceeae0
69536 F20101114_AACGEA li_r_Page_075.jp2
2b7febbad3e42664f2862b7c9fe31b43
204ce78a3822874708b505623df573e446673c4e
F20101114_AACGDL li_r_Page_126.tif
1acd29e4c8b255b34b5b17d35e68981b
43cc223738de1ef798cc511a3e80846ff2924a61
F20101114_AACHGN li_r_Page_086.tif
2e759913c99fa3516d7cc872bcb85a55
00daf1c593c1c300f616134fbebc1ea3d6b91825
69096 F20101114_AACHHD li_r_Page_009.pro
db6dca26c46ed18b6e84953ef1ed21d4
5cb5dd3dd630bd6f190ee65eb82939151036c9f3
74044 F20101114_AACGCY li_r_Page_070.jp2
5f3d820446c4fe3c42a1a5190cc52681
4e6553dd70f9ac9f2e32c61f9068ef4aca25aa90
2113 F20101114_AACGEB li_r_Page_051.txt
92381d10dae8ce9b28571f17b90c033a
9d428ed9d1d400e65b999a1d373a7423dc4a221a
43551 F20101114_AACGDM li_r_Page_082.jpg
0767fadeba03cc1ba4d24dd08ba09c36
8907913e6a43b3fd10fa317410a908e51fc189a1
F20101114_AACHGO li_r_Page_092.tif
eb823405f0419052fcc54b888f21fabb
82db22968698133c24e3571fee1bd422ae7c5112
43377 F20101114_AACHHE li_r_Page_010.pro
19d04039efda128d4a31ed7a9eaaf915
6dabf3eb390dd4d29e4975eca4a2aea7dc941190
55145 F20101114_AACGCZ li_r_Page_060.pro
e606225a0975e5d7019786e0e4c137a8
cebd90c71f81b664c79cd36639aa5707b8f3ba68
70126 F20101114_AACGEC li_r_Page_059.jpg
570371c115602eee99b582088e22b482
2401e22ac5ef5630fef7a89924e01eb392e4d53b
40784 F20101114_AACGDN li_r_Page_102.jpg
33688d44cad3b7af5abef81f2760b89a
e97f219e9cb7d09b01faf37640d762cf1649cee6
F20101114_AACHGP li_r_Page_093.tif
109c6d49dcfa23dbed83120f6f63e4cb
565cce12aca3c740c2781388e0472e80cf87f576
23584 F20101114_AACHHF li_r_Page_011.pro
09ef6149ab2e3c0d0f18aa13f903ee41
7b9674750f3626cd0a8444f34fe82df2fc24a04a
50281 F20101114_AACGED li_r_Page_076.jpg
ec79194161e2e2e8b445c60b521f7f10
d8c8e01acc81f8137ae7ce33d6550083b5b5a5c5
7200 F20101114_AACGDO li_r_Page_125thm.jpg
7c709ce42cf454361bb8af9cbce6bedd
53f8250ea6c8b4b5e9582b7f6dab7a935c7ebf9e
F20101114_AACHGQ li_r_Page_097.tif
5f28502978b4e553e9ab7a2974a242ef
fbfa779c4508c3701e39841afa09a55296cff7b9
54063 F20101114_AACHHG li_r_Page_013.pro
bd81720b2071dc09efcd7a58a1bebc82
ef4e5872f6c3051eea73936792a833e9b96d4296
67262 F20101114_AACGEE li_r_Page_019.jpg
adffdcec06fd43b46a34d712cb1962f4
a2a6c0980a79e7d8fdb802c25b97512ac8bdac21
4857 F20101114_AACGDP li_r_Page_002.jp2
5436ac03cfaf6c266724c03c30d8dfb5
23bd45827189b4cb16c613b8d32b640d534ad60e
F20101114_AACHGR li_r_Page_098.tif
b9ef98e1f3f2d9131c7808aadb30f802
ccf274c962bb7824bce548154bd5d205890b4c3f
51074 F20101114_AACHHH li_r_Page_016.pro
46cb80a94fefbbde205d8c237736bd3f
038090f06a357f728866f27618942f0a43cdf327
538188 F20101114_AACGEF li_r_Page_088.jp2
35a3ec6ad2e523bc18804cbbdb747d7a
f57601c20d85b9bed4c6609fafe76e81361abd36
F20101114_AACGDQ li_r_Page_007.tif
a295fdbf148812e1250eda7e654fbdf5
f1366a626be87dcbab5b8504a880e4dba3a29356
F20101114_AACHGS li_r_Page_103.tif
1cfa05caf36954a1f638870b8a1d19c7
21f9c5ae0e7f78aa681f093370adadb32ede78ec
46066 F20101114_AACHHI li_r_Page_021.pro
abbb423096b96acc9f8eb9993a035fd4
bf814b5e5ea9cbf2ae1fbc55ab5ab1966d20f632
F20101114_AACHGT li_r_Page_104.tif
202e0c2cbd109914ae11410c7303d3fc
7a0d69b0a18be47db57ad75481a81c1ed713242e
23975 F20101114_AACGEG li_r_Page_049.QC.jpg
72811f805d178e951a0f40cbe49836d7
8d50cf72f516f631f0bbe92a3fa4976381733e22
24164 F20101114_AACGDR li_r_Page_015.QC.jpg
9c8b3e67c1284ccb4012d308e30ea953
ae5a48cf135f0b6f138b2d8646983e8dc9567485
F20101114_AACHGU li_r_Page_107.tif
e866731e4697b78651979aee7e86e638
e7325e3ee69bbaf12350ca1d5f7be2d91121047e
108470 F20101114_AACGDS li_r_Page_014.jp2
efae0d4e356daf692cc2059bdca413d9
1d18df246bfe6a48bc85ef845242ffc8eb80d928
43356 F20101114_AACHHJ li_r_Page_025.pro
f4ad94680ad8840c26453357b94f7607
01e0296580cfbf8bc0fb77a0344591de468e26c5
F20101114_AACHGV li_r_Page_113.tif
f2d5aa9ea025c0a99d83c86b245b6ca6
ddf495214166e078e7f4f9002a7c0699bb73e38e
43767 F20101114_AACGEH li_r_Page_054.jpg
7308e09e402074d0e0afb163f3562d59
27d3028c4d95e0a6f26923f745cd471c95a28a15
9892 F20101114_AACGDT li_r_Page_103.QC.jpg
394a66c40c5f05e845f279a3b8465c83
e966f188382f39c0d65607c0abbd97435e90894c
45524 F20101114_AACHHK li_r_Page_026.pro
490a045f474d8141bbb5420aa0cbf212
a0f8ce3ed399283d19acf74d373f865526893b51
F20101114_AACHGW li_r_Page_115.tif
909c6d2e20d54253f8ae3741c19a6358
29b7ff3489464d5b7c8cbd7c5689ede682ed8abf
F20101114_AACGEI li_r_Page_020.tif
354d9a8fb9435c43e6c10e9c8717ad05
3115dfd5ebb96bf95a7179a734c7cfb863c7f074
22843 F20101114_AACGDU li_r_Page_059.QC.jpg
532c7b7fa9130b2f5dce159d3d99ccfc
738ff635bf74e13f1e8dcb89250c22b78cfd8941
53934 F20101114_AACHIA li_r_Page_100.pro
f336e17d158953eda95761253cf38e24
056eae286992dc604068440de29e0e989381e95f
48534 F20101114_AACHHL li_r_Page_029.pro
c28256c7cc6423b9d064c466df188a62
53a262a8b5fcbc7e99acde5a56ceae001386e0ec
F20101114_AACHGX li_r_Page_117.tif
8a5fc34d6acdb9974a3e7ec16400533a
982123a03fc4ec679d994a1c7bb31359fa614a11
F20101114_AACGEJ li_r_Page_058.pro
3765a21a5e4c09140ebd75893f7e0f96
b9e36ea7704d21fe60b5cea9a88a355cabeec483
1881 F20101114_AACGDV li_r_Page_044.txt
3456948bc18c92c0c91e58a1b879b2e5
12339e51993542d3bec657d095189ff18620fbe4
25715 F20101114_AACHIB li_r_Page_102.pro
191760e899bb266d396cc1a866932285
36275c1b383a629eaeef92fe6e6554b584133dbd
31364 F20101114_AACHHM li_r_Page_031.pro
8f2c342bad6e4adf232a6bd3dde0879e
680502cf835adb8b4e3225cc97f654c67bdb113b
F20101114_AACHGY li_r_Page_124.tif
0e91d2fb943cf4c8de8537a6596cd5b6
7d56b7e894b61e91a4a4d44e9ad84696e79b861d
26236 F20101114_AACGEK li_r_Page_091.QC.jpg
449e3bdfdd16c715d26004d018e7f8d9
e6c812a01523a990e44e970b7ea2058474b8b2b7
114784 F20101114_AACGDW li_r_Page_098.jp2
a6004fb2be1e49ad705a3d7eab7d8cbc
6d434bc50eed2c1c790e86e4f62050c3cff9c3c1
1660 F20101114_AACHIC li_r_Page_103.pro
ddff7cb9c1fad518ec2bbb064daee0d3
2d4112b3f221ea72cccacd5690ce8a8cef1b2ded
53848 F20101114_AACHHN li_r_Page_034.pro
2f10d1f15af52abf035ed742175c8cc2
75e8f7fc598aefe7c2e7fdfe0dcf7a2b53dc419c
F20101114_AACHGZ li_r_Page_129.tif
274327c3c2dc7ad7179e34270ac35530
9de955472e3ddf5b28149dcc831f96bdefbd05d9
44843 F20101114_AACGFA li_r_Page_012.pro
fca89fed2e5fbe682b683922e2aeb460
e7b53b7eb52043f5b098ff8c2e5ad328aeeeade5
92419 F20101114_AACGEL li_r_Page_005.pro
bbf7fbd20d19acb5b4283787441c962b
66dd6a523fc2d7a24afba90de42923e90b76a0ad
F20101114_AACGDX li_r_Page_047.tif
c798d952a66f6e9739b09b3d025161c7
b525fa3db881eeb3040c4f3794723aa854b130c3
8979 F20101114_AACHID li_r_Page_105.pro
87895de96220efe545a024f35056aa12
dc02dfdc74d37fc7cf29b7ccfeb6aef95244113e
39998 F20101114_AACHHO li_r_Page_041.pro
a5a10bae247582a8ad2e2126ea4599e5
921772ab640149c70d7b3dcd7b3a807f70a28503
138404 F20101114_AACGFB li_r_Page_125.jp2
0b4e619f7fbb90aeb714f5deaf28d0c4
94e824507e7a9842c4d4d3d15128e0967718ebd9
17208 F20101114_AACGEM li_r_Page_061.QC.jpg
aa58cdd664229065523fdbc511dc8ef7
17a209e08e4110541e63acade66d7a48e20e7c46
6246 F20101114_AACGDY li_r_Page_066thm.jpg
ddaedc378cae47160ec093fcf520d0b9
3494411e03106174fa45d10661dcf337c4607a7c
6339 F20101114_AACHIE li_r_Page_106.pro
77d3c2cb6dbbdcf2871e77a4c1caf888
b17cf835da4b33e4f5784d074e61035eab440518
34886 F20101114_AACHHP li_r_Page_044.pro
da2deca689635c3b9275c97d35a7e774
b736549fde232fc2dec68baa0c2215687a6f3e94
F20101114_AACGFC li_r_Page_088.tif
93463f674e03255debc87c63cf11a9ee
9f71c121a6d908e9d4fc6ba2e37f8ba189f63ba2
13739 F20101114_AACGEN li_r_Page_116.QC.jpg
2bb06176cd290e1f6910daf87504b5da
f204bd95279287e7f6ab98211acf1063ddc9422e
2007 F20101114_AACGDZ li_r_Page_059.txt
a53af0469ab0d4d528549ddb58d3595e
f1896c7fdcd4463c110d578018b87cf4fa40a340
20683 F20101114_AACHIF li_r_Page_107.pro
dbcb8ca7c71a40ab9509ae7ac08375de
301c07458467333596ba2b5e37687c4e279cafe3
44335 F20101114_AACHHQ li_r_Page_050.pro
859cc952baf6b015991040c5aa7a4c25
e392b69409efd0dcc6aed973e8b4ad8ec3ad2cec
817790 F20101114_AACGFD li_r_Page_103.jp2
8aa186313e20dc5ade686ceec12381a9
2c9b1a1c4053a118b92e953801dd841cb85e8cab
18508 F20101114_AACGEO li_r_Page_088.pro
d93b75a10b39743170dbfee1ac8d5758
923b51c60bd02b2543b54d1236e030813f400b06
8690 F20101114_AACHIG li_r_Page_108.pro
1a3d2cfb002f947b17e7e6e3ef42fb4c
38f8d969afcc88226d9ffc021a889481970b9750
48360 F20101114_AACHHR li_r_Page_059.pro
20dffb78002ca36693d12c535fe8792f
7a926fd81e8de23cc3f285e492084ec661196308
2135 F20101114_AACGFE li_r_Page_091.txt
d7e041ff400c762f48d830b8cb33b8d2
fd55bf3f3d8f740ae823b71a4fb49ec6132142a5
26593 F20101114_AACGEP li_r_Page_125.QC.jpg
5069e0ad98e1086648c8ec872e36cc69
89915a343f88cf60a89aa99bd615eb22b813376e
21348 F20101114_AACHIH li_r_Page_113.pro
3ac0127c33ec9d3bf2ebae0906d6ab47
94480533fad5654560ec6474d4e037576642fb72
46232 F20101114_AACHHS li_r_Page_066.pro
f34d81b7c2b53dfc12311d2745d56384
ae8013d911281b2c46c0317215cc51fb653df0f1
2130 F20101114_AACGFF li_r_Page_037.txt
5b42087b15d7bba7ba2d9dc45dbe3b6f
98429529ddcdce9441b0025072b646edd7f71627
6425 F20101114_AACGEQ li_r_Page_017thm.jpg
558b24e1f0f4874c5c62e7681b0f56ad
9f50b2fa0e3d49dc197e9cab50655f8f3dc2c979
36243 F20101114_AACHII li_r_Page_115.pro
1250c74e55ccc93ceacfba3e4dff8b59
ef3c6260837dc3205e6c7cb32c67019aec49a4fe
6344 F20101114_AACHHT li_r_Page_074.pro
29a0b906514818b597f102d20a79582d
69d3675fe4dec77ee278ff8bf94e2ca24d1db03d
24271 F20101114_AACGFG li_r_Page_046.QC.jpg
ada79a553dc27e5b7eaf472f51ffd62c
555577db9cb0a22733d8bc43b57eddca89e3a502
26425 F20101114_AACGER li_r_Page_094.QC.jpg
7b89c8943ee32586b631044362667958
5f207c14cc9211a47f4a08d752513e25afc543c2
27551 F20101114_AACHIJ li_r_Page_118.pro
e2b8a8fe7a194797ac4e6ee37a69fffd
38e2f8a980f45ebdbcb1fa42e43d3ea0498a29d3
15289 F20101114_AACHHU li_r_Page_077.pro
9b0f1e46db3693cd5f4ca28257563be0
2c6bb52078a87b2207e684581d9d0d31f19e1ee6
2020 F20101114_AACGFH li_r_Page_042.txt
d8b311080f9d02ecc27dc001f1f32579
6c6981cd0bf5ee5c72e707a1a30fd442bf7b3a67
6522 F20101114_AACGES li_r_Page_090thm.jpg
0f743b8e215342c6222c4bac912aa3d7
7290b550a8b03780dc4fa0552366c76b1ca9950c
17466 F20101114_AACHHV li_r_Page_083.pro
3539f3d32098f91d13db7c15bcf5cb06
b651804b45261f691ca0a5b82d45c341f830d60c
78464 F20101114_AACGET li_r_Page_094.jpg
6c9b197fba8fb7b9205dacfe0c1b400e
ed70e9eca6362a6f7f9766c1b122cc654e562701
12556 F20101114_AACHIK li_r_Page_122.pro
ce7fee80c1eafdcfeff078928cf2efa7
74be5436b8611a08e8cc9a150049905e5bf3558e
21112 F20101114_AACHHW li_r_Page_084.pro
6a11298fe19089d6815f71f4a41f3eb6
ba41aace6858f00e34aecf952dbf993aaa4f43b0
F20101114_AACGFI li_r_Page_019.tif
6402b34f6e27e964e891debfbe98249d
88271c8546f1155190333f07b1e01954f19cc003
2009 F20101114_AACGEU li_r_Page_065.txt
c2ad3792a5795c73e94cf97e3da69389
258f7e00b9d240ad70a3b49b8af21daf54b5cc54
2080 F20101114_AACHJA li_r_Page_055.txt
e65eca59584ec63993aed886f3c8569e
1980c1f4c3484bbcfc938bde038998c63f311c93
57012 F20101114_AACHIL li_r_Page_123.pro
062a1922b0d438532dd2fa6fd0b7292e
81a2199006a9239ae06905643192b627c30f0af9
20420 F20101114_AACHHX li_r_Page_086.pro
b7112161f6cd60d3bac87d66e9a99b8d
6f12eba52e7aecfeced8c3116aa3313bfc1f433f
F20101114_AACGFJ li_r_Page_122.tif
8d89d4127afc99fa332e30fcbef1701d
0c7e0ec1d17b1a16867a4479b89d6567fde214ed
76080 F20101114_AACGEV li_r_Page_096.jpg
aa0c6d0f8e6a832faa5f9adf2caf59c9
f75b6c0358f4861b54e5d234cd14f5418cfe84b7
401 F20101114_AACHJB li_r_Page_057.txt
c93781748efcb5553b0ebbb7e9395823
6060e25a76d308722bf7d35d4a0c434a87f1e058
35809 F20101114_AACHIM li_r_Page_134.pro
34fcc94278f9440c7ce2f4e0d871b5b2
02130c26e10e24bd65f81ffc7d46f234658fc110
50665 F20101114_AACHHY li_r_Page_090.pro
a8fdc3bc5e53ce7e7f92137304dae0bc
012ff1a87abe0163a70e7a599b4f748f18f4cf0d
114607 F20101114_AACGFK li_r_Page_051.jp2
750673c953647f10aa0a96431fa04012
1da399b29cc33bed15bb69861f2789bf74b8a748
10615 F20101114_AACGEW li_r_Page_087.QC.jpg
60279fc27165639cef7dfe5adb3f0234
dae5710e2a2688ba9f8ffc875c6b365205238aef
2169 F20101114_AACHJC li_r_Page_060.txt
3faee76fc7d845a9359b7700917912e4
f9d352625d242fc2b72b9e5f182ee9a1acbde2b5
3779 F20101114_AACHIN li_r_Page_006.txt
ea6f642d505e0adb00feb7fbabaca70c
6ce84a2ed2f1ae0d8df9eb7bf5262d31e7f6fea8
54384 F20101114_AACHHZ li_r_Page_091.pro
70e432586f9b9be879c822bf3a26d372
32b1da5347b3dc92777090e0a7925534403b20db
F20101114_AACGFL li_r_Page_118.tif
9a9010a8c8beceea6b14a411bfe7f62b
c341e48209fdb8d5e65243ddb1477388bd230548
2078 F20101114_AACGEX li_r_Page_046.txt
7d4636e7e8c4e3d1763f75e16b43cbf0
ae61487e86542945d2179e36fcd8780fa67f73c0
53029 F20101114_AACGGA li_r_Page_032.pro
a3f74957bfe65a7cc39d8e719fd4a065
2e316668892c325761beb3dbdb21616998ad6992
2149 F20101114_AACHJD li_r_Page_063.txt
af44ac1a03d7837762ecb3cadb25a7bf
87b00c1f72448b9c07f1027f2860cd6ce2b60193
1740 F20101114_AACHIO li_r_Page_008.txt
eb181734421ebf86923ccf6bf3b9af60
95f05b86cbf27b84dc349cc3330082c4aed72d99
F20101114_AACGFM li_r_Page_084.tif
86c65e50ba7ec55659d1b91b26477496
cdc51803c838f3ba9bf2d102186c4ea902dc45d4
F20101114_AACGEY li_r_Page_065.tif
f898060e470999e496d3ef0dc801117f
3edded78f1583fe3bad1adddd7c07a04d3a8d32b
F20101114_AACGGB li_r_Page_021.tif
9d6d7cf78956762f02faf653605ac9eb
42926ee495764931344194fa36910ec1c758fe50
1936 F20101114_AACHJE li_r_Page_064.txt
d99d1355eb1b939afdb75651be14a5ea
a393adbb9de87293ce60089c36a5b54e847395ff
2124 F20101114_AACHIP li_r_Page_013.txt
a79c619644b19a1724badcab04541104
393ade49dfc5411b261900e2750910072d49d833
17657 F20101114_AACGFN li_r_Page_087.pro
0eb0c67351e52e254cee6cc4cf09f001
b76263b77b0d8ed0247db1a61ef629ac4cb91419
54972 F20101114_AACGEZ li_r_Page_075.jpg
154fdc388d7632c72540b404acae3e7d
bf7092f22059fc0b8b36b4b57aa4d73c52fb9147
15776 F20101114_AACGGC li_r_Page_109.pro
2884d4c1a68e3a6993552535a5e1028f
ed2575ac95fcd657bba574e759cc37e6f0ffbe43
2144 F20101114_AACHJF li_r_Page_068.txt
67834a378e5e3c755675b20c1f638b78
331ee86f590f5956553f19e520d7ae533719998c
F20101114_AACHIQ li_r_Page_018.txt
33cd49e5eedebc4c6cf35c70f188d94e
89631e3e1ead62a813db7dda4035b5e7dffd935b
1434 F20101114_AACGFO li_r_Page_085.txt
e4694877961fa5234ccd6db5cdb87b3f
0047e5113d2423bcb77214d6596c0470d7926dd0
F20101114_AACGGD li_r_Page_056.tif
7d95699977d60c2c6e1fc1b0afe9583c
4bf3216436322247c76f1c3f8515c2e3af0085f6
1751 F20101114_AACHJG li_r_Page_069.txt
25a391564affb43d8adbbf5cf96a995d
cc6ed1dafb3d65315506777e5a946218acad0722
F20101114_AACHIR li_r_Page_019.txt
9ce772a6ed49b32eef7edd604cef7d3e
922a3113209b9785235ab29a06c7f45a0655eccf
56001 F20101114_AACGFP li_r_Page_035.pro
36e5aaf57582af034b6c81612fe0c098
f401605c5481da5a8bb428297307167921ca4ca8
7334 F20101114_AACGGE li_r_Page_091thm.jpg
afbdc9c6d3ca1c04198bce548de56f86
db7276720dc4fe85707d6708727b09a3405bf98d
2402 F20101114_AACHJH li_r_Page_075.txt
dd213e7a19132997b77b1d4f8aba668c
db3817f15d7c1949df00f9dc9e714228e7a33b24
2050 F20101114_AACHIS li_r_Page_020.txt
5628d5cf320dd8db45fbc0c2779a79f9
3e062bdeedac65115044640e75b7ca6295c52c20
43881 F20101114_AACGFQ li_r_Page_040.pro
5564c02328d829a33ef3fd5ceced4177
2d506f7c9fe6edbfb8e085526bdfc3ab74d575c3
676 F20101114_AACGGF li_r_Page_119.txt
ee40a87ab1f7f26c1c5b6a48ba5db021
b54a2c58e808ce222ee682b9372079dbc592bfa5
F20101114_AACHJI li_r_Page_076.txt
7290b05639332f6ddd6d0963044961f5
10e5c78302db97be8c952dcf03e256a52090ffab
2098 F20101114_AACHIT li_r_Page_032.txt
a30378505c39bcc57e72bff3e4d1af19
e2efe94fad3bc766b3b535e983b79b062d49ee52
F20101114_AACGGG li_r_Page_079.tif
b8374134b7382be66551606719ff53d7
f716f55a6d5454ed5ca11c2f2190cb3cce8eb311
F20101114_AACGFR li_r_Page_006thm.jpg
2451dca13f99bc0a182e7d3005b3b771
7bf7df14c08e26dae975020d2de0be01cf578b0f
392 F20101114_AACHJJ li_r_Page_080.txt
f2e82cec6c6c5559367919e2ae036d4a
33aca07984bf077a04b776c1c6cf5410169ad1b4
2062 F20101114_AACHIU li_r_Page_033.txt
6a105a94a812f6b7257d496f967ff1ec
1da203563aabf60088b63ecab15e66a1043871e8
124 F20101114_AACGGH li_r_Page_103.txt
83fbf79a4681ee710b6f4bd4cc780546
794a53245c71569b9363758d180e53e5f9cc10f3
108969 F20101114_AACGFS li_r_Page_022.jp2
7c789eaa615ca219eff2fbe07bc2a7d6
605fe3ab2cc7a3641eb08747991d76eade3a7597
1002 F20101114_AACHJK li_r_Page_088.txt
844bfa7c54aa83fea0c8bda9da201a48
1183b11aa02ab329bd7cd63eb8e01cb102453aa0
2126 F20101114_AACHIV li_r_Page_034.txt
bcc694e5fe471cab141dc99189f1a6b2
ff07507b272dadadde436c3b0b08a4fadd7f3c85
84256 F20101114_AACGGI li_r_Page_133.jpg
f0a7d931ab45bd5bca711e2530c1cd84
5832f54de7aeb2efc43c05effc6647dcacd83c80
6800 F20101114_AACGFT li_r_Page_049thm.jpg
7015d99a7178f0fce81613c8f60ffbcb
2eccf94e891439f81640f1eaade5c8f8027fdc7a
2111 F20101114_AACHIW li_r_Page_038.txt
106be8d4d0339c6b35637abe1d3fd566
c75a30be398ebc5f1ab4112f871ad863ed201cc0
2193 F20101114_AACGFU li_r_Page_094.txt
8116c564541ae639d563f09cdb33633a
e3458e4947d1b75d8bd0054bcd54d99c6613b926
3363 F20101114_AACHKA li_r_Page_082thm.jpg
a40c98be82779a8ea9f26ffe735aebf3
526d48a0690b268811f136b7b94fb686445ec74b
2085 F20101114_AACHJL li_r_Page_090.txt
f4474dd04970dfd739e0fa7c24dad120
2159d4d9b5f61bfb2ec0705b251e1f2426b51d3b
673 F20101114_AACHIX li_r_Page_039.txt
caa61c133b14e8e794586ac3209b41b9
9c399489ae519312d9613d8c661edba3a5feb65a
F20101114_AACGGJ li_r_Page_128.tif
527e260d631ad7f10a66694d632b27d3
32607788671653de49268981c5cfce87739785f3
2056 F20101114_AACGFV li_r_Page_040.txt
8a918cda02bdeaaabfec33e53d22820c
e8030f0dab41589d5301fc8cb2360f696f32d450
3556 F20101114_AACHKB li_r_Page_058thm.jpg
c78ecb9550e576402d86d0c3cb4b2772
221055d03b4ce35f6e82ecfd3c31437401f85fef
2164 F20101114_AACHJM li_r_Page_097.txt
e124b3064fa0b47ea0b3fff62b4c37f9
88e4d635a5f1a3371c814a4b0d33505291c6ec87
1884 F20101114_AACHIY li_r_Page_043.txt
3821c772c0f35c9d2ce73982100db01f
bf12bc023dd652680dcbe92643625608d3220b32
87 F20101114_AACGGK li_r_Page_002.txt
975b2e6b1243e5065067389a44c94f2e
5af74dbac0bbe3b88a49afda7efc01965ddebaf6
6988 F20101114_AACGFW li_r_Page_093thm.jpg
a31591040ec23933c0db1d71178dd1f1
acd9022c1431a9b5e0eb94305432508a6f9ec2f8
3809 F20101114_AACHKC li_r_Page_101thm.jpg
bca5387ba256406725e0c7deacfe5019
210dbc910c7b0892c17af74b5c495aa51c89afb4
948 F20101114_AACHJN li_r_Page_101.txt
98dda58124ed8627ea4fdc2f639cc57b
5491a350d7674264d615d07af3d2d5f1371bd13f
1665 F20101114_AACHIZ li_r_Page_048.txt
7b723490f368a906a4c95c2754397bd2
74d0eeb0dbe2a8be1a90f6f9681b6e10efe6486d
4018 F20101114_AACGHA li_r_Page_056thm.jpg
4f3e68c588b5b95fb680a487e9193091
dc66af8e21cd9e7d0b3931ce2e0f528cb49760ca
95154 F20101114_AACGGL li_r_Page_010.jp2
52931c2a3b24c4e734105167b38a3740
fcd8c9446249538737bec698d1c8a093233003cd
103814 F20101114_AACGFX li_r_Page_059.jp2
5959098fa792e745d3a871701e30652a
53acbca504012c4aef814b0b9916d3de6d1b32ec
3033 F20101114_AACHKD li_r_Page_002.QC.jpg
b3b6b41b93e385a1367c427d125f7170
17a2c3674549472006bf0354a3d8ae93b7bd9f03
784 F20101114_AACHJO li_r_Page_109.txt
19e550c4015a956b9dc467f8798ebe85
c4b01dfab4c4306fee064315852f13dc21695f70
50362 F20101114_AACGHB li_r_Page_042.pro
2ccfcc56d67f81a8580f4c7fc78aff3e
003ad0535b1a6a0b7febfe3c69b41aed355a80f6
58098 F20101114_AACGGM li_r_Page_131.pro
c30f02867734dfad56cc602f2e963678
c393c319088a04e096380292bd114385e9fd7e6a
4381 F20101114_AACGFY li_r_Page_114thm.jpg
a9b1e4a3768a27e49d774e986858465e
a2b13404a3d6ec413ffe488bd28d7f9e62db1f7f
25633 F20101114_AACHKE li_r_Page_110.QC.jpg
a7746050e1731a8e4b1b90548a414cb1
1ed2ecd10cdedb6e5f363408c4f3f58f21bd30c4
2216 F20101114_AACHJP li_r_Page_110.txt
d471f70f01c12206c67357d3e3fde173
5e91b82e14e7c86f999cbdeba60dc176a1b35ba9
77645 F20101114_AACGHC li_r_Page_111.jpg
f200d140a947239ab16134eb8a994523
dc63a13931962ebfb4392432c021a90ed560c7b1
31692 F20101114_AACGGN li_r_Page_122.jp2
d1a48238a8f220febbda76c266f42df0
5d32af95093e5913174debc3682bd19251c6a577
F20101114_AACGFZ li_r_Page_090.tif
1fbf317031a88d042fa29e4eac5d3118
60566721de370961fb1c4a676414fbef3bf20a2a
18495 F20101114_AACHKF li_r_Page_073.QC.jpg
4a7d432f47abc32df2e92cc0651b4b51
25c869dbaa0de97fd844eff8b2bb0c47f4a42b3f
2155 F20101114_AACHJQ li_r_Page_111.txt
ce0aa8050c1f888a289f7077c7e3e80c
bf770479d4a9f3d23884d146153d2d34ba63b8a3
3540 F20101114_AACGHD li_r_Page_113thm.jpg
1392ba6bbfb26228cf0972aaf65e0051
97a65d01194d3c385eea3d7569907d3679430bfb
42708 F20101114_AACGGO li_r_Page_069.pro
4c8a56d51b4d53cbb9b184ec333556d5
7e1fd24d21196a42b326436e04ed6c89748d1d89
4244 F20101114_AACHKG li_r_Page_116thm.jpg
ce369d3573d3ede18acbf630328fa15b
7ea3c1b634dd7b54e9321744e0b75c989c08ff5e
2110 F20101114_AACHJR li_r_Page_112.txt
f20442f5e6cfd921381f7ce62ad78e88
41cf5d0833418569d5ac068daa9556532f5a803f
54350 F20101114_AACGHE li_r_Page_037.pro
6c005794c9d5ce72c174a03b0a5a3179
33918e383256c3e2854b58daafeae035336ef736
F20101114_AACGGP li_r_Page_022.tif
6261beb77f2a2db0a5c0e1c141228f56
34279f429ca46460b518caafb4e6c597fcc0164c
198350 F20101114_AACHKH UFE0022519_00001.xml
201dbf6ef43dda08f9e7e69dec488e86
6de860c3f4347473269d7f761e3d5af62efb148d
1217 F20101114_AACHJS li_r_Page_114.txt
e8115289908c55a75e573f506839ee0a
789554fb34bba5ecffda0bb5289454903a20e97f
12512 F20101114_AACGHF li_r_Page_011.QC.jpg
503b4c5ebadbb62086c10fa0a928e054
4feaee5e00ed94dac04a9dd75fba8ecd01332f34
22646 F20101114_AACGGQ li_r_Page_021.QC.jpg
943c52c8f4759ff31bb8bee974866d0c
0a87fd0a85224d56fd6a1c36ed1599a201c50d87
1355 F20101114_AACHKI li_r_Page_002thm.jpg
bc593bef9f76f1f6caf752e4feadbec3
3e19810a96fd3f33484a9b4c9d81c1bc91fd87a0
1308 F20101114_AACHJT li_r_Page_118.txt
52e6b5ea213d387478198f517e49103a
3511b1abb13606a4c92aa3121bb41afd0df4cf86
F20101114_AACGHG li_r_Page_134.tif
6ab8a201e01033f62415924595daa680
994dba9283f76d4d289f2dace91cbfde735caeae
60031 F20101114_AACGGR li_r_Page_124.pro
32da06a98c5b5073db3ba89cb3f6b1a3
53439b368d74bc579071aa6d00c4604cd800869c
1689 F20101114_AACHKJ li_r_Page_003thm.jpg
d310432bf1a2764b162c6c04b8fcc18f
d1f22aea9cc22a97c759c5c99eed900d5702db31
668 F20101114_AACHJU li_r_Page_122.txt
2bb700f4f1257496c0f421a3845abb9b
7243ab6e53ce8f76ef5e507eb7e013b3f50edd47
6569 F20101114_AACGHH li_r_Page_015thm.jpg
0fadabee2e6567904f7e66d2b854ee19
9604cc58ff91eae607a879c3d3c8b022d4933c49
F20101114_AACGGS li_r_Page_052.txt
c7515b750932d9eee5a7f224b7562bcb
6a48c4a0dfd6a1b339a76d026f7876ca654918e8
14250 F20101114_AACHKK li_r_Page_004.QC.jpg
538fb25680a1c39c349cb4311fa50fba
ceca52839a78baa20a86d6fd488fc4e636ebc185
2586 F20101114_AACHJV li_r_Page_125.txt
ff6a03613cb7fa94499dc3b337143ae5
1d0ef20fff6c71d3dc7cf930c967ceef54f743a7
F20101114_AACGHI li_r_Page_009.tif
d4dfa3030376623e0349c17d7b5c90f5
4f3eb776502a3c71167c8ed50701ed8c47cec4a4
10522 F20101114_AACGGT li_r_Page_083.QC.jpg
f44e2ed705e3785ea7f32e20ebecb7bb
d3359c2a02a1030a5cde0d8e9f6a08bcc2828f48
23659 F20101114_AACHKL li_r_Page_005.QC.jpg
f1b74a8ccb8ccb32f2b625c236a81f10
17c9ce72e0dbb88080cf55abdbbe864b760ee2c4
2440 F20101114_AACHJW li_r_Page_132.txt
86f01bdc31ac34ad7fa8bfc344827907
7a775bfea71248b167ae690a3e5bcbbb9bc9ca32
F20101114_AACGHJ li_r_Page_001.jp2
e7422f89b42e8be8886ee33ca93c6c9e
0c5bc8823cb9236bd06703176c313ecb856b7aeb
F20101114_AACGGU li_r_Page_123.tif
6dc762555bde481a1c433e0011afdc00
395e1ce59b7aeb4ffba4e737c4abd819c2f25c2f
7111 F20101114_AACHLA li_r_Page_038thm.jpg
4ef1fdac2b67d335e726e4793da580cd
6f32803ab46fc1cd02e772a01beb934367ecd0ed
2349 F20101114_AACHJX li_r_Page_133.txt
9baa1a88b4d684740f6f4d8e5b35f050
b874b2dc0df3287fe2671d77064e3d466211d624
6534 F20101114_AACGGV li_r_Page_007.QC.jpg
b88f20f895addf895d728e264d9c03be
b5378715e8a0bb8091c23a32e30419f8c6fc69b3
2945 F20101114_AACHLB li_r_Page_039thm.jpg
2103d2e1ba1ec47771f58749e2456517
0924e4dfcdad813c4e3af8bdcc4617b0dc4fd9f6
6680 F20101114_AACHKM li_r_Page_009thm.jpg
8d15d66dc976bfb448876e042aff8f2d
15ffd608998517b7eca396b302b92005a0d428cd
859248 F20101114_AACHJY li_r.pdf
c7fe0ae626564e80419bf6a8b4de45d4
32ba64aedae5a8417df144a4b8c84362d744ca9a
2204 F20101114_AACGHK li_r_Page_035.txt
e19471ce332c4a332c88c377b8daa5ae
7d0b28f78eee7f481951254a329d96d975340142
6583 F20101114_AACGGW li_r_Page_131thm.jpg
76631ea734688dd14a2929df43f88e4c
2ca5919fbd734440612080ec29df24973373b5e7
6153 F20101114_AACHLC li_r_Page_040thm.jpg
1fc3690a83fea964c6907699c927e10b
d5f6920c7ded0a59fda63d9d41526e0655d64741
5821 F20101114_AACHKN li_r_Page_010thm.jpg
084aa4b73c9058eddf2f0f74aad02043
0ebafdeb076df48ccfd3661f271114854d46d70d
13860 F20101114_AACHJZ li_r_Page_120.QC.jpg
c010b06501694a82d250006e9c238714
a47ba3fc388f0437653cb064c576d6bb439efb9f
10750 F20101114_AACGHL li_r_Page_085.QC.jpg
6b080eed90431b4e3b3d4cbf978f6e4e
66d6353f0a73855d1b0a16a884eaa7336c8272b5
105941 F20101114_AACGGX li_r_Page_029.jp2
477901e9be6d006b4c2d7e5a5596a991
f5ca00aade5451b120d17f0a1faf690e1b13e30c
2284 F20101114_AACGIA li_r_Page_123.txt
a2d08869af985652344c441b2a955c39
245cdf08720acc689a70aca9345a1fda295d62df
5940 F20101114_AACHLD li_r_Page_041thm.jpg
fdda0aa70d1d5aa0a45f992f5e661404
cdaa7edb43228e04ac9248056743388770cb42de
6664 F20101114_AACHKO li_r_Page_016thm.jpg
f419d4409acf904cf0dd60001df71ed4
aefa9bedcdb202aead2eed608da92de67d9f7b6f
F20101114_AACGHM li_r_Page_135.tif
4d8654e167556b36b2715fab50819ee8
c7cc2ab628fb8ac9f60bea24207ca8fa37ffde12
11947 F20101114_AACGGY li_r_Page_007.pro
141ef58ef8657051d4e26d81d682e207
6ee2294976c03ab3e2caacdcad5c15e4eabe3d1f
7014 F20101114_AACGIB li_r_Page_013thm.jpg
057ed14151023f74b6fc6783539650dd
a16b4b429d3e7e804016c14552cd47102172ec79
23678 F20101114_AACHLE li_r_Page_042.QC.jpg
f6ef59292e31f28a450b3ed2c718b444
8d3443780a1f0bc0e4bab0d66cafb5bdd77c3deb
6519 F20101114_AACHKP li_r_Page_019thm.jpg
01b82d9b6df46328009d40c9a9baf929
47af7e6024b95e8161bb5b973948fe19fc486533
1127 F20101114_AACGHN li_r_Page_120.txt
c3413a46de1a3c44bf5649814397cb1b
02e15d50c2da0efe116a32cdc40ec60ec47063a2
F20101114_AACGGZ li_r_Page_127.tif
3be24668ba1c26fbac71bce343731cc0
6c2a6b59af847d0e7e154dc2f755291131d3a555
7039 F20101114_AACGIC li_r_Page_051thm.jpg
9f2e45d4a559e45f63ad9e62b1198f96
089696d3cb1ea290879288fd5aacd8955e71dbd7
6732 F20101114_AACHLF li_r_Page_042thm.jpg
e5e43f914df85368c125e2ba2ca3c4ee
6a9c550fb544f81f7c99394906dddbd900ca7722
6804 F20101114_AACHKQ li_r_Page_020thm.jpg
605c670a03250dcfbe9a9725996c0f09
7855d2f48d3950cca15fe1cc7f4a4ee8a4f42a2d
4483 F20101114_AACGHO li_r_Page_134thm.jpg
48baf2829d49ea9f5625e35d7a5a09ba
2a547e88bd4560d4ef8f3244061e6417e8a9ba58
260473 F20101114_AACGID li_r_Page_105.jp2
2a00ed8d41b4aef70b35516a74407bf1
4149c59cca7383ecaf89f1e1ab5aaeec42d14f6f
5817 F20101114_AACHLG li_r_Page_043thm.jpg
d57d0efd0205281061722c69e88e1439
f3a034c804eadc3a1d6599f4cc96d1fec852d886
22103 F20101114_AACHKR li_r_Page_026.QC.jpg
d46cb64dd150435baab667ecc84c17fd
2488cb486b0ab50a63e889c0eca0a0940214f33e
35832 F20101114_AACGHP li_r_Page_076.pro
712e3253025ec9f8007dbcdd5ed347d3
1d67fb2dc2ded1e1f47907377e0d0fa2fe8cdeb5
3830 F20101114_AACGIE li_r_Page_106thm.jpg
d7b17495f1deba1ccf2bfa85f423ef36
28c53255783b34e9269ce3f0e89a9bff8d73c0d1
5310 F20101114_AACHLH li_r_Page_044thm.jpg
ca705030e88a092c3d5f2704339f6c8a
860f677aeb5c23384dd6700f425949a95b7359a0
20181 F20101114_AACHKS li_r_Page_027.QC.jpg
bade39925327b4cd6b9954b8b8391f86
0741eb6169efe434519037f85d91ce5bf26f217a
1975 F20101114_AACGHQ li_r_Page_014.txt
c48307b31d927576edbdd98997273fc8
20049a45016624902c3543d6c5a2e09babd14229
77688 F20101114_AACGIF li_r_Page_063.jpg
8baae572186089b81153f5cd67abb57a
257dd9a342876761eba8807f49a7cc711c605e83
20548 F20101114_AACHLI li_r_Page_047.QC.jpg
b46659c2e2faf8227fc83ca521b7c907
149a92001625c15922e7b1f047ce6dfda1d387ac
5910 F20101114_AACHKT li_r_Page_027thm.jpg
dcddf107c889ad5fdc7b9f9f643ce1b7
037775c3eb2554c16cd71dcca47051b0cf1a1673
55809 F20101114_AACGHR li_r_Page_094.pro
4392f262fc652e3ecc9324245788d7c8
9fbdcceead6cba51e8eb505ff2c8c60566f8e4c4
11285 F20101114_AACGIG li_r_Page_113.QC.jpg
af0a1d40ee32a867c64bf50c8605b370
7e999ea24ca8ce9a8e5794ba7944b2b3ab402634
22206 F20101114_AACHLJ li_r_Page_050.QC.jpg
a94e259348978e885a15f864e6604713
d80aa300a28e6acfb32c98bb3018ce1d25cfd8bb
23107 F20101114_AACHKU li_r_Page_029.QC.jpg
26504acb73952a270e1e6ee15735bc65
9e9b7c0d22902be7d31a07f775193cc9c660a60f
36783 F20101114_AACGHS li_r_Page_104.jpg
056fdfa8f224ba6f84f905e972ab7006
7268e1a83062e8835635ce4d87d1d0187cdb9fb1
13551 F20101114_AACGIH li_r_Page_118.QC.jpg
72cd0b32d00f935a1aedb5875c08542c
26eccbdee8cbdf4cf85f00364576e9a6ca22ae14







SPATIOTEMPORAL FILTERING METHODOLOGY FOR SINGLE-TRIAL
EVENT-RELATED POTENTIAL COMPONENT ESTIMATION





















By

RUIJIANG LI


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008


































2008 Ruijiang Li
































To all scientists and researchers, who have lived in pursuit of knowledge, and have
dedicated themselves to the advancement of science









ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor Dr. Jose C. Principe, for his great

inspiration and encouragement throughout the course of my research. Not just that. He has really

become a mentor and guide during pivotal times of my life, which I would have to say

regretfully that I did not take full advantage of. One could ask for no more from such an advisor.

I wish to thank the members of my committee, Dr. John Harris, Dr. Jianbo Gao, and Dr.

Mingzhou Ding, for their valuable time and interest in serving on my supervisory committee, as

well as their comments, which helped improve the quality of this dissertation. I am grateful for

Dr. Andreas Keil's expertise on psychology as well as his support, which made our collaboration

fruitful.

I would like to thank my friends and colleagues at the Computational NeuroEngineering

Laboratory. They have made my stay in Florida during the past four years an enjoyable

experience. Last but not least, I wish to thank my parents, who raised me up. Without them, all is

in vain.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

LIST OF TABLES .............. ...... ........................ ...............

LIST OF FIGURES ................................... .. .... ..... ................. .9

A B S T R A C T ............ ................... ............................................................ 10

CHAPTER

1 INTRODUCTION ............... ................. ........... .............................. 12

1.1 B asic C concepts of the E R P ................................... ............ .....................................12
1.1.1 G generation of the ERP .............................. ............................................. 12
1.1.2 The ERP C om ponents.................... ....... ................................. ............... 13
1.2 Estim action of the ERP....................................................................... ............... 16

2 SINGLE-TRIAL ERP ESTIMATION ............ ..... ........ ................... 19

2.1 Single-Trial ERP Estimation Using Single-Channel Recording ..............................19
2.1.1 Tim e-Invariant D igital Filtering ............................................. ............... 20
2.1.2 Time-Varying Wiener Filtering ............................... ...............20
2.1.3 A adaptive Filtering ......... ..................................... ...... .. .. ........ .... 21
2.1.4 K alm an Filtering ......... .......... ............... ........ ...... .......... .............. 22
2.1.5 Subspace Projection and Regularization......................................................22
2.1.6 Parametric Modeling.......................................................... 23
2.1.7 Other Methods Using Single-Channel Recording ..........................................24
2.2 Single-Trial ERP Estimation Using Multi-Channel Recording..............................25
2.2.1 Generative EEG M odel ................................................................................ 25
2.2.2 What Is a Spatial Filter and What Can It Do? ....................................... 27
2.3 Review of Spatiotemporal Filtering Methods.................. ............ ................28
2.3.1 Principal Component Analysis (PCA)............................................................28
2.3.2 Independent Component Analysis (ICA)........................................................ 32
2.3.3 Spatiotemporal Filtering Methods for the Classification Problem ..................36

3 NEW SPATIOTEMPORAL FILTERING METHODOLOGY: BASICS.............................40

3.1 Spatial Filter as a Noise Canceller in the Spatial Domain ...........................................40
3.2 D term inistic A approach ......... ......... ............... .. ........................... ............... 42
3.2.1 Finding Peak Latency ................................................. ............................. 42
3.2.2. Finding Scalp Topography and Peak Amplitude................... .............. 44
3.3 Stochastic A approach ............................................ ................... ........ 46
3.4 Sim ulation Study ................. .... .... .. .................................................... .. ... 48
3.4.1. Gamma Function as a Template for ERP Component ........... ...............49









3.4.2. Generation of Simulated ERP Data ....................................... ............... 49
3.4.3 Case Study I: Comparison with Other Methods .........................................50
3.4.4 Case Study II: Effects of Mismatch............... .. ..... .................... 53

4 ENHANCEMENTS TO THE BASIC METHOD.............. ... .............. .............. 59

4.1 Iteratively R efined Tem plate ............................................... ............................ 59
4.2 R egularization .............................61................................................61
4.2.1 Constrained O ptim ization ........................................................ ............... 61
4.2.2 Unconstrained Optim ization ................................................... ................. 64
4.3 Robust Estim action: the CIM M etric ......................................................... ......... 65
4.4 Bayesian Formulations of the Topography Estimation .............................................68
4.4.1 M odel 1: A dditive N oise M odel ............................................. ............... 69
4.4.2 Model 2: Normalized Additive Noise Model .......................................... 70
4.4.3 M odel 3: O original M odel ............................................... ........................ 71
4.4.4 Comparison among the Three M odels.................................. ............... 71
4.4.5 O line E stim action ................. ........... .. .. .......... .............. .............. 72
4.5 Explicit Compensation for Temporal Overlap of Components ...................................72

5 APPLICATIONS TO COGNITIVE ERP DATA ...................................... ............... 90

5.1 O ddball Target D etection.................................................. ............................... 90
5.1.1 M materials and M ethods............................................................ .....................90
5.1.2 E stim action R results ......................... .... ..................... .... .. ........... 92
5 .1.3 D iscu ssio n s ................................................................9 8
5.2 H abituation Study .......................... .......... .. ......... .............. .. 99
5.2 .1 M materials and M ethods............................................................ .....................99
5.2.2 E stim action R results ......................... .... ..................... .... .. ........... 99

6 CONCLUSIONS AND FUTURE RESEARCH ........................................................110

6 .1 C o n c lu sio n s ................................................... ................. ................ 1 10
6.2 Future Research ............................ .................. .......... .. ...... ....... 12

APPENDIX

A PROOF OF VALIDITY OF THE PEAK LATENCY ESTIMATION IN (3-10) ................114

B GAMMA FUNCTION AS AN APPROXIMATION FOR MACROSCOPIC
ELECTRIC FIELD .................. .................. .............................. ...... ... .... 115

C DERIVATION OF THE UPDATE RULE FOR THE CONSTRAINED
OPTIM IZA TION PR OBLEM ............................................. ...................... ............... 116

D DERIVATION OF THE UPDATE RULE FOR THE UNCONSTRAINED
OPTIM IZA TION PR OBLEM ............................................. ...................... ............... 117

E MAP SOLUTION FOR THE ADDITIVE MODEL.................................. .................118









F NORMALIZED ADDITIVE NOISE MODEL 1: MAP SOLUTION..............................120

G NORMALIZED ADDITIVE NOISE MODEL 2: MAP SOLUTION..............................122

L IST O F R E F E R E N C E S ......... .................................... ......................................................... 123

B IO G R A PH IC A L SK E T C H ......... .............................................................. ........................... 135









LIST OF TABLES


Table page

3-1 Latency estimation: mean and standard deviation ......... ................. ...................55

3-2 Amplitude estimation: mean and standard deviation............................................. 55

3-3 Scalp topography estimation: correlation coefficient.................... ..................55

3-4 Effects of mismatch I: SNR = -20dB............................ .... .................................56

3-5 Effects of mismatch II: SNR = -10dB ............................................... ................... 56

4-1 Latency estimation: mean and standard deviation...... ............................75

4-2 Amplitude estimation: mean and standard deviation.............................. ............... 75

4-3 Scalp topography estimation: correlation coefficient............. ..... ..................75

4-4 Estimation results for the iteratively refined template method............. ................76

4-5 Estimation with MCC for the mismatch case at SNR = OdB.................. ............ 76

4-6 Estimation with MCC for the mismatch case at SNR = -20dB ......................................76

4-7 Amplitude estimation for three Bayesian models.................................. ............... 77

4-8 Scalp topography estimation for three Bayesian models............................................77

5-1 Correlation statistics for the 4 subjects: Scalp topography estimation .........................102

5-2 Regression statistics for response time and estimated amplitude.................................. 102









LIST OF FIGURES


Figure page

3-1 Gamma functions with different shapes and scales. .................................. ............... 57

3-2 Waveforms of synthetic and presumed ERP component................................................58

4-1 Mean and standard deviation of the estimated amplitude under different SNR
c o n d itio n s.......................................................................... .. 7 8

4-2 The waveforms of the synthetic component, presumed template and refined template
under 4 SNR conditions. ........................................ ........ .. ....... ...... 79

4-3 Waveforms of two overlapped components used in regularization ..............................80

4-4 Scalp topography of two overlapped ERP components used in regularization ................81

4-5 Amplitude and scalp topography estimation I with regularization (constrained
optim ization) under 3 SNR conditions.. ......................... ............................................82

4-6 Amplitude and scalp topography estimation II with regularization (constrained
optim ization) under 3 SN R conditions ........................................ ......................... 84

4-7 Amplitude and scalp topography estimation I with regularization (unconstrained
optimization) under 3 SNR conditions. ........................................ ......................... 86

4-8 Amplitude and scalp topography estimation II with regularization (unconstrained
optim ization) under 3 SNR conditions. ........................................ ......................... 88

5-1 Pictures used in the experiment as stimuli................................. ...............103

5-2 Cost function in (3.9) versus time lag for different regularization parameters for
subject #2 ........... ... ...................... ....... .......... 104

5-3 Estimated pdf of time lags corresponding to local minima of the cost function in (3-
9) using the Parzen windowing pdf estimator with a Gaussian kernel size of 4.2.. ........105

5-4 Scalp topographies for the four subjects.............. ......... .. ......... ........... ............... 106

5-5 Scatter plot of the response time versus the estimated amplitude for each single trial
for the four subjects. ............................................... ..............107

5-6 Estimated scalp topography for mixed and habituation phase......................................108

5-7 Estimated amplitude for mixed and habituation phase .................................................109









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SPATIOTEMPORAL FILTERING METHODOLOGY FOR SINGLE-TRIAL
EVENT-RELATED POTENTIAL COMPONENT ESTIMATION


By

Ruijiang Li

December 2008

Chair: Jose Principe
Major: Electrical and Computer Engineering

Event-related potential (ERP) is an important technique for the study of human cognitive

function. In analyzing ERP, the fundamental problem is to extract the waveform specifically

related to the brain's response to the stimulus from electroencephalograph (EEG) measurements

that also contain the spontaneous EEG, which may be contaminated by artifacts. A major

difficulty for this problem is the low (typically negative) signal-to-noise ratio (SNR) in EEG

data. The most widely used tool analyzing ERP has been to average EEG measurements over an

ensemble of trials. Ensemble averaging is optimal in the least square sense provided that the ERP

is a deterministic signal. However, over four decades of research have shown that the nature of

ERP is a stochastic process. In particular, the latencies and the amplitudes of the ERP

components can have random variation between repetitions of the stimulus. Under these

circumstances, estimation of the ERP on a single-trial basis is desirable. Traditional single-trial

estimation methods only consider the time course in a single channel of the EEG. With the

advent of dense electrode EEG, a number of spatiotemporal filtering methods have been

proposed for the single-trial estimation of ERP using multiple channels.









In this work, we introduce a new spatiotemporal filtering method for the problem of

single-trial ERP component estimation. The method relies on modeling of the ERP component

local descriptors (latency and amplitude) and thus is tailored to extract faint signals in EEG. The

model allows for both amplitude and latency variability in the actual ERP component. The

extracted ERP component is constrained through a spatial filter to have minimal distance (with

respect to some metric) in the temporal domain from a template ERP component. The spatial

filter may be interpreted as a noise canceller in the spatial domain. Study with simulated data

shows the effectiveness of the proposed method to signal to noise ratios down to -10 dB. The

method is also tested in real ERP data from cognitive experiments where the ERP are known to

change, and corroborate experimentally the expected behavior.









CHAPTER 1
INTRODUCTION

1.1 Basic Concepts of the ERP

Event-related potential (ERP) is an important and well-established technique for

neuroscientists and psychologists to study human cognitive function. In this section, we briefly

review some of the basic facts and concepts related to ERP generation and analysis (Coles et al.

1995).

1.1.1 Generation of the ERP

When a pair of electrodes are attached to the surface of the human scalp and connected to a

differential amplifier, the output of the amplifier reveals a pattern of voltage variation over time.

This voltage variation is known as the electroencephalograph (EEG). The amplitude of the

normal EEG varies between approximately +100 /V and most of the EEG frequency contents

range between 0.5Hz and 40Hz. Here, we do not review the recording techniques of EEG

(Ruchkin 1987).

If we present a stimulus to a human subject while recording the EEG, we can define a

period of time (an epoch or a trial) where some of the EEG components are time-locked to the

stimulus. Within this epoch, there may be voltage changes that are specifically related to the

brain's response to the stimulus. These voltage changes constitute the event-related potential, or

ERP.

Although it is not completely understood how the measurements at the scalp relate to the

underlying brain activity, the following points appear to be clear and are generally accepted

(Scherg and Picton 1991, Wood 1987). First, ERP recorded from the scalp represents net

electrical fields associated with the activity of sizeable neuron populations. These neuron

populations act as current sources whose electrical fields propagate to the entire scalp through









volume conduction. Second, the individual neurons that compromise such a population must be

synchronously active and have a certain geometric configuration to produce measurable

potentials at the scalp. In particular, these neurons must be configured in such a way (usually in a

parallel orientation) that their individual fields summate to yield a dipolar field.

Therefore, the ERP recorded at the scalp is selective of the totality of the brain activity.

This is advantageous in that the resultant measurements would otherwise be so complex as to be

difficult or impossible to analyze. On the other hand, we should also be aware that there are

certainly numerous functionally important neural processes that cannot be detected by the ERP

technique.

1.1.2 The ERP Components

The issue of ERP components has aroused much controversy among the ERP research

community, particularly the question of the definition of an ERP component. Suppose for the

moment that we have obtained the ERP using some method. A simple way to define a

component is to focus on some feature of the resulting waveform (for instance, a peak or trough),

and this feature becomes the component of interest. Some common features include the

amplitude and latency parameters of a particular peak or trough.

A major problem with the simple approach mentioned above is component overlap, both

spatially and temporally. Since the brain is a conducting medium, activity generated in one

spatial location may be propagated through the brain tissue and appear at other locations. Thus,

the waveform we observe by measuring the voltage at the scalp may well be attributed to a

variety of different sources in different spatial locations of the brain. One consequence of volume

conduction is that there need be no direct correspondence between the timing of the distinctive

features of an ERP waveform (peaks and troughs) and the temporal characteristics of the

underlying neural systems. For instance, an ERP peak with a latency of 300 ms, might reflect the









activity of a single neural generator maximally active at that time, or the combined activity of

two (or more) neural generators, maximally active before and after 300 ms, but with fields

summating to a maximum at that time.

Due to these ambiguities surrounding the interpretation of peaks and troughs in ERP

waveforms, other definitions for ERP components have been proposed. Naatanen and Picton

(1987) adopted what might be called the physiological approach to component definition. They

proposed that a defining characteristic of an ERP component is its anatomical source within the

brain. According to this definition, to measure a particular ERP component, we must have a

method of identifying the contributing sources. Dochin (1979, 1981) adopted what might be

called the functional approach to ERP component definition, which is concerned more with the

information processing operations with which a particular component is correlated. According to

this definition, it is entirely possible for a component to be identified with a particular feature of

the waveform that reflects the activity of multiple generators within the brain, so long as these

generators constitute a functionally homogeneous system (Coles, et al.1995).

Although the above physiological and psychological approaches to component definition

seem to be counteractive, for many investigators it is more appropriate to combine both

approaches. A classical approach to component definition, was proposed by Dochin et al. (1978).

They argued that an ERP component should be defined by a combination of its polarity, its

characteristic latency, its distribution across the scalp and its sensitivity to characteristic

experimental manipulations. Notice that polarity and scalp distribution imply a consistency in

physiological source, while latency and sensitivity imply a consistency in psychological

function.









ERP components can be broadly classified into two types: exogenous and endogenous

components. Characteristics of the exogenous components (amplitude, latency and scalp

projection) largely depend on the physical properties of sensory stimuli, such as their modality

and intensity. On the other hand, endogenous components largely depend on the nature of the

subject's interaction with the stimulus. These components vary as a function of such factors as

attention, task relevance and the nature of the information processing required by the stimulus.

The dichotomy of the exogenous-endogenous distinction turned out to be an oversimplified

version of the reality. Many early 'sensory' components have been shown to be modifiable by

cognitive manipulations (e.g., attention) and many of the later 'cognitive' components have been

shown to be influenced by the physical attributes of stimulus (e.g., modality).

In what follows, we briefly discuss one particular well-known ERP component, the P300.

For a comprehensive review on other well-known components, we refer to Coles, et al. 1995.

The P300 is probably the most important and the most studied component of the ERP. It

was first described in the 1960s by Sutton et al. (1965). The P300 is evoked by a task known as

the odd-ball paradigm. During this task a series of one type of frequent stimuli is presented to the

experimental subject. A different type of non-frequent (target) stimulus is also presented. The

task of the subject is to react to the presence of target stimulus by a given motor response,

typically by pressing a button, or just by mental counting to the target stimuli. Virtually any

sensory modality (auditory, visual, somatosensory, olfactory) can be used to elicit the P300

response. (Polich 1999). The shape and latency of the P300 differs with each modality. This

indicates that the sources generating the P300 differ and depend on the stimulus modality

(Johnson 1989).









There are several theories on the neural processes underlying the origin of the P300. The

most cited and most criticized theory was proposed by Donchin and Coles (1988a, b). According

to their theory, the P300 reflects a process of context or memory updating by which the current

model of the environment is modified as a function of incoming information. Several

investigators (e.g., Johnson 1986) have pointed out that the P300 does not appear to be a unitary

component, and instead may represent the activity of a widely distributed system which may be

more or less coupled depending on the situation. More information about the underlying neural

systems is required before a consensus is attained about the functional significance of this

component. For a more recent review on the research of P300, we refer to Linden, 2005.

1.2 Estimation of the ERP

The fundamental problem in the analysis of ERP is to extract the signals that are brain's

specific response to the stimulus from the EEG measurements that also contain 'noise'. By noise,

or the background EEG, we mean the electrical activities from heart, muscles and eye

movements as well as the spontaneous brain activities that are not related to brain's response to

the stimulus.

A major difficulty with the extraction of ERP is that, in most cases, ERP signals are small

(on the order of microvolts) relative to the background EEG (on the order of tens of microvolts)

in which they are embedded. For this reason, it is necessary to employ signal processing

techniques to estimate the ERP signals in the presence of noise.

By far the most commonly used technique has been the averaging of the EEG

measurements over an ensemble of time-locked epochs. This is optimal in the mean square

sense, given the assumption that the ERP is a deterministic signal time-locked to the stimulus

and the additive background EEG is zero-mean and uncorrelated with the ERP.









However, for over four decades it has been evident that the nature of ERP is more or less

random. In particular, the amplitudes and latencies of the peaks in the ERP can have random

variations between repetitions of the stimuli (Brazier 1964). In addition, the variations may be

trend-like and the mean of the amplitudes and the latencies can change across the trials. Under

these circumstances, the information regarding to these variations in ERP is lost through

averaging. Furthermore, the average waveform may not, in fact, resemble the actual ERP

waveform that is recorded in an individual trial. The resulting estimates for the ERP, therefore,

may not correspond to the underlying neural processes and inference about the cognitive

function may be misleading.

Estimation of the ERP on a single-trial basis is desired for the situations when the peak

amplitude and latency of a particular component change significantly across trials. A major

difficulty with single-trial ERP estimation is again the very low signal-to-noise-ratio (SNR) in

the single-trial EEG, typically lower than -10dB.

Statistically speaking, the average ERP, or the sample mean, is an example of the use of

the first order statistics, where only the first order moment of the population parameters is

estimated. The next obvious improvement is to use the second order statistics, i.e., covariance

analysis. The most common approach is to form an estimator (filter) with which the unwanted

contribution of the background EEG can be filtered out. To find such an estimator, some models

or assumptions are imposed on the ERP and background EEG concerning their respective second

order statistics. The estimator that satisfies the minimum mean square criterion can then be

derived. The performance of the estimator then largely depends on how realistic these

assumptions are.









For historical reasons, these traditional single-trial estimation methods only consider the

time course of a single recording channel in the EEG. In some cases, simple ERP components,

e.g., the brainstem auditory evoked potentials, can be adequately examined using a single

channel. However, for most ERPs, simultaneous recording from multiple electrode locations is

necessary to disentangle overlapping ERP components on the basis of their topographies, to

recognize the contribution of artifactual potentials to the ERP waveform, and to measure

different components in the ERP that may be optimally recorded at different scalp sites (Picton et

al. 2000).

Today, high-density EEG can simultaneously record scalp potentials in up to 256

electrodes. This increased number of sensors and thus increased spatial resolution has created a

need for signal processing methods that can simultaneously analyze the time series of multiple

channels. Recently, various methods have been proposed for single-trial analysis that linearly

combine the time series in multiple channels to generate a representation of the observed data

that is easier to interpret (Chapman and McCrary 1995, Makeig et al. 1996, Parra et al. 2002).

This linear projection combines the information from all the available sensors into a single

channel with reduced interference from other neural sources and may provide a better estimate of

the underlying neural activity than the EEG measurements in a single channel. The linear

projection, in this sense, may be called a 'spatial filter' and these methods can be generally called

spatiotemporal filtering methods.

We will review these and other single-trial estimation methods in detail in Chapter 2.









CHAPTER 2
SINGLE-TRIAL ERP ESTIMATION

In this chapter, we review the existing methods for single-trial ERP estimation. The

methods are broadly categorized into two classes: those based on single-channel EEG recording

and those based on multi-channel EEG recording. Methods based on single-channel recording

rely solely on the modeling of temporal characteristics of the ERP and EEG, while methods

based on multi-channel recording investigate both the spatial and temporal characteristics of the

ERP and EEG, and are termed with the general notion of spatiotemporal filtering methods.

2.1 Single-Trial ERP Estimation Using Single-Channel Recording

For the estimation of ERP using a single channel, all the available information is contained

in a single time series x(t): the EEG recording at a certain electrode. We assume that the

measurements consist of two parts: the signal of interest (ERP) and additive noise (background

EEG), denoted by s(t) and n(t), respectively. The observation model for the EEG can be

written as:

x(t) = s(t) + n(t) (2-1)

In this review, the EEG measurements for a single trial x is a finite-length vector with

elements sampled from the original continuous-time waveform. When the time series in (2-1) are

interpreted as stochastic processes, x becomes a random vector. Its length equals the number of

samples in one trial. In vector form, the observation model is:

x = s + n (2-2)

Here we do not attempt to give an exhaustive review on the topic of single-trial ERP

estimation with a single channel. For other reviews on the general ERP estimation problem, we

refer to Aunon et al. (1981), Ruchkin (1987), McGillem and Aunon (1987), Silva (1993),

Karjalainen (1997).









2.1.1 Time-Invariant Digital Filtering

Digital filtering is a good place to start for time series analysis. The simplest approach is

to design digital filters that have a desired frequency response. Ruchkin and Glaser (1978) used

simple moving average FIR filters to estimate ERP on a single trial basis. More complicated ones

may be designed to estimate some particular component such as P300 (Farwell, et al. 1993).

Wiener filtering may also be used to estimate single-trial ERP, provided that the power spectra of

the ERP and background EEG can be estimated appropriately (Aunon and McGillem 1975,

Cerutti, et al. 1987).

The major problem with linear time-invariant filtering is the fact that the ERP is typically

a transient and smooth waveform with no periodicity. The spectrum for this kind of signal is not

defined properly. Consequently, the spectra of the ERP and background EEG (if they are

estimated) were usually found to overlap significantly (Krieger et al. 1995, Spreckelsen and

Bromm 1988, Steeger et al. 1983). Thus the application of digital filters with constant frequency

response is not expected to give desirable results in most cases. The effects of digital filtering are

studied in (Ruchkin and Glaser 1978, Maccabee, et al. 1983, Nishida et al. 1993).

2.1.2 Time-Varying Wiener Filtering

When the optimal Wiener solution is computed for single-trial EEG data, the filtering

becomes time-varying with respect to each trial. In general, these methods necessitate some

analytic model for the ERP. Yu and McGillem (1983) introduced what was called the time-

varying minimum mean square error filter. A crucial task for their method is to obtain a good

estimate for the cross-covariance between the ERP and measurements. Under the assumption

that the ERP and background EEG are uncorrelated, the cross-covariance becomes the auto-

covariance of the ERP signal itself. The ERP is parametrically modeled by a superposition of the









components with random location and amplitude. The parameters for the ERP are then calculated

from the Wiener solution on a single trial basis.

2.1.3 Adaptive Filtering

The use of adaptive filtering for the analysis of single-trial ERP, particularly the use of

the least mean square (LMS) algorithm (Widrow 1985), was extensively studied during the

1980s (Madhavan et al. 1984, 1986; Vila et al. 1986; Thakor 1987; Doncarli 1988). For these

methods, the measurement x(t) is selected as the desired signal, and several choices are

proposed for the input signal.

Thakor (1987) is probably the one of the most cited works among the adaptive filtering

methods for ERP estimation. From the principles of adaptive noise cancellation, Thakor

proposed a novel way of choosing reference and primary inputs. Two sets of single-trial

measurements x& (t), x, (t) (i, j being the trial index) serve as the reference and primary inputs,

respectively. The idea is to estimate the primary input with a set of delayed version of the

reference input on which some form of ensemble averaging is performed. The criticism of

Thakor's work is summarized in (Madhavan 1988), where the author asserted that if the ERP

signal is assumed to be identical across trials, the above approach does not provide any signal-to-

noise ratio improvement and distorts the signal at frequencies where signal and noise power

spectra overlap.

Madhavan (1992) proposed a modified adaptive line enhancement method. In this method

the pre-stimulus EEG data are adaptively modeled with an autoregressive (AR) model, which is

then used to filter the post-stimulus EEG data. The notion of 'modified' means that a non-

adaptive filter is used to process the post-stimulus data.









2.1.4 Kalman Filtering

Al-Nashi (1986) adopted the Kalman filtering approach for the ERP estimation problem. It

is assumed that the ERP can be modeled as a deterministic signal with additive random noise.

The additive noise is assumed to be an autoregressive moving average (ARMA) process and

another ARMA model is used for the background EEG. The scalar Kalman filter is then used to

predict the single-trial ERP. The basic assumption for Al-Nashi's approach is that the difference

between the single-trial ERP and the ensemble average is a stationary process. This is not

consistent with (Ciganek 1969), which found that the differences are usually larger in late

components than in early components.

Liberati et al. (1991) model the single-trial ERP as a time-varying AR process using the

ensemble average data and model the background EEG as a stationary AR process using the pre-

stimulus data. The AR parameters are then used to create the state and observation equations

with the ERP as the unknown states. The single-trial ERP is then estimated using the Kalman

filter equations (Kalman 1960).

2.1.5 Subspace Projection and Regularization

The subspace projection approach starts with the linear observation model:

x= s + n = HO + n (2-3)

where, 0 contains the parameters to be estimated and the ERP signal is constrained to lie in the

subspace spanned by some basis vectors, namely the columns of the matrix H.

If the ERP is assumed to consist of positive and negative humps, sampled Gaussian

functions may be a good choice for the basis vectors instead of a generic basis (e.g.,

polynomials). An alternative is to choose the eigenvectors of the EEG data autocorrelation

matrix that correspond to the first few largest eigenvalues. This is motivated by the fact that the

eigenvectors constitute a basis set with the minimum number of basis vectors that are required to









model the ERP, assuming that the ERP spans nearly the same subspace with the EEG

measurements. The least square solution with this basis set is equivalent to the principal

component regression approach (Lange 1996, Karjalainen 1997).

The above two basis sets may be combined into a single criterion, with the Gaussian basis

vectors modeling the ERP, and the subspace spanned by the eigenvectors representing the prior

information about the problem. This leads to the subspace regularization method, which is

closely related to the Bayesian mean square estimation (Karjalainen et al. 1999). The ERP may

also be estimated recursively using Kalman filtering (Karjalainen et al. 1996).

2.1.6 Parametric Modeling

There is one type of parametric models, which uses damped sinusoids as basis function for

the modeling of single-trial ERP. The model for the ERP with additive noise can be written as:

p
x(t)= p, sin(ot)+ n(t) (2-4)
i=1

Estimation of the parameters A, p,, 0, is a nonlinear problem, which can be solved with

an approximation method called Prony's method (Marple 1987). Its use with generalized

singular value decomposition was proposed by Hansson and Cedholt (1990) and Gansler and

Hansson (1991). The Prony's method was utilized by Hansson et al. (1996) for the estimation of

single-trial ERP and robust performance was achieved for EEG data with SNR>10dB. A more

recent improvement, called piecewise Prony's method was proposed by Garoosi and Jansen

(2000) to deal with nonstationary characteristics of the sinusoids.

Another well studied tool that can be used for the analysis of single-trial ERP is the

wavelet transform (Daubechies 1992). Wavelets provide a tiling of time-frequency space that

gives a balance between time and frequency resolution and they can represent both smooth

signals and singularities. This makes them suitable models for the analysis of transient and









nonstationary signals like the ERP (Thakor 1993; Schiff et al. 1994; Samar 1995; Coifman 1996;

Basar et al. 1999; Effern et al. 2000; Quian and Garcia 2003).

The idea is based on a technique called 'wavelet shrinkage' or 'wavelet denoising', which

can automatically select an appropriate subset of basis functions and the corresponding wavelet

coefficients. This relies on the property that natural signals, such as images, neural activity, can

be represented by a sparse code compromising only a few large wavelet coefficients. Gaussian

noise, on the other hand, compromises a full set of wave coefficients whose size depends on the

noise variance. By shrinking these noise coefficients to zero using a thresholding procedure

(Donoho and Johnstone 1994), one can denoise data.

However, the application of the wavelet method to single-trial ERP analysis requires some

form of ensemble averaging in order to derive an 'optimal' wavelet basis set that is tuned to the

ERP signal. Sometimes the appropriate selection of the ERP ensemble may be a difficult task

due to the effects of internal and external experimental parameters (Effern et al. 2000).

2.1.7 Other Methods Using Single-Channel Recording

Some methods exist that try to explicitly estimate the latencies of the single-trial ERP

components. A simple approach is to use cross-correlation of the signal with a template

waveform and find the maximum point of the correlation (Gratton et al. 1989). Pham et al.

(1987) applied a maximum likelihood (ML) method to estimate the latencies of ERP assuming a

constant shape and amplitude. The ML method was extended in (Jaskowski and Verleger 1999)

incorporating variable amplitude into consideration.

Truccolo et al. (2003) developed a Bayesian inference framework for estimation of single-

trial multicomponent ERP termed differentially variable component analysis(dVCA). Each

component is assumed to have a trial-invariant waveform with trial-dependent amplitude scaling

factors and latency shifts. A Maximum a Posteriori solution of this model is implemented via an









iterative algorithm from which the component's waveform, single-trial amplitude scaling factors

and latency shifts are estimated. The method works well for relatively low-frequency and large-

amplitude event-related components.

2.2 Single-Trial ERP Estimation Using Multi-Channel Recording

The use of multi-channel recording for the estimation of single-trial ERP gave rise to a

number of spatiotemporal filtering methods. These methods assume, either explicitly or

implicitly, a generative EEG model, which we will introduce in the next section. We then explain

what is meant by a 'spatial' filter and illustrate what it can do for us in estimating ERP on a

single-trial basis. A review on existing spatiotemporal filtering methods is furnished in the

following section, where we concentrate on methods that are based on well-known statistical

principles.

2.2.1 Generative EEG Model

We start with the 'neural generator' assumption of EEG data, i.e., neuron populations in

cortical and subcortical brain tissues act as current sources (Caspers et al. 1980, Sams 1984).

Within the EEG frequency range (below 100Hz), brain tissues can be assumed to be primarily a

resistive medium (Reilly 1992). Thus, according to Ohm's law, the electrical potentials collected

at each sensor (channel) as a result of volume conduction, is basically a linear combination of

neural current sources (and non-neural artifacts). The linear generative model for EEG data can

be written in matrix form:

X=A-S (2-5)

or:

N
X = a,s, (2-6)

where, N is the number of current sources.
where, N is the number of current sources.









We denote the single-trial EEG data with a D x T matrix X, with D channels and

T samples; S is a Nx T matrix with each row s, representing the time course of the current

density of the i-th current source; A is an unknown Dx N matrix. Strictly speaking, the number

of the neural current sources N is necessarily much larger than the number of channels D. It is

usually assumed that the numbers of sources and sensors are equal for the purpose of

convenience.

The column vector a, of the matrix A represents the projection of the i-th current source

to each sensor at the scalp and is called the forward model associated with the source. This scalp

projection is generally unknown and depends on the location and orientation of the dipolar

current source as well as the conductivity distribution of the underlying brain tissues, skull, skin

and electrodes (Parra et al. 2005). Thus, if the scalp projection can be estimated, it may provide

us some further evidence to the neurophysiological significance of the corresponding estimated

source.

An equivalent way to write the generative EEG model (2-5) is to use the notation of time

series:

x(t)= A.s(t) (2-7)

where, x(t)= [x,(t), ., xD (t)] is a column vector representing the EEG recordings in D


channels; s(t) = [s (t), ..., s (t)] is a column vector representing the time course of the current

density of the sources. A is the same matrix defined as before.

There is some degeneracy in the model, i.e., the scaling factor of the current source s, and

its corresponding scalp projection a,. In this case, either the current sources or the scalp

projections are constrained to have unit power to avoid ambiguity. We wish to point out that









there is no ambiguity whatsoever if we want to extract or eliminate from the EEG data the

contribution of the i-th current source, i.e., X' I asT

2.2.2 What Is a Spatial Filter and What Can It Do?

To illustrate our point, we begin with a simple example. Suppose we measure the EEG

from 3 electrodes where only 3 sources are present. Using the time series notation (the numbers

are selected for illustration purposes):

x, (t)=s, (t)+ 2 s t)+ s,(t)
x () = 2.1(t) + S2 (t) + s (t) (2-8)
x (t) = s,(t)+ s (t) + 2 s(t)

We would like to recover each of the three sources using the EEG measurements from all

the available sensors. Sine the measurements are linearly related to the current sources, we

speculate that this could be done by linearly combining all the EEG measurements through a

weight vector w:

D
y(t) = w'x(t)= x, (t) (2-9)
i=1

In fact, if we select the weight vector in (2-9) as: w' = [-1,3, -1], we will get,

y(t) = -x, (t) + 3. x (t)- x (t) = 4s, (t) (2-10)

which is exactly the first current source with a scaling factor. The other two sources can be

recovered by using the weight vectors: [3, -1, 1], [-1,-, 1,3] respectively.

Of course the above example is simplistic, because in reality, there are certainly numerous

current sources simultaneously active in the brain and the number of sensors is usually up to a

few hundred. We also do not know any of the elements of the matrix A in general. However, the

following point should be clear: by combining the EEG measurements from multiple channels

with a simple weight vector, we are able to recover (or estimate) many sources of interest that









could not be recovered using a single channel. In principle, the rejection of the interference can

be perfect, as shown in (2-10) if the coefficients are known.

The weight vector w in (2-9), which operates on the EEG measurements in the sensor

space, is called a spatialfilter. Just like a filter operating in the time domain, a spatial filter can

have either low-pass or high-pass characteristics in the spatial frequency domain. For instance, a

spatial filter summing the measurements from a group of neighboring sensors have a low-pass

characteristic; the use of a single channel recording corresponds to a high-pass spatial filter with

an 'impulse' response attenuating the data from all the other channels to zero.

The selection of the spatial filter w is usually based on some constraints or desired

characteristics of the output y(t). Different constraints will generally lead to different methods

of extracting the outputs. Loosely speaking, maximum power of the outputs leads to principal

component analysis (PCA); statistical independence among the outputs leads to independent

component analysis (ICA); and maximum difference between the outputs leads to linear

discriminant analysis (LDA). We will review these and other spatiotemporal filtering methods in

the following section.

2.3 Review of Spatiotemporal Filtering Methods

In this section, we provide a review on existing spatiotemporal filtering methods for single-

trial ERP analysis. Particularly, we focus on two popular and well established methods, namely,

principal component analysis (PCA), independent component analysis (ICA).

2.3.1 Principal Component Analysis (PCA)

From the early days of cognitive ERP research, principal component analysis (PCA) was

already proposed as a linear, multivariate data-reduction approach (Donchin, 1966). Since then,

PCA has been one of most widely used tools among psychologists for ERP analysis (Glaser and









Ruchkin, 1976; Donchin and Heffley, 1978; Mocks and Verleger 1991; Chapman and McCrary

1995; van Boxtel 1998). By identifying unique variance patterns in a given set of ERP data, PCA

decomposes the variance structure of the observed data into a set of latent variables that ideally

correspond to the individual ERP components. In the ERP research community, these latent

variables are usually called 'factors' instead of 'components' to avoid confusion with the ERP

components.

Among the vast literatures on PCA applied to ERP analysis, one classical method using the

PCA-Varimax strategy, is particularly popular and is the primary analytic tool for many ERP

researchers (Gaillard and Ritter 1983). The method treats the recorded potential at a given time

of the EEG epoch as variables. The domain of the observations is taken to be the Cartesian

product of the recording channels, experimental conditions, participants. Suppose we have T

samples in a given EEG epoch, D recording channels, C experimental conditions and P

participants. The data matrix for this method has a dimension of T x (D x C x P). This particular

arrangement of the data matrix leads to the so-called temporal PCA approach, which gives

orthogonal factors (eigenvectors of the covariance matrix). The PCA solution is then followed by

the Varimax rotation (Kaiser, 1955). The Varimax rotation is an orthogonal rotation that aims to

maximize the values that are large for a factor and minimize the values that are small (by

maximizing the fourth power the factor). This corresponds to the 'maximum compactness'

criterion, which will make the new factors have a small number of large values and a large

number of zero (or small) values. This is reasonable for ERP estimation because for the most

part, ERP components appear to be monophasic and compact in time.









It is easy to see that the above PCA-Varimax approach is a spatiotemporal filtering method.

We denote X,p as the single-trial EEG data defined in (2-5) from c-th experimental condition

and p-th participant. Then the covariance matrix is:

1
C= XT X (2-11)


where,

1 PC
X =-P X, (2-12)
PCp=l c=l

Formally, PCA is equivalent to the singular value decomposition (SVD) of the data matrix

defined in (2-12), which is the average EEG data matrix for all experimental conditions and all

participants. Suppose we have the SVD of X as follows:

X = UVT (2-13)

where, U, V are orthogonal matrices of dimension D x D and T x T, and contain the left-

singular and right-singular vectors, respectively. Y contains the singular values of the data

matrix. Equation (2-13) can be equivalently written as:

UTX= _VT (2-14)

The Varimax rotation procedure simply adds another D x D orthogonal matrix R

multiplied on the both sides of (2-14):

RUTX = RV'T (2-15)

The right side of (2-15) is a DxT matrix, whose rows can be seen as the factors extracted

by the PCA-Varimax approach.

We define a Dx D matrix:









T
W1
W= : =RUT (2-16)
T
wD_

We further denote:


y1
Y= : = RV'T (2-17)

LYD

Thus, we have:

Y=W-X (2-18)

or:


y, =wI.X, for i= 1,...,D (2-19)

This is the familiar form for the spatial filter defined in (2-9), which is now written in

matrix form. Clearly, the matrix W is an orthogonal matrix. So PCA finds a number of (D)

outputs that are uncorrelated with the constraint that the spatial filters are orthogonal. On the

other hand, the PCA-Varimax method searches for outputs that are maximally compact in time

while still constraining the spatial filters to be orthogonal.

In the context of the generative EEG model, PCA basically assumes that there are equal

number of sources and channels. If we multiply W 1 on both sides of (2-18), we get:

X = W 'Y (2-20)

Thus, the rows of the output matrix Y contain the time course of the current sources, while

the columns of the matrix W-1 constitute the scalp projections of the corresponding sources.

This means that the scalp projections for the underlying current sources are orthogonal to each

other, which is a highly dubious assumption.









Due to the above problem, an oblique rotation like Promax (Hendrickson and White 1964)

has been proposed as a post-processing stage after Varimax, to relax the orthogonality constraint

on the scalp projections. Studies with both simulated and real dataset have shown that temporal

PCA with Promax extracted markedly more accurate ERP components (Dien 1998).

An alternative approach to the popular temporal PCA is the spatial PCA (Duffy et al. 1990,

Donchin 1997, Spencer, et al. 1999), which treats the recorded potential at a given channel as

variables. The EEG data matrix is formed with channel as one dimension, and time by

experimental condition by participant as the other dimension. The same rotation procedures

follow as in temporal PCA. However, spatial PCA still assumes orthogonality of the scalp

projections.

Two other well-documented problems for the PCA approach are the misallocation of

variance (Wood and McCarthy 1984) and the issue of latency jitter. Dien (1998) using extensive

simulations, has argued that spatial PCA as a complement to temporal PCA, together with

parallel analysis (Horn, 1965) to identify noise factors, and oblique rotation to allow for

correlated factors, can address these and other shortcomings of PCA.

More recent developments include a combined spatial and temporal PCA approach that is

successfully applied to real ERP data extracting known ERP components (Spencer et al. 1999,

2001). Dien et al. (2005) have presented a standard protocol to optimize the performance of PCA

when it is applied to ERP datasets, recommending the use of covariance matrix over correlation

matrix, and Promax rotation over Varimax rotation, etc.

2.3.2 Independent Component Analysis (ICA)

ICA was originally proposed to solve the blind source separation (BSS) problem, to

recover a number of source signals after they are linearly mixed and observed in a number of

sensors, while assuming as little as possible about the mixing process and the individual sources









(Comon 1994). The most basic ICA model assumes linear and instantaneous mixing, which

means that the source signals arrive at the sensors without time delay and are mixed in the

sensors linearly with other source signals. This basic ICA model naturally fits into the generative

EEG model in (2-5), which we repeat here, assuming that the sources and measurements are

random vectors:

x = As (2-21)

In the ICA literature, the observation x is called 'mixtures', and the unknown matrix A is

called 'mixing matrix'. Usually it is assumed that the number of sources is equal to the number

of sensors. In this case, the mixing matrix becomes a square matrix.

The key assumption used in ICA to solve the BSS problem is that the time courses of the

sources are as statistically independent as possible. Statistical independence means that the joint

probability density function (pdf) of the outputs factorizes. For the linear instantaneous BSS

problem, the solution is in the form of a square 'demixing' matrix W, specifying spatial filters

that linearly invert the mixing process. If the mixing matrix is invertible, the outputs should be

identical to the original source signals, except for scaling and permutation indeterminacies

(Comon 1994):

y = Wx (2-22)

There are a multitude of algorithms that have been proposed to solve the basic ICA

problem, among which, Infomax (Bell and Sejnowski 1995, Lee et al. 1999), FastICA

(Hyvarinen and Oja, 1997), JADE (Cardoso, 1999), SOBI (Belouchrani et al. 1997) are probably

the most widely used. Some algorithms are based on the canonical information-theoretic contrast

function for ICA, i.e., mutual information, or its approximations (Infomax, FastICA, etc.); others

utilize higher-order statistics of the data (e.g., forth-order cumulant) to perform source separation









(JADE); still others make use of the difference in the temporal spectra of the source signals

(SOBI). For a more detailed review on ICA and its applications to BSS problems, we refer to the

following: Cardoso (1998); Hyvarinen et al. (2001); Roberts and Everson (2001); Cichocki and

Amari (2002); James and Hesse (2004); Choi et al. (2005). Review papers comparing different

ICA algorithms and their relationships are also available: Hyvarinen (1999); Lee et al. (2000).

We notice that while PCA requires the spatial filters to be orthogonal, here in the case of

ICA, there is no more constraint on the spatial filters (or the demixing matrix W). On the other

hand, while PCA only uses second-order statistics (the covariance matrix), to decorrelate

outputs, ICA imposes a much stronger condition, i.e., independence on the outputs. The fact that

ICA tries to factorize the joint pdf of the outputs implies that all the higher-order statistics (HOS)

are taken into consideration by ICA. This means that for non-Gaussian data, the structures

contained in HOS (e.g., kurtosis), while totally ignored by PCA, may be captured by ICA. Since

many natural signals are non-Gaussian distributed (e.g., speech signals usually follow a

Laplacian distribution), ICA may be more suitable for this and other applications than PCA.

Since Makeig et al. (1996) published their seminal paper on the application of ICA to EEG

data, there have been numerous studies during the last decade dedicated to this research topic

(Makeig et al. 1997, 1999, 2002, 2004; Vigario et al. 1998, 2000; Jung et al. 1999, 2000, 2001;

Delorme et al. 2002, 2003, 2007; Debener et al. 2005). Until now, ICA and its variants still

remains a powerful tool for the analysis of EEG and ERP data.

The application of ICA to the study of EEG data requires that the following conditions be

satisfied: (1), statistical independence of all the underlying neural source signals; (2), their linear

instantaneous mixing at the sensors; (3), the stationarity of the mixing process.









Since most of the energy in EEG data lies below 100Hz, the quasistatic approximation of

Maxwell equations holds. So there is (virtually) no propagation delay of the electrical potentials

from the neuronal sources to the sensors through volume conduction. Thus the assumption of

instantaneous mixing is valid. The linearity of the mixing follows from the Maxwell equations as

well. The stationarity of the mixing process corresponds to a constant mixing matrix. For the

dipole source model, this means that the dipolar neuronal sources should have fixed locations

and orientations. Although there is no reason to believe that these neuronal sources are spatially

fixed over time, for those that are involved in a specific information processing task and

therefore are of interest to ERP researchers, they should at least have a relatively stable

configuration or a stable scalp projection, which is congruent with the definition of ERP

components as proposed by Fabiani et al. (1987).

We have seen that conditions (2) and (3) are approximately valid for EEG data. The most

debatable and perhaps perplexing condition is the first one: statistical independence of all the

underlying neural source signals. The independence criterion applies solely to the amplitudes of

the source signals, and does not correspond to any consideration of the morphology or

physiology of the neural structures. However, the different nature of the sources originated from

completely different mechanisms often yields signals that appear to be statistically independent.

Particularly, analysis of the distributions of artifacts such as the cardiac cycle, ocular activity has

shown the statistical independence assumption approximately holds (Vigario 2000).

Although ICA continues to be a useful tool for EEG and ERP analysis, there are also some

limitations to it. First, ICA can decompose up to (or at most) D sources from data recorded at D

scalp electrodes (D may be ranged from several dozen to a few hundred). On one hand, the

researcher has to analyze the extracted D components one by one (including the time course and









scalp projection), which is laborious when is D large and the results are subject to interpretation.

If he/she chooses to analyze only a part of the all the components, the subsequent analysis is

correlated with the retention criterion. (Note that PCA also has this problem). On the other hand,

the effective number of statistically independent signals contributing to scalp EEG is almost

certainly much larger than the number of electrodes D. Using simulated EEG data, Makeig et al.

(2000) has found that given a large number of sources with a limited number of available

channels, ICA algorithm can accurately identify a few relatively large sources but fails to

reliably extract smaller and briefly active sources. This suggests that ICA decomposition in high

dimensional space is an ill-posed problem.

Second, the assumption of statistical independence used by ICA is violated when the

training dataset is too small or separate topographically distinguishable phenomena nearly

always co-occur in the data (Li and Principe, 2006). In the latter case, simulations show that ICA

may derive a single component accounting for the co-occurring phenomena, along with

additional components accounting for their separate activities (Makeig et al. 2000).

These limitations imply that the results obtained with ICA must be validated by researchers

using behavioral and/or physiological evidence before their functional significance can be

correctly interpreted. Current research on applications of ICA is focused on incorporating

domain-specific knowledge into the ICA framework. Recently there has been work on

combining ICA with the Bayesian approach (Tsai, et al. 2006) or with the regularization

technique (Hesse and James, 2006).

2.3.3 Spatiotemporal Filtering Methods for the Classification Problem

It is worthwhile to point out a related but different approach, which is the (supervised)

single-trial EEG classification problem. It is generally less difficult than the (unsupervised)

single-trial estimation problem in the sense that the availability of label information for









classification facilitates learning. Many spatiotemporal filtering methods have been proposed for

the single-trial EEG classification problem, which include, but not limited to, common spatial

patterns (CSP) (Ramoser et al., 2000), common spatio-spectral patterns (CSSP) (Lemm et al.,

2005), linear discriminant analysis (Parra et al., 2002), bilinear discriminant component analysis

(Dyrholm et al., 2007).

The common spatial patterns method was initially proposed by Koles et al. (1990) to classify

normal versus abnormal EEG (Koles et al. 1994). The method has been used for single-trial EEG

classification in brain-computer interface (BCI) systems (Muller-Gerking et al. 1999; Ramoser et

al. 2000).

Given the single-trial EEG data for two different experimental conditions, the CSP method

decomposes the EEG data into spatial patterns, which maximize the difference between the two

conditions. The spatial filters are designed such that the variances of the outputs are optimal (in

the least-square sense) for the discrimination of the two conditions. This is realized by

simultaneously diagonalizing the two covariance matrices of the EEG associated with the two

experimental conditions. The two resulting diagonal matrices (containing the eigenvalues for the

two covariance matrices) add up to the identity matrix. Thus, the spatial filters that give the (n,

an integer) largest variance in their outputs (associated with the largest eigenvalues) for one

condition, will accordingly give the (n) smallest variance in their outputs for the other condition;

and vice versa. It is along these directions that the largest differences between the two conditions

lie. In (Miller-Gerking et al. 1999), the CSPs are called the source distribution matrix

(equivalent to the mixing matrix in ICA), and the spatially filtered outputs are claimed to be the

source signals, although the EEG data were temporally band-pass filtered between 8-30Hz prior

to analysis. Para et al. (2005) showed that the simultaneous diagonalization of the covariance









matrices is equivalent to the generalized eigenvalue decomposition, and according to Parra and

Sajda (2003), the CSP method is in fact estimating the independent components of the

temporally filtered EEG data.

The original CSP method does not take into account the temporal information of the filtered

EEG data. In light of this, Lemm et al. (2005) proposed an algorithm called common spatio-

spectral pattern (CSSP), which utilized the method of delay embedding and extended the CSP

algorithm to the state space (with only one tap-delay). Dornhege et al. (2006) further improved

the CSSP algorithm by optimizing an arbitrary finite-impulse response (FIR) filter within the

CSP framework. The overfitting of the spectral filter is controlled by a regularizing sparsity

constraint.

The CSP method and its variants all use the relevant oscillatory brain activity for EEG

classification. Sometimes it is more appropriate to use coherent evoked potentials (of low-pass

nature) instead. Para et al. (2002) proposed a spatiotemporal filtering method using conventional

linear discrimination to compute the optimal spatial filters for single-trial detection in EEG.

Specifically, the search for the optimal spatial filter given the single-trial EEG data as in (2-7), is

based on constraining the output y(t) to be maximally discriminating between two different

experimental conditions. The optimality criterion is restricted to a pre-specified time interval,

i.e., the time corresponding to a number of samples prior to an explicit button push. After finding

the optimal spatial filter using conventional logistic regression, the output is averaged within that

period of time to obtain a more robust feature. The detection performance is then evaluated using

receiver operating characteristic (ROC) analysis on a single-trial basis.

Unlike other conventional methods such as ICA, where the scalp projections are given

directly by inverting the demixing matrix containing all the spatial filters, here since there is only









one spatial filter and one output, other techniques have to be sought in order to estimate the scalp

projection associated with that output. Parra et al. did this by projecting the EEG data to the

discriminating output y(t) assuming that the output is uncorrelated with all other brain sources,

and found that the discrimination model captured information directly related to the underlying

cortical activity. The method was improved in (Luo and Sajda, 2006), where the pre-specified

time interval is allowed to be different and optimized for each EEG channel. This effectively

defines a discrimination trajectory in the EEG sensor space.









CHAPTER 3
NEW SPATIOTEMPORAL FILTERING METHODOLOGY: BASICS

In this chapter, we propose a new spatiotemporal filtering method for single-trial ERP

estimation. The method relies on modeling of the ERP component descriptors and thus is tailored

to extract small signals in EEG. The model allows for both amplitude and latency variability in

the actual ERP component. We constrain through a spatial filter w the extracted ERP

component to have minimal distance (with respect to some metric) in the temporal domain from

a presumed ERP component. Note that we do not constrain the entire ERPs, but instead a single

ERP component. We maintain the point in the next section that the spatial filter may be

interpreted as a noise canceller in the spatial domain. We then introduce two approaches for the

proposed method: the deterministic approach and the stochastic approach.

3.1 Spatial Filter as a Noise Canceller in the Spatial Domain

Since the method deals with one ERP component at a time, we wish to distinguish between

'signal' and 'noise' instead of using the general term 'sources'. To accommodate this distinction,

we rewrite the generative EEG model in (2-6) as follows:

N
X=a-ST + Zb,n f (3-1)


where, s is the time course of the ERP component to be extracted, n, denotes noise in general.

The distinction between 'signal' and 'noise' is somewhat arbitrary, e.g., when P300 is the signal

of interest, N100 will become noise in the model. Note that for notational convenience, we have

assumed the effective number of sources to be N+1.

The EEG model in (3-1) can in turn be rewritten as:

N
X = ca-.s+so ,b .no, (3-2)
i=1









where, ao, so, bo,, no, are the normalized versions (with respective to some norm, e.g., 12)

of their counterparts in (3-1) and o, > o, ,. The scalars o,,o, may be seen respectively as the

overall contributions of the signal and noise to the single-trial EEG data. In the case of

independent noise, we may define the SNR for the single-trial EEG data (note that it is different

from SNR in a single channel.) as:


SNR = 20log J 2 (3-3)


The vectors ao, b', represent the scalp topography of the corresponding signal and noise.

For a meaningful ERP component, it must have a stable scalp topography a,. Thus, we may

assume that a, is fixed for all trials. We also assume that the waveform of a particular ERP

component so dimensionlesss) remains the same for all trials, although its amplitude o, may

change across trials.

Next, we claim the following lemma, which is basically a direct consequence of the linear

generative EEG model in (3-1).

Lemma: There exists a spatial filter w, that will completely reject the interference from the

first D -1 largest noise in the output when it is applied to the single-trial EEG data, if,

det[a b, ... bD-1] (3-4)

Further, the extracted ERP component will approach the actual ERP component if,

-, >> O (3-5)

Such a spatial filter w can be found by taking the first row of the inverse of the matrix in (3-

4). Note that (3-4) implies that,

angle(a,b,) ->,i=1,...,D-1 and, angle(b,,b,) -0,1<- i








which means that at least the scalp topography of the source and the first D -1 noise should

not be the same or very similar to each other from a computational perspective.

Here, we wish to stress the point that the spatial filter specified in the above lemma may be

interpreted as a noise canceller in the spatial domain. It may or may not be the optimal spatial

filter for enhancing the SNR in the extracted component. In addition, the SNR enhancement due

to the spatial filter increases monotonically with the number of channels (electrodes) if the EEG

data were measured in those channels. This means that the more channels we use to record the

EEG, the higher SNR we will get (theoretically) in the extracted component. Note that the

lemma is an existence statement, it does not tell us how to find such a 'optimal' spatial filter.

This will be the subject of the next two sections.

3.2 Deterministic Approach

Most ERP components are monophasic waveforms with compact support in time. The

morphology of the waveform can be considered relatively fixed due to the common

cytoarchitecture of the neocortex and similar neuron populations, but may vary in both its peak

latency and amplitude from trial to trial. Based on this, we assume that a particular ERP

component can be modeled by a fixed dimensionless template (e.g. no physical unit), in the

temporal domain, denoted by so(l) (where / is the unknown peak latency), multiplied by an

unknown and possibly variable amplitude o- across trials. We attempt to estimate the variable

peak latency and amplitude on a single-trial basis.

3.2.1 Finding Peak Latency

Since we do not know the peak latency in a single trial, we denote the template as so(r)

with a variable time lag parameter r and slide it one lag a time to search for the peak latency.

The search for the optimal filter w could be realized by minimizing some distance measure









between the spatially filtered output w' X and the assumed waveform s (r) for the particular

ERP component. We propose the following cost function based on second-order statistics (SOS):

min W -X-so(r)' 2 (3-7)
W 2

Note that the above optimization is with respect to w only, with r fixed. The optimal

solution for w is given by:

w(r)= (XXT) .X.so(r) (3-8)

Obviously, the optimal spatial filter w depends on which ERP component is to be extracted,

and also is a function of the variable time lag r. From (3-7) and (3-8), we obtain the cost solely

as a function of the time lag r :


J(r)= s(r.X' (XX') X -I (3-9)


The peak latency of the ERP component can be set as the time lag where the local minimum

of J(r) occurs within the meaningful range of peak latencies (T, ) for that particular component

(provided that its waveform is monophasic) i.e.,

= argmin J(r) (3-10)
IGTs

The estimated ERP component is then (this need not be normalized):

y,(/)= X' .w(/) (3-11)

It can be shown that under certain conditions, the solution in (3-10) is identical to the true

peak latency of the ERP component (Appendix A). Exact match between the modeled and actual

ERP component is not a necessary condition for the solution in (3-10) to be correct. For instance,

it is easy to show that when both components are symmetric waveforms, then (3-10) also gives

the correct latency.









3.2.2. Finding Scalp Topography and Peak Amplitude

In the following, we make the index for trial number k explicit. Denote the estimated ERP

component for k -th trial by (the peak latency depends on the trial number):

Yk () =-kYk (4) (3-12)

We can absorb the scalar ok into a variable scalp topography:

ak = kao (3-13)

In order to estimate the unknown scalp topography and amplitude of the ERP component,

we assume that the ERP component is uncorrelated with all the noise sources. Replacing the

dimensionless ERP component in (3-2) by its estimate in (3-11) and multiplying yk (l) on both

sides of (3-2), we will get an estimate for the single-trial scalp topography (the cross terms

T
n,, yk k() vanish because of the uncorrelatedness assumption):


a Xk'sk(k)
ka =(4 (3-14)
Yk (skT)Ysk (1k)


Taking the normalized version (note that ak is in Volts) we have,


ak (3-15)


Ideally, the normalized aok should be the same as the dimensionless scalp topography ao.

However, in low-SNR EEG data, the above estimation in (3-14) is very poor, due to the

interference from background activity in the finite-sample data. To estimate the scalp topography

for a stable ERP component, we propose the following cost function:

K
min a-a k 2 (3-16)
ao k=









This corresponds to a maximum likelihood (ML) estimator for a, under the assumption that

each entry of the normalized single-trial scalp topography is an independent identically

distributed (i.i.d.) Gaussian random variable. The optimal solution for (3-16) is a simple average

of the estimated single-trial scalp topographies for all K trials. Taking the normalized version, the

following estimate for a, is obtained:

1K K (3-17)
no = K n-'ok fiK [ ok (3-17)
k= k=1 2

Notice that (3-17) is in fact a weighted average of the estimates in (3-14). We also point out

that (3-17) is different from summing up directly (3-14) for all trials since the peak latency

parameter is involved and it changes from trial to trial.

In the ideal case, the two vectors a, and ak are identical except for a scaling factor, which

is exactly the unknown amplitude ak associated with the ERP component in the k -th trial.

Replacing their respective estimates in (3-14), we can find 0k using again a SOS criterion:

min aik --ka (3-18)


Simple calculation leads to the following estimate for the amplitude:

=k oT k (3-19)

This estimate involves information from all the available channels. In order to eliminate the

indeterminacies of the linear generative EEG model, we set the peak amplitude as the maximum

of the estimated ERP component in (3-12), i.e., k ybk (k). (All the amplitudes in the rest of the

paper refer to this quantity). Accordingly, the contribution of the ERP component to the EEG

data may be computed by:

X, = r yg k) (3-20)









These estimates for the scalp topography, peak latencies and amplitudes of ERP

component can be used to analyze its psychological significance on a single-trial basis.

Note that we do not directly compute the amplitude from the estimated component, nor do

we measure it in any single channel. Instead, the amplitude is computed in (3-19) indirectly

through an inner product of two scalp topographies, which involves information from all

available channels. These estimates for the scalp topography, peak latencies and amplitudes of

ERP component can then be used to analyze their psychological significance on a single-trial

basis.

We wish to point out that in contrast to all the spatiotemporal methods mentioned before,

where only one, representative spatial filter (or matrix consisted of spatial filters), is computed

given all the EEG data, here the optimal spatial filter is computed on a single-trial basis, i.e.,

given a single-trial EEG data matrix, we can get a spatial filter, as in (3-8). The reason we did it

in this way is that, we believe that in theory the optimal spatial filter that is designed to extract

small signals should change from trial to trial. Note in (3-2) the 'noise' sources are sorted in

decreasing order of their power. It is likely that the noisy sources that have relatively large power

change drastically across trials. In effect, this will change the configuration in (3-2) and

accordingly, the optimal spatial filter will also change.

3.3 Stochastic Approach

In the deterministic approach, the ERP component is considered to be a deterministic signal

except for a random latency and amplitude across trials. It does not take into account the intrinsic

error in the modeling of the ERP component itself. Here we propose a stochastic approach for

the spatiotemporal filtering method. The idea is to constrain the extracted ERP component to be

close to the presumed component with respect to some statistic.









We still use the generative EEG model as in (3-1), but here both the signal and the noise are

interpreted as stochastic processes. The approach utilizes the following observation model to

search for the optimal spatial filter:

XTw = + v (3-21)

where, s is the actual ERP component and v is the observation noise appearing in the

spatially filtered EEG data. They are both random vectors with each entry as a sample within a

certain time interval from the corresponding stochastic processes.

We assume that the ERP component is generated from the following additive model:

s = g+u (3-22)

where, g is a fixed signal with a certain morphology serving as the template for the ERP

component, and u represents the model uncertainties of the ERP component and is assumed to

be independent of the observation noise v.

Given the above model in (3-21) and (3-22), the optimal spatial filter may be found by

maximizing the log-likelihood of the filtered EEG data y = X'w :

max L(w) = log p(y I g) (3-23)

or,

max L(w) = log p(u + v) (3-24)

The log-likelihood function has a simple form under the assumption that u and v are zero-

mean Gaussian stochastic processes with covariance matrices C, CD respectively, and they are

independent of each other, since u + v is nothing but another Gaussian stochastic process with

covariance matrix CM + CD. In this case, maximizing the log likelihood of the (transformed)

observed data given the observation model and the template of the ERP component yields the

following solution for the optimal spatial filter:









w, = argmin ('w- g)1 (C + CD 1 (X'w -g) (3-25)
w

This is a quadratic form of w, so it has a closed-from solution:

w = (X-1X) -.X lg (3-26)

where,

Y = CM + C (3-27)

The estimate for the ERP component is then:

Yo = X wo (3-28)

We point out that (3-25) suggests that in the Gaussian assumption, observational

uncertainties and model uncertainties simply combine by addition of their respective covariance

matrices. We also note that if CM and CD are both chosen as identity matrix (i.e., incorporating

the least amount of a priori information into the model), then the solution in (3-26) essentially

reduces to (3-8) in the deterministic approach. The optimality of the spatial filter will depend on

how we choose the two covariance matrices.

The estimation of the scalp topography and amplitude follow the same procedures as

described in (3-15) (3-20).

3.4 Simulation Study

We present in this section a simulation study with synthetic data and real EEG data

recorded from subjects during a passive picture-viewing experiment. The goal of the simulation

study is to evaluate the latency and amplitude precision of synthetically generated transients

immersed in real EEG background with different SNRs and waveform mismatching conditions.









3.4.1. Gamma Function as a Template for ERP Component

Lange et al. (1996) have used a Gaussian function as the template for an ERP component.

Here, we prefer the Gamma function for the shape of the synthetic ERP component because this

is a very flexible function for waveform modeling and has been used extensively in

neurophysiological modeling (Koch et al., 1983; Patterson et al., 1992). Freeman(1975) argued

that the macroscopic EEG electrical field is created from spike trains by a nonlinear generator

with a second-order linear component with real poles. According to this model, the impulse

response of the system is a monophasic waveform with a single mode, where the rising time

depends on the relative magnitude of the two real poles (Appendix B). This may be

approximated by a Gamma function, which is expressed by:

g(t) = c tk exp(-t / 0), t >0 (3-29)

where, k > 0 is a shape parameter, 0 > 0 is a scale parameter and c is a normalizing

constant. The Gamma function is a monophasic waveform with the mode at t = (k -1)0, (k > 1).

It has a short rise time and a longer tail for small k, and approximates a symmetric waveform for

large k. This makes it a good candidate for modeling both early and late ERP components that

tend to be symmetric in the early components and have longer tails in the late components.

Figure 3-1 shows four Gamma functions with different shape and scale parameters.

3.4.2. Generation of Simulated ERP Data

EEG data were recorded from subjects during a passive picture-viewing experiment,

consisting of 12 alternating phases: the habituation phase and mixed phase. Each phase has 30

trials. During the 30 trials of the habituation phase, the same picture was repeatedly presented 30

times. During the mixed phase, the 30 pictures are all different. Each trial lasts 1600 ms, and

there is 600 ms pre-stimulus, and 1000 ms post-stimulus.









The scalp electrodes were placed according to the 128-channel Geodesic Sensor Nets

standards. All 128 channels were referred to channel Cz and were digitally sampled for analysis

at 250Hz. A bandpass filter between 0.01Hz and 40Hz was applied to all channels, which were

then converted to average reference. Ocular artifacts (eye movement) were corrected with EOG

recordings.

The scalp topography of the synthetic ERP component is chosen as the normalized P300

scalp topography from another study (Li, et al., 2006). The simulation data were created by

taking the superposition of the 600ms (150 samples) prestimulus data from 120 trials as the

background EEG data and the scalp projected Gamma waveform as a proxy for the ERP

component.

The SNR levels given the background EEG data can be easily adjusted by modifying the

normalizing constant c in (17). We define the SNR given the single-trial EEG data as:


SNR= 20logT (3-30)


Note that since the 'actual' ERP component is a nonstationary signal, the magnitude o, is

taken be its peak amplitude. This is different from the conventional definition of SNR. The SNR

levels given the background EEG data can be easily adjusted by modifying the normalizing

constant c in (3-29).

3.4.3 Case Study I: Comparison with Other Methods

We will test the performance of the proposed spatiotemporal filtering method at varying

SNR levels (from -20dB to 12dB), where we have access to the 'actual' (synthetic) ERP

component. Two scenarios will be investigated: one where there is an exact match between the

synthetic and the ERP component template and the other where there is a mismatch between the









two components. For the case of exact match, we use the parameters k = 3, 0 = 13 for both ERP

components. For the mismatch case, the synthetic ERP component remains the same, but the

template has a different waveform with parameters k = 5, 0 = 6. Figure 3-2(a) shows the

waveforms of the two components for the mismatch case. We fixed the peak latency of the

synthetic component at 200ms for all the 120 trials.

For Woody's filter, we selected channel Pz for analysis, and use the initial ensemble average

as the template, avoiding the iterative update on the template (since the true latency is fixed, this

is the best-case scenario for Woody's filter). We search around the true latency within 100ms for

maximum correlation. We estimate the peak amplitude by taking the average of the peak value

and its two adjacent values (corresponding to a noncausal low pass filter with cut-off frequency

of 12Hz). For spatial PCA, we select the eigenvector which has the maximal correlation with the

P300 scalp topography (Note this is an ideal case for PCA, since in reality we do not know

exactly the true scalp topography, nor the exact time course).

The simulation results are summarized in Table 3-1, 3-2 and 3-3, which show the

estimation mean and standard deviation for the estimated latency and the ratio between

estimated and true peak amplitude, as well as the correlation coefficient between estimated and

true scalp topography. Since PCA does not give an explicit estimate for latency and amplitude,

we will omit the latency estimation and only compute the amplitude ratio between the estimated

ERP component and the synthetic ERP component.

First, we note that the single-trial estimation of the peak latency is very stable in the case of

exact match. Notably, for EEG data with SNR higher than 4dB, the method estimates the

latency correctly for all the trials. Second, the amplitude estimate for the exact case is also

stable but is more variable than the latency estimation. The mean amplitude approaches to one









and the standard deviation decreases to zero as the SNR increases. We may say that in the case

of exact match between the model and the component, the estimator for the amplitude is

asymptotically unbiased and asymptotically consistent with increasing SNR.

The mismatch between the model and the generated component effectively introduces a

bias in the estimation of the latency for realistic SNRs. From Table 3-1, we can see that the

mean latency approaches to 188ms, yielding a bias of around 3 samples and the standard

deviation is around 9ms (around 2 samples). However, the mean does converge to its true value

(200ms) and the variance does approach to zero with increasing SNR, although very slowly. For

instance, for SNR as high as 40dB and 60dB, the estimated latency has a mean-std statistic of

193 4ms and 200 + 0.7ms, respectively. Therefore, empirically we can see that the estimation

for latency under the mismatch case is also asymptotically unbiased and asymptotically

consistent with increasing SNR. The mismatch of components introduced a bias in the

estimation of the amplitude for realistic SNRs, which is partly due to the difference in the

waveforms of the synthetic component and template. The estimated amplitude has a statistic of

1.0458 0.0075 and 1.0015 0.0002 at a SNR of 40dB and 60dB, respectively. Thus in the

case of mismatch, the estimator for the amplitude is also asymptotically unbiased and

asymptotically consistent with respect to SNR, although the convergence is much slower than

the exact match case. Of course, the estimation variance does increase notably as SNR

decreases. But as evident from Table 4-2, our method at -12dB still gives a estimation variance

smaller than Woody's method at OdB.

Table 3-1 and 3-2 clearly show the advantage of using spatial information, in contrast to the

Woody filter based on single-channel analysis. Specifically, the estimation variance of Woody

filter for both latency and amplitude are much larger for realistic (negative) SNR conditions.









PCA overestimates the amplitude of the ERP component for low SNR data. In contrast to

our method, PCA gives a statistically significant bias even at OdB from the baseline at 12dB (p-

value less than 0.0001). This means that varying SNR (below OdB) imposes a serious problem

on the application of PCA in low SNR conditions.

Finally, the simulated mismatch of components affects the estimation of the scalp

topography negligibly for the proposed method when SNR is higher than -10dB. In fact, the

estimation for scalp topography with mismatch with our method at -20 dB is comparable to

PCA at -4 dB.

3.4.4 Case Study II: Effects of Mismatch

The second simulation concerns the effects of the mismatch on the estimation, specifically

mismatch in the spread parameter which is the most important. We use the same synthetic

component as before and vary the spread parameter with K fixed. Fig. 3-2(b) shows the

waveforms of the synthetic and 3 of the templates, including a Gaussian with a spread of 20.

The results are summarized in Table 3-4 and 3-5, for SNR = -20dB and -10dB, respectively. We

have included a new quantification for the amplitude estimation: coefficient of variation (CV),

which is defined as the ratio of the standard deviation to the mean of a random variable. It is

used as a measure of dispersion of the estimated amplitude (since its true mean is fixed at 1).

From Table 3-4 and 3-5, we can see that at the same SNR level, the mean and variance of

the estimated amplitude systematically change with respect to the spread parameter, i.e., larger

spread parameter gives smaller amplitude. The degree of variability in the amplitude estimation

(measured by coefficients of variation) for mismatch cases exceeds the exact match case by less

than 10% except at -10dB for a spread parameter of 7. In some cases, CV is even smaller than

the exact match case, which gives a better estimate, but only in terms of the amplitude. This is

important because although the estimated amplitudes differ in the mismatch case, as long as we









use the same template, these amplitudes on average will always be magnified or attenuated by a

constant factor at a certain SNR level. Intuitively, this means that given a fixed template and

varying SNR (>-20dB), the dominant source of variability in amplitude estimation mainly

comes from the estimation variance (not bias) and this variability (measured by CV) is well-

bounded from above for a range of spread parameters. Therefore, these amplitude estimates

may still be effectively compared across experimental conditions as long as the same

(meaningful) template is used and the SNR of the ERP data does not fall below -20dB.

However, the mismatch clearly introduces a bias in the latency estimation, which may be as

large as 50ms in absolute terms. This may or may not be significant depending on the

applications. We can also see from the two tables that choosing a higher spread parameter will

lead to a slightly better estimation for the scalp topography. But of course, this comes at the cost

of much worse latency estimation.









Table 3-1. Latency estimation: mean and standard deviation
SNR (dB) Woody Exact match


191 60
190 + 60
190 + 61
191 + 63
195 + 60
195 + 54
199 + 45
199 + 32
200 + 15


202 10
201 +7
201 5
200 3
200 2
200 + 1
200 0
200 0
200 0


True latency: 200ms.


Table 3-2. Amplitude estimation: mean and standard deviation
SNR (dB) Woody PCA Exact match Mismatch
-20 -0.22 9.32 10.7 2.12 1.68 2.20 2.22 2.93
-16 0.07 + 5.95 6.80 + 1.35 1.34 + 1.29 1.71 + 1.75
-12 0.40 3.84 4.33 0.86 1.18 0.77 1.49 1.05
-8 0.36 2.60 2.78 0.55 1.10 +0.47 1.34 0.64
-4 0.53 + 1.75 1.86 + 0.36 1.06 + 0.29 1.26 + 0.40
0 0.64+ 1.14 1.35 0.23 1.04 0.18 1.23 0.25
4 0.80 0.74 1.13 0.13 1.02 0.12 1.20 +0.16
8 0.90 0.45 1.05 0.08 1.01 0.07 1.19 0.10
12 0.98 0.24 1.02 0.05 1.01 0.05 1.18 0.07
True amplitude: 1

Table 3-3. Scalp topography estimation: correlation coefficient
SNR (dB) PCA Exact match Mismatch
-20 0.568 0.829 0.759
-16 0.579 0.910 0.854
-12 0.601 0.959 0.926
-8 0.649 0.982 0.966
-4 0.759 0.993 0.986
0 0.906 0.997 0.994
4 0.979 0.999 0.998
8 0.996 1.000 0.999
12 1.000 1.000 1.000


Mismatch


190 + 16
189 + 14
188 11
188 + 10
188 + 10
187 + 10
187 + 10
188 + 10
188 9









Table 3-4. Effects of mismatch I: SNR = -20dB
Spread
Sprea Latency(ms) Amplitude
parameter
7 150 20 2.32 3.0;
9 175 16 2.18 2.7L
11 191 + 12 1.93 2.5
13 202 10 1.68 2.2
15 215 11 1.43 1.8;
17 228 + 15 1.24 1.6(
19 246 + 18 1.07 + 1.41
20 (Gaussian) 210 + 22 2.14 2.89


1


)
7
)


Coefficient of
Variation
1.30
1.25
1.30
1.31
1.31
1.29
1.31
1.35


CC. of scalp
topography
0.68
0.73
0.79
0.83
0.87
0.88
0.87
0.76


Table 3-5. Effects of mismatch II:
Spread
Spread Latency(ms)
parameter
7 151 21
9 175 16
11 190 8
13 200 4
15 211 5
17 220 7
19 229 9
20 (Gaussian) 208 11


SNR = -10dB


Amplitude

1.30 + 0.92
1.34 + 0.78
1.24 + 0.69
1.14 + 0.60
1.03 + 0.53
0.94 0.46
0.86 + 0.41
1.30 + 0.78


Coefficient of
Variation
0.71
0.58
0.55
0.53
0.51
0.49
0.47
0.60


CC. of scalp
topography
0.93
0.95
0.96
0.97
0.98
0.98
0.99
0.95














0.9

0.8

0.7

0.6

0.5
E
0.4


0.3- /

0.2

0.1

0 -----
0 2 4 6 8 10
time


- K=3,theta=2
K=5,theta=l
K=2,theta=2
K=9,theta=1


12 14 16 18 20


Figure 3-1 Gamma functions with different shapes and scales.














presumed ERP component
synthetic ERP component


) 0.6
0.4
E
8 0.4


time (ms)


K=3, theta=13
K=3, theta=7
K=3, theta=19
S Gaussian


B


Figure 3-2 Waveforms of synthetic and presumed ERP component. A) Synthetic component
Gamma: K= 3, 0 = 13; presumed component: Gamma: K= 5, 0 = 6. B) Synthetic
component Gamma: K= 3, 0 = 13; presumed components: Gamma: K= 3, 0 = 7, and
0 = 19, Gaussian with a spread of 20.









CHAPTER 4
ENHANCEMENTS TO THE BASIC METHOD

In chapter 3, we developed the basic spatiotemporal filtering method for single-trial ERP

component estimation. In this chapter, we will consider some modifications to the basic method.

Some serves as a heuristic post-processing technique (Section 4.1: iteratively refined template);

some aims to deal with large salient EEG artifacts (Section 4.2: robust estimation); some utilizes

the apriori knowledge on the scalp topography of the ERP component (Section 4.3:

regularization); some provides alternative formulations for our previous results and also derives

new ones (Section 4.4: Bayesian formulations of the topography estimation); still others try to

deal with the interference from other overlapping ERP components (Section 4.5: explicit

compensation for temporal overlap). The details are presented below.

4.1 Iteratively Refined Template

The deterministic approach of our estimation in Chapter 3 uses a fixed waveform for the

template, regardless of the SNR. The stochastic approach incorporates some degree of variability

in the template, but it is still implicit. We would like to explicitly utilize the posterior information

from the data to update or refine our apriori assumed template. Intuitively, this should improve

our estimation at least for high SNR conditions.

We use the estimated scalp topography ao as a spatial filter. The output is optimal in the

sense that it has the largest correlation coefficient with the actual component with the

uncorrelated noise assumption. (Of course, we use the estimate as a proxy for the true

topography. Note that it is different from w). The refined template is the ensemble average of

spatially filtered data, with ao as the filter. The results are shown in Table 4-1, 4-2 and 4-3 (with

one iteration of refining).









We can see that for the latency estimation, the refined template method has a larger bias

below -12dB than the original template, but improves quickly and approaches to the exact match

case for positive SNR conditions. For the amplitude estimation, the refined template consistently

beats the original template for all SNR conditions in terms of both bias and variance. It also

approaches to the exact match case for positive SNR conditions. Also see Figure 4-1. For the

scalp projection estimation, the refined template is worse below -8dB but is very close to the

exact match case above -8dB.

Figure 4-2 shows the waveforms of the synthetic component, original template and refined

template at 4 SNR conditions: -20dB, -12dB, OdB and 12dB. The spatially filtered ensemble

average is still quite erratic below -10dB. That possibly accounts for the worse performance of

refined template for very low SNR conditions. So, there exists a critical SNR below which, using

the spatially filtered ensemble average will probably worsen performance (in terms of scalp

topography, but not amplitude) compared with the original template. For this particular data set,

it would be safe to use a refined template as long as the SNR is above -10dB.

With 1 iteration, there comes 2, 3 and infinity. The natural question then is: Will it

converge? If so, what does it converge to? Theoretically, these are difficult questions. Aside

from the variable latency parameter, the scalp topography is still nonlinearly related to the data.

However, we can experimentally determine the limiting results. These are shown in Table 4-4 for

3 SNR conditions. The algorithm converges within 10 (sometimes 2) iterations. Compared with

the first iteration, the estimation for latency and scalp projection barely changes, but there is a

reduced bias and variance in terms of amplitude for negative SNR conditions.

Of course, all these results are a function of the mismatch in the waveforms and the number

of trials we use to compute the ensemble average (if we had 1000 trials instead of 120, it would









be a different story). At this point, it seems that, what matters the most to deal with negative

SNR is to accumulate more data, either in space or in time.

4.2 Regularization

Sometimes, we have some a priori knowledge of the scalp topography of the ERP

component. For instance, the P300 component usually has a large positive projection around Pz

area. In these cases, we should utilize that information and incorporate it into our model.

4.2.1 Constrained Optimization

Assuming that the ERP component latency has been estimated, we attempt to minimize the

following cost function with respect to the amplitude a and scalp topography a (which is

constrained to have a unit norm in 12):

argmin X-c-a-sT +A a-a,, (4-1)
,a F

S.t., a2 =1

where, A is a regularization parameter and a, is a normalized vector representing the a

priori knowledge of the scalp topography of the ERP component. F denotes the Frobenius

norm of a matrix. The reason that we chose this norm will be evident later: the minimization of

this norm gave the same solution as (3.14), which was derived with the uncorrelatedness

assumption.



s s
In Appendix C we derive a fix-point update equation: sas (4-2)
Aa, + oXs
a= -- --
Aa0 + cXs 2

Particularly, when A = 0 (no regularization), the optimal solution becomes:










Xs
sTs
s s (4-3)
Xs
a=--


When A 0o the optimal solution is:

aT Xs
C = s "sS (4-4)



When A takes intermediate values, the scalp topography is a weighted average of the two

extreme case solutions.

This is a "real" single-trial estimation scheme in that the amplitude is computed from one

single-trial data matrix. For the case of no regularization, it is different from our original

formulation, where the amplitude is the inner product between Xs and the normalized average

scalp topographies from all the trials. The original formulation makes the reasonable assumption

that the ERP component has a fixed scalp topography and utilizes that information.

We demonstrate the effectiveness of regularization to deal with the interference from other

overlapping (possibly unknown, if they are all known, we can explicitly compensate for that- See

Section 4.5) ERP components. Specifically, we will investigate the effect of regularization on the

estimation of amplitude and scalp topography under well controlled conditions.

We assume that there are 2 overlapping ERP components and their latencies are fixed and

known. Their waveforms are shown in Figure 4-3. They are both Gamma functions with the

same parameters K = 3, 0 = 10, with a peak interval of 80ms. The two ERP components have a

correlation coefficient of 0.36. We use templates that are exactly matched with the synthetic

components. These components are projected to spontaneous EEG data to generate simulated









ERP data. Their scalp topographies are shown in the Figure 4-4. We use that for a0 (exact a

priori knowledge).

We then find the optimal solution of c for a given A. The fixed point update always

converges within 2 steps. We summarize the results for 3 SNR conditions (12dB, -12dB, -

20dB), shown in Figure 4-5. For high SNR (12dB) data, regularization brings little difference.

Since the overlapping ERP components have the exact opposite scalp topography, it is expected

that the estimated amplitude is smaller than 1 for high SNR data. It converges to around 0.42 and

0.41 for component 1 and 2 for large A, respectively. Note the huge bias in the estimated

amplitude for low SNR (especially -20dB) without regularization. But it converges to as small as

0.69 and 0.27 for component 1 and 2 for large A, respectively. This demonstrates the necessity

of regularization for the constrained optimization problem. A reasonable regularization

parameter for all the SNR conditions is between 104 and 105, where an unbiased estimation for

amplitude could be achieved. Also notice that there is a hump for the standard deviation of the

estimated amplitude. Interestingly, this is near the reflection point of the mean amplitude. In

practice, this can give us some hint for finding a reasonable regularization parameter. Note that

for these choices of A, the correlation coefficient of scalp topography is already very close to 1.

This is an example when the overlapping components have negative correlation on the scalp.

What about positive correlated components? Figure 4-6 shows the results for the same

components, except that now they have the same scalp topography. The estimated amplitude is

expected to be larger than 1 for high SNR data. It converges to 1.6 for both components for large

A. Note the huge bias for low SNR (especially -20dB). It converges to 1.8 and 1.7 for

component 1 and 2 for large A, respectively. This also demonstrates the benefits of

regularization for the constrained optimization problem: using a sufficiently large regularization









parameter in this case can reduce the bias of the amplitude estimation, while the variance are not

affected very much. As expected, for large A, the correlation coefficient of scalp topography

converges to 1.

4.2.2 Unconstrained Optimization

Parallel with the above constrained optimization, we can also frame the problem into an

unconstrained optimization one.

argmin X-a.s' +A a/a 2 -a, 1 (4-5)


Note that the optimization variable a contains the amplitude parameter as well as the

topography information. As shown in Appendix D, a fixed-pointed update can be obtained:

Xs+Aa0 ||a||
a = (4-6)
s sAa a/|a -

Particularly, when A = 0 (no regularization), the optimal solution is the same as our original

solution in (3-14). When A -> oc, the optimal solution is not unique (any scaled version of a,

can be a solution). The problem becomes ill-posed.

We have obtained the regularized solution for single-trial scalp topography. The same

procedures for estimating the amplitude follow: take the average of normalized single-trial scalp

topography as our estimate for the overall scalp topography, then the amplitude for a particular

trial is just the inner product between this vector and the corresponding scalp topography.

We test the performance of regularization under the same conditions as in the constrained

optimization. We find the optimal solution for a given A. Fixed point update usually converges

within 10 steps.

We summarize the results for 3 SNR conditions (12dB, -12dB, -20dB) shown in Figure 4-7

and 4-8. As in constrained optimization, for high SNR (12dB) data, regularization has little effect









on the estimation. The amplitude converges to around the same values as before (0.4 for opposite

topography and 1.6 for the same topography). The variance is not affected much for all the SNR

conditions, either. As expected, the scalp topography gets monotonically better as A increases.

The difference is that the estimated amplitude mean increases monotonically (except for a

small interval) with increasing regularization parameter A. This translates to a larger bias

(particularly for low SNR data) when the overlapping components have positively correlated

topographies. The estimated amplitude becomes unstable for large A. There is no evidence at

this point that regularization can benefit the estimation for general scalp topography

configurations (both positive and negative topography correlations). So the unconstrained

optimization formulation need not be regularized, at least for overlapping components with

positively correlated topographies. This lends support to our original solution in (3.14), which is

exactly the unregularized solution to the unconstrained optimization problem here.

We also point out that, unlike the constrained optimization problem, here we utilize the

reasonable assumption that an ERP component has a fixed scalp topography. So it does not

suffer from the huge bias problem in the constrained optimization. For instance, at -20dB without

regularization, the estimated mean amplitude is around 2.3 and 1.9 for the two components with

positively correlated topographies, a modest increase from 1.6 at 12dB.

4.3 Robust Estimation: the CIM Metric

We have seen in Section 4.2 that the estimation for single-trial scalp topography in (3.14)

can be found equivalently from the minimization the following criterion:

arg min Xk -ak, ykT (4-7)


where, F denotes the Frobenius norm.









It is evident from (3-14) that the estimate for single-trial scalp topography bears a linear

relationship with the EEG data. Because of the noisy nature of EEG (particularly large salient

artifacts), this gives a noisy estimate for the single-trial scalp topography (with large variance)

and in turn translates into a noisy estimate for the single-trial amplitude in (3-18).

We would like to derive a robust estimator in order to reduce the effects of large EEG

artifacts. We can replace the Frobenius norm in (4-7) with other norms (e.g., li norm) or metrics.

Here we will consider a special metric: correntropy induced metric (CIM) proposed by Liu et al

(2007).

First we introduce what is called correntropy. Given two scalar random variables X and Y,

correntropy is defined as:

Vh(X,Y) = E[kh(X Y)] (4-8)

where, k(X -Y) is the Gaussian kernel (h is the kernel size),


kh(X Y) exp (4-9)
j2h 2h2)

The correntropy function is a localized similarity measure in the joint probability space,

which is controlled by the bandwidth parameter h (also called kernel size in kernel methods). It

induces a metric (CIM) in the sample space which behaves like the 12 norm when the sample

point is close to the origin (relative to the kernel size); when the sample point gets further apart

from the origin, the metric is similar to the 11 norm and eventually saturates and approaches to

the 10 norm (Liu, 2007). As such, CIM practically incorporates the 12 norm as a special case (if h

is chosen to be sufficiently large).

Minimization of CIM is equivalent to the maximum correntropy criterion (MCC). It has

been shown that MCC has a close relation to M-estimation (Huber, 1981) and since correntropy









is inherently insensitive to outliers, MCC is especially suitable for rejecting impulsive noise (Liu,

2007).

Now, treating each entry of the matrix in (4-7) as a realization of a random variable, we can

write our new cost function as:

argmin Xk ak yk T (4-10)


The nuisance parameter h (kernel size) should be tuned to the data (most notably to the

standard deviation). Here, we use the Silverman's rule as a baseline to quantify different values

of kernel sizes that we use in the simulation. It is given by h = 1.06dN 02, where N is the number

of samples, and d the standard deviation of the data (Silverman, 1986).

While minimization of the Forbenius norm has a closed-form solution, minimization of CIM

does not. So we have to search for a local minimum, using the standard gradient descent method.

The convergence to a certain local minimum is guaranteed by adopting a stopping criterion that

the change in the correlations of the estimate and the MSE solution between the previous and

current iteration is less than 106.

Table 4-5 and 4-6 summarize the estimation results for 2 SNR conditions (0dB and -20dB)

for the mismatch case. We also include the MSE solution for comparison. The results are mixed:

MCC gives a slightly higher variance than MSE for the estimation of amplitude, but the bias is

marginally reduced; it also gives a more accurate estimate for the scalp topography. We also

notice that at OdB, the results of MCC barely change from MSE. Intuitively, when the SNR

becomes sufficiently large, the optimal MCC solution should converge to the MSE solution (both

agree with the true values of the parameters).

These differences in the results are by no means statistically significant. We venture 2

reasons why the MCC results do not change very much from MSE.









First, the EEG data are already preprocessed and relatively clean. Large artifacts have

already been removed. The resulting distribution is not far from Gaussian. So MSE should give a

solution already close to optimal. MCC has its edge when there is large noise, especially impulse

noise. Strictly speaking, it is only optimal (in the sense of maximum likelihood) for one

particular type of distribution, just as MSE is strictly optimal for Gaussian distribution. It is not

clear that how these two criteria compare when the data distribution changes in a neighborhood

of their optimal ones. In reality (when EEG data are usually preprocessed and artifacts are

removed), there is no reason to believe that MCC will outperform MSE uniformly.

Another less compelling reason concerns the optimization process associated with MCC.

The initial condition of MCC is set to be the MSE solution (starting a random initial condition

will seldom beat MSE). When the kernel size is small, the performance surface is highly

irregular, so the optimization will never go far from MSE solution (it is stuck around the local

minimum near the MSE solution). When kernel size is sufficiently large, it is easy to see that

MCC approaches to MSE. Only intermediate values of kernel size will produce somewhat

different results from MSE. This is seen in both SNR levels, though less evident for OdB.

There are two other cost functions in the estimation of amplitude that use the MSE criterion,

i.e., (3.14) and (3.16), which can also be replaced by MCC. Li et al (2007) have tested its

performance and the improvement was shown to be marginal.

In practice, one has to weigh the small improvement in the performance of MCC against its

high computational cost (and no guarantee of convergence to global optima).

4.4 Bayesian Formulations of the Topography Estimation

The amplitude estimation consists of 3 steps. First we estimate the single-trial scalp

topography ak (either by the uncorrelated assumption or equivalently through the minimization









of the Frobenius norm of Xk -akk ). Then we compute the normalized scalp topography a, as

the normalized version of a weighted average of the single-trial scalp topography ak (the

weights being their respective 12 norm). The third step is the minimization of the 12 norm

ak oa 2, which gives the optimal single-trial amplitude as the inner product between ak and

a,.

We have seen that the first step gives a near optimal solution (as opposed to CIM) if the

EEG data have been cleaned. We have also shown that in the third step, using other metrics (e.g.,

CIM) gives a marginally better results than using the 12 norm. Here we will investigate other

alternatives to the estimation of the normalized scalp topography a, in the second step in (3.17).

We maintain that after estimating the single-trial scalp topography in the first step, we treat

ak as known and given. Then we ask the question: what is the best estimate for the normalized

scalp topography a, given ak for K trials. This is a divide-and-conquer approach and simplifies

matters.

Naturally the problem is best formulated in a Bayesian framework. Of course, the

formulation will depend on the model we assume for the data. Next, we will present three

different models and compare their performance.

4.4.1 Model 1: Additive Noise Model

ak = oa +uk (4-11)

where, ok is the single-trial amplitude, and uk is the error (model uncertainty).

This model is consistent with our linear generative EEG model. We shall assume conditional

independence between a I o-,, a and a, I c, ao, for any i j. Of course, we also assume o, and









oa are independent for any i j, and they are all independent of the normalized scalp

topography a,.

We wish to maximize aposteriori probability (MAP):

arg max p(a,, O... a,...aK) (4-12)
ao,,i OK

It can be shown (Appendix E) that the MAP solution occurs when ao is the normalized

eigenvector of the matrix A corresponding to the largest eigenvalue, where,

K K
A= iAk = akk (4-13)
k=l k=l

Now we have solved the MAP problem under model 1. But, there is a weakness: the model

error has a constant covariance across all trials. This is a dubious assumption. Intuitively, when

the data are noisy, the variance in uk will increase accordingly. We should somehow

"normalize" the data in our model. This leads to model 2.

4.4.2 Model 2: Normalized Additive Noise Model

= ao +uk (4-14)
Ok

The left hand side is in fact a proxy for aok in our linear generative EEG model, except that

it may not be normalized. uk is again zero-mean i.i.d. Gaussian noise, but now with unit norm.

The model can also be written as: ak = ok aOk + k uk.

Again, it can be shown (Appendix F) that the MAP solution occurs when ao is the

normalized eigenvector of the matrix B corresponding to the largest eigenvalue, where,


B=: ak k (4-15)
k=l k=l ak ak









Note that the components Bk is a "normalized" version of Ak in that the trace of Bk is

always 1. We also note that, we can treat ok only as a normalizing factor, not necessarily as the

single-trial "amplitude". In this case, we separate the estimation of ok and a,. As before, we can

compute the amplitude by: ak = aoT ak.

4.4.3 Model 3: Original Model

Our last model is actually simpler. We do not include the unknown amplitude in this stage

and only consider the estimation of a,. The posteriori probability is simply

arg max p(ao I a,...a) .
ao


Given the model: k = a + uk, Appendix G shows that the MAP solution actually


coincides with our original solution in (3.17). The single-trial amplitude estimation is the same as

before.

4.4.4 Comparison among the Three Models

We compare the amplitude and scalp topography estimation for the three models. The results

are summarized in Table 4-7 and 4-8. We can see that model 3 (original solution) consistently

gave the best results among the three under different SNR conditions. The topography estimation

with model 1 is poor at negative SNR conditions. Model 2 is an obvious improvement. Note that

its amplitude estimation has a huge bias and variance for low SNR data. This may be due to the

improper prior we assign to ok (it assign large probability to large values) in the derivation of

the MAP estimator. We also included the amplitude estimation with the traditional inner product.

There is a significant improvement in both the bias and variance, especially in low SNR

conditions.










4.4.5 Online Estimation

Sometimes we wish to know the single-trial parameters after recording each trial. It is then

necessary to obtain an on-line estimation method. We assume that the stimulus onset time is

known to us. Again, the problem is best formulated in a Bayesian framework in order to utilize

all the information in previous trials. In fact, it is trivial given the above analysis. Here, we adopt

model 3.

After the first trial, set: ai = a/ al2 1 = al 2

K
At each trial, we store the running average: ck = ak/la 2
k=1

When finishing recording trial K+1, we update the topography estimate:


a(K+l) K +aK+l aK+ 2
o cK+ a+/aK+1



and the amplitude for the newly recorded trial: K+1 = aK+1 K a1)

4.5 Explicit Compensation for Temporal Overlap of Components

In developing our basic method in Chapter 3, we assumed that ERP components are

uncorrelated with each other. In reality, this is seldom satisfied. Because ERP components have

relatively stable waveforms and latency, when they overlap in time, there will generally be a

nonzero correlation among them. Here we are mainly concerned with the situation where ERP

components overlap in time, but latency jitter across trials is relatively small.

Consider two ERP components overlapping in time (it is easy to generalize to multiple

components). We assume that the time courses of the components s, and s2 are known from

physiological knowledge. We also assume that the latency is given or can be estimated, e.g.,

from ensemble average, and it is relatively fixed. For the purpose of amplitude estimation,









latency jitter is considered here as a minor issue compared with the possibly heavy overlap of

components. Note that, in the case of two overlapping (correlated) components, we lose the

ability to estimate the latency simply from the cost function in (3-9).

If we assume that all other components are uncorrelated with these two components (or have

negligible temporal overlap with them), then we can compensate for the correlation (due to

temporal overlap) to get an unbiased estimate for both components' amplitudes. We write the

linear generative EEG model in this case:

N
X=a s, +a2 s2 2T + bnf (4-16)


We wish to estimate the scalp topographies for the two overlapping components. With the

uncorrelated assumption, we can get two set of equations:

{a, s1 s + a2 *2T *S =X X(4-17)
1 (4-17)
a1 sT S2+a2 2 S 2 = X-s2

Solve for a, and a2:

D2X-s1 CX.s2
al ID2 (4-18)

DX s2 CX-s,
2 DD 2 _C2


where, D s = s, s, (i = 1, 2), and C = s, .s2 = s2T S .

If C = 0, we have the same solution as before:

X-s
a, = ,i=1,2 (4-19)
s -s,

The procedures for estimating the amplitude are the same as before. The above technique

assumes that accurate estimates of the waveforms of the components are available, since the

cross-correlation in (4-18) depends on the tails of the overlapping components. This restricts its









applicability in practice. But if the researchers believe that the ERP components are heavily

overlapped and are fairly certain of their waveforms, this technique should serve as a first

attempt to reduce the bias in the estimation.









Table 4-1. Latency estimation: mean and standard deviation
SNR (dB) Exact match Mismatch
-20 202 +10 190+ 16
-16 201 7 189 14
-12 201 5 188 11
-8 200 3 188 10
-4 200 2 188 10
0 200+ 1 187 10
4 200 0 187 10
8 200 0 188 10
12 200 +0 188 9
True latency: 200ms.


Refined template
182 12
183 13
187 12
191 + 12
197 6
199 4
200 + 1
200 + 1
200 + 1


Table 4-2. Amplitude estimation: mean and standard deviation
SNR (dB) Exact match Mismatch Refined template
-20 1.68 + 2.20 2.22 + 2.93 2.17 + 2.73
-16 1.34 1.29 1.71 + 1.75 1.61 + 1.76
-12 1.18 + 0.77 1.49 + 1.05 1.33 1.05
-8 1.10 + 0.47 1.34 0.64 1.18 0.59
-4 1.06 + 0.29 1.26 + 0.40 1.11 + 0.34
0 1.04 + 0.18 1.23 + 0.25 1.06 + 0.20
4 1.02 0.12 1.20 + 0.16 1.04 0.12
8 1.01 + 0.07 1.19 + 0.10 1.02 + 0.07
12 1.01 + 0.05 1.18 + 0.07 1.01 + 0.05
True amplitude: 1

Table 4-3. Scalp topography estimation: correlation coefficient
SNR (dB) Exact match Mismatch Refined template
-20 0.829 0.759 0.644
-16 0.910 0.854 0.772
-12 0.959 0.926 0.891
-8 0.982 0.966 0.963
-4 0.993 0.986 0.987
0 0.997 0.994 0.995
4 0.999 0.998 0.998
8 1.000 0.999 0.999
12 1.000 1.000 1.000









Table 4-4. Estimation results for the iteratively refined template method

SNR Latency estimation Amplitude estimation Scalp topography
(dB) Refined Iterative Refined Iterative Refined Iterative
template Refined template Refined template Refined
-20 182 12 183 12 2.17 2.73 1.83 2.36 0.644 0.644
-12 187 12 187 12 1.33 1.05 1.23 1.01 0.891 0.879
0 199 4 199 2 1.06 0.20 1.06 0.20 0.995 0.995

Table 4-5. Estimation with MCC for the mismatch case at SNR = OdB
Kernel size Correlation coefficient of
(multiples of h) Amp e Scalp projection
0.5 1.22 + 0.25 0.994
1 1.22 0.26 0.994
2 1.22 0.27 0.995
5 1.22 0.26 0.995
10 1.22 0.25 0.994
20 1.23 + 0.25 0.994
MSE 1.23 + 0.25 0.994

Table 4-6. Estimation with MCC for the mismatch case at SNR = -20dB
Kernel size Correlation coefficient of
Amplitude .
(multiples of h) Amp e Scalp projection
0.5 2.22 + 2.95 0.758
1 2.21 + 3.05 0.754
2 2.17 3.18 0.786
5 2.18 3.14 0.789
10 2.21 3.00 0.770
20 2.22 + 2.93 0.759
MSE 2.22 + 2.94 0.759









Table 4-7. Amplitude estimation for three Bayesian models
Model 3
SNR (dB) Model 1 Model 2 Model 2 (2) (origin
(original)
-20 0.38 + 1.30 12.2 + 378 1.95 3.79 2.22 2.93
-12 0.86 + 0.85 9.96 60 1.42 1.34 1.49 1.05
0 1.20 + 0.26 1.66 + 0.26 1.23 + 0.25 1.23 + 0.25
12 1.18 0.07 1.21 0.07 1.18 0.07 1.18 0.07

Table 4-8. Scalp topography estimation for three Bayesian models
SNR (dB) Model 1 Model 2 Model 3 (original)


0.188
0.558
0.971
0.998


0.642
0.841
0.994
1.000


0.759
0.926
0.994
1.000
















exact match exact match
mismatch ,\ mismatch
refined template refined template
25




2 2




\ 15
E 2



1 5 1




05




0
-20 -15 -10 -5 0 5 10 15 -20 -15 -10 -5 0 5 10 15
SNR(dB) SNR(dB)




Figure 4-1 Mean and standard deviation of the estimated amplitude under different SNR

conditions. The refined template method approaches to the exact match case for high

SNR conditions.


























A 5so 100 150o 5o 100 150o
sample sample


10 10
-- synthetic -- synthetic
8 \ presumed 8 / \\ presumed
refined / \ -- refined


4 4





C -2 -2 D
S 0 50 100 150 0 50 100 150
sample sample


Figure 4-2 The waveforms of the synthetic component, presumed template and refined template
under 4 SNR conditions. A) -20dB. B) -12dB. C) OdB. D) 12dB. The refined template
appears erratic for low SNR and approaches to the synthetic component for high
SNR.













9/

8

7-

6

5

4

3-

2-



0
0 50 100 150



Figure 4-3 Waveforms of two overlapped components used in regularization. The correlation
coefficient between the two waveforms is around 0.36.












0.2


0.15


0.1


0.05


O

-0.05


-0.1


-0.15 -


-0.2
0 20 40 60 80 100 120 140
channel number


Figure 4-4 Scalp topography of two overlapped ERP components used in regularization.


















- mean
-- std


102 10 104 10 10o
REG PARAMETER LAMDA


102 10 104 105 10
REG PARAMETER LAMDA


S0999


8 0 998
S85
0 998
8 0 997
5
0 997
-
107 0996
5 102


0 999
o 0999
%6
o 0999
S4
0999
2
8 0999
0998
8
0 998
107 6 102


10 10 10 10
REG PARAMETER LAMDA


10 10 10 10
REG PARAMETER LAMDA


A
Figure 4-5 Amplitude and scalp topography estimation I with regularization (constrained
optimization) under 3 SNR conditions. A) 12dB. B) -12dB. C) -20dB. The
overlapping ERP components have the exact opposite scalp topography. So the
estimated amplitude is smaller than 1 for high SNR data. It converges to 0.42 and
0.41 for component 1 and 2 for large A, respectively. Notice the huge bias for low

SNR (especially -20dB) without regularization. But it converges to as small as 0.69
and 0.27 for component 1 and 2 for large A, respectively.





























102 10 10 10 10
REG PARAMETER LAMDA


mean /
095
09
t
o 0 85
S08
o /


8 075

07J

107 065
102 10 104 105 10 107
REG PARAMETER LAMDA


0 95

S09


S0 85


1d 1 10d 10 10 o 10
REG PARAMETER LAMDA


08
1i 10 10 1 Id 17
REG PARAMETER LAMDA


10 10 10 10 10 10
REG PARAMETER LAMDA


10 10 10 10 10 10'
REG PARAMETER LAMDA

C

Figure 4-5 Continued


09

08
8
8 07

06

05

04-----------------
102 10 104 105 10 10
REG PARAMETER LAMDA


1

09

08
8
8 07
06

05
05


04
10d 0d 1id 10d d d1
REG PARAMETER LAMDA


- -w""


$




















mean
std


05




REG PARAMETER LAMDA


S 0999
9



S0 999
0999
8
0999
8
107 0 999
7 102


10 10 10 10
REG PARAMETER LAMDA


mean
std


REG PARAMETER LAMDA


1

1
o
S 1


L 0999
0 999

o 0999
9
0999
9
0 999
IJ 9 16i


10 10 10 10
REG PARAMETER LAMDA


A

Figure 4-6 Amplitude and scalp topography estimation II with regularization (constrained

optimization) under 3 SNR conditions. A) 12dB. B) -12dB. C) -20dB. The

overlapping ERP components have the same scalp topography. So the estimated

amplitude is larger than 1 for high SNR data. It converges to 1.6 for both components

for large A. Notice the huge bias for low SNR (especially -20dB) without

regularization. It converges to 1.8 and 1.7 for component 1 and 2 for large 1,

respectively.























5




102 10-REG---RAMETE1---------
10 10 10d 10 10
REG PARAMETER LAMDA


0 -
102 10 104 105 10
REG PARAMETER LAMDA


mean
std
6\
5

4

3

2

10 10 10o 10 10 10
REG PARAMETER LAMDA




mean
\
6 std
5

4
\
3

2


10 10 104 10 106 10
REG PARAMETER LAMDA

C

Figure 4-6 Continued


std
099

0 0 98

S097

0 96

S107 095
102 10 10 10 10 10
REG PARAMETER LAMDA



mean /
std a

S0995



S099



0 985
10 1i2 10 10 10 10 10
REG PARAMETER LAMDA


0 95
o /

| 09

o 85
08

O
12 10i 1 0 10 1 10
REG PARAMETER LAMDA




0 99

o 098

S097

S096

8 095

0 94

093
10 10o 10 105 10 10
REG PARAMETER LAMDA

















I J


102 10 104 10
REG PARAMETER LAMDA


12 10f 10 10
REG PARAMETER LAMDA


S0999


S0 998
S85
0 998

8 0 997
5
S0997
-
106 0996
5 102


0 999
8
o 0999
%6
o 0999
S4
0999
2
S0999
0998
__ 8
0 998
100 6 102


10 10 10
REG PARAMETER LAMDA


10 10 10
REG PARAMETER LAMDA


A

Figure 4-7 Amplitude and scalp topography estimation I with regularization (unconstrained
optimization) under 3 SNR conditions. A) 12dB. B) -12dB. C) -20dB. The

overlapping ERP components have the exact opposite scalp topography. So the
estimated amplitude is smaller than 1 for high SNR data. The estimated amplitude

mean generally increases with increasing A while the variance is not affected much.

It becomes unstable for large A.


















mean
std /


10 10 10L
REG PARAMETER LAMDA


0 95

E0

S085
08
o 08

8 075

07-

10 065-
102


10 10 10
REG PARAMETER LAMDA


S095


S09


o 085


Mean
3 std
2

2

15
1-




0 --------------------------
10 103 104 10 10o
REG PARAMETER LAMDA







mean
6 std







12 1 1 1 1
1RE

0 ~ ~ --i -----------------


REG PARAMETER LAMDA







6 std


2



10? 10" 104 105
REG PARAMETER LAMDA

C

Figure 4-7 Continued


10 10 10
REG PARAMETER LAMDA


10 10 10
REG PARAMETER LAMDA


o 06

05


10 10 10
REG PARAMETER LAMDA


3

25


E
S12




























1i 10 10
REG PARAMETER LAMDA


-- mean
std







REG PARAMETER LAMDA


o
0 999
88


0 999
0 8


106 0999
7 1C0




1


0 0999

S0 999
& 9

0999
9
0999
9 1
0 999
,f 9 102


10 10 10
REG PARAMETER LAMDA















10R 10P 10L
REG PARAMETER LAMDA


A

Figure 4-8 Amplitude and scalp topography estimation II with regularization (unconstrained

optimization) under 3 SNR conditions. A) 12dB. B) -12dB. C) -20dB. The

overlapping ERP components have the same scalp topography. So the estimated

amplitude is larger than 1 for high SNR data. The estimated amplitude mean

generally increases with increasing A while the variance is not affected much. It

becomes unstable for large A.































103 10 10
REG PARAMETER LAMDA


S099


0 098
o
S097
o

0 96


0 95
102


10 10 10
REG PARAMETER LAMDA


mean


0
102 10 104 105 10
REG PARAMETER LAMDA





10
mean
8 std 0

6

4




0 ---------------------------

10 103 10 10 106
REG PARAMETER LAMDA


10 103 104 105
REG PARAMETER LAMDA

C

Figure 4-8 Continued


0 995
8
o

0 099


0 985
1o2


0 95


| 09


085



08
1o2


0 99

o 098

8 097

S096

8 095

0 94

0 93
102


10 10 10
REG PARAMETER LAMDA




















10" 104 10"
REG PARAMETER LAMDA
R E





























103 104 105
REG PARAMETER LAMDA









CHAPTER 5
APPLICATIONS TO COGNITIVE ERP DATA

In this chapter, we apply the spatiotemporal filtering method proposed in Chapter 3 to the

single-trial ERP estimation problem in two different experiments. The first application is an

oddball target detection task with different pictures as stimuli, where the difficulty of the task or

saliency of the stimuli leads to decreased P300 amplitude. The second one is the habituation

study where the subjects were repeatedly presented identical pictures and the amplitude of

certain ERP components is expected to decrease rapidly with respect to trials.

5.1 Oddball Target Detection

5.1.1 Materials and Methods

Because we were interested in single-trial, single-subject analyses of amplitude and latency,

we selected 4 participants that met a minimum signal-to-noise ratios based on their averaged

ERPs, from a pilot study (n=8) on implicit content processing during feature selection. They

were right-handed according to the Edinburgh Handedness Questionnaire and all had normal or

corrected vision.

Stimuli consisted of pictures from the International Affective Picture System, depicting

adventure scenes, emotionally neutral social interactions, erotica, attack scenes, and mutilations.

Their color content was manipulated such that they contained only shades of green or shades of

red, and for each, color brightness was systematically manipulated to yield one bright and one

dim version (Figure 5-1). All pictures were presented for 200 ms on the center of a 21-inch

monitor, situated 1.5 m in front of the subjects. From this viewing distance the checkerboards

subtended 4.0 deg. x 4.0 deg. of visual angle. A fixation cross was always present, even when no

picture was presented on the screen. Target stimuli (p = 0.25) were defined for each experimental

block (see below) by a combination of color and brightness.. All pictures were presented in









randomized order, with an inter-stimulus-interval varying randomly between 1000 to 1500 ms in

4 blocks of 120 trials each. One block lasted 7 min. on average. At the beginning of each block

subjects were instructed to attend either to the bright/dark green or red pictures and to press the

space bar of the computer keyboard when they detected a target. The target color and brightness

were designated in counter-balanced order. Furthermore, the responding hand was changed half

way through the experiment, and the sequence of hand usage was counterbalanced across

subjects. Subjects were also instructed to avoid blinks and eye-movements and to maintain gaze

onto the central fixation cross. Practice trials were provided for each subject for each condition

to make sure that every subject had fully understood the task.

EEG was recorded continuously from 257 electrodes using an Electrical GeodesicsTM

(EGI) EEG system and digitized at a rate of 250 Hz, using Cz as a recording reference.

Impedances were kept below 50 kM, as recommended for the Electrical Geodesics high input-

impedance amplifiers. A subset of EGI net electrodes located at the outer canthi as well as above

and below the right eye was used to determine horizontal and vertical Electrooculogram (EOG).

All channels were preprocessed on-line by means of 0.1 Hz high-pass and 100 Hz low-pass

filtering.

Epochs of 1000 ms (280 ms pre-, 720 ms post-stimulus) were obtained for each picture

from the continuously recorded EEG, relative to picture onset. The mean voltage of a 120-msec

segment preceding startle probe onset was subtracted as the baseline. In a first step, data were

low-pass filtered at a frequency of 40 Hz (24 dB / octave) and then submitted to the procedure

proposed by (Junghofer et al., 2000), which uses statistical parameters to exclude channels and

trials that are contaminated with artifacts. This procedure resulted in rejection of trials that were

contaminated with artifacts (including ocular artifacts). Artifacts were also evaluated by visual









inspection and respective trials were rejected. Recording artifacts were first detected using the

recording reference (i.e. Cz). Subsequently, global artifacts were detected using the average

reference and distinct sensors from particular trials were removed interactively, based on the

distribution of their mean amplitude, standard deviation and maximum slope. Data at eliminated

electrodes were replaced with a statistically weighted spherical spline interpolation from the full

channel set. The mean number of approximated channels across conditions and subjects was 20.

With respect to the spatial arrangement of the approximated sensors, it was ensured that the

rejected sensors were not located within one region of the scalp, as this would make interpolation

for this area invalid. Spherical spline interpolation was used throughout both for approximation

of sensors and illustration of voltage maps (Junghofer et al., 1997).

Single epochs with excessive eye-movements and blinks or more than 30 channels

containing artifacts in the time interval of interest were discarded. The validity of this procedure

was further tested by visually inspecting the vertical and horizontal EOG as computed from a

subset of the electrodes that were part of the electrode net. Subsequently, data were

arithmetically transformed to the average reference, which was used for all analyses. After

artifact correction an average of 69 % of the trials were retained in the analyses. The present

analysis highlighted the most reliable signal available in this feature-based target identification

task, which is the P300 component in response to a target stimulus (defined by a combination of

color and brightness, irrespective of picture content). Thus, all subsequent analyses focused on

amplitude and latency estimates for single trials belonging to the target condition.

5.1.2 Estimation Results

The present study illustrates the application of the method for a single late potential

component. In reality, we do not know a priori how many ERP components there are in a single-









trial recording, nor do we know exactly when they occur. However, we may be able to estimate

these values from single-trial EEG data in the data analysis session.

This is a good time to mention one technical requirement of our latency estimation. The

single-trial latency is estimated from the cost function in (3.9), which involves the inversion of

the matrix XX In reality, this matrix is usually ill-conditioned for dense-array EEG data (it

will certainly be rank-deficient if there are any bad channels which were linearly interpolated

from other channels.). This poses a computational problem in practice. Thus, the solution in (3.9)

somehow has to be regularized. Here, we adopt a simple approach and add a regularization term

AI (A > 0 ) to the matrix XX' before taking the matrix inversion operation. The regularization

parameter A acted as a smoother to the cost function in (3.9). Generally, the solution is rather

irregular without regularization, leading to too many local minima and spurious candidates for

single-trial latencies due to large noise. With increasing A, the cost function becomes smoother.

This is clearly seen in Fig. 5-2, which shows the cost function in (3.9) for four different A, for a

particular trial from subject 2. With a smooth cost function, we can avoid the dilemma of

choosing the right latency from too many candidates.

Now we have to select an appropriate value (or a meaningful range) for the regularization

parameter A. A good value for A is one that achieves a balance between two extremes: too few

and too many local minima. The idea is this: for a particular A, we group all the candidates for

single-trial latencies (time lags corresponding to local minima) together and perform ID density

estimation on these candidates. We count the number of modes (peaks) from the estimated

probability density function (pdf). If this number is close to the average number of candidates for

each trial, then the regularization parameter A is at least internally consistent. Otherwise, it will

contradict with itself and should not be used.









We illustrate our point using the results from one subject. Fig. 5-3 shows the estimated pdf

of the candidates for single-trial latency from 200ms up to 600ms after stimulus onset when the

regularization parameter A equals 10 5. We used the Parzen windowing pdf estimator (Parzen,

1962) with a Gaussian kernel size of 4.2. The kernel size was selected according to Silverman's

rule (Silverman, 1986), which is given by h = 1.06cN 02, where Nis the number of samples,

and a is the standard deviation of the data. The number of peaks depends on the kernel size, but

we found that a kernel size between 0.5h and 2h will give the same number of peaks in the

estimated pdf for this data. We can see that the pdf consists of 4 modes (peaks) after 200ms of

the stimulus onset. There are 418 local minima and 102 trials in total, so the average number of

local minima for each trial is about 4.1 (very close to the number of peaks in estimated pdf). This

indicates that =10 5 gives an internally consistent estimate for latency.

We can repeat the above procedures for a wide range of regularization parameters and

compute the ratio of the number of peaks in estimated pdf to the average number of local minima

for each trial. For instance, the ratio was computed as around 4.75, 1.03, 0.96, 0.74 for the 4

regularization parameters in Fig. 5-2 respectively. Clearly, the first and last regularization

parameter should not be used since they generate self-contradictory results. It is interesting to

note that for a wide range of regularization parameters (from 10 5 to 100), the results are quite

similar. This can also be seen from Fig.5-2, where both cost functions display 4 local minima

and all time lags are near to their counterparts. For practical purposes, we can select any value

from this range as a regularization parameter.

We were primarily interested in the P300 component, preferably the largest one. From the

ensemble average, we know that the maximum ERP occurred around 380ms after stimulus onset.

In Fig. 5-3, the estimated pdf displays a mode around 420ms. Thus, we searched around this









latency and set the single-trial peak latency as the one that was closest to it. The mode of latency

is 360ms, 420ms and 400ms for the other three subjects, respectively.

We should point out that since there is about 1 local minimum per mode, the search need not

be around the true mode for latency (we do not know this anyway). The results would be almost

the same as long as the estimated mode is not skewed to its two neighboring true modes.

Figure 5-4 shows the scalp topographies for the four subjects plotted using EEGLAB

(Delorme et al., 2004). As expected for a P300 topography, it has a large positive topography

around the Pz area. To evaluate the single-trial estimation of the scalp topography, we compute

the correlation between the single-trial scalp topography in (3.15) and the overall normalized

scalp topography (3.17). For comparison, we also compute the correlation between the single-

trial scalp topography in (3.15) and the scalp topography obtained from ensemble average for

each subject. We name these two correlations r, and r2 respectively. Statistical inference based

directly on the correlation itself is difficult since its distribution is complicated. A popular

approach is to first apply the Fisher Z transformation to correlation and then do inference on the

transformed variable. The Fisher Z transform is given by (Fisher, 1915):

Z = 0.51n (5-1)
(1-r)

Z has a simpler distribution and it converges more quickly to a normal distribution. We can

calculate the mean and confidence interval of Z based on the correlation, if we assume that the

estimation error in (3-15) is a normal distribution. The statistics of the correlation can be easily

obtained from the inverse transform of (5-1).

The results are summarized in Table 5-1. We can see that there is a moderate amount of

correlation between the single-trial and overall scalp topography (the average mean correlation

for 4 subjects is around 0.40) although the mean correlation is lower for subject 3 at around 0.20.









There is a small degradation in mean correlation when the overall scalp topography is computed

from the ensemble average. This is expected since the estimate in (3-17) is close to the ensemble

averaged estimate. The correlation between these two estimates for the four subjects are: 0.80,

0.85, 0.89, 0.79 respectively.

To evaluate the effectiveness of the single-trial amplitude estimation, we related our

estimates to a behavioral measure of target identification: response time in target trials. Response

time was selected because task difficulty was relatively low, and therefore error rate did not

show pronounced variability, with only limited numbers of misses (mean of 3.9 % across 4

participants) and false alarms (mean of 1.2 % across 4 participants). Thus, response time was

used as a measure of target identification, with short response times indicating facilitated

discrimination and long response times indicating difficulties with identification in a given trial.

Using these measures, we were interested in the relationship between P300 amplitude and

response time, expecting that trials in which participants found discrimination relatively easy

(short RT trials) should be associated with greater P300 amplitude, which also indicates

successful encoding of the target features and preparation for responding to a target that has been

identified.

There seems to be little relationship between the response time and estimated single-trial

peak latency. The correlation coefficients between these two for the four subjects are: 0.022, -

0.248, 0.168 and 0.093 respectively. However, there were reliable negative correlations between

the response time and estimated single-trial amplitude. Figure 5 shows the scatter plot of the

response time versus the estimated amplitude for each single trial for the four subjects. To

evaluate the statistical significance of the results, we performed linear regression on the response

time and estimated single-trial amplitude for the four subjects. The results are summarized in









Table 5-2. the negative slope parameter estimated from linear regression is statistically

significant under a significance level of 0.05 for all the four subjects, which supports our

hypothesis that larger amplitude correspond to smaller response time, and vice versa.

To compare our results with conventional methods, we calculated the average P300

amplitude at channel 100 for subject #2. This was simply the average single-trial amplitude times

the 100-th entry of the scalp topography in (3.17). It was found to be 22. mV, compared with the

17.3mV from the ensemble average ERP. Taking into account of the possible latency jitter of

P300, the true amplitude could be only larger than 17.3mV. Therefore, we obtained an upper

bound of 28% on the positive bias of our average P300 estimate in channel 100. The coefficient

of variation, which is defined as the ratio of the standard deviation to the mean of a positive

random variable, is used as a measure of dispersion of the estimated amplitude and it was found

to be around 0.60. This compares favorably with 0.79 obtained using the simple peak-picking

method around its ensemble average peak at 400ms. Although the gain may seem small, we

should keep in mind that this variation will incorporate the estimation error as well as that of the

underlying change in P300 amplitude itself, because there are systematic changes in P300

amplitude as suggested above. So the estimation variance of our method is reduced by a factor of

at least 1.7 from the peak-picking method. For instance, if one half of the total variance of our

method came from the underlying P300 amplitude, this roughly means that our method reduced

the estimation variance by a factor of 2.5 (assuming additive and uncorrelated estimation error).

Of course, the comparison would be much more direct and informative if the P300 amplitude

was expected to remain constant.

All the above results were obtained using a fixed Gamma template with k = 11, 0 =5. If we

change the template, specifically, the spread parameter 0, the estimated amplitude will also









change. However, we found that the amplitude estimation is only slightly affected by this

change. For instance, the average estimated P300 amplitude in channel 100 for subject #2 was

around 20.5mV when 0 = 1 (this is too small for P300, rise time 40ms) and was around 23.8mV

when 0 = 8 (this is too large, rise time 320ms). There is less than 8% change from the result

(22. mV) obtained with the original template with 0 = 5. This agrees with our earlier findings

using simulated ERP data (Li et al., 2008).

5.1.3 Discussions

As a straightforward test of the present method, we examined the relationship between target

detection performance and features of the P300 component evoked by the targets in an oddball

task with rare targets varying in terms of their salience on a trial-by-trial basis. In the present

case, we replicated and extended a standard result in target detection studies in the visual

domain: When target identification is made difficult or saliency is reduced (e.g., by presenting

many targets in succession, Gonsalvez and Polich, 2002), P300 amplitude often decreases

(Polich et al., 1997). This pattern has been interpreted as reflecting reduced resource allocation to

a given target stimulus (Keil et al., 2007). Notably, previous work in this area has typically relied

on averages across all trials of an experimental condition, or on block by condition averages

across many trials (for a review, see Kok, 2001). The present results suggest that the relationship

between response time and P300 amplitude in feature-based attention task is of a continuous

nature, rather than a consequence of a bimodal function separating easy and hard trials. The

sensitivity of the method was sufficient to demonstrate this linear relationship on a single-subject

level, which is often desirable in clinical studies. In a similar manner, other research questions

will benefit from the ability to examine hypotheses as to the time course and distribution of

single brain responses, in terms of their magnitude and latency.









5.2 Habituation Study


5.2.1 Materials and Methods

EEG data were recorded from subjects during a passive picture-viewing experiment,

consisting of 12 alternating phases: the habituation phase and mixed phase. Each phase has 30

trials. During the 30 trials of the habituation phase, the same picture was repeatedly presented 30

times. During the mixed phase, the 30 pictures are all different. Each trial lasts 1600 ms, and

there is 600 ms pre-stimulus, and 1000 ms post-stimulus.

The scalp electrodes were placed according to the 128-channel Geodesic Sensor Nets

standards. All 128 channels were referred to channel Cz and were digitally sampled for analysis

at 250Hz. A bandpass filter between 0.01Hz and 40Hz was applied to all channels, which were

then converted to average reference. To correct for vertical and horizontal ocular artifacts, an eye

movement artifact movement correction procedure (Gratton et al., 1983) was applied to EEG

recordings.

5.2.2 Estimation Results

We assume that the entire ERP may be decomposed into several monophasic components

with compact support. We will estimate their parameters (amplitude and latency) one by one,

using the Gamma template as in the simulation study. The present study illustrates the

application of the method for a single late potential component, and the Gamma is not adapted.

Its parameters are selected based on neurophysiology plausibility and are set as k = 5, 0 = 6,

corresponding to a rise time of 96ms.

In reality, we do not know a priori exactly how many components there are in a single-

trial, nor do we know when they occur. However, we may be able to estimate these values from

single-trial EEG data in the data analysis session. Following the same procedures in Section

5.1.2, we identified 5 distinct peaks after 300ms of the stimulus onset. Assuming that the error in









the latency estimation is equally biased and independent from trial to trial and since there are

also about 5 local minima for each trial, we conjecture that these peaks correspond to the

latencies of 5 distinct components. These components, which may have different origins, are

likely to compromise the Late Positive Potentials (LPP). According to Codispoti et al.(2006), the

grand-average of LPP is maximal around 400ms to 500ms after stimulus. We will concentrate on

the component with a latency of 500ms to exemplify the methodology. We search between

440ms and 560ms (which correspond to the two neighboring local minima) and set the

component latency as the local minimum closest to 500ms.

To avoid the influence of EEG outliers from unexpected artifacts, we reject those trials with

3 times or larger amplitude of the minimum one. This will eliminate 14 trials from the total of

360 trials (rejection rate: 4%). The same rejection criterion was applied to the other two subjects,

leading to the rejection of 8 (2%) and 33 (9%) trials, respectively.

Figure 5-6 shows the results of estimated scalp topography of the LPP component for 3

subjects. They are similar in the sense that all show large projections in the posterior area. The

difference with subject 2 is that the scalp topography shifts its strength a bit to the occipital area.

It may be that the pictures shown to the 3 subjects caused some emotional bias. It is also possible

that the SNR of the ERP data is too low to allow for a stable estimate of the scalp projection

across subjects (note that in habituation phase, the LPP amplitude decreases quickly with the trial

index).

Figure 5-7 shows the results of estimated amplitude of the LPP component for 3 subjects.

Each point in Fig. 5-7 stands for the average amplitude over 6 trials with the same index in the

same phase (habituation or mixed). It is clear that for the habituation phase, the amplitude

diminishes rapidly with the trial index, while for the mixed phase, the amplitude does not show









significant decay. To make the figure more intuitive, we also include the best fit (in the least

square sense) to the estimated amplitude for both habituation and mixed phase. We fitted a

straight line for the mixed phase, while an exponential curve was fitted to the estimated

amplitude of the habituation phase. The fitted exponential curve for the habituation phase has a

time constant of around 1.5 trials, which suggests that after 3 or 4 trials, the LPP amplitude

decreases close to zero. We estimate the SNR for the mixed phase at around -4.1dB. Similar

results were obtained with the ERP data from 2 other subjects as shown in Fig. 5-7 (B) and (C).

The fitted exponential curves for the habituation phase for these 2 subjects has a time constant of

around 1.5 and 2.0 trials, respectively. The SNR of the mixed phase for these 2 subjects are

estimated to be around -7.6dB and -2.4dB respectively.









Table 5-1. Correlation statistics for the 4 subjects: Scalp topography estimation

Subject Sample 2
#size Confidence interval Confidence interval
Mean Mean
(95%) (95%)
1 98 0.520 [0.362, 0.649] 0.393 [0.215, 0.546]
2 85 0.367 [0.167, 0.538] 0.304 [0.097, 0.486]
3 74 0.198 [-0.027, 0.404] 0.200 [-0.025, 0.406]
4 65 0.446 [0.233, 0.619] 0.359 [0.133, 0.551]



Table 5-2. Regression statistics for response time and estimated amplitude
Subjec Sample R Slope Confidence
SCorrelation R sope t statistic p value
t # size square estimate interval (95%)


98 0.440 0.194 -0.712 -4.80 <0.0001
85 0.539 0.291 -0.310 -5.84 <0.0001
74 0.263 0.069 -0.105 -2.31 0.012
65 0.351 0.123 -0.108 -2.97 0.002


[-1.007, -0.418]
[-0.416, -0.205]
[-0.197, -0.014]
[-0.181, -0.036]


1
2
3
4




































Figure 5-1 Pictures used in the experiment as stimuli










x 10
35
-A
3
25

S1 5


A 200 250 300


350 400 450 500 550 600
time lag (ms)


200 250 300 350 400 450 500 550 600 B
time lag (ms)


3 95r


S 200 250 300 350 400 450 500 550 600
time lag (ms)


a 3 85
S38
3 75
37
3 65
36-
200


250 300 350 400 450 500 550 600 D
time lag (ms)


Figure 5-2 Cost function in (3.9) versus time lag for different regularization parameters for
subject #2. A) A = 106 B) A = 10 5. C) A = 100. D) A = 102. Regularization
parameter that is too small led to ragged cost function and spurious latency estimates;
Regularization parameter that is too large led to over-smoothed cost function and
missed candidates for latency.


VV


_,_J











x 10-3
3



2.5-



2



S1.5

1 \





0.5



0
100 200 300 400 500 600 700
time lag(ms)

Figure 5-3 Estimated pdf of time lags corresponding to local minima of the cost function in (3-9)
using the Parzen windowing pdf estimator with a Gaussian kernel size of 4.2.
Regularization parameter A = 10 .







p

I


2


SIi -- I
Ii i
C D
Figure 5-4 Scalp topographies for the four subjects. A) Subject 1. B) Subject 2. C) Subject 3. D)
Subject 4.

















S300
C
E 200

S100

0


A 1300


350 400 450 500 550 600 650
Response time (ms)


200

o10
150 0 o



S50 o





250 300 350 400 450 500 550 600 650 B
Response time (ms)


150

E 100

E50
I
W,


400 500 600 700 800 900 1000
Response time (ms)


-50-
200 300 400 500 600 700 800 900 1000
Response time (ms)


Figure 5-5 Scatter plot of the response time versus the estimated amplitude for each single trial
for the four subjects. A) Subject 1. B) Subject 2. C) Subject 3. D) Subject 4. Note that
the estimated amplitude is with respect to the EEG data in all the channels as a whole.
There appears to be a negative relationship between the response time and the
estimated amplitude.






































107















C26
25




C 16


os1
S015



(/ {o 2






02




0 25


















B
02
0 15
















n I02





















C


Figure 5-6 Estimated scalp topography for mixed and habituation phase. A) Subject 1. B)
Subject 2. C) Subject
J021


0 I5





(0 1




7-02







Figure 5-6 Estimated scalp topography for mixed and habituation phase. A) Subject 1. B)

Subject 2. C) Subject 3.





















80 +

----- --- --------
60




00

20 0 0
20 0


0 5 10 15
tnal index


10 15
trial index


20 25 30


20 25 30


Figure 5-7 Estimated amplitude for mixed and habituation phase. A) Subject 1. B) Subject 2. C)

Subject 3. Note that the LPP amplitude decreases with trials.


0 Habituation
Mixed
-- expotential it
----- linear fit


20 25


Habituation
Mixed
expotential fit
linear fit
- -


E 60


- 40
E


Habituation
Mixed
expotential fit
linear fit






t


10 15
trial index









CHAPTER 6
CONCLUSIONS AND FUTURE RESEARCH

6.1 Conclusions

Traditional ERP analysis has relied on ensemble average over a large number of trials to

deal with the typically low SNR environments in EEG data. To analyze ERP on a single event

basis, we have introduced a new spatiotemporal filtering method for the problem of single-trial

ERP estimation. Our method relies on explicit modeling of ERP components (not the full ERP

waveform), and its output is limited to local descriptors (amplitude and latency) of these

components. The reason that we model the ERP components instead of the full ERP waveform is

to exploit the localization of scalp projection for each single ERP component, which is

impossible to do for the entire ERP. Indeed, note that the ensemble ERP in different channels

usually have different morphology because there are multiple neural sources originating from

different locations of the brain that give rise to different scalp projections. Since one spatial filter

can extract effectively only one scalp projection, in order to utilize the spatial information in a

meaningful way, only a component based analysis is viable. Concentrating only on latency and

amplitude of each component together with optimal spatial filtering presents an alternative to

deal with the negative SNR. Moreover, since these are in fact the features of importance in

cognitive studies, the methodology has the same descriptive power of traditional approaches.

The proposed methodology can be seen as a generalization of Woody's filter (Woody 1967)

in the spatial domain for latency estimation. It also obtains an explicit expression for amplitude

estimation on a single-trial basis. By design, the method is especially suitable to extract ERP

features in the spontaneous EEG activity, in contrast to PCA and ICA which work best for

reliable (large) signals. Another distinction is that, unlike most methods based on PCA and ICA,

our method utilizes explicitly the timing information, as well as the spatial information. The









methodology as presented is based on least squares, but it can be further extended to robust

estimation (Li et al. 2007) for better results.

Using simulated ERP data, we have shown that although the mismatch between the

presumed and synthetic ERP components introduces a bias for both latency and amplitude

estimation, the bias for the latency is relatively small and the estimated amplitudes are still

comparable across experimental conditions for ERP data with a SNR higher than -20dB.

Furthermore, the mismatch of components has minimal influence on the estimation of scalp

projection. These all compare favorably with some of the popular methods (Li et al., 2008).

Despite its advantages over traditional methods, there are still some issues with our

spatiotemporal filtering method. First it is based on the linear generative EEG model in (3.1).

While this greatly simplifies the analysis, it may not be adequate to fully describe the complex

information processing in the brain. One weak link of the method is that it requires an explicit

template that is unknown apriori. Mismatch between the template and the true ERP component

waveform brings both bias and larger variance to the estimation of the latency and amplitude that

increases with decreasing SNR (Li, et al. 2008). It may be desirable to be able to adapt the

template while estimating the model parameters. Another weakness of the method is the

assumption of statistical uncorrelatedness among all the ERP components in deriving (3.14).

With monophasic waveforms, this is equivalent to the condition that all the ERP components do

not overlap in time (but overlap in space is allowed), which is seldom satisfied in practice.

Temporal overlap will bring bias to the amplitude estimation and poses a serious problem for the

latency estimation, since it works effectively only for monophasic waveforms that are well

separated in time. When there is heavy overlap among multiple components (e.g., P300 and

possibly other unknown late components), the peak latency estimation based on (3.10) may fail.









Therefore, care must be taken not to over-interpret the results of single-trial estimates. A

crucial factor for amplitude estimation is a reasonably low SNR (>-20dB). This may not be

satisfied for some ERP components under certain experimental conditions. Our ability to infer

the template accurately, which are selected heuristically from real data, deteriorates with

decreasing SNR. As a rule of thumb, we would recommend against the use of the present method

for data with SNR less than -15dB.

6.2 Future Research

The use of a parametric template (Gamma function) provides the flexibility to change the

shape and scale parameters continuously. However, this introduces undesirable bias when there

is a mismatch between the template and true ERP component. Using a stochastic formulation,

our method may be extended to a noisy template model and potentially the two nuisance Gamma

parameters may be extracted from the data also for best fit.

It is almost certain that activations of different ERP components overlap in time. If this is the

case, the temporal overlap will introduce a bias to the estimation of single-trial scalp projection,

because the derivation in (3.15) assumes the uncorrelatedness between the ERP component and

all the other sources (including the overlapping ERP components) and unlike the background

EEG, these overlapping components are coherent in all the trials. This bias, together with the

estimation variance due to finite-sample data, constitute the two main sources of error in the

estimation of the scalp projection. Note that this in turn will influence the estimation of the

amplitude. In Chapter 4, we have proposed an explicit procedure to compensate for the

overlapping effects for the amplitude and topography estimation. This assumes that the latency is

(relatively) fixed and a fairly accurate knowledge of the shape of all the overlapping ERP

components. When this is not the case (particularly when we wish to find the latency change









from trial to trial in the presence of component overlap), we need to come up with new

procedures to compensate for the overlapping issue.

The current method considers the single-trial amplitude of an ERP component as i.i.d. data.

It may be advantageous to take into account the dynamics of certain properties of the component

with respect to the trial index. For instance, we expect that during the habituation phase, the

amplitude of LPP components diminishes rapidly with the number of trials. Using regularization

techniques, this apriori information may be incorporated into the proposed single-trial

estimation method to provide more stable estimate for the amplitude. In the end, the evolution of

the single-trial amplitude with trial index may be inferred with more resolution and more

confidence.









APPENDIX A
PROOF OF VALIDITY OF THE PEAK LATENCY ESTIMATION IN (3-10)

We justify the use of the time lag corresponding to the local minimum of J(r) in (3-10) as
the peak latency. Given the single-trial data matrix X, the peak latency of an ERP component
coincides with the local minimum of J(r) if the following conditions are satisfied:

(1), the presumed component so and the actual component s have the same morphology;

(2), n -sso(r) = 0, for i 1,...,N, and r e Ts. (the signal and noise are uncorrelated);
(3), X is full rank.
Proof: The optimal spatial filter is given by (3-8). We plug it into (3-7) and get:

J(r) = so(-r)' .-(C-I) s= o(-r) .(C -C -C+I)S (Z-)

where, C= X XX') X. note that CC = C, so,

J(r)= -so(r-)" C.so(r) +so(r)s (r)= -[so(r-) s .(a'R la)+so(r) S (-)

where, R = X X' is a positive definite matrix independent of the time lag r.
With the constraint that s (r)' s (r) = const, the minimum of the cost function J(r) is

achieved when so(r)Ts achieves its maximum, since a'R 'a is positive. This happens when r

coincides with the peak latency / of the actual ERP component s.









APPENDIX B
GAMMA FUNCTION AS AN APPROXIMATION FOR MACROSCOPIC ELECTRIC FIELD

The macroscopic electrical field is created from spike trains by a nonlinear generator with a

second-order linear component with real poles (Freeman 1975). Suppose that the transfer

function of the second-order system with stable real poles a, b is:

1
H(s) =
(s a)(s b)

where, without loss of generality: b < a < 0.


Then the impulse response in the time domain is: h(t) = 1 (eat ebt).
a-b

1
This is also a monophasic waveform with a single mode at t = In(b /a). The rising
a-b

time depends on the relative magnitude of the two real poles. The impulse response can be

expanded:


h(t) 1 1 (-b
h(t)= ebt.(e (a-b)t ebt. [(ab)t]
a-b a-b n-, n!

Thus we can see that the impulse response is a sum of infinite weighted Gamma functions.

However, it is always possible to find a few dominant terms around the mode, where,

t, (a b) = ln(b / a). If we knew the values for a, b, we can choose the shape parameter K of the

Gamma function as the largest term, i.e., the integer part of ln(b/a).

A special case is when the system has two identical poles. Then, the impulsive response is

exactly modeled by a single Gamma function with K = 1, 0 = -1/a. This is also approximately

true when the magnitude of one pole is much larger than the other one.









APPENDIX C
DERIVATION OF THE UPDATE RULE FOR THE CONSTRAINED OPTIMIZATION
PROBLEM

Using one Lagrange multiplier, we convert the constrained optimization problem in (4-1) to

an unconstrained optimization problem.

argmin X-c-a-s +- a- +a 2\ | 1)
(T,a

Note that,

X-c-a-s =2 Tr (x-c-a-sT).(X _X- a.asT)

= Tr (XX) 2Tr (Xsa) + C2Tr (assa' )

= Tr (XX' )- 2cTr (a'Xs) + a2'sTr (aar)
= Tr (XX) 2coa'Xs + oC2s s a.a

Setting the gradient of the Lagrangian function to 0 with respect to a, a, p respectively, we

find that the following set of equations holds:

20ss a a 2cXs + 2A (a ao) + 2/a = 0
-2a Xs +2ss- a a = 0
{allu -1= 0

Solving for a, c, we have:

a'Xs
sTs
LAa + CXs
Aao + Xs 2

This is not a closed-form solution for the optimal values. However, it can be effectively used

as a fixed point update to iteratively find the optimal values.









APPENDIX D
DERIVATION OF THE UPDATE RULE FOR THE UNCONSTRAINED OPTIMIZATION
PROBLEM

The unconstrained optimization problem is:

argmin X-a-s +A- a/ a -a 2
a

First we note that:

a/a2 "[ 21(/ a0) -(l/a ao)

= a a/|aI| + ao ao -2a a,/ a 2
= 2-2aao/ a 2

Taking the derivative to 0, we get,

2s's*a-2Xs+2 (a'aaoa/ a| -ao/a ) = 0

Or equivalently,

SXs + Aa0/ lll
a=
sTs+ Aa0Ta/ ai||.

This is not a closed-form solution for a. However, it can be effectively used as a fixed point

update to iteratively find the optimal values.










APPENDIX E
MAP SOLUTION FOR THE ADDITIVE MODEL

With the assumptions indicated in Section 4.4, the posteriori probability can be rewritten as:

K
p(ao,o ...-K al .aK) 1 =l (ak jk,ao)p(k)P(ao) (al...aK)
k=l

Given the model, maximization of the posterior probability is equivalent to:

K
arg max j p(ak ao, k)p(cak)P(ao)
a,o- -K k=1

We assume a uniform (flat) apriori distribution for ok. Since ao is constrained to have a


unit norm, its a priori distribution is a Dirac delta function: (1 ao 12) -. If we assume that uk is


zero-mean i.i.d. Gaussian noise with the same covariance matrix d2I across all the trials,

maximization of the posterior probability can be further simplified:

K K
arg max log j p(ak ao,, k)p(a,)= arg max ak (- ak a )+ log0 ( ao -1)
ao, U uK k=l ao,- 'K k=l

This can be converted to a constrained optimization problem:

K
min J Iak- ka
k=l

S.t.

Ia02 =1


Setting the derivative to zero, a necessary condition for minimum is: o- = ak o ak.

Plug this into the above cost function. We have:











K 2 K
J= ak -oT ak oz k ao o
k=l k=l

Zak (-a ao -ao a) ka
k=1
K
ak T a. ao T )ak
k-I
k=1
K K
k T ak a k k T a
k=l k=l


The first term does not depend on ao, so MAP is equivalent to:


arg max af A ao,
ao


S.t.


ao 2=1

K K
where A = Ak = k a ak a The matrix A is symmetric, so it can be diagonalized.
k=l k=1


The maximum occurs when ao is the normalized eigenvector of the matrix A corresponding to


the largest eigenvalue.









APPENDIX F
NORMALIZED ADDITIVE NOISE MODEL 1: MAP SOLUTION

Following the same rationale in model 1, we attempt to derive the MAP solution for the model 2.

The difference is in the covariance matrix of conditional probability: p(ak I ao, k) N(0, rk21)


So, p(ak ac,)= exp- a,-a |/2)
(2z )D 0_k k k akao

The MAP becomes:

K
arg max j p(ak a,,k)p(k)p(ao)
ao,01 (K k=1

=argmax ( ak,/k a )+logS(a, 2 i)
ao,,0i K k=1


where, we have used the prior distribution for Uk: p(k) = k D

This is an improper prior (but still a prior for Bayesian inference). It is mainly motivated by

analytical tractability. Similarly as in model 1, we can write the following constrained

optimization problem:

K
argmin Yak,/,-a
ao, k k=l

S.t.

Ilaoz =1


We first find a necessary condition for minimum: o- = a kT ak/ao ak .

T
Let Bk = akak Note that Bk is idempotent, i.e., BkBk = Bk
ak *ak

Plug this into the above cost function. We have:











K
argmin (I-Bk)ao,
ao k=1
K
= arg min a' (I Bk )a
ao k=l
K
= argmin K- Y a) Bka,
ao k=1

= arg max a/ Bao
ao

K K akT
where, B= B, k aa-
k=l k=l k ak


Again, the maximum occurs when ao is the normalized eigenvector of the matrix B


corresponding to the largest eigenvalue.










APPENDIX G
NORMALIZED ADDITIVE NOISE MODEL 2: MAP SOLUTION


Given the model: k = a + u, we attempt to find the MAP solution for the posteriori



probability.

arg max p(a I a,...a )
ao
K
= arg maxlog j p(ak ao)p(ao)
ao k=1
K
=argmax (- ak,/a, -a |:)+log( ao 2-1)
ao k-1

The equivalent constrained optimization problem is:

K 2
argmax ak/ak 2-0 2
ao k=1

S.t.


a2o =

This can be solved in a straightforward way:

K
ak /ll41 :
k=1
oa K
ak ak 2
k=1 2









LIST OF REFERENCES


Al-Nashi H. (1986). A maximum likelihood method for estimating EEG evoked potentials. IEEE
Transactions on Biomedical Engineering, 33:1087-1095.

Aunon J.I. and McGillem C.D. (1975). Techniques for processing single evoked potentials.
Proceedings San Diego Biomedical Symposium, 211-218

Aunon J.I., McGillem C.D., and Childers D.G.. (1981). Signal processing in evoked potential
research: averaging and modeling. CRC Critical Reviews in Biomedical Engineering, 4:323-
367.

Basar E, Demiralp T, Schurmann M, Basar-Erglu C, Ademoglu A. (1999). Oscillatory brain
dynamics, wavelet analysis, and cognition. Brain Lang;66:146-183.

Bell A., Sejnowski T., (1995). An Information Approach to Blind Separation and Blind
Deconvolution, Neural Computation., vol. 7: 1129-1159.

Belouchrani A., Abed-Meraim K., Cardoso J-F, Moulines E., (1997). A Blind Source Separation
Technique Based on Second-Order Statistics, IEEE Transaction on Signal Processing,
vol.45, 434-444.

Brazier. M.A.B. (1964). Evoked responses recorded from the depths of the human brain. Annals
of New York Academy of Sciences, 112: 33-59.

Bruin K.J., Kenemans J.L., Verbaten M.N., Van der Heijden A.H., (2000) Habituation: an event-
related potential and dipole source analysis study. Intional Journal of Psychophysiol. 36,
199-209.

Cardoso J.-F. (1998). Blind signal separation: statistical principles, In Proceedings of the IEEE,
special issue on blind identification and estimation, R.-W. Liu and L. Tong editors. Vol.
9(10), 2009-2025.

Cardoso J.-F. (1999). High-Order Contrasts for Independent Component Analysis. Neural
Computation, vol. 11, 157-192.

Caspers, H., Speckmann, E.-J., & Lehmenkuihler, A. (1980). Electrogenesis of cortical DC
potentials. (eds. H. H. Komhuber & L. Deecke), Progress in brain research: Vol. 54.
Motivation, motor and sensory processes of the brain: Electrical potentials, behavior and
clinical use 3-15. Amsterdam: Elsevier.

Cerutti S., Bersani V., Carrara A., and Liberati D. (1987). Analysis of visual evoked potentials
through Wiener filtering applied to a small number of sweeps. Journal on Biomedical
Engineering, 9:3-12.









Chapman, R., McCrary, J., (1995). EP component identification and measurement by principal
components analysis. Brain Cognition 27 (3), 288-310. (Erratum in: Brain Cogn. 28 (3)
342, 1995.)

Choi S., Cichocki A., Park H. -M., and Lee S. -Y. (2005). Blind source separation and
independent component analysis: A review. Neural Information Processing Letters and
Review, vol. 6, no. 1, 1-57.

Cichocki A. and Amari S. (2002). Adaptive Blind Signal and Image Processing: Learning
Algorithms and Applications. New York: Wiley.

Ciganek Z. (1969). Variability of the human visual evoked potential: normative data.
Electroencephalography and Clinical Neurophysiology, 27:35-42.

Codispoti M., Ferrari V., Bradley M., (2006). Repetitive picture processing: autonomic and
cortical correlates. Brain Research. 1068: 213-220.

Coifman R.R., Wickerhauser MY. Wavelets, (1996). adapted waveforms and denoising. Clinical
Neurophysiology Supplement; 45:57-78.

Coles MGH, Rugg MD. (1995). Event-related brain potentials: an introduction.
Electrophysiology of mind: event-related brain potentials and cognition. (ed. Rugg MD,
Coles MGH), Oxford University Press. Oxford, UK. 1-26.

Comon P. (1994). Independent component analysis, a new concept? SignalProcessing, vol. 36,
287-314.

Cuthbert, B.N., Schupp, H. T., Bradley, M.M., and Lang, P.J. (1996). Affective picture viewing:
Task and stimulus effects on startle P3 and blink. Psychophysiology, 33.

Daubechies. I. (1992). Ten Lectures on Wavelets. SIAM.

Debener S, Makeig S, Delorme A, Engel AK, (2005). What is novel in the novelty oddball
paradigm? Functional significance of the novelty P3 event-related potential as revealed by
independent component analysis, Experimental Brain Research. Cognitive Brain Research.
22:309-321.

Delorme A, Makeig S, Fabre-Thorpe M, Sejnowski TJ, (2002). From single-trial EEG to brain
area dynamics, Neurocomputing, 44-46: 1057-1064.

Delorme, A., Makeig, S. (2003) EEG changes accompanying learning regulation of the 12-Hz
EEG activity. IEEE Transactions on Rehabilitation Engineering, 11(2), 133-136.

Delorme, A., Sejnowski, T., Makeig, S. (2007) Improved rejection of artifacts from EEG data
using high-order statistics and independent component analysis. Neuroimage, 34, 1443-1449









Dien J. (1998). Addressing misallocation of variance in principal components analysis of event-
related potentials. Brain Topography; 11(1):43-55.

Dien J, Beal DJ, Berg P. (2005). Optimizing principal components analysis of event-related
potentials: Matrix type, factor loading weighting, extraction, and rotations. Clinical
Neurophysiology; 116:1808-25.

Doncarli C. and Goerig L. (1988). Adaptive smoothing of evoked potentials: a new approach.
Proceedings 10th Annual Confereence IEEE-EMBS, New Orleans, Lousiana, 1152-1153

Donchin E. (1966). A multivariate approach to the analysis of average evoked potentials. IEEE
Transaction on Biomedical Engineering; 13(3): 131-9.

Donchin E, Heffley EF. (1978). Multivariate analysis of event-related potential data: a tutorial
review. In: Otto DA, editor. Multidisciplinary perspectives in event-related brain potential
research. Proceedings of the fourth international congress on event-related slow potentials
of the brain (EPIC IV), Hendersonville, NC, April 4-10, 1976. Washington, DC: The
Office; 555-572.

Donchin, E., Ritter, W., McCallum, C. (1978). Cognitive psychophysiology: The endogenous
components of the ERP. Brain event-relatedpotentials in man (ed. E. Callaway, P. Tueting,
S. Koslow), Academic Press, New York. 349-441.

Donchin, E. (1979). Event-related brain potentials: a tool in the study of human information
processing. Evoked potentials and behavior (ed. H. Begleiter), Plenum, New York. 13-75.

Donchin, E. (1981). Surprise! surprise? Psychophysiology, 18, 493-513.

Donchin, E. and Coles, M. G. H. (1988a). Is the P300 component a manifestation of context
updating? Behavioral and Brain Sciences, 11, 355-72.

Donchin, E. and Coles, M. G. H. (1988b). On the conceptual foundations of cognitive
psychophysiology. Behavioral and Brain Sciences, 11, 406- 17.

Donchin, E., Spencer, K.M. and Dien, J. (1997). The varieties of deviant experience: ERP
manifestations of deviance processors. In: G.J.M. van Boxtel and K.B.E. Bocker (Eds.),
Brain and Behavior: Past, Present, and Future, Tilburg: Tilburg University Press,: 67-91.

Donoho D.L. and Johnstone I.M. (1994). Ideal spatial adaptation by wavelet shrinkage.
Biometrika, 81:425-455.

Dornhege G., Blankertz B., Krauledat M., Losch F., Curio G., Miller K.-R. (2006). Combined
optimization of spatial and temporal filters for improving Brain-Computer Interfacing. IEEE
Transactions on Biomedical Engineering, 53(11), 2274-2281.









Duffy, F.H., Jones, K., Bartels, P., Albert, M., McAnulty, G.B. and Als, H. (1990). Quantified
neurophysiology with mapping: Statistical inference, exploratory and confirmatory data
analysis. Brain Topography,3(1): 3-12.

Dyrholm M, Christoforou C., Parra L.C., (2007). Bilinear discriminant component analysis,
Journal ofMachine Learning Research, vol. 8, 1097-1111

Effern A, Lehnertz K, Fernandez G, Grunwald T, David P, Elger CE. (2000). Single trial
analysis of event related potentials: non-linear de-noising with wavelets. Clinical
Neurophysiology; 11:2255-2263.

Fabiani, M., Gratton, G., Karis, D., & Donchin, E. (1987). Definition, identification and
reliability of measurement of the P300 component of the event-related brain potential. In P.
K. Ackles, J. R. Jennings, & M. G. H. Coles (Eds.) Advances in psychophysiology, Vol. 2
(pp. 1-78). Greenwich, CT : JAI Press.

Farwell L.A., Martinerie J.M., Bashore T.R., Rapp P.E., and Goddard P.H. (1993). Optimal
digital filters for long-latency components of the event-related brain potential.
Psychophysiology, 30:306-315.

Fisher, R.A., (1915). Frequency distribution of the values of the correlation coefficient in
samples of an indefinitely large population. Biometrika, 10, 507-521.

Freeman W., (1975). Mass activation in the nervous system. Academic Press, New York

Gaillard AWK, Ritter WK. (1983). Tutorials in event-related potential research: endogenous
components. Amsterdam: North-Holland Publishing Company.

Gansler T. and Hansson M. (1991). Estimation of event-related potentials with GSVD.
Proceedings 13th Annual Conference. IEEE-EMBS, 423-424.

Garoosi V. and Jansen B. (2000). Development and evaluation of the piecewise prony method
for evoked potential analysis. IEEE Transaction on Biomedical Engineering. 47(12): 1549-
1554.

Glaser EM, Ruchkin DS. (1976). Principles of neurobiological signal analysis. New York:
Academic Press.

Gonsalvez CL, Polich J (2002) P300 amplitude is determined by target-to-target interval.
Psychophysiology 39:388-396

Gratton, G., Coles, M.G.H., Donchin, E., (1983). A new method for off-line removal of ocular
artifact. Electroencephalography Clinical Neurophysiology. 55, 468 484

Gratton G., Kramer A.F., Coles M.G.H., and Donchin E. (1989). Simulation studies of latency
measures of components of the event-related brain potential. Psychophysiology, 26:233-248.










Hansson M. and Cedholt T. (1990). Estimation of event related potentials. Proceedings 12th
Annual Conference IEEE-EMBS, 901-902.

Hansson M., Gansler T., and Salomonsson G. (1996). Estimation of single event-related
potentials utilizing the prony method. IEEE Transactions on Biomedical Engineering,
43(10):973-981.

Hendrickson AE, White PO. (1964). Promax: a quick method for rotation to oblique simple
structure. Br J Stat Psychol; 17:65-70.

Hesse C.W. and James C.J., (2006). On Semi-Blind Source Separation using Spatial Constraints
with Applications in EEG Analysis, IEEE Transactions on Biomedical Engineering. 53(12):
2525-2534.

Horn JL. (1965). A rationale and test for the number of factors in factor-analysis. Psychometrika;
30(2):179-85.

Huber P., (1981). Robust Statistics. New York: Wiley.

Hyvarinen A., Oja E. (1997). A Fast Fixed-Point Algorithm for Independent Component
Analysis, Neural Computation, vol. 9: 1483-1492.

Hyvaerinen A., (1999). Survey on independent component analysis. Neural Computation. Sur.,
vol. 2: 94-128.

Hyvarinen A., Karhunen J. and Oja E. (2001). Independent Component Analysis. New York:
Wiley

James, C. and Hesse, C. (2004). Independent component analysis for biomedical signals.
Physiological Measurement, 26, (1), 15-39.

Jaskowski P. and Verleger R. (1999). Amplitudes and latencies of single-trial ERP's estimated
by a maximum-likelihood method. IEEE Transactions on Biomedical Engineering,
46(8):987-993.

Johnson, R., Jr. (1986). A triarchic model of P300 amplitude. Psychophysiology, 23, 367 384.

Johnson R. (1989). Developmental evidence for modality-dependent P300 generators: a
normative study. Psychophysiology 26: 651-667.

Jung T-P, Makeig S, Westerfield M, Townsend J, Courchesne E, and Sejnowski TJ, (1999).
Analyzing and Visualizing Single-trial Event-related Potentials. Advances in Neural
Information Processing Systems, 11:118-124.









Jung T-P, Makeig S, Lee T-W, McKeown M.J., Brown G., Bell, A.J. and Sejnowski TJ, (2000).
Independent Component Analysis of Biomedical Signals, The 2ndlnt'l Workshop on
Independent Component Analysis and Signal Separation, 633-644.

Jung T-P, Makeig S, McKeown M.J., Bell, A.J., Lee T-W, and Sejnowski TJ, (2001). Imaging
Brain Dynamics Using Independent Component Analysis, Proceedings of the IEEE,
89(7):1107-1122.

Junghofer M, Elbert T, Leiderer P, Berg P, Rockstroh B (1997) Mapping EEG-potentials on the
surface of the brain: a strategy for uncovering cortical sources. Brain Topography 9:203-217

Junghofer M, Elbert T, Tucker DM, Rockstroh B (2000) Statistical control of artifacts in dense
array EEG/MEG studies. Psychophysiology 37:523-532

Kaiser HF. (1958). The Varimax criterion for analytic rotation in factor analysis. Psychometrika.
23:187-200.

Kalman R.E. (1960). A new approach to linear filtering and prediction problems. Transactions
ASME, Journal ofBasic Engineering. 82D:35-45.

Karjalainen P.A., Kaipio J.P., Koistinen A.S., and Karki T. (1996). Recursive Bayesian
estimation of single trial evoked potentials. Proceedings 18th Annual Conference IEEE-
EMBS, Amsterdam

Karjalainen PA. (1997). Regularization and Bayesian Methods for Evoked Potential Estimation.
Ph.D. Thesis, Kuopio University Publications C. Natural and Environmental Sciences 61

Karjalainen P.A., Kaipio J.P., Koistinen A.S., Vauhkonen M. (1999). Subspace regularization
method for the single-trial estimation of evoked potentials. IEEE Transaction on Biomedical
Engineering 46(7):849-60.

Keil A, Bradley MM, Junghoefer M, Russmann T, Lowenthal W, Lang PJ (2007) Cross-modal
Attention Capture by Affective Stimuli: Evidence from Event-Related Potentials. Cognitive,
Affective, & Behavioral Neuroscience 7:18-24

Koch C., Poggio T., Torre V., (1983). Nonlinear interactions in a dendritic tree: localization,
timing, and role in information processing. Proceedings NationalAcademy of Science USA
80:2799-2802

Koles ZJ, Lazar MS, Zhou SZ. (1990). Spatial patterns underlying population differences in the
background EEG. Brain Topography; 2(4):275-284.

Koles ZJ, Lind JC, Flor-Henry P. (1994). Spatial patterns in the background EEG underlying
mental disease in man. Electroenceph Clinical Neurophysiology, 91:319-328.









Kok A., (2001) On the utility of P3 amplitude as a measure of processing capacity.
Psychophysiology 38:557-577

Krieger S., Timmer J., Lis S., and Olbrich H.M. (1995). Some considerations on estimating
event-related brain signals. Journal of Neural Trans Gen Sect, 99(1-3): 103-129.

Lang, P., Bradley, M. M., and Cuthbert, B. N. (1997). Motivated attention: Affect, activation,
and action. In P. Lang, R. F. Simons, & M. Balaban (Eds.), Attention and orienting: Sensory
and motivationalprocesses. Hillsdale, NJ: Erlbaum. 97-136.

Lang PJ, Bradley MM, Cuthbert BN (2005) International Affective Picture System: Technical
Manual and Affective Ratings. In. Gainesville, FL: NIMH Center for the Study of Emotion
and Attention

Lange D.H. (1996). Variable single-trial evoked potential estimation via principal component
identification. Proceedings 18th Annual Conference IEEE-EMBS, Amsterdam.

Lee T. W., Girolami M., and Sejnowski T. J., (1999). Independent component analysis using an
extended infomax algorithm for mixed sub-Gaussian and superGaussian sources. Neural
Computation., vol. 11, 417-441.

Lee T.W., Girolami M., Bell A. J., and Sejnowski T. J., (2000). A unifying information-theoretic
framework for independent component analysis. Computing Math Application., vol. 39, 1-
21.

Lemm S., Blankertz B., Curio G., and Miller K.-R., (2005). Spatio-spectral filters for improved
classification of single trial EEG, IEEE Transactions on Biomedical Engineering, vol. 52,
no. 9, 1541-1548.

Li R., Principe J., (2006). Blinking Artifact Removal in Cognitive EEG Data using ICA,
International Conference of Engineering in Medicine and Biology Society, 6273-6278.

Li R., Principe J.C., Bradley M., Ferrari V., (2007) Robust single-trial ERP estimation based on
spatiotemporal filtering. Proceedings IEEE EMBS Conference. 5206-5209

Li R., Principe J.C., Bradley M., Ferrari V., (2008) A spatiotemporal filtering methodology for
single-trial ERP component estimation. IEEE Transactions on Biomedical Engineering, In
Press

Liberati D., Bertolini L., and Colombo D. C. (1991). Parametric method for the detection of
inter- and intrasweep variability in VEP processing. Medical andBiological Engineering
and Computing, 29:159-166,.

Linden D. (2005). The P300: Where in the brain is it produced and what does it tell us?
Neuroscientist, 11(6): 563-576.









Liu W., Pokharel P., Principe J., (2007). Correntropy: Properties and Applications in Non-
Gaussian Signal Processing, IEEE Transactions on Signal Processing, Vol. 55, Issue 11

Luo A. and Sajda P. (2006) Learning discrimination trajectories in EEG sensor space:
application to inferring task difficulty, Journal ofNeural Engineering, 3 (1) L1-L6.

Maccabee P.J., Pinkhasov E. I., and Cracco R. Q. (1983). Short latency evoked potentials to
median nerve stimulation: Effect of low-frequency filter. Electroencephalography and
Clinical Neurophysiology, 55:34-44.

Madhavan G. P., Bruin H. de, and Upton A.R.M. (1984). Evoked potential processing and
pattern recognition. Proceedings 6th Annual Conference IEEE-EMBS, 699-702.

Madhavan G. P., Bruin H. de, and Upton A. R. M. (1986) Improvements to adaptive noise
cancellation. Proceedings 8th Annual Conference IEEE-EMBS, Fort Worth, TX, 482-486.

Madhavan G. P. (1988). Comments on adaptive filtering of evoked potentials. IEEE
Transactions on Biomedical Engineering, 35:273-275.

Madhavan G. P. (1992). Minimal repetition evoked potentials by modified adaptive line
enhancement. IEEE Transactions on Biomedical Engineering, 39(7):760-764.

Makeig, S., Bell, A., Jung, T., Sejnowski, T., (1996). Independent component analysis of
electroencephalographic data. Advances in Neural Information Processing Systems, vol. 8.
MIT Press, 145-151.

Makeig S, Jung T-P, Bell AJ, and Sejnowski TJ, (1997). Blind Separation of Auditory Event-
related Brain Responses into Independent Components. Proceedings ofNationalAcademy of
Sciences, 94:10979-10984.

Makeig S, Westerfield M, Jung T-P, Covington J, Townsend J, Sejnowski TJ, and Courchesne E,
(1999). Functionally Independent Components of the Late Positive Event-Related Potential
during Visual Spatial Attention," Journal ofNeuroscience, 19: 2665-2680.

Makeig S., Jung T.-P., Ghahremani D., and Sejnowski T. J. (2000). Independent component
analysis of simulated ERP data. in Hum. High. Func. I: Adv. Meth., T. Nakada, Ed.

Makeig S, Westerfield M, Jung T-P, Enghoff S, Townsend J, Courchesne E, Sejnowski TJ.
(2002). Dynamic brain sources of visual evoked responses. Science, 295:690-694.

Makeig S, Delorme A, Westerfield M, Jung T-P, Townsend J, Courchesne E, Sejnowski TJ.
(2004). Electroencephalographic brain dynamics following visual targets requiring manual
responses, PLOS Biology, 2(6):747-762.

Marple S. Lawrence Jr. (1987). Digital Spectral Analysis with Applications. Prentice Hall,
Englewood Cliffs.










McGillem C.D. and Aunon J.I. (1987). Analysis of event-related potentials, chapter 5, Elsevier
Science Publisher. 131-169.

Mocks J, Verleger R. (1991). Application of principal component analysis to event-related
potentials. In Weitkunat R., editor. Multivariate methods in biosignal analysis: Digital
Biosignal Processing. Amsterdam: Elsevier; 399-458.

Miuller-Gerking J., Pfurtscheller G., and Flyvbj erg H., (1999). Designing optimal spatial filters
for single-trial EEG classification in a movement task, Clinical Neurophysiology, 110: 787-
798.

Naatanen, R. and Picton, T. W. (1987). The N 1 wave of the human electric and magnetic
response to sound: a review and an analysis of the component structure. Psychophysiology,
24, 375-425.

Nishida S., Nakamura M., and Shibasaki H. (1993). Method for single-trial recording of
somatosensory evoked potentials. Journal on Biomedical Engineering, 150:257-262.

Parra, L.C., Alvino, C., Tang, A., Pearlmutter, B., Young, N., Osman, A., Sajda, P., (2002).
Linear spatial integration for single-trial detection in encephalography. Neurolmage 17,
223- 230

Parra, L.C., Sajda, P., (2003). Blind source separation via generalized eigenvalue decomposition.
Journal ofMachine Learning Research. 4, 1261-1269.

Parra, L., Spence, C., Gerson, A., Sajda, P., (2005). Recipes for the linear analysis of EEG.
Neuroimage, 28, 326-341.

Parzen, E., (1962) On estimation of a probability density function and mode. The Annals of
Mathematical Statistics, Vol. 33, No. 3. 1065-1076

Patterson R. D., Robinson K., Holdsworth J., McKeown D., Zhang C., and Allerhand M. H.,
(1992) Complex sounds and auditory images, In Auditory Physiology and Perception, (Eds.)
Y Cazals, L. Demany, K.Horner, Pergamon, Oxford, 429-446

Pham D. T., Mocks J., Kohler W., and Gasser T. (1987). Variable latencies of noisy signals:
Estimation and testing in brain potential data. Biometrika, (74) 525-533.

Picton TW, Bentin S, Berg P, Donchin E, Hillyard SA, Johnson R Jr, Miller GA, Ritter W,
Ruchkin DS, Rugg MD, Taylor MJ. (2000). Guidelines for using human event-related
potentials to study cognition: recording standards and publication criteria. Psychophysiology.
37(2):127-52









Polich J, Alexander JE, Bauer LO, Kuperman S, Morzorati S, O'Connor SJ, Porjesz B,
Rohrbaugh J, Begleiter H (1997) P300 topography of amplitude/latency correlations. Brain
Topogr 9:275-282

Polich J. (1999) P300 in clinical alications. Electroencephalography: basic principles, clinical
alications and relatedfields (Eds. E. Niedermayer and F. Lopes de la Silva). Urban and
Schwartzenberger, Baltimore-Munich. 1073-1091.

Quian Quiroga R., Garcia H. (2003). Single-trial event-related potentials with wavelet denoising.
Clinical Neurophysiology. 114: 376-390.

Ramoser, H., Mueller-Gerking, J., Pfurtscheller, G., (2000). Optimal spatial filtering of single
trial EEG during imagined hand movement. IEEE Transaction on Rehabilitation
Engineering. 8 (4), 441- 446.

Reilly, J., (1992). Applied Bioelectricity. Springer. New York

Roberts S. and Everson E. (2001) Independent Component Analysis: Principles and Practice
Cambridge: Cambridge University Press

Ruchkin D.S. and Glaser E.M. (1978). Simple digital filters for examining CNV and P300 on a
single trial basis. (ed. Otto D.A.) Multidisciplinary perspectives on event-related brain
potential research, U.S. Government Printing Office, Washington DC. 579-581.

Ruchkin D.S. (1987). Measurement of event-related potentials. In Human Event-Related
Potentials, volume 3 of Handbook ofElectroencephalography and Clinical
Neurophysiology, Elsevier. 7-44.

Samar VJ, Swartz KP, Raghuveer MR. (1995). Multiresolution analysis of event related
potentials by wavelet decomposition. Brain Cognition; 27:398-438.

Sams, M., Alho, K., Naatanen, R. (1984). Short-term habituation and dishabituation of the
mismatch negativity of the ERP. Psychophysiology, 21, 434-441.

Scherg, M. and Picton, T. W. (1991). Separation and identification of event-related potential
components by brain electric source analysis. Event-related brain research, EEG Sul. 42,
(ed. C. H. Brunia, G. Mulder, and M. N. Verbaten), Elsevier, Amsterdam. 24-37.

Schiff SJ, Aldrouby A, Unser M, Sato S. (1994). Fast wavelet transform of EEG. Electroenceph
Clinical Neurophysiology; 91:442-455.

Silva F.H. Lopes da. (1993). Event-related potentials: Methodology and quantification.
Electroencephalography: Basic principles, clinical applications and related fields,
Williams & Wilkins,877-886.









Silverman B.W. (1986), Density Estimation for Statistics and Data Analysis. Chapman and Hall,
London

Spencer KM, Dien J, Donchin E. (1999). A componential analysis of the ERP elicited by novel
events using a dense electrode array. Psychophysiology;36:409-14.

Spencer KM, Dien J, Donchin E. (2001). Spatiotemporal analysis of the late ERP responses to
deviant stimuli. Psychophysiology; 38(2):343-58.

Spreckelsen M. and Bromm B. (1988). Estimation of single-evoked cerebral potentials by means
of parametric modeling and filtering. IEEE Transactions on Biomedical Engineering,
35:691-700.

Steeger G.H., Herrmann 0., and Spreng M. (1983). Some improvements in the measurements of
variable latency acoustically evoked potentials in human EEG. IEEE Transactions on
Biomedical Engineering, 30:295-303.

Sutton, S., Braren, M., Zubin, J., and John, E. R. (1965). Evoked potential correlates of stimulus
uncertainty. Science, 150, 1187-8.

Tang A., Pearlmutter B., Malaszenko N., Phung D., Reeb B., (2002) Independent components of
magnetoencephalography: localization. Neural Computation. 14 (8), 1827- 1858

Thakor N.V. (1987). Adaptive filtering of evoked potentials. IEEE Transactions on Biomedical
Engineering, 34(1):6-12.

Thakor N, Xin-rong G, Yi-Chun S, Hanley D. (1993). Multiresolution wavelet analysis of
evoked potentials. IEEE Transactions on Biomedical Engineering; 40:1085-1094.

Truccolo, W., Knuth, K.H., Shah, A., Schroeder, C., Bressler, S.L., Ding, M. (2003) Estimation
of single-trial multi-component ERPs: differentially Variable Component Analysis (dVCA).
Biological Cybernetics, 89, 426-438.

Tsai AC, Liou M, Jung TP, Onton JA, Cheng PE, Huang CC, Duann JR, Makeig S. (2006).
Mapping single-trial EEG records on the cortical surface through a spatiotemporal modality.
Neuroimage. 32(1): 195-207.

van Boxtel GJM. (1998). Computational and statistical methods for analyzing event-related
potential data. Behavioral Research Methods Instrumental Computation. 30(1): 87-102.

Vigario R., Jousmaki V., Hamalainen M., Hari R., and Oja E., (1998). Independent component
analysis for identification of artifacts in magnetoencephalographic recordings, Proceedings
NIPS, Cambridge, MA, MIT Press, 229-235.









Vigario R, Sarela J, Jousmaki V, Hamalainen M, Oja E. (2000). Independent component
approach to the analysis of EEG and MEG recordings. IEEE Transaction on Biomedical
Engineering. 47(5):589-593.

Vila C.E. Da, Welch A.J., and Rylander H.G. III. (1986). Adaptive estimation of single evoked
potentials. Proceedings 8th Annual Conference IEEE-EMBS, 406-409.

Wan Eric A. and Nelson Alex T., (1996). Dual Kalman Filtering Methods for Nonlinear
Prediction, Estimation, and Smoothing, in Advances in Neural Information Processing
Systems 9.

Widrow B. and Steams S. D. (1985). Adaptive Signal Processing. Englewood Cliffs, NJ:
Prentice-Hall.

Wood CC, McCarthy G. (1984). Principal component analysis of event-related potentials:
Simulation studies demonstrate misallocation of variance across components.
Electroencephalography Clinical Neurophysiology. 59:249-60.

Wood, C. C. (1987). Generators of event-related potentials. A textbook of clinical
neurophysiology (ed. A. M. Halliday, S. R. Butler, and R. Paul), Wiley, New York. 535-567.

Woody C.D., (1967) Characterization of an adaptive filter for the analysis of variable latency
neuroelectric signals. Medical and Biological Engineering and Computing, 5:539-553

Yu K. and McGillem C.D. (1983). Optimum filters for estimating evoked potential waveforms.
IEEE Transactions on Biomedical Engineering, 30:730-737.









BIOGRAPHICAL SKETCH

Ruijiang Li was born in Shandong, China. He received the B.S. degree in automation with

emphasis on systems and control in 2004, from Zhejiang University, Hangzhou, China. Since

2004, he has been working toward his Ph.D. at the Electrical and Computer Engineering

Department at the University of Florida, under the supervision of Jose C. Principe. His current

research interests include statistical signal processing, machine learning and their applications in

biomedical engineering.





PAGE 1

1 SPATIOTEMPORAL FILTERING METHODOLOGY FOR SINGLE-TRIAL EVENT-RELATED POTENTIAL COMPONENT ESTIMATION By RUIJIANG LI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008

PAGE 2

2 2008 Ruijiang Li

PAGE 3

3 To all scientists and researchers, who ha ve lived in pursuit of knowledge, and have dedicated themselves to the advancement of science

PAGE 4

4 ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor Dr. Jose C. Principe, for his great inspiration and encouragement thr oughout the course of my research. Not just that. He has really become a mentor and guide during pivotal times of my life, which I would have to say regretfully that I did not take full advantage of. One could ask for no more from such an advisor. I wish to thank the members of my committ ee, Dr. John Harris, Dr. Jianbo Gao, and Dr. Mingzhou Ding, for their valuable time and interest in serving on my supervisory committee, as well as their comments, which helped improve the quality of this dissertation. I am grateful for Dr. Andreas Keils expertise on psychology as well as his support, which made our collaboration fruitful. I would like to thank my friends and colle agues at the Computational NeuroEngineering Laboratory. They have made my stay in Florida during the pa st four years an enjoyable experience. Last but not least, I wish to thank my parents, who raised me up. Without them, all is in vain.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............10 CHAPTER 1 INTRODUCTION..................................................................................................................12 1.1 Basic Concepts of the ERP..........................................................................................12 1.1.1 Generation of the ERP.......................................................................................12 1.1.2 The ERP Components........................................................................................13 1.2 Estimation of the ERP..................................................................................................16 2 SINGLE-TRIAL ERP ESTIMATION...................................................................................19 2.1 Single-Trial ERP Estimation Usi ng Single-Channel Recording.................................19 2.1.1 Time-Invariant Digital Filtering........................................................................20 2.1.2 Time-Varying Wiener Filtering.........................................................................20 2.1.3 Adaptive Filtering..............................................................................................21 2.1.4 Kalman Filtering................................................................................................22 2.1.5 Subspace Projection and Regularization............................................................22 2.1.6 Parametric Modeling..........................................................................................23 2.1.7 Other Methods Using Single-Channel Recording.............................................24 2.2 Single-Trial ERP Estimation Usi ng Multi-Channel Recording...................................25 2.2.1 Generative EEG Model......................................................................................25 2.2.2 What Is a Spatial Filter and What Can It Do?...................................................27 2.3 Review of Spatiotempor al Filtering Methods..............................................................28 2.3.1 Principal Component Analysis (PCA)...............................................................28 2.3.2 Independent Component Analysis (ICA)...........................................................32 2.3.3 Spatiotemporal Filtering Methods for the Classification Problem....................36 3 NEW SPATIOTEMPORAL FILTERI NG METHODOLOGY: BASICS.............................40 3.1 Spatial Filter as a Noise Can celler in the Spatial Domain...........................................40 3.2 Deterministic Approach...............................................................................................42 3.2.1 Finding Peak Latency........................................................................................42 3.2.2. Finding Scalp Topography and Peak Amplitude...............................................44 3.3 Stochastic Approach....................................................................................................46 3.4 Simulation Study..........................................................................................................48 3.4.1. Gamma Function as a Template for ERP Component.......................................49

PAGE 6

63.4.2. Generation of Simulated ERP Data...................................................................49 3.4.3 Case Study I: Comparison with Other Methods................................................50 3.4.4 Case Study II: Effects of Mismatch...................................................................53 4 ENHANCEMENTS TO THE BASIC METHOD..................................................................59 4.1 Iteratively Refined Template.......................................................................................59 4.2 Regularization..............................................................................................................61 4.2.1 Constrained Optimization..................................................................................61 4.2.2 Unconstrained Optimization..............................................................................64 4.3 Robust Estimation: the CIM Metric.............................................................................65 4.4 Bayesian Formulations of the Topography Estimation...............................................68 4.4.1 Model 1: Additive Noise Model........................................................................69 4.4.2 Model 2: Normalized Additive Noise Model....................................................70 4.4.3 Model 3: Original Model...................................................................................71 4.4.4 Comparison among the Three Models...............................................................71 4.4.5 Online Estimation..............................................................................................72 4.5 Explicit Compensation for Tem poral Overlap of Components...................................72 5 APPLICATIONS TO COGNITIVE ERP DATA..................................................................90 5.1 Oddball Target Detection.............................................................................................90 5.1.1 Materials and Methods.......................................................................................90 5.1.2 Estimation Results.............................................................................................92 5.1.3 Discussions........................................................................................................98 5.2 Habituation Study........................................................................................................99 5.2.1 Materials and Methods.......................................................................................99 5.2.2 Estimation Results.............................................................................................99 6 CONCLUSIONS AND FUTURE RESEARCH..................................................................110 6.1 Conclusions................................................................................................................110 6.2 Future Research.........................................................................................................112 APPENDIX A PROOF OF VALIDITY OF THE PEAK LATENCY ESTIMATION IN (3-10)................114 B GAMMA FUNCTION AS AN APPROXIMATION FOR MACROSCOPIC ELECTRIC FIELD...............................................................................................................115 C DERIVATION OF THE UPDATE RULE FOR THE CONSTRAINED OPTIMIZATION PROBLEM..............................................................................................116 D DERIVATION OF THE UPDATE RU LE FOR THE UNCONSTRAINED OPTIMIZATION PROBLEM..............................................................................................117 E MAP SOLUTION FOR THE ADDITIVE MODEL............................................................118

PAGE 7

7 F NORMALIZED ADDITIVE NOISE MODEL 1: MAP SOLUTION.................................120 G NORMALIZED ADDITIVE NOISE MODEL 2: MAP SOLUTION.................................122 LIST OF REFERENCES.............................................................................................................123 BIOGRAPHICAL SKETCH.......................................................................................................135

PAGE 8

8 LIST OF TABLES Table page 3-1 Latency estimation: mean and standard deviation.............................................................55 3-2 Amplitude estimation: mean and standard deviation.........................................................55 3-3 Scalp topography estimation: correlation coefficient........................................................55 3-4 Effects of mismatch I: SNR = -20dB.................................................................................56 3-5 Effects of mismatch II: SNR = -10dB...............................................................................56 4-1 Latency estimation: mean and standard deviation.............................................................75 4-2 Amplitude estimation: mean and standard deviation.........................................................75 4-3 Scalp topography estimation: correlation coefficient........................................................75 4-4 Estimation results for the iterati vely refined template method..........................................76 4-5 Estimation with MCC for the mismatch case at SNR = 0dB.............................................76 4-6 Estimation with MCC for the mismatch case at SNR = -20dB.........................................76 4-7 Amplitude estimation for three Bayesian models..............................................................77 4-8 Scalp topography estimation for three Bayesian models...................................................77 5-1 Correlation statistics for the 4 subj ects: Scalp topography estimation............................102 5-2 Regression statistics for response time and estimated amplitude....................................102

PAGE 9

9 LIST OF FIGURES Figure page 3-1 Gamma functions with differe nt shapes and scales...........................................................57 3-2 Waveforms of synthetic and presumed ERP component...................................................58 4-1 Mean and standard deviation of the es timated amplitude under different SNR conditions..................................................................................................................... ......78 4-2 The waveforms of the synthetic component, presumed template and refined template under 4 SNR conditions.....................................................................................................79 4-3 Waveforms of two overlapped comp onents used in regularization...................................80 4-4 Scalp topography of two overlapped ERP co mponents used in regularization.................81 4-5 Amplitude and scalp topography estimati on I with regularization (constrained optimization) under 3 SNR conditions..............................................................................82 4-6 Amplitude and scalp topography estimati on II with regularization (constrained optimization) under 3 SNR conditions..............................................................................84 4-7 Amplitude and scalp topography estimati on I with regularization (unconstrained optimization) under 3 SNR conditions..............................................................................86 4-8 Amplitude and scalp topography estimati on II with regularization (unconstrained optimization) under 3 SNR conditions..............................................................................88 5-1 Pictures used in the experiment as stimuli.......................................................................103 5-2 Cost function in (3.9) versus time lag fo r different regularization parameters for subject #2..................................................................................................................... ....104 5-3 Estimated pdf of time lags corresponding to local minima of the cost function in (39) using the Parzen window ing pdf estimator with a Gaussian kernel size of 4.2..........105 5-4 Scalp topographies for the four subjects..........................................................................106 5-5 Scatter plot of the response time versus the estimated amplitude for each single trial for the four subjects. .......................................................................................................107 5-6 Estimated scalp topography for mixed and habituation phase.........................................108 5-7 Estimated amplitude for mixed and habituation phase....................................................109

PAGE 10

10 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SPATIOTEMPORAL FILTERING METHODOLOGY FOR SINGLE-TRIAL EVENT-RELATED POTENTIAL COMPONENT ESTIMATION By Ruijiang Li December 2008 Chair: Jose Principe Major: Electrical and Computer Engineering Event-related potential (ERP) is an importa nt technique for the study of human cognitive function. In analyzing ERP, the fundamental probl em is to extract the waveform specifically related to the brains response to the stimulus from electroen cephalograph (EEG) measurements that also contain the spontaneous EEG, whic h may be contaminated by artifacts. A major difficulty for this problem is the low (typically negative) signal-to-noi se ratio (SNR) in EEG data. The most widely used tool analyzing ERP has been to average EEG measurements over an ensemble of trials. Ensemble aver aging is optimal in the least s quare sense provided that the ERP is a deterministic signal. However, over four deca des of research have sh own that the nature of ERP is a stochastic process. In particular the latencies and the amplitudes of the ERP components can have random variation between repetitions of the stimulus. Under these circumstances, estimation of the ERP on a single-tr ial basis is desirable. Traditional single-trial estimation methods only consider the time course in a single channel of the EEG. With the advent of dense electrode EEG a number of spatiotemporal filtering methods have been proposed for the single-trial estimation of ERP using multiple channels.

PAGE 11

11 In this work, we introduce a new spatiotemporal filtering method for the problem of single-trial ERP component estimation. The met hod relies on modeling of the ERP component local descriptors (latency and amplitude) and thus is tailored to extract faint signals in EEG. The model allows for both amplitude and latency variability in the act ual ERP component. The extracted ERP component is cons trained through a spat ial filter to have minimal distance (with respect to some metric) in the temporal domai n from a template ERP component. The spatial filter may be interpreted as a noi se canceller in the spatial do main. Study with simulated data shows the effectiveness of the proposed method to signal to noise ratios down to -10 dB. The method is also tested in real ERP data from cognitive experiments where the ERP are known to change, and corroborate experiment ally the expected behavior.

PAGE 12

12 CHAPTER 1 INTRODUCTION 1.1 Basic Concepts of the ERP Event-related potential (ERP) is an impor tant and well-estab lished technique for neuroscientists and psychologists to study human c ognitive function. In this section, we briefly review some of the basic facts and concepts rela ted to ERP generation and analysis (Coles et al. 1995). 1.1.1 Generation of the ERP When a pair of electrodes are attached to th e surface of the human scalp and connected to a differential amplifier, the output of the amplifier reveals a pattern of volta ge variation over time. This voltage variation is known as the electroencephalograph (EEG). The amplitude of the normal EEG varies between approximately V and most of the EEG frequency contents range between 0.5Hz and 40Hz. Here, we do not review the recordi ng techniques of EEG (Ruchkin 1987). If we present a stimulus to a human subj ect while recording the EEG, we can define a period of time (an epoch or a trial) where some of the EEG components are time-locked to the stimulus. Within this epoch, there may be voltage changes that are specifically related to the brains response to the stimulus. These voltage changes constitute the event-related potential, or ERP. Although it is not completely understood how the measurements at the scalp relate to the underlying brain activity, the following points app ear to be clear and are generally accepted (Scherg and Picton 1991, Wood 1987). First, ERP recorded from the scalp represents net electrical fields associated with the activity of sizeable neuron populations. These neuron populations act as current sources whose electrical fields pr opagate to the entire scalp through

PAGE 13

13 volume conduction. Second, the individual neurons that compromise such a population must be synchronously active and have a certain geom etric configuration to produce measurable potentials at the scalp. In particul ar, these neurons must be configur ed in such a way (usually in a parallel orientation) that their individual fields summa te to yield a dipolar field. Therefore, the ERP recorded at the scalp is se lective of the totality of the brain activity. This is advantageous in that the resultant measur ements would otherwise be so complex as to be difficult or impossible to analyze. On the othe r hand, we should also be aware that there are certainly numerous functionally important neural processes that cannot be detected by the ERP technique. 1.1.2 The ERP Components The issue of ERP components has aroused much controversy among the ERP research community, particularly the ques tion of the definition of an ERP component. Suppose for the moment that we have obtained the ERP usi ng some method. A simple way to define a component is to focus on some feature of the resu lting waveform (for instance, a peak or trough), and this feature becomes the component of interest. Some common features include the amplitude and latency parameters of a particular peak or trough. A major problem with the simple approach mentioned above is component overlap, both spatially and temporally. Since the brain is a conducting medium, activ ity generated in one spatial location may be propagate d through the brain tissue and app ear at other locations. Thus, the waveform we observe by measuring the voltage at the scalp may well be attributed to a variety of different sources in different spatial locations of the brain. One consequence of volume conduction is that there need be no direct correspondence between the timing of the distinctive features of an ERP waveform (peaks and troughs ) and the temporal characteristics of the underlying neural systems. For instance, an ERP p eak with a latency of 300 ms, might reflect the

PAGE 14

14 activity of a single neural generator maximally ac tive at that time, or the combined activity of two (or more) neural generators, maximally act ive before and after 300 ms, but with fields summating to a maximum at that time. Due to these ambiguities surrounding the interpretation of peaks and troughs in ERP waveforms, other definitions for ERP components have been proposed. Naatanen and Picton (1987) adopted what might be called the physiol ogical approach to component definition. They proposed that a defining characte ristic of an ERP component is its anatomical source within the brain. According to this defin ition, to measure a particular ER P component, we must have a method of identifying the contributing sources. Dochin (1979, 1981) adopted what might be called the functional approach to ERP component definition, which is concerned more with the information processing operations with which a pa rticular component is correlated. According to this definition, it is entirely possible for a componen t to be identified with a particular feature of the waveform that reflects the activity of multiple gene rators within the br ain, so long as these generators constitute a functionally ho mogeneous system (Coles, et al.1995). Although the above physiological and psychological approaches to component definition seem to be counteractive, for many investigat ors it is more appropr iate to combine both approaches. A classical approach to component definition, was pr oposed by Dochin et al. (1978). They argued that an ERP component should be defined by a combination of its polarity, its characteristic latency, its dist ribution across the scalp and its sensitivity to characteristic experimental manipulations. Noti ce that polarity and scalp distri bution imply a consistency in physiological source, while latency and sensit ivity imply a consistency in psychological function.

PAGE 15

15 ERP components can be broadly classified into two types: exogenous and endogenous components. Characteristics of the exogenous components (amplitude, latency and scalp projection) largely depend on the physical properties of sensory stimuli, such as their modality and intensity. On the other hand, endogenous com ponents largely depend on the nature of the subjects interaction with the stimulus. These co mponents vary as a function of such factors as attention, task relevance and the nature of the information proc essing required by the stimulus. The dichotomy of the exogenous-endogenous distin ction turned out to be an oversimplified version of the reality. Many early sensory components have been shown to be modifiable by cognitive manipulations (e.g., attention) and many of the later cognitive components have been shown to be influenced by the physical attributes of stimulus (e.g., modality). In what follows, we briefly discuss one particular well-known ERP component, the P300. For a comprehensive review on other well-known components, we refer to Coles, et al. 1995. The P300 is probably the most important and th e most studied compone nt of the ERP. It was first described in the 1960s by Sutton et al (1965). The P300 is evoked by a task known as the odd-ball paradigm. During this task a series of one type of fre quent stimuli is presented to the experimental subject. A different type of non-frequent (target) stimulus is also presented. The task of the subject is to react to the presence of target s timulus by a given motor response, typically by pressing a button, or just by mental counting to the target stimuli. Virtually any sensory modality (auditory, visual, somatosensor y, olfactory) can be us ed to elicit the P300 response. (Polich 1999). The shape and latency of the P300 differs with each modality. This indicates that the sources generating the P 300 differ and depend on the stimulus modality (Johnson 1989).

PAGE 16

16 There are several theories on the neural pro cesses underlying the origin of the P300. The most cited and most criticized theory was proposed by Donchin and Coles (1988a, b). According to their theory, the P300 reflect s a process of context or memo ry updating by which the current model of the environment is modified as a function of incoming information. Several investigators (e.g., Johnson 1986) have pointed out th at the P300 does not appear to be a unitary component, and instead may represent the activity of a widely distributed system which may be more or less coupled depending on the situati on. More information about the underlying neural systems is required before a consensus is atta ined about the functiona l significance of this component. For a more recent review on the research of P300, we refer to Linden, 2005. 1.2 Estimation of the ERP The fundamental problem in the analysis of ER P is to extract the si gnals that are brains specific response to the stimulus from the EEG meas urements that also cont ain noise. By noise, or the background EEG, we mean the electrical activities from hear t, muscles and eye movements as well as the spontaneous brain activiti es that are not related to brains response to the stimulus. A major difficulty with the extraction of ERP is that, in most cases, ERP signals are small (on the order of microvolts) relative to the b ackground EEG (on the order of tens of microvolts) in which they are embedded. For this reason, it is necessary to employ signal processing techniques to estimate the ERP si gnals in the presence of noise. By far the most commonly used techni que has been the averaging of the EEG measurements over an ensemble of time-locked epochs. This is optimal in the mean square sense, given the assumption that the ERP is a de terministic signal time-locked to the stimulus and the additive background EEG is zeromean and uncorrelated with the ERP.

PAGE 17

17 However, for over four decades it has been evid ent that the nature of ERP is more or less random. In particular, the amplitudes and latenc ies of the peaks in the ERP can have random variations between repetitions of the stimuli (Brazier 1964). In addition, the variations may be trend-like and the mean of the amplitudes and the latencies can change across the trials. Under these circumstances, the information regarding to these variations in ERP is lost through averaging. Furthermore, the average waveform may not, in fact, resemble the actual ERP waveform that is recorded in an individual trial. The resulting estimates for the ERP, therefore, may not correspond to the underlying neural processes and inferen ce about the cognitive function may be misleading. Estimation of the ERP on a single-trial basis is desired for the situations when the peak amplitude and latency of a particular component change significantly across trials. A major difficulty with single-trial ERP estimation is agai n the very low signal-to-noise-ratio (SNR) in the single-trial EEG, typically lower than -10dB. Statistically speaking, the aver age ERP, or the sample mean, is an example of the use of the first order statistic s, where only the first order moment of the population parameters is estimated. The next obvious improvement is to use the second order statis tics, i.e., covariance analysis. The most common approach is to form an estimator (filter) with which the unwanted contribution of the background EEG can be filtered out. To find such an estimator, some models or assumptions are imposed on the ERP and b ackground EEG concerning their respective second order statistics. The estimator that satisfies th e minimum mean square criterion can then be derived. The performance of the estimator th en largely depends on how realistic these assumptions are.

PAGE 18

18 For historical reasons, these traditional sing le-trial estimation met hods only consider the time course of a single recordi ng channel in the EEG. In some cases, simple ERP components, e.g., the brainstem auditory evoked potentials, can be adequately examined using a single channel. However, for most ERPs, simultaneous recording from multiple electrode locations is necessary to disentangle overl apping ERP components on the ba sis of their topographies, to recognize the contribution of ar tifactual potentials to the ER P waveform, and to measure different components in the ERP that may be optima lly recorded at different scalp sites (Picton et al. 2000). Today, high-density EEG can simultaneously record scalp potentials in up to 256 electrodes. This increased number of sensors and thus increased spatial resolution has created a need for signal processing methods that can simu ltaneously analyze the time series of multiple channels. Recently, various methods have been pr oposed for single-trial analysis that linearly combine the time series in multiple channels to generate a representation of the observed data that is easier to interpret (Chapman and Mc Crary 1995, Makeig et al. 1996, Parra et al. 2002). This linear projection combines the information from all the available sensors into a single channel with reduced interferen ce from other neural sources and may provide a better estimate of the underlying neural activity th an the EEG measurements in a single channel. The linear projection, in this sense, may be called a spatial filter and these methods can be generally called spatiotemporal filtering methods. We will review these and othe r single-trial estimation methods in detail in Chapter 2.

PAGE 19

19 CHAPTER 2 SINGLE-TRIAL ERP ESTIMATION In this chapter, we review the existing methods for single-trial ERP estimation. The methods are broadly categorized into two classes: those based on single-channel EEG recording and those based on multi-channel EEG recording. Methods based on single-channel recording rely solely on the modeling of temporal char acteristics of the ERP and EEG, while methods based on multi-channel recording inve stigate both the spatial and temporal characteristics of the ERP and EEG, and are termed with the genera l notion of spatiotemporal filtering methods. 2.1 Single-Trial ERP Estimation Using Single-Channel Recording For the estimation of ERP using a single channel, all the available information is contained in a single time series () x t: the EEG recording at a certain electrode. We assume that the measurements consist of two parts: the signal of interest (ERP) and additive noise (background EEG), denoted by () st and () nt, respectively. The observation model for the EEG can be written as: ()()() x tstnt (2-1) In this review, the EEG measurements for a single trial x is a finite-length vector with elements sampled from the origin al continuous-time waveform. When the time series in (2-1) are interpreted as stochastic processes, x becomes a random vector. Its length equals the number of samples in one trial. In vector form, the observation model is: x=s+n (2-2) Here we do not attempt to give an exhaus tive review on the topi c of single-trial ERP estimation with a single channe l. For other reviews on the gene ral ERP estimation problem, we refer to Aunon et al. (1981), Ruchkin (1987), McGille m and Aunon (1987), Silva (1993), Karjalainen (1997).

PAGE 20

202.1.1 Time-Invariant Digital Filtering Digital filtering is a good place to start for time series analysis. The simplest approach is to design digital filters that have a desired fre quency response. Ruchkin and Glaser (1978) used simple moving average FIR filters to estimate ER P on a single trial basis. More complicated ones may be designed to estimate some partic ular component such as P300 (Farwell, et al. 1993). Wiener filtering may also be used to estimate sing le-trial ERP, provided that the power spectra of the ERP and background EEG can be estimated appropriately (Aunon and McGillem 1975, Cerutti, et al. 1987). The major problem with linear time-invariant filt ering is the fact that the ERP is typically a transient and smooth waveform w ith no periodicity. The spectrum for this kind of signal is not defined properly. Consequently, the spectra of the ERP and background EEG (if they are estimated) were usually found to overlap significantly (Krieger et al. 1995, Spreckelsen and Bromm 1988, Steeger et al. 1983). Thus the application of digita l filters with constant frequency response is not expected to give desirable results in most cases. Th e effects of dig ital filtering are studied in (Ruchkin and Glaser 1978, Maccabee, et al. 1983, Nishida et al. 1993). 2.1.2 Time-Varying Wiener Filtering When the optimal Wiener solution is computed for single-trial EEG data, the filtering becomes time-varying with respect to each trial. In general, these methods necessitate some analytic model for the ERP. Yu and McGillem (1983) introduced what was called the timevarying minimum mean square error filter. A cr ucial task for their method is to obtain a good estimate for the cross-covariance between the ERP and measurements. Under the assumption that the ERP and background EEG are uncorrelate d, the cross-covariance becomes the autocovariance of the ERP signal itself. The ERP is parametrically modeled by a superposition of the

PAGE 21

21 components with random location and amplitude. The parameters for the ERP are then calculated from the Wiener solution on a single trial basis. 2.1.3 Adaptive Filtering The use of adaptive filtering for the analysis of single-trial ERP, particularly the use of the least mean square (LMS) algorithm (Widrow 1985), was ex tensively studied during the 1980s (Madhavan et al. 1984, 1986; Vila et al. 1986; Thakor 1987; Doncarli 1988). For these methods, the measurement () x t is selected as the desired signal, and several choices are proposed for the input signal. Thakor (1987) is probably the one of th e most cited works among the adaptive filtering methods for ERP estimation. From the princi ples of adaptive noise cancellation, Thakor proposed a novel way of choosing reference a nd primary inputs. Two sets of single-trial measurements (),()ij x txt (, ij being the trial index) serve as the reference and primary inputs, respectively. The idea is to estimate the primar y input with a set of delayed version of the reference input on which some form of ensemble averaging is performe d. The criticism of Thakors work is summarized in (Madhavan 1988) where the author asse rted that if the ERP signal is assumed to be identical across trials, the above approach does not provide any signal-tonoise ratio improvement and dist orts the signal at frequencie s where signal and noise power spectra overlap. Madhavan (1992) proposed a modified adaptive line enhancement method. In this method the pre-stimulus EEG data are adaptively modeled with an autoregressive (AR) model, which is then used to filter the post-stimulus EEG da ta. The notion of modified means that a nonadaptive filter is used to process the post-stimulus data.

PAGE 22

222.1.4 Kalman Filtering Al-Nashi (1986) adopted the Kalman filtering approach for the ERP estimation problem. It is assumed that the ERP can be modeled as a deterministic signal with additive random noise. The additive noise is assumed to be an auto regressive moving average (ARMA) process and another ARMA model is used for the background EEG The scalar Kalman filter is then used to predict the single-trial ERP. The basic assumption for Al-Nashis approach is that the difference between the single-trial ERP and the ensemble av erage is a stationary process. This is not consistent with (Ciganek 1969), which found that the differences are usually larger in late components than in early components. Liberati et al. (1991) model the single-trial ERP as a time-varying AR process using the ensemble average data and mode l the background EEG as a stati onary AR process using the prestimulus data. The AR parameters are then used to create the state and observation equations with the ERP as the unknown states. The single-trial ERP is then estimated using the Kalman filter equations (Kalman 1960). 2.1.5 Subspace Projection and Regularization The subspace projection approach starts with the linear observation model: H x=s+n=+n (2-3) where, contains the parameters to be estimated a nd the ERP signal is constrained to lie in the subspace spanned by some basis vectors, namely the columns of the matrix H. If the ERP is assumed to consist of positive and negative humps, sampled Gaussian functions may be a good choice for the basis vectors instead of a generic basis (e.g., polynomials). An alternative is to choose the eigenvectors of the EEG data autocorrelation matrix that correspond to the firs t few largest eigenvalues. This is motivated by the fact that the eigenvectors constitute a basis se t with the minimum number of ba sis vectors that are required to

PAGE 23

23 model the ERP, assuming that the ERP span s nearly the same subspace with the EEG measurements. The least square solution with th is basis set is equivalent to the principal component regression approach (Lange 1996, Karjalainen 1997). The above two basis sets may be combined in to a single criterion, w ith the Gaussian basis vectors modeling the ERP, and the subspace span ned by the eigenvectors representing the prior information about the problem. This leads to the subspace regularization method, which is closely related to the Bayesian mean square estimation (Karjalainen et al. 1999). The ERP may also be estimated recursively using Kalman filtering (Karjalainen et al. 1996). 2.1.6 Parametric Modeling There is one type of parametric models, which uses damped sinusoids as basis function for the modeling of single-trial ERP. The model for th e ERP with additive noise can be written as: 1()sin()()p t iii i x tAtnt (2-4) Estimation of the parameters ,,iiiA is a nonlinear problem, which can be solved with an approximation method called Pronys method (Marple 1987). Its use with generalized singular value decomposition was proposed by Ha nsson and Cedholt (1990) and Gansler and Hansson (1991). The Pronys method was utilized by Hansson et al. (1996) for the estimation of single-trial ERP and robust performance was ach ieved for EEG data with SNR>10dB. A more recent improvement, called piecewise Pronys me thod was proposed by Garoosi and Jansen (2000) to deal with nonstationary characteristics of the sinusoids. Another well studied tool that can be used for the analysis of single-trial ERP is the wavelet transform (Daubechies 1992 ). Wavelets provide a tiling of time-frequency space that gives a balance between time and frequency re solution and they can represent both smooth signals and singularities. This makes them suita ble models for the anal ysis of transient and

PAGE 24

24 nonstationary signals like th e ERP (Thakor 1993; Schiff et al. 1994; Samar 1995; Coifman 1996; Basar et al. 1999; Effern et al. 2000; Quian and Garcia 2003). The idea is based on a technique called wavel et shrinkage or wavel et denoising, which can automatically select an appropriate subset of basis functions and the corresponding wavelet coefficients. This relies on the property that natura l signals, such as images, neural activity, can be represented by a sparse code compromising onl y a few large wavelet coefficients. Gaussian noise, on the other hand, compromises a full set of wave coefficients whose size depends on the noise variance. By shrinking these noise coeffi cients to zero using a thresholding procedure (Donoho and Johnstone 1994), one can denoise data. However, the application of the wavelet method to single-trial ERP anal ysis requires some form of ensemble averaging in order to derive an optimal wavelet basis set that is tuned to the ERP signal. Sometimes the appropriate selection of the ERP ensemble may be a difficult task due to the effects of internal and exte rnal experimental parameters (Effern et al. 2000). 2.1.7 Other Methods Using Single-Channel Recording Some methods exist that try to explicitly estimate the latencies of the single-trial ERP components. A simple approach is to use cr oss-correlation of the signal with a template waveform and find the maximum point of the correlation (Gratton et al. 1989). Pham et al. (1987) applied a maximum likelihood (ML) method to estimate the latencies of ERP assuming a constant shape and amplitude. The ML method was extended in (Jaskowski and Verleger 1999) incorporating variable amp litude into consideration. Truccolo et al. (2003) devel oped a Bayesian inference framework for estimation of singletrial multicomponent ERP termed differentially variable component analysis(dVCA). Each component is assumed to have a trial-invariant waveform with trial-dependent amplitude scaling factors and latency shifts. A Maxi mum a Posteriori solution of this model is implemented via an

PAGE 25

25 iterative algorithm from which the components waveform, single-trial amplitude scaling factors and latency shifts are estimated. The method wo rks well for relatively low-frequency and largeamplitude event-related components. 2.2 Single-Trial ERP Estimation Using Multi-Channel Recording The use of multi-channel recording for the esti mation of single-trial ERP gave rise to a number of spatiotemporal filt ering methods. These methods a ssume, either explicitly or implicitly, a generative EEG model, which we will introduce in the next section. We then explain what is meant by a spatial filter and illustrate what it can do for us in estimating ERP on a single-trial basis. A review on existing spatiote mporal filtering methods is furnished in the following section, where we concentrate on me thods that are based on well-known statistical principles. 2.2.1 Generative EEG Model We start with the neural ge nerator assumption of EEG data i.e., neuron populations in cortical and subcortical brain tissues act as current sources (Caspers et al. 1980, Sams 1984). Within the EEG frequency range (below 100Hz), br ain tissues can be assu med to be primarily a resistive medium (Reilly 1992). Thus, according to Oh ms law, the electrical potentials collected at each sensor (channel) as a result of volume co nduction, is basically a linear combination of neural current sources (and non-ne ural artifacts). The linear generative model for EEG data can be written in matrix form: X=AS (2-5) or: 1N T ii iX=as (2-6) where, N is the number of current sources.

PAGE 26

26 We denote the single-trial EEG data with a DT matrix X with D channels and Tsamples; S is a NT matrix with each row T is representing the time course of the current density of the i -th current source; A is an unknown DN matrix. Strictly speaking, the number of the neural current sources N is necessarily much larger than the number of channels D It is usually assumed that the numbers of sources and sensors are equal for the purpose of convenience. The column vector ia of the matrix A represents the projection of the i -th current source to each sensor at the scalp and is called the forward model associated with the source. This scalp projection is generally unknown and depends on the location and orientation of the dipolar current source as well as the conductivity distri bution of the underlying brai n tissues, skull, skin and electrodes (Parra et al. 2005). Thus, if the scalp projection can be estimated, it may provide us some further evidence to the neurophysiological significance of the corresponding estimated source. An equivalent way to write the generative EEG model (2-5) is to us e the notation of time series: ()() tt xAs (2-7) where, 1()(),,()T Dtxtxt x is a column vector representing the EEG recordings in D channels; 1()(),,()T Ntstst s is a column vector representi ng the time course of the current density of the sources. A is the same matrix defined as before. There is some degeneracy in the model, i.e ., the scaling factor of the current source T is and its corresponding scalp projection ia. In this case, either the current sources or the scalp projections are constrained to ha ve unit power to avoid ambiguity. We wish to point out that

PAGE 27

27 there is no ambiguity whatsoever if we want to extract or eliminate from the EEG data the contribution of the i -th current source, i.e., iT iiXas. 2.2.2 What Is a Spatial Filter and What Can It Do? To illustrate our point, we begin with a si mple example. Suppose we measure the EEG from 3 electrodes where only 3 sources are presen t. Using the time series notation (the numbers are selected for illustration purposes): 1123 2123 3123()()2()() ()2()()() ()()()2() x tststst x tststst x tststst (2-8) We would like to recover each of the three so urces using the EEG measurements from all the available sensors. Sine the measurements ar e linearly related to the current sources, we speculate that this could be done by linearl y combining all the EEG measurements through a weight vector w: 1()()()D T ii i y ttwxtwx (2-9) In fact, if we select the weight vector in (2-9) as: [1,3,1]TT w, we will get, 1231()()3()()4() y txtxtxtst (2-10) which is exactly the first current source with a scaling factor. The other two sources can be recovered by using the weight vectors: [3,1,1],[1,1,3]TT respectively. Of course the above example is simplistic, be cause in reality, there are certainly numerous current sources simultaneously active in the brain and the number of sensors is usually up to a few hundred. We also do not know an y of the elements of the matrix A in general. However, the following point should be clear: by combining the EEG measurements fr om multiple channels with a simple weight vector, we are able to r ecover (or estimate) many sources of interest that

PAGE 28

28 could not be recovered using a sing le channel. In principle, the re jection of the interferences can be perfect, as shown in (2-10) if the coefficients are known. The weight vector w in (2-9), which operates on the EEG measurements in the sensor space, is called a spatial filter Just like a filter operating in th e time domain, a spatial filter can have either low-pass or high-pass characteristics in the spatial fr equency domain. For instance, a spatial filter summing the measurements from a group of neighboring se nsors have a low-pass characteristic; the use of a single channel recording corresponds to a high-pass spatial filter with an impulse response attenuating the data from all the other channels to zero. The selection of the spatial filter w is usually based on some constraints or desired characteristics of the output () yt. Different constraints will generally lead to different methods of extracting the outputs. Loosely speaking, ma ximum power of the output s leads to principal component analysis (PCA); statistical indepe ndence among the outputs leads to independent component analysis (ICA); and maximum difference between the outputs leads to linear discriminant analysis (LDA). We will review these and other spatiotemporal filtering methods in the following section. 2.3 Review of Spatiotemporal Filtering Methods In this section, we provide a review on exis ting spatiotemporal filtering methods for singletrial ERP analysis. Particularly, we focus on tw o popular and well established methods, namely, principal component analysis (PCA), independent component analysis (ICA). 2.3.1 Principal Component Analysis (PCA) From the early days of cognitive ERP resear ch, principal component analysis (PCA) was already proposed as a linear, multivariate data -reduction approach (Donchin, 1966). Since then, PCA has been one of most widely used tools among psychologists for ERP analysis (Glaser and

PAGE 29

29 Ruchkin, 1976; Donchin and Heffley, 1978; Mo cks and Verleger 1991; Chapman and McCrary 1995; van Boxtel 1998). By identifying unique vari ance patterns in a given set of ERP data, PCA decomposes the variance structure of the observed data into a set of latent variables that ideally correspond to the individual ERP components. In the ERP research community, these latent variables are usually called fact ors instead of components to avoid confusion with the ERP components. Among the vast literatures on PCA applied to ERP analysis, one classical method using the PCA-Varimax strategy, is particularly popular an d is the primary analytic tool for many ERP researchers (Gaillard and Ritter 1983). The method treats the recorded potential at a given time of the EEG epoch as variables. The domain of the observations is taken to be the Cartesian product of the recording channels, experimental conditions, participants. Suppose we have T samples in a given EEG epoch, D recording channels, C experimental conditions and P participants. The data matrix for this method has a dimension of TDCP This particular arrangement of the data matrix leads to the so-called temporal PCA approach, which gives orthogonal factors (eigenvectors of the covariance matrix). The PCA solution is then followed by the Varimax rotation (Kaiser, 1955). The Varimax rotation is an orthogonal rotation that aims to maximize the values that are large for a factor and minimize the values that are small (by maximizing the fourth power the factor). Th is corresponds to the maximum compactness criterion, which will make the new factors have a small number of large values and a large number of zero (or small) values. This is reas onable for ERP estimation because for the most part, ERP components appear to be monophasic and compact in time.

PAGE 30

30 It is easy to see that the above PCA-Varimax ap proach is a spatiotemporal filtering method. We denote ,cpX as the single-trial EEG data defined in (2-5) from c -th experimental condition and p -th participant. Then the covariance matrix is: 21TD CXX (2-11) where, 111PC cp pcPCXX (2-12) Formally, PCA is equivalent to the singular value decomposition (SVD) of the data matrix defined in (2-12), which is the average EEG data matrix for all experimental conditions and all participants. Suppose we have the SVD of X as follows: TUV X (2-13) where, UV are orthogonal matrices of dimension DD and TT and contain the leftsingular and right-singular vectors, respectively. contains the singular values of the data matrix. Equation (2-13) can be equivalently written as: TTUV X (2-14) The Varimax rotation procedure simply adds another DD orthogonal matrix R multiplied on the both sides of (2-14): TT R URV X (2-15) The right side of (2-15) is a DT matrix, whose rows can be seen as the factors extracted by the PCA-Varimax approach. We define a DD matrix:

PAGE 31

311 T T T D R U w W w (2-16) We further denote: 1 T T T D R V y Y y (2-17) Thus, we have: YWX (2-18) or: ,TT ii y wX for 1,, iD (2-19) This is the familiar form for the spatial filter defined in (2-9), which is now written in matrix form. Clearly, the matrix W is an orthogonal matrix. So PCA finds a number of (D) outputs that are uncorrelate d with the constraint that the sp atial filters are orthogonal. On the other hand, the PCA-Varimax method searches for outputs that are maximally compact in time while still constraining the spat ial filters to be orthogonal. In the context of the generative EEG model, PCA basically assumes that there are equal number of sources and channels. If we multiply 1 W on both sides of (2-18), we get: 1 XWY (2-20) Thus, the rows of the output matrix Y contain the time course of the current sources, while the columns of the matrix 1 W constitute the scalp projections of the corresponding sources. This means that the scalp projections for th e underlying current source s are orthogonal to each other, which is a high ly dubious assumption.

PAGE 32

32 Due to the above problem, an oblique rota tion like Promax (Hendrickson and White 1964) has been proposed as a post-processing stage after Varimax, to relax the orthogonality constraint on the scalp projections. St udies with both simulated and real dataset have shown that temporal PCA with Promax extracted markedly more accurate ERP components (Dien 1998). An alternative approach to the popular temporal PCA is the spatial PCA (Duffy et al. 1990, Donchin 1997, Spencer, et al. 1999), which treats the recorded potential at a given channel as variables. The EEG data matrix is formed with channel as one dimension, and time by experimental condition by participant as the ot her dimension. The same rotation procedures follow as in temporal PCA. However, spatia l PCA still assumes orthogonality of the scalp projections. Two other well-documented problems for th e PCA approach are the misallocation of variance (Wood and McCarthy 1984) a nd the issue of latency jitter. Dien (1998) using extensive simulations, has argued that spa tial PCA as a complement to temporal PCA, together with parallel analysis (Horn, 1965) to identify noise factors, and oblique rotation to allow for correlated factors, can address thes e and other shortcomings of PCA. More recent developments include a combined spatial and temporal PC A approach that is successfully applied to real ERP data extracting known ERP components (Spencer et al. 1999, 2001). Dien et al. (2005) have presented a st andard protocol to optimi ze the performance of PCA when it is applied to ERP datasets, recommendi ng the use of covariance matrix over correlation matrix, and Promax rotation over Varimax rotation, etc. 2.3.2 Independent Component Analysis (ICA) ICA was originally proposed to solve th e blind source separation (BSS) problem, to recover a number of source signa ls after they are linearly mi xed and observed in a number of sensors, while assuming as little as possible a bout the mixing process and the individual sources

PAGE 33

33 (Comon 1994). The most basic ICA model assu mes linear and instantaneous mixing, which means that the source signals arrive at the se nsors without time delay and are mixed in the sensors linearly with other source signals. This basic ICA model naturally fits into the generative EEG model in (2-5), which we repeat here, a ssuming that the sources and measurements are random vectors: x=As (2-21) In the ICA literature, the observation x is called mixtures, and the unknown matrix A is called mixing matrix. Usually it is assumed th at the number of sources is equal to the number of sensors. In this case, the mixing matrix becomes a square matrix. The key assumption used in ICA to solve the B SS problem is that the time courses of the sources are as statistically independent as possibl e. Statistical independence means that the joint probability density function (pdf) of the outputs factorizes. For the linear instantaneous BSS problem, the solution is in the form of a square demixing matrix W, specifying spatial filters that linearly invert the mixing pr ocess. If the mixing matrix is invertible, the outputs should be identical to the original source signals, excep t for scaling and permutation indeterminacies (Comon 1994): y Wx (2-22) There are a multitude of algorithms that ha ve been proposed to solve the basic ICA problem, among which, Infomax (Bell and Sejnowski 1995, Lee et al. 1999), FastICA (Hyvarinen and Oja, 1997), JADE (C ardoso, 1999), SOBI (Belouchrani et al. 1997) are probably the most widely used. Some algorithms are base d on the canonical information-theoretic contrast function for ICA, i.e., mutual information, or it s approximations (Infomax, FastICA, etc.); others utilize higher-order statistics of the data (e.g., fo rth-order cumulant) to perform source separation

PAGE 34

34 (JADE); still others make use of the difference in the temporal spectra of the source signals (SOBI). For a more detailed review on ICA and it s applications to BSS problems, we refer to the following: Cardoso (1998); Hyvarinen et al. (2001); Roberts and Ever son (2001); Cichocki and Amari (2002); James and Hesse (2004); Choi et al. (2005). Review papers comparing different ICA algorithms and their relationships are also available: Hy varinen (1999); Lee et al. (2000). We notice that while PCA requires the spatial fi lters to be orthogonal, here in the case of ICA, there is no more constraint on the spatial filters (or the demixing matrix W). On the other hand, while PCA only uses secondorder statistics (the covari ance matrix), to decorrelate outputs, ICA imposes a much stronger condition, i.e., independence on the outputs. The fact that ICA tries to factorize the joint pdf of the outputs implies that all the higher-order statistics (HOS) are taken into consideration by ICA. This mean s that for non-Gaussian data, the structures contained in HOS (e.g., kurtosis), while totally ignored by PCA, may be captured by ICA. Since many natural signals are non-Gau ssian distributed (e.g., spe ech signals usually follow a Laplacian distribution), ICA may be more suitable for this and other applications than PCA. Since Makeig et al. (1996) published their seminal pape r on the application of ICA to EEG data, there have been numerous studies during th e last decade dedicated to this research topic (Makeig et al. 1997, 1999, 2002, 2004; Vigario et al. 1998, 2000; Jung et al. 1999, 2000, 2001; Delorme et al. 2002, 2003, 2007; Debener et al. 2005). Until now, ICA and its variants still remains a powerful tool for the analysis of EEG and ERP data. The application of ICA to th e study of EEG data requires th at the following conditions be satisfied: (1), statistical independence of all the underlying neural source si gnals; (2), their linear instantaneous mixing at the sensors; (3), the stationarity of the mixing process.

PAGE 35

35 Since most of the energy in EEG data lies below 100Hz, the quasistatic approximation of Maxwell equations holds. So there is (virtually) no propagation delay of the electrical potentials from the neuronal sources to the sensors thr ough volume conduction. Thus the assumption of instantaneous mixing is valid. The linearity of the mixing follows from the Maxwell equations as well. The stationarity of the mixing process co rresponds to a constant mixing matrix. For the dipole source model, this means that the dipola r neuronal sources should have fixed locations and orientations. Although there is no reason to believe that thes e neuronal sources are spatially fixed over time, for those that are involved in a specific in formation processing task and therefore are of interest to ERP researchers, they should at least ha ve a relatively stable configuration or a stable scalp projection, wh ich is congruent with the definition of ERP components as proposed by Fabiani et al. (1987). We have seen that conditions (2) and (3) ar e approximately valid for EEG data. The most debatable and perhaps perplexing condition is the first one: statistical independence of all the underlying neural source signals. Th e independence criterion applies solely to the amplitudes of the source signals, and does not correspond to any consideration of the morphology or physiology of the neural structures However, the different nature of the sources originated from completely different mechanisms often yields signals that appear to be statistically independent. Particularly, analysis of the dist ributions of artifacts such as th e cardiac cycle, ocular activity has shown the statistical inde pendence assumption approximately holds (Vigario 2000). Although ICA continues to be a useful tool fo r EEG and ERP analysis, there are also some limitations to it. First, ICA can decompose up to (or at most) D sources from data recorded at D scalp electrodes (D may be ranged from several dozen to a few hundred). On one hand, the researcher has to analyze the extracted D components one by one (including the time course and

PAGE 36

36 scalp projection), which is laborious when is D large and the results are subject to interpretation. If he/she chooses to analyze onl y a part of the all the component s, the subsequent analysis is correlated with the retention criterion. (Note that PCA also has this problem). On the other hand, the effective number of statistically independent signals contributing to scalp EEG is almost certainly much larger than the number of electrodes D. Using simulated EEG data, Makeig et al. (2000) has found that given a large number of sources with a limited number of available channels, ICA algorithm can accurately identify a few relatively large sources but fails to reliably extract smaller and briefly active sources. This suggests that ICA decomposition in high dimensional space is an ill-posed problem. Second, the assumption of sta tistical independence used by ICA is violated when the training dataset is too small or separate topographi cally distinguishable phenomena nearly always co-occur in the data (Li and Principe, 2006) In the latter case, simulations show that ICA may derive a single component accounting for the co-occurring phenomena, along with additional components accounting for their separate activities (Makeig et al. 2000). These limitations imply that the results obtaine d with ICA must be va lidated by researchers using behavioral and/or physiological eviden ce before their functional significance can be correctly interpreted. Current research on app lications of ICA is focused on incorporating domain-specific knowledge into the ICA fram ework. Recently there has been work on combining ICA with the Bayesian approach (Tsai, et al. 2006) or with th e regularization technique (Hesse and James, 2006). 2.3.3 Spatiotemporal Filtering Method s for the Classification Problem It is worthwhile to point out a related but different approa ch, which is the (supervised) single-trial EEG classification pr oblem. It is generally less difficult than the (unsupervised) single-trial estimation problem in the sense that the availability of label information for

PAGE 37

37 classification facilitates learning. Many spatiote mporal filtering methods have been proposed for the single-trial EEG classification problem, which include, but not limited to, common spatial patterns (CSP) (Ramoser et al., 2000), common sp atio-spectral patterns (CSSP) (Lemm et al., 2005), linear discriminant analysis (Parra et al., 2002), bilinear di scriminant component analysis (Dyrholm et al., 2007). The common spatial patterns met hod was initially proposed by Koles et al. (1990) to classify normal versus abnormal EEG (Koles et al. 1994). The method has been used for single-trial EEG classification in brain-computer in terface (BCI) systems (Mller-Gerking et al. 1999; Ramoser et al. 2000). Given the single-trial EEG data for two diffe rent experimental conditions, the CSP method decomposes the EEG data into spatial patterns which maximize the difference between the two conditions. The spatial filters are designed such that the variances of the outputs are optimal (in the least-square sense) for the discriminati on of the two conditions. This is realized by simultaneously diagonalizing the two covariance matrices of th e EEG associated with the two experimental conditions. The two resulting diagonal matrices (containing the eigenvalues for the two covariance matrices) add up to the identity ma trix. Thus, the spatial filters that give the (n, an integer) largest variance in their outputs (ass ociated with the largest eigenvalues) for one condition, will accordingly give the (n) smallest variance in their ou tputs for the other condition; and vice versa. It is along these directions that the la rgest differences between the two conditions lie. In (Mller-Gerking et al. 1999), the CSPs are called th e source distribution matrix (equivalent to the mixing matrix in ICA), and the spatially filtered outputs are claimed to be the source signals, although the EEG data were tempor ally band-pass filtered between 8-30Hz prior to analysis. Para et al. (2005) showed that the simultaneous diagonalization of the covariance

PAGE 38

38 matrices is equivalent to the generalized eige nvalue decomposition, and according to Parra and Sajda (2003), the CSP method is in fact es timating the independent components of the temporally filtered EEG data. The original CSP method does not take into account the temporal information of the filtered EEG data. In light of this, Lemm et al. (2005) proposed an algor ithm called common spatiospectral pattern (CSSP), which utilized the me thod of delay embedding and extended the CSP algorithm to the state space (with only one tap-delay). Dornhege et al. (2006) futher improved the CSSP algorithm by optimizing an arbitrary fin ite-impulse response (F IR) filter within the CSP framework. The overfitting of the spectral f ilter is controlled by a regularizing sparsity constraint. The CSP method and its variants all use the relevant oscill atory brain activity for EEG classification. Sometimes it is more appropriate to use coherent evoked potentials (of low-pass nature) instead. Para et al. (2002) proposed a spatiotemporal filtering method using conventional linear discrimination to compute the optimal spatial filters for single-trial detection in EEG. Specifically, the search for the optimal spatial filter given the single-trial EEG data as in (2-7), is based on constraining the output () yt to be maximally discrimi nating between two different experimental conditions. The optimality criterion is restricted to a pre-specified time interval, i.e., the time corresponding to a number of samples prior to an explicit button push. After finding the optimal spatial filter using c onventional logistic regr ession, the output is av eraged within that period of time to obtain a more robust feature. Th e detection performance is then evaluated using receiver operating characteristic (ROC ) analysis on a single-trial basis. Unlike other conventional methods such as ICA, where the scalp projections are given directly by inverting the demixing ma trix containing all the spatial filters, here since there is only

PAGE 39

39 one spatial filter and one output, other techniques ha ve to be sought in order to estimate the scalp projection associated with that output. Parra et al. did this by projecting the EEG data to the discriminating output () y t assuming that the output is uncorre lated with all other brain sources, and found that the discrimination model captured information direc tly related to the underlying cortical activity. The method was improved in (Luo and Sajda, 2006), where the pre-specified time interval is allowed to be different and opt imized for each EEG channel. This effectively defines a discrimination traject ory in the EEG sensor space.

PAGE 40

40 CHAPTER 3 NEW SPATIOTEMPORAL FILTERI NG METHODOLOGY: BASICS In this chapter, we propose a new spatio temporal filtering method for single-trial ERP estimation. The method relies on modeling of the ER P component descriptors and thus is tailored to extract small signals in EEG. The model allows for both amplitude and latency variability in the actual ERP component. We constrain through a spatial filter w the extracted ERP component to have minimal distance (with respect to some metric) in the temporal domain from a presumed ERP component. Note that we do not constrain the entire ERPs, but instead a single ERP component. We maintain the point in the next section that the spatial filter may be interpreted as a noise canceller in the spatial domain. We then introduce two approaches for the proposed method: the deterministic appr oach and the stochastic approach. 3.1 Spatial Filter as a Noise Ca nceller in the Spatial Domain Since the method deals with one ERP component at a time, we wish to distinguish between signal and noise instead of using the general term sources. To accommodate this distinction, we rewrite the generative EEG model in (2-6) as follows: 1N TT ii iX=as+bn (3-1) where, s is the time course of the ER P component to be extracted, in denotes noise in general. The distinction between signal a nd noise is somewhat arbitrary, e.g., when P300 is the signal of interest, N100 will become noise in the model. Note that for notational convenience, we have assumed the effective number of sources to be 1N The EEG model in (3-1) can in turn be rewritten as: 1N TT s ooioioi i X=as+bn (3-2)

PAGE 41

41 where, oa, os, oib, oin are the normalized versions (with respective to some norm, e.g., 2l) of their counterparts in (3-1) and 1ii The scalars s i may be seen respectively as the overall contributions of the signal and noise to the single-trial EEG data. In the case of independent noise, we may define the SNR for the si ngle-trial EEG data (note that it is different from SNR in a single channel.) as: 2 120logN si iSNR (3-3) The vectors oa, 'ib, represent the scalp topography of the corresponding signal and noise. For a meaningful ERP component, it must have a stable scalp topography oa. Thus, we may assume that oa is fixed for all trials. We also assume that the waveform of a particular ERP component os (dimensionless) remains the same for all trials, although its amplitude s may change across trials. Next, we claim the following lemma, which is ba sically a direct consequence of the linear generative EEG model in (3-1). Lemma: There exists a spatial filter w, that will completely reject the interference from the first 1D largest noise in the output when it is applied to the single-trial EEG data, if, 11det0D abb (3-4) Further, the extracted ERP component will approach th e actual ERP component if, s D (3-5) Such a spatial filter wcan be found by taking the first row of the inverse of th e matrix in (34). Note that (3-4) implies that, (,)0,1,...,1iangleiDab and, (,)0,11ijangleijD bb (3-6)

PAGE 42

42 which means that at leas t the scalp topography of th e source and the first 1 D noise should not be the same or very similar to each other from a computational perspective. Here, we wish to stress the point that the spa tial filter specified in the above lemma may be interpreted as a noise canceller in the spatial dom ain. It may or may not be the optimal spatial filter for enhancing the SNR in the extracted co mponent. In addition, the SNR enhancement due to the spatial filter increases m onotonically with the number of ch annels (electrodes) if the EEG data were measured in those channels. This mean s that the more channels we use to record the EEG, the higher SNR we will get (theoretically) in the extracted component. Note that the lemma is an existence statement, it does not tell us how to find such a optimal spatial filter. This will be the subject of the next two sections. 3.2 Deterministic Approach Most ERP components are monophasic waveform s with compact support in time. The morphology of the waveform can be consider ed relatively fixed due to the common cytoarchitecture of the neocorte x and similar neuron populations, but may vary in both its peak latency and amplitude from trial to trial. Based on this, we assume that a particular ERP component can be modeled by a fixed dimensionless template (e.g. no physical unit), in the temporal domain, denoted by ()ols(where l is the unknown peak late ncy), multiplied by an unknown and possibly variable amplitude s across trials. We attempt to estimate the variable peak latency and amplitude on a single-trial basis. 3.2.1 Finding Peak Latency Since we do not know the peak latency in a single trial, we denote the template as ()o s with a variable time lag parameter and slide it one lag a time to search for the peak latency. The search for the optimal filter w could be realized by mini mizing some distance measure

PAGE 43

43 between the spatially filtered output T wX and the assumed waveform ()o s for the particular ERP component. We propose the following cost f unction based on second-ord er statistics (SOS): 2 2min()TT owwXs (3-7) Note that the above optimi zation is with respect to w only, with fixed. The optimal solution for w is given by: 1()()()T o wXXXs (3-8) Obviously, the optimal spatial filter w depends on which ERP component is to be extracted, and also is a function of the variable time lag From (3-7) and (3-8), we obtain the cost solely as a function of the time lag : 2 1 2()()TTT oJ sXXXXI (3-9) The peak latency of the ERP component can be set as the time lag where the local minimum of () J occurs within the meaningful range of peak latencies (S ) for that particular component (provided that its wavefo rm is monophasic) i.e., argmin()SlJ (3-10) The estimated ERP component is then (this need not be normalized): ()()T sll y Xw (3-11) It can be shown that under cert ain conditions, the solution in (310) is identical to the true peak latency of the ERP component (Appendix A) Exact match between the modeled and actual ERP component is not a necessary condition for the solution in (3-10) to be correct. For instance, it is easy to show that when bot h components are symmetric wavefo rms, then (3-10) also gives the correct latency.

PAGE 44

443.2.2. Finding Scalp Topogr aphy and Peak Amplitude In the following, we make the index for trial number k explicit. Denote the estimated ERP component for k-th trial by (the peak latenc y depends on the trial number): ()()kkkskkll yy (3-12) We can absorb the scalar k into a variable scalp topography: kko aa (3-13) In order to estimate the unknown scalp topograp hy and amplitude of the ERP component, we assume that the ERP component is uncorrela ted with all the noise sources. Replacing the dimensionless ERP component in (3-2) by it s estimate in (3-11) and multiplying () s kkly on both sides of (3-2), we will get an estimate for th e single-trial scalp t opography (the cross terms ()T oiskkl ny vanish because of the uncorrelatedness assumption): () ()()kskk k T s kkskkl ll Xy a yy (3-14) Taking the normalized version (note that ka is in Volts ) we have, k ok T kk a a aa (3-15) Ideally, the normalized oka should be the same as the dimensionless scalp topography oa However, in low-SNR EEG data, the above esti mation in (3-14) is ve ry poor, due to the interference from background activit y in the finite-sample data. To estimate the scalp topography for a stable ERP component, we pr opose the following cost function: 2 2 1 minoK ook k aaa (3-16)

PAGE 45

45 This corresponds to a maximum likelihood (ML) estimator for oa under the assumption that each entry of the normalized single-trial scal p topography is an independent identically distributed (i.i.d.) Gaussian random variable. The optimal so lution for (3-16) is a simple average of the estimated single-trial scalp topographies for all K trials. Taking the normalized version, the following estimate for oa is obtained: 11 211 KK ookok kkKKaaa (3-17) Notice that (3-17) is in fact a weighted averag e of the estimates in (3-14). We also point out that (3-17) is different from summing up directly (3-14) for all trials since the peak latency parameter is involved and it ch anges from trial to trial. In the ideal case, the two vectors oa and ka are identical except for a scaling factor, which is exactly the unknown amplitude k associated with the ERP component in the k-th trial. Replacing their respective estimates in (3-14), we can find k using again a SOS criterion: 2 2 minkkkoaa (3-18) Simple calculation leads to the following estimate for the amplitude: T kok aa (3-19) This estimate involves information from all the available channels. In order to eliminate the indeterminacies of the linear generative EEG mo del, we set the peak amplitude as the maximum of the estimated ERP component in (3-12), i.e., ()kskkl y (All the amplitudes in the rest of the paper refer to this quantity). Accordingly, th e contribution of the ER P component to the EEG data may be computed by: () s soskklXay (3-20)

PAGE 46

46 These estimates for the scalp topography, peak latencies and amplitudes of ERP component can be used to analyze its psychol ogical significance on a single-trial basis. Note that we do not directly compute the amplitude from the estimated component, nor do we measure it in any single channel. Instead, th e amplitude is computed in (3-19) indirectly through an inner product of two scalp topograp hies, which involves information from all available channels. These estimates for the scal p topography, peak latencies and amplitudes of ERP component can then be used to analyze their psychological significance on a single-trial basis. We wish to point out that in contrast to all the spatiotem poral methods mentioned before, where only one, representative spatia l filter (or matrix consisted of spatial filters), is computed given all the EEG data, here the optimal spatial f ilter is computed on a single-trial basis, i.e., given a single-trial EEG data matrix, we can get a spatial filter, as in (3-8). The reason we did it in this way is that, we believe th at in theory the optimal spatial f ilter that is designed to extract small signals should change from trial to trial. Note in (3-2) the noise sources are sorted in decreasing order of their power. It is likely that the noisy sources that have relatively large power change drastically across trials. In effect, th is will change the conf iguration in (3-2) and accordingly, the optimal spatial filter will also change. 3.3 Stochastic Approach In the deterministic approach, the ERP componen t is considered to be a deterministic signal except for a random latency and amplitude across tria ls. It does not take into account the intrinsic error in the modeling of the ERP component itself. Here we propose a stochastic approach for the spatiotemporal filtering method. The idea is to constrain the extracted ERP component to be close to the presumed component with respect to some statistic.

PAGE 47

47 We still use the generative EEG model as in (3-1 ), but here both the signal and the noise are interpreted as stochastic processes. The approach utilizes the following observation model to search for the optimal spatial filter: T Xws+v (3-21) where, s is the actual ERP component and v is the observation noi se appearing in the spatially filtered EEG data. They are both random vectors with each entry as a sample within a certain time interval from the corresponding stochastic processes. We assume that the ERP component is ge nerated from the following additive model: s= g +u (3-22) where, g is a fixed signal with a certain morphol ogy serving as the template for the ERP component, and u represents the model uncertainties of the ERP component and is assumed to be independent of th e observation noise v. Given the above model in (3-21) and (3-22) the optimal spatial filter may be found by maximizing the log-likelihood of the filtered EEG data T y Xw : max()log(|)Lp w yg (3-23) or, max()log()Lp wu+v (3-24) The log-likelihood function has a simple form under the assumption that u and v are zeromean Gaussian stochastic processes with covariance matrices M DCC respectively, and they are independent of each other, since u+v is nothing but another Gaussi an stochastic process with covariance matrix M D CC In this case, maximizing the log likelihood of the (transformed) observed data given the observati on model and the template of the ERP component yields the following solution for the optimal spatial filter:

PAGE 48

48 1argminT TT oMD wwXw g CCXw g (3-25) This is a quadratic form of w, so it has a closed-from solution: 1 11 T o wX XX g (3-26) where, M D CC (3-27) The estimate for the ERP component is then: T oo y Xw (3-28) We point out that (3-25) suggests that in the Gaussi an assumption, observational uncertainties and model uncertainties simply co mbine by addition of thei r respective covariance matrices. We also note that if M C and D C are both chosen as identity matrix (i.e., incorporating the least amount of a priori information into the model), then the solution in (3-26) essentially reduces to (3-8) in the deterministic approach. Th e optimality of the spatial filter will depend on how we choose the two covariance matrices. The estimation of the scalp topography and amplitude follow the same procedures as described in (3-15) (3-20). 3.4 Simulation Study We present in this section a simulation st udy with synthetic data and real EEG data recorded from subjects during a passive pictureviewing experiment. The goal of the simulation study is to evaluate the latency and amplitude precision of synt hetically generated transients immersed in real EEG backgr ound with different SNRs and wa veform mismatching conditions.

PAGE 49

49 3.4.1. Gamma Function as a Template for ERP Component Lange et al. (1996) have used a Gaussian function as the template for an ERP component. Here, we prefer the Gamma function for the shap e of the synthetic ERP component because this is a very flexible function for waveform modeling and has been used extensively in neurophysiological modeling (Koch et al., 1983 ; Patterson et al., 1992). Freeman(1975) argued that the macroscopic EEG electri cal field is created from spik e trains by a nonlinear generator with a second-order linear com ponent with real poles. Accordi ng to this model, the impulse response of the system is a monophasic waveform with a single mode, where the rising time depends on the relative magnitude of the tw o real poles (Appendix B). This may be approximated by a Gamma function, which is expressed by: 1()exp(/), 0kgtcttt (3-29) where, 0 k is a shape parameter, 0 is a scale parameter and c is a normalizing constant. The Gamma function is a monophasic waveform with the mode at (1),(1)tkk It has a short rise time and a longer tail for small k, and approximates a symmetric waveform for large k. This makes it a good candidate for modeling both early and late ERP components that tend to be symmetric in the ear ly components and have longer ta ils in the late components. Figure 3-1 shows four Gamma functions with different shape and scale parameters. 3.4.2. Generation of Simulated ERP Data EEG data were recorded from subjects dur ing a passive pictureviewing experiment, consisting of 12 alternating phases: the habitu ation phase and mixed phase. Each phase has 30 trials. During the 30 trials of the habituation pha se, the same picture was repeatedly presented 30 times. During the mixed phase, the 30 pictures are all different. Each trial lasts 1600 ms, and there is 600 ms pre-stimulus and 1000 ms post-stimulus.

PAGE 50

50 The scalp electrodes were placed according to the 128-channel Geodesic Sensor Nets standards. All 128 channels were referred to channel Cz and were digitally sampled for analysis at 250Hz. A bandpass filter between 0.01Hz and 40H z was applied to all channels, which were then converted to average reference. Ocular ar tifacts (eye movement) were corrected with EOG recordings. The scalp topography of the synthetic ERP comp onent is chosen as the normalized P300 scalp topography from another st udy (Li, et al., 2006). The si mulation data were created by taking the superposition of the 600ms (150 samples) prestimulus data from 120 trials as the background EEG data and the scalp projecte d Gamma waveform as a proxy for the ERP component. The SNR levels given the background EEG data can be easily adjusted by modifying the normalizing constant c in (17). We define the SNR gi ven the single-trial EEG data as: 20log ()/s TSNR TrT XX (3-30) Note that since the actual ERP component is a nonstati onary signal, the magnitude s is taken be its peak amplitude. This is different from the conventional definition of SNR. The SNR levels given the background EEG data can be easily adjusted by modifying the normalizing constant c in (3-29). 3.4.3 Case Study I: Comparison with Other Methods We will test the performance of the proposed spatiotemporal filtering method at varying SNR levels (from -20dB to 12dB), where we have access to the actual (synthetic) ERP component. Two scenarios will be investigated: on e where there is an exact match between the synthetic and the ERP component template and th e other where there is a mismatch between the

PAGE 51

51 two components. For the case of exact match, we use the parameters 3, 13 k for both ERP components. For the mismatch case, the synthetic ERP component remains the same, but the template has a different waveform with parameters 5, 6k Figure 3-2(a) shows the waveforms of the two components for the mismat ch case. We fixed the peak latency of the synthetic component at 200ms for all the 120 trials. For Woodys filter, we selected channel Pz for analysis, and use the initial ensemble average as the template, avoiding the iter ative update on the templa te (since the true late ncy is fixed, this is the best-case scenario for Woodys filter). We search around the true la tency within 100ms for maximum correlation. We estimate the peak amplitude by taking the average of the peak value and its two adjacent values (corresponding to a noncausal low pass filter with cut-off frequency of 12Hz). For spatial PCA, we select the eigenv ector which has the maximal correlation with the P300 scalp topography (Note this is an ideal cas e for PCA, since in reality we do not know exactly the true scalp topography, nor the exact time course). The simulation results are summarized in Table 3-1, 3-2 and 3-3, which show the estimation mean and standard deviation for the estimated latency and the ratio between estimated and true peak amplitude, as well as the correlation coefficient between estimated and true scalp topography. Since PCA does not give an explicit estimate for latency and amplitude, we will omit the latency estimation and only com pute the amplitude ratio between the estimated ERP component and the synthetic ERP component. First, we note that the single-trial estimation of the peak latency is very stable in the case of exact match. Notably, for EEG data with SN R higher than 4dB, the method estimates the latency correctly for all the tria ls. Second, the amplitude estimate for the exact case is also stable but is more variable than the latency es timation. The mean amplitude approaches to one

PAGE 52

52 and the standard deviation decreases to zero as the SNR increases. We may say that in the case of exact match between the model and the component, the estimator for the amplitude is asymptotically unbiased and asymptotically consistent with increasing SNR. The mismatch between the model and the generated component effectively introduces a bias in the estimation of the latency for realistic SNRs. From Table 3-1, we can see that the mean latency approaches to 188ms, yielding a bias of around 3 samples and the standard deviation is around 9ms (around 2 sa mples). However, the mean does converge to its true value (200ms) and the variance does approach to zero with increasing SNR, although very slowly. For instance, for SNR as high as 40dB and 60dB, the es timated latency has a mean-std statistic of 193 4ms and 200 0.7ms, respectively. Therefore, empirically we can see that the estimation for latency under the mismatch case is also asymptotically unbiased and asymptotically consistent with increasing SNR. The mismatch of components introduced a bias in the estimation of the amplitude for realistic SNRs, which is partly due to the difference in the waveforms of the synthetic component and templa te. The estimated amplitude has a statistic of 1.0458 0.0075 and 1.0015 0.0002 at a SNR of 40dB and 60dB, respectively. Thus in the case of mismatch, the estimator for the amplitude is also asymptotically unbiased and asymptotically consistent with respect to SNR, although the convergence is much slower than the exact match case. Of course, the estimation variance does increase notably as SNR decreases. But as evident from Table 4-2, our method at -12dB still gives a estimation variance smaller than Woodys method at 0dB. Table 3-1 and 3-2 clearly show the advantage of using spatial information, in contrast to the Woody filter based on single-channel analysis. Specifically, the estimation variance of Woody filter for both latency and amplitude are much larger for realistic (negative) SNR conditions.

PAGE 53

53 PCA overestimates the amplitude of the ERP co mponent for low SNR data. In contrast to our method, PCA gives a statistica lly significant bias even at 0dB from the baseline at 12dB (pvalue less than 0.0001). This means that varyi ng SNR (below 0dB) imposes a serious problem on the application of PCA in low SNR conditions. Finally, the simulated mismatch of compone nts affects the estimation of the scalp topography negligibly for the proposed method wh en SNR is higher than -10dB. In fact, the estimation for scalp topography with mismatch w ith our method at -20 dB is comparable to PCA at -4 dB. 3.4.4 Case Study II: Effects of Mismatch The second simulation concerns the effects of the mismatch on the estimation, specifically mismatch in the spread parameter which is the most important. We use the same synthetic component as before and vary the spread parameter with K fixed. Fig. 3-2(b) shows the waveforms of the synthetic and 3 of the template s, including a Gaussian with a spread of 20. The results are summarized in Table 3-4 and 35, for SNR = -20dB and -10dB, respectively. We have included a new quantification for the amplit ude estimation: coefficient of variation (CV), which is defined as the ratio of the standard de viation to the mean of a random variable. It is used as a measure of dispersion of the estimated amplitude (since its true mean is fixed at 1). From Table 3-4 and 3-5, we can see that at the same SNR leve l, the mean and variance of the estimated amplitude systematically change with respect to the spread parameter, i.e., larger spread parameter gives smaller amplitude. The de gree of variability in the amplitude estimation (measured by coefficients of variation) for mismatch cases exceeds the exact match case by less than 10% except at -10dB for a spread parameter of 7. In some cases, CV is even smaller than the exact match case, which gives a better estimate, but only in terms of the amplitude. This is important because although the estimated amplitudes differ in the mismatch case, as long as we

PAGE 54

54 use the same template, these amplitudes on average will always be magnified or attenuated by a constant factor at a certain SNR level. Intuitiv ely, this means that given a fixed template and varying SNR (>-20dB), the dominant source of variability in amplitude estimation mainly comes from the estimation variance (not bias) and this variability (measured by CV) is wellbounded from above for a range of spread parameters. Therefore, these amplitude estimates may still be effectively compared across expe rimental conditions as long as the same (meaningful) template is used and the SNR of the ERP data does not fall below -20dB. However, the mismatch clearly introduces a bias in the latency estimation, which may be as large as 50ms in absolute terms. This may or may not be signif icant depending on the applications. We can also see from the two tabl es that choosing a highe r spread parameter will lead to a slightly bette r estimation for the scalp topography. But of course, this comes at the cost of much worse latency estimation.

PAGE 55

55 Table 3-1. Latency estimation: mean and standard deviation SNR (dB) Woody Exact match Mismatch -20 191 60 202 10 190 16 -16 190 60 201 7 189 14 -12 190 61 201 5 188 11 -8 191 63 200 3 188 10 -4 195 60 200 2 188 10 0 195 54 200 1 187 10 4 199 45 200 0 187 10 8 199 32 200 0 188 10 12 200 15 200 0 188 9 True latency: 200ms. Table 3-2. Amplitude estimation: mean and standard deviation SNR (dB) Woody PCA Exact match Mismatch -20 -0.22 9.32 10.7 2.12 1.68 2.20 2.22 2.93 -16 0.07 5.95 6.80 1.35 1.34 1.29 1.71 1.75 -12 0.40 3.84 4.33 0.86 1.18 0.77 1.49 1.05 -8 0.36 2.60 2.78 0.55 1.10 0.47 1.34 0.64 -4 0.53 1.75 1.86 0.36 1.06 0.29 1.26 0.40 0 0.64 1.14 1.35 0.23 1.04 0.18 1.23 0.25 4 0.80 0.74 1.13 0.13 1.02 0.12 1.20 0.16 8 0.90 0.45 1.05 0.08 1.01 0.07 1.19 0.10 12 0.98 0.24 1.02 0.05 1.01 0.05 1.18 0.07 True amplitude: 1 Table 3-3. Scalp topography esti mation: correlation coefficient SNR (dB) PCA Exact match Mismatch -20 0.568 0.829 0.759 -16 0.579 0.910 0.854 -12 0.601 0.959 0.926 -8 0.649 0.982 0.966 -4 0.759 0.993 0.986 0 0.906 0.997 0.994 4 0.979 0.999 0.998 8 0.996 1.000 0.999 12 1.000 1.000 1.000

PAGE 56

56 Table 3-4. Effects of mi smatch I: SNR = -20dB Spread parameter Latency(ms) Amplitude Coefficient of Variation CC. of scalp topography 7 150 20 2.32 3.02 1.30 0.68 9 175 16 2.18 2.74 1.25 0.73 11 191 12 1.93 2.51 1.30 0.79 13 202 10 1.68 2.20 1.31 0.83 15 215 11 1.43 1.87 1.31 0.87 17 228 15 1.24 1.60 1.29 0.88 19 246 18 1.07 1.41 1.31 0.87 20 (Gaussian) 210 22 2.14 2.89 1.35 0.76 Table 3-5. Effects of mi smatch II: SNR = -10dB Spread parameter Latency(ms) Amplitude Coefficient of Variation CC. of scalp topography 7 151 21 1.30 0.92 0.71 0.93 9 175 16 1.34 0.78 0.58 0.95 11 190 8 1.24 0.69 0.55 0.96 13 200 4 1.14 0.60 0.53 0.97 15 211 5 1.03 0.53 0.51 0.98 17 220 7 0.94 0.46 0.49 0.98 19 229 9 0.86 0.41 0.47 0.99 20 (Gaussian) 208 11 1.30 0.78 0.60 0.95

PAGE 57

57 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 timeamplitude K=3,theta=2 K=5,theta=1 K=2,theta=2 K=9,theta=1 Figure 3-1 Gamma functions with different shapes and scales.

PAGE 58

58 A 0 100 200 300 400 500 600 -0.2 0 0.2 0.4 0.6 0.8 1 time (ms)amplitude presumed ERP component synthetic ERP component B 0 100 200 300 400 500 600 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K=3, theta=13 K=3, theta=7 K=3, theta=19 Gaussian Figure 3-2 Waveforms of synthe tic and presumed ERP compone nt. A) Synthetic component Gamma: K= 3, 13; presumed component: Gamma: K= 5, 6. B) Synthetic component Gamma: K= 3, 13; presumed components: Gamma: K= 3, 7, and 19, Gaussian with a spread of 20.

PAGE 59

59 CHAPTER 4 ENHANCEMENTS TO THE BASIC METHOD In chapter 3, we developed the basic spatio temporal filtering method for single-trial ERP component estimation. In this chapter, we will co nsider some modifications to the basic method. Some serves as a heuristic post-processing techni que (Section 4.1: iterativ ely refined template); some aims to deal with large salient EEG artifacts (S ection 4.2: robust estimation); some utilizes the a priori knowledge on the scalp topography of the ERP component (Section 4.3: regularization); some provides alte rnative formulations for our prev ious results and also derives new ones (Section 4.4: Bayesian formulations of the topography es timation); still others try to deal with the interference from other overl apping ERP components (Section 4.5: explicit compensation for temporal overlap). The details are presented below. 4.1 Iteratively Refined Template The deterministic approach of our estimation in Chapter 3 uses a fixed waveform for the template, regardless of the SNR. The stochastic approach incorporates so me degree of variability in the template, but it is still im plicit. We would like to explicitly utilize the posterior information from the data to update or refine our a priori assumed template. Intuitively, this should improve our estimation at least for high SNR conditions. We use the estimated scalp topography oa as a spatial filter. The output is optimal in the sense that it has the largest correlation coe fficient with the actual component with the uncorrelated noise assumption. (Of course, we use the estimate as a proxy for the true topography. Note that it is different from w ). The refined template is the ensemble average of spatially filtered data, with oa as the filter. The results are show n in Table 4-1, 4-2 and 4-3 (with one iteration of refining).

PAGE 60

60 We can see that for the latency estimation, th e refined template method has a larger bias below -12dB than the original template, but impr oves quickly and approaches to the exact match case for positive SNR conditions. For the amplitude estimation, the refined template consistently beats the original template for all SNR conditions in terms of both bias and variance. It also approaches to the exact match case for positiv e SNR conditions. Also see Figure 4-1. For the scalp projection estimation, the refined template is worse below -8dB but is very close to the exact match case above -8dB. Figure 4-2 shows the waveforms of the synthetic component, original template and refined template at 4 SNR conditions: -20dB, -12dB, 0dB and 12dB. The spatially filtered ensemble average is still quite erratic below -10dB. That possibly accounts for the worse performance of refined template for very low SNR conditions. So, there exists a critical SNR below which, using the spatially filtered ensemble average will probably worsen performance (in terms of scalp topography, but not amplitude) compared with the orig inal template. For this particular data set, it would be safe to use a refined template as long as the SNR is above -10dB. With 1 iteration, there comes 2, 3 and infinity. The natural question then is: Will it converge? If so, what does it converge to? Theoretic ally, these are difficult questions. Aside from the variable latency parameter, the scalp topography is still nonlinearly related to the data. However, we can experimentally determine the li miting results. These are shown in Table 4-4 for 3 SNR conditions. The algorithm co nverges within 10 (sometimes 2) iterations. Compared with the first iteration, the estimation for latency and scalp projection barely ch anges, but there is a reduced bias and variance in terms of amplitude for negative SNR conditions. Of course, all these results are a function of the mismatch in the waveforms and the number of trials we use to compute the ensemble aver age (if we had 1000 trials instead of 120, it would

PAGE 61

61 be a different story). At this point, it seems that, what matters the most to deal with negative SNR is to accumulate more data, either in space or in time. 4.2 Regularization Sometimes, we have some a priori knowledge of the scalp topography of the ERP component. For instance, the P300 component us ually has a large posit ive projection around Pz area. In these cases, we should utilize that information and in corporate it into our model. 4.2.1 Constrained Optimization Assuming that the ERP component latency has been estimated, we attempt to minimize the following cost function with respect to the amplitude and scalp topography a (which is constrained to have a unit norm in l2): 2 2 0 2 ,argminT FaXasaa (4-1) S.t., 21 a where, is a regularization parameter and 0a is a normalized vector representing the a priori knowledge of the scalp topog raphy of the ERP component. F denotes the Frobenius norm of a matrix. The reason that we chose this norm will be evident later: the minimization of this norm gave the same solution as (3.14) which was derived with the uncorrelatedness assumption. In Appendix C we derive a fix-point update equation: 0 0 2T T aXs ss aXs a aXs (4-2) Particularly, when 0 (no regularization), the optimal solution becomes:

PAGE 62

62 2 2T Xs ss Xs a Xs (4-3) When the optimal solution is: 00T T aXs ss aa (4-4) When takes intermediate values, the scalp t opography is a weighted average of the two extreme case solutions. This is a real single-trial estimation scheme in that the amplitude is computed from one single-trial data matrix. For the case of no regul arization, it is differe nt from our original formulation, where the amplitude is the inner product between Xs and the normalized average scalp topographies from all the trials. The original formulati on makes the reasonable assumption that the ERP component has a fixed scalp topography and utilizes that information. We demonstrate the effectiveness of regulariza tion to deal with the interference from other overlapping (possibly unknown, if th ey are all known, we can explic itly compensate for thatSee Section 4.5) ERP components. Specifically, we w ill investigate the effect of regularization on the estimation of amplitude and scalp topogr aphy under well controlled conditions. We assume that there are 2 overlapping ERP components and their la tencies are fixed and known. Their waveforms are shown in Figure 4-3. They are both Gamma functions with the same parameters3,10 K with a peak interval of 80ms. The two ERP components have a correlation coefficient of 0.36. We use template s that are exactly matched with the synthetic components. These components are projected to spontaneous EEG data to generate simulated

PAGE 63

63 ERP data. Their scalp topogra phies are shown in the Figur e 4-4. We use that for 0a (exact a priori knowledge). We then find the optimal solution of for a given The fixed point update always converges within 2 steps. We summarize the results for 3 SNR condi tions (12dB, -12dB, 20dB), shown in Figure 4-5. For hi gh SNR (12dB) data, regularizat ion brings little difference. Since the overlapping ERP compon ents have the exact opposite s calp topography, it is expected that the estimated amplitude is smaller than 1 for high SNR data. It converges to around 0.42 and 0.41 for component 1 and 2 for large respectively. Note the huge bias in the estimated amplitude for low SNR (especially -20dB) without regularization. Bu t it converges to as small as 0.69 and 0.27 for component 1 and 2 for large respectively. This demonstrates the necessity of regularization for the constrained optim ization problem. A reas onable regularization parameter for all the SNR conditions is between 104 and 105, where an unbiased estimation for amplitude could be achieved. Also notice that th ere is a hump for the sta ndard deviation of the estimated amplitude. Interestingly, this is near the reflection point of the mean amplitude. In practice, this can give us some hint for finding a reasonable regul arization parameter. Note that for these choices of the correlation coefficient of scalp t opography is already very close to 1. This is an example when the overlapping compon ents have negative correlation on the scalp. What about positive correlated components? Figure 4-6 shows the results for the same components, except that now they have the same scalp topography. The estimated amplitude is expected to be larger than 1 for high SNR data. It converges to 1.6 for both components for large Note the huge bias for low SNR (especially -20dB). It converges to 1.8 and 1.7 for component 1 and 2 for large respectively. This also de monstrates the benefits of regularization for the constrained optimization pr oblem: using a sufficiently large regularization

PAGE 64

64 parameter in this case can reduce the bias of the amplitude estimation, while the variance are not affected very much. As expected, for large the correlation coefficient of scalp topography converges to 1. 4.2.2 Unconstrained Optimization Parallel with the above constrained optimizati on, we can also frame the problem into an unconstrained optimization one. 22 0 2 2argminT FaXasaaa (4-5) Note that the optimization variable a contains the amplitude parameter as well as the topography information. As show n in Appendix D, a fixed-pointed update can be obtained: 0 2 3 0 2 TT Xsaa a ssaaa (4-6) Particularly, when 0 (no regularization), the optimal solution is the same as our original solution in (3-14). When the optimal solution is not unique (any scaled version of 0a can be a solution). The pr oblem becomes ill-posed. We have obtained the re gularized solution for single-trial scalp topography. The same procedures for estimating the amplitude follow: ta ke the average of normaliz ed single-trial scalp topography as our estimate for the overall scalp t opography, then the amplitude for a particular trial is just the inner product between this vector and the corr esponding scalp topography. We test the performance of regularization unde r the same conditions as in the constrained optimization. We find the optimal solution for a given Fixed point update usually converges within 10 steps. We summarize the results for 3 SNR conditi ons (12dB, -12dB, -20dB) shown in Figure 4-7 and 4-8. As in constrained optimization, for high SNR (12dB) data, regulariz ation has little effect

PAGE 65

65 on the estimation. The amplitude converges to around the same valu es as before (0.4 for opposite topography and 1.6 for the same topography). The va riance is not affected much for all the SNR conditions, either. As expected, the scal p topography gets monot onically better as increases. The difference is that the estimated amplitude mean increases monot onically (except for a small interval) with increasing regularization parameter This translates to a larger bias (particularly for low SNR data) when the overlapping components have positively correlated topographies. The estimated amplitude becomes unstable for large There is no evidence at this point that regularization can benef it the estimation for general scalp topography configurations (both positive and negative t opography correlations). So the unconstrained optimization formulation need not be regulariz ed, at least for overlapping components with positively correlated topographies. This lends support to our original solution in (3.14), which is exactly the unregularized solution to th e unconstrained optimization problem here. We also point out that, unlik e the constrained optimization problem, here we utilize the reasonable assumption that an ERP component has a fixed scalp topography. So it does not suffer from the huge bias problem in the constrai ned optimization. For inst ance, at -20dB without regularization, the estimated mean amplitude is around 2.3 and 1.9 for the two components with positively correlated topographies, a modest increase from 1.6 at 12dB. 4.3 Robust Estimation: the CIM Metric We have seen in Section 4.2 that the estima tion for single-trial scalp topography in (3.14) can be found equivalently from the minimization the following criterion: 2argminkT kkk FaXay (4-7) where, F denotes the Frobenius norm.

PAGE 66

66 It is evident from (3-14) th at the estimate for single-tria l scalp topography bears a linear relationship with the EEG data. Because of the noisy nature of EEG (particularly large salient artifacts), this gives a noisy estimate for the single-trial scalp topogra phy (with large variance) and in turn translates into a noisy estimate for the single-trial amplitude in (3-18). We would like to derive a robust estimator in order to reduce the effects of large EEG artifacts. We can replace the Frobenius norm in (4-7) with other norms (e.g., l1 norm) or metrics. Here we will consider a special metric: co rrentropy induced metric (CIM) proposed by Liu et al (2007). First we introduce what is called corrent ropy. Given two scalar random variables X and Y, correntropy is defined as: (,)()hhVXYEkXY (4-8) where, ()hkXY is the Gaussian kernel (h is the kernel size), 2 21() ()exp 2 2hXY kXY h h (4-9) The correntropy function is a localized similar ity measure in the join t probability space, which is controlled by the bandwidth parameter h (also called kernel size in kernel methods). It induces a metric (CIM) in the sample space which behaves like the 2l norm when the sample point is close to the origin (relative to the kern el size); when the sample point gets further apart from the origin, the metric is similar to the 1l norm and eventually saturates and approaches to the 0l norm (Liu, 2007). As such, CIM practically incorporates the 2l norm as a special case (if h is chosen to be sufficiently large). Minimization of CIM is equivalent to the maximum correntropy criterion (MCC). It has been shown that MCC has a close relation to M-estimation (Huber, 1981) and since correntropy

PAGE 67

67 is inherently insensitive to outliers, MCC is esp ecially suitable for reject ing impulsive noise (Liu, 2007). Now, treating each entry of the matrix in (4-7) as a realization of a random variable, we can write our new cost function as: 2argminkT kkk CIMaXay (4-10) The nuisance parameter h (kernel size) should be tuned to the data (most notably to the standard deviation). Here, we use the Silvermans rule as a baseline to quantify different values of kernel sizes that we use in the simulation. It is given by 0.21.06hdN where N is the number of samples, and d the standard deviation of the data (Silverman, 1986). While minimization of the Forbenius norm has a closed-form solution, minimization of CIM does not. So we have to search for a local minimum, using the st andard gradient descent method. The convergence to a certain local minimum is guaranteed by adopting a stopping criterion that the change in the correlations of the estimate and the MSE so lution between the previous and current iteration is less than 610. Table 4-5 and 4-6 summarize the estimation re sults for 2 SNR conditions (0dB and -20dB) for the mismatch case. We also include the MSE solution for comp arison. The results are mixed: MCC gives a slightly higher variance than MSE fo r the estimation of amplitude, but the bias is marginally reduced; it also gives a more accurate estimate for the scalp topography. We also notice that at 0dB, the results of MCC barely change from MSE. Intuitively, when the SNR becomes sufficiently large, the optimal MCC so lution should converge to the MSE solution (both agree with the true values of the parameters). These differences in the results are by no m eans statistically significant. We venture 2 reasons why the MCC results do not change very much from MSE.

PAGE 68

68 First, the EEG data are already preprocessed and relatively clean. Large artifacts have already been removed. The resul ting distribution is not far from Gaussian. So MSE should give a solution already close to optimal. MCC has its edge when there is large noise, especially impulse noise. Strictly speaking, it is only optimal (in the sense of maximum likelihood) for one particular type of distribution, ju st as MSE is strictly optimal for Gaussian distribution. It is not clear that how these two criteria compare when the data distribution changes in a neighborhood of their optimal ones. In reality (when EEG da ta are usually preprocessed and artifacts are removed), there is no reason to believe that MCC will outperform MSE uniformly. Another less compelling reason concerns the optimization process associated with MCC. The initial condition of MCC is set to be the MSE solution (starting a random initial condition will seldom beat MSE). When the kernel size is small, the performance surface is highly irregular, so the optimization will never go fa r from MSE solution (it is stuck around the local minimum near the MSE solution). When kernel size is sufficiently large, it is easy to see that MCC approaches to MSE. Only intermediate values of kernel size will produce somewhat different results from MSE. This is seen in both SNR levels, though less evident for 0dB. There are two other cost functions in the estima tion of amplitude that use the MSE criterion, i.e., (3.14) and (3.16), which can also be replaced by MCC. Li et al (2007) have tested its performance and the improvement was shown to be marginal. In practice, one has to weigh the small improve ment in the performan ce of MCC against its high computational cost (and no guarant ee of convergence to global optima). 4.4 Bayesian Formulations of the Topography Estimation The amplitude estimation consists of 3 steps. First we estimate the single-trial scalp topography ka (either by the uncorrelated assumption or equivalently through the minimization

PAGE 69

69 of the Frobenius norm of kkk Xa y ). Then we compute the normalized scalp topography oa as the normalized version of a weighted aver age of the single-trial scalp topography ka (the weights being their respective l2 norm). The third step is the minimization of the l2 norm 2kko aa, which gives the optimal single-trial amplitude as the inner product between ka and oa We have seen that the first step gives a n ear optimal solution (as opposed to CIM) if the EEG data have been cleaned. We have also shown that in the third step, using other metrics (e.g., CIM) gives a marginally better results than using the l2 norm. Here we will investigate other alternatives to the estimation of the normalized scalp topography oa in the second step in (3.17). We maintain that after estima ting the single-trial s calp topography in the first step, we treat ka as known and given. Then we ask the question: what is the best es timate for the normalized scalp topography oa given ka for K trials. This is a divide-and -conquer approach and simplifies matters. Naturally the problem is best formulated in a Bayesian framework. Of course, the formulation will depend on the model we assume for the data. Next, we will present three different models and compare their performance. 4.4.1 Model 1: Additive Noise Model kkok aau (4-11) where, k is the single-trial amplitude, and ku is the error (model uncertainty). This model is consistent with our linear gene rative EEG model. We shall assume conditional independence between |,iio aa and |,jjo aa for any ij Of course, we also assume i and

PAGE 70

70 j are independent for any ij and they are all independen t of the normalized scalp topography oa. We wish to maximize a posteriori probability (MAP): 111 ,...argmax(,...|...)oKoKKp aaaa (4-12) It can be shown (Appendix E) that the MAP solution occurs when oa is the normalized eigenvector of the matrix A corresponding to the largest eigenvalue, where, 11KK T kkk kk AAaa (4-13) Now we have solved the MAP problem under mode l 1. But, there is a weakness: the model error has a constant covariance across all trials. This is a dubious assumption. Intuitively, when the data are noisy, the variance in ku will increase accordingly. We should somehow normalize the data in our m odel. This leads to model 2. 4.4.2 Model 2: Normalized Additive Noise Model k ok k a au (4-14) The left hand side is in fact a proxy for oka in our linear generative EEG model, except that it may not be normalized. ku is again zero-mean i.i.d. Gaussian noise, but now with unit norm. The model can also be written as: kkokkk aau Again, it can be shown (Appendix F) that the MAP solution occurs when oa is the normalized eigenvector of the matrix B corresponding to the largest eigenvalue, where, 11T KK kk k T kk kk aa BB aa (4-15)

PAGE 71

71 Note that the components kB is a normalized version of kA in that the trace of kB is always 1. We also note that, we can treat k only as a normalizing factor not necessarily as the single-trial amplitude. In this case, we separate the estimation of k and oa As before, we can compute the amplitude by: T kok aa 4.4.3 Model 3: Original Model Our last model is actually simpler. We do not include the unknown amplitude in this stage and only consider the estimation of oa The posteriori probability is simply 1argmax(|...)ooKpaaaa. Given the model: 2k ok k a au a Appendix G shows that the MAP solution actually coincides with our original solution in (3.17). Th e single-trial amplitude estimation is the same as before. 4.4.4 Comparison among the Three Models We compare the amplitude and scalp topography estimation for the three models. The results are summarized in Table 4-7 and 4-8. We can s ee that model 3 (original solution) consistently gave the best results among the three under di fferent SNR conditions. The topography estimation with model 1 is poor at negative SNR conditions. Model 2 is an obvious improvement. Note that its amplitude estimation has a huge bias and variance for low SNR data. This may be due to the improper prior we assign to k (it assign large probability to large values) in the derivation of the MAP estimator. We also incl uded the amplitude estimation with the traditiona l inner product. There is a significant improvement in both the bias and variance, especially in low SNR conditions.

PAGE 72

72 4.4.5 Online Estimation Sometimes we wish to know the si ngle-trial parameters after recording each trial. It is then necessary to obtain an on-line estimation method. We assume that the stimulus onset time is known to us. Again, the problem is best formulated in a Bayesian framewor k in order to utilize all the information in previous trials. In fact, it is trivial given the above analysis. Here, we adopt model 3. After the first trial, set: (1) 11 2o aaa, 11 2 a At each trial, we store the running average: 2 1K kkk kcaa When finishing recording trial K+1, we update the topography estimate: 11 (1) 2 11 2 2KKK K o KKK caa a caa, and the amplitude for the newly recorded trial: (1) 11TK KKo aa 4.5 Explicit Compensation for Temporal Overlap of Components In developing our basic method in Chapte r 3, we assumed that ERP components are uncorrelated with each other. In reality, this is seldom satisfied. Because ERP components have relatively stable waveforms and latency, when th ey overlap in time, there will generally be a nonzero correlation among them. Here we are ma inly concerned with the situation where ERP components overlap in time, but latency ji tter across trials is relatively small. Consider two ERP components overlapping in time (it is easy to generalize to multiple components). We assume that th e time courses of the components 1s and 2s are known from physiological knowledge. We also as sume that the latency is give n or can be estimated, e.g., from ensemble average, and it is relatively fi xed. For the purpose of amplitude estimation,

PAGE 73

73 latency jitter is considered here as a minor i ssue compared with the possibly heavy overlap of components. Note that, in the case of two overl apping (correlated) comp onents, we lose the ability to estimate the latency simply from the cost function in (3-9). If we assume that all other components are un correlated with these tw o components (or have negligible temporal overlap with them), then we can compensate for the correlation (due to temporal overlap) to get an unbi ased estimate for both components amplitudes. We write the linear generative EEG model in this case: 1122 1N TTT ii iX=as+asbn (4-16) We wish to estimate the scalp topographies for the two overlapping components. With the uncorrelated assumption, we can get two set of equations: 1112211 1122222 TT TT ass+assXs ass+assXs (4-17) Solve for 1a and 2a : 212 1 2 12 121 2 2 12DC DDC DC DDC XsXs a XsXs a (4-18) where, ,(1,2)T iiiDi ss and 1221 TTC ssss If 0C, we have the same solution as before: ,1,2i i T iii Xs a ss (4-19) The procedures for estimating the amplitude ar e the same as before. The above technique assumes that accurate estimates of the waveform s of the components are available, since the cross-correlation in (4-1 8) depends on the tails of the overl apping components. This restricts its

PAGE 74

74 applicability in practice. But if the research ers believe that the ER P components are heavily overlapped and are fairly certain of their wave forms, this technique should serve as a first attempt to reduce the bias in the estimation.

PAGE 75

75 Table 4-1. Latency estimation: mean and standard deviation SNR (dB) Exact match Mismatch Refined template -20 202 10 190 16 182 12 -16 201 7 189 14 183 13 -12 201 5 188 11 187 12 -8 200 3 188 10 191 12 -4 200 2 188 10 197 6 0 200 1 187 10 199 4 4 200 0 187 10 200 1 8 200 0 188 10 200 1 12 200 0 188 9 200 1 True latency: 200ms. Table 4-2. Amplitude estimation: mean and standard deviation SNR (dB) Exact match Mismatch Refined template -20 1.68 2.20 2.22 2.93 2.17 2.73 -16 1.34 1.29 1.71 1.75 1.61 1.76 -12 1.18 0.77 1.49 1.05 1.33 1.05 -8 1.10 0.47 1.34 0.64 1.18 0.59 -4 1.06 0.29 1.26 0.40 1.11 0.34 0 1.04 0.18 1.23 0.25 1.06 0.20 4 1.02 0.12 1.20 0.16 1.04 0.12 8 1.01 0.07 1.19 0.10 1.02 0.07 12 1.01 0.05 1.18 0.07 1.01 0.05 True amplitude: 1 Table 4-3. Scalp topography esti mation: correlation coefficient SNR (dB) Exact match Mismatch Refined template -20 0.829 0.759 0.644 -16 0.910 0.854 0.772 -12 0.959 0.926 0.891 -8 0.982 0.966 0.963 -4 0.993 0.986 0.987 0 0.997 0.994 0.995 4 0.999 0.998 0.998 8 1.000 0.999 0.999 12 1.000 1.000 1.000

PAGE 76

76 Table 4-4. Estimation results for the iteratively refined template method Latency estimation Amplitude estimation Scalp topography SNR (dB) Refined template Iterative Refined Refined template Iterative Refined Refined template Iterative Refined -20 182 12 183 12 2.17 2.73 1.83 2.36 0.644 0.644 -12 187 12 187 12 1.33 1.05 1.23 1.01 0.891 0.879 0 199 4 199 2 1.06 0.20 1.06 0.20 0.995 0.995 Table 4-5. Estimation with MCC for the mismatch case at SNR = 0dB Kernel size (multiples of h) Amplitude Correlation coefficient of Scalp projection 0.5 1.22 0.25 0.994 1 1.22 0.26 0.994 2 1.22 0.27 0.995 5 1.22 0.26 0.995 10 1.22 0.25 0.994 20 1.23 0.25 0.994 MSE 1.23 0.25 0.994 Table 4-6. Estimation with MCC for the mismatch case at SNR = -20dB Kernel size (multiples of h) Amplitude Correlation coefficient of Scalp projection 0.5 2.22 2.95 0.758 1 2.21 3.05 0.754 2 2.17 3.18 0.786 5 2.18 3.14 0.789 10 2.21 3.00 0.770 20 2.22 2.93 0.759 MSE 2.22 2.94 0.759

PAGE 77

77 Table 4-7. Amplitude estimation for three Bayesian models SNR (dB) Model 1 Model 2 Model 2 (2) Model 3 (original) -20 0.38 1.30 12.2 378 1.95 3.79 2.22 2.93 -12 0.86 0.85 9.96 60 1.42 1.34 1.49 1.05 0 1.20 0.26 1.66 0.26 1.23 0.25 1.23 0.25 12 1.18 0.07 1.21 0.07 1.18 0.07 1.18 0.07 Table 4-8. Scalp topography estima tion for three Bayesian models SNR (dB) Model 1 Model 2 Model 3 (original) -20 0.188 0.642 0.759 -12 0.558 0.841 0.926 0 0.971 0.994 0.994 12 0.998 1.000 1.000

PAGE 78

78 -20 -15 -10 -5 0 5 10 15 1 1.5 2 2.5 SNR (dB)mean exact match mismatch refined template -20 -15 -10 -5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 SNR (dB)standard deviation exact match mismatch refined template Figure 4-1 Mean and standard deviation of the estimated amplitude under different SNR conditions. The refined template method appr oaches to the exact match case for high SNR conditions.

PAGE 79

79 Figure 4-2 The waveforms of the synthetic compone nt, presumed template and refined template under 4 SNR conditions. A) -20dB. B) -12dB. C) 0dB. D) 12dB. The refined template appears erratic for low SNR and approach es to the synthetic component for high SNR. A B C D 0 50 100 150 -10 -5 0 5 10 sample synthetic presumed refined 0 50 100 150 -6 -4 -2 0 2 4 6 8 10 sample synthetic presumed refined 0 50 100 150 -2 0 2 4 6 8 10 sample synthetic presumed refined 0 50 100 150 -2 0 2 4 6 8 10 sample synthetic presumed refined

PAGE 80

80 0 50 100 150 0 1 2 3 4 5 6 7 8 9 10 Figure 4-3 Waveforms of two overlapped compone nts used in regulari zation. The correlation coefficient between the two waveforms is around 0.36.

PAGE 81

81 0 20 40 60 80 100 120 140 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 channel numbertopography Figure 4-4 Scalp topography of tw o overlapped ERP components used in regularization.

PAGE 82

82 A Figure 4-5 Amplitude and scalp topography es timation I with regularization (constrained optimization) under 3 SNR conditions. A) 12dB. B) -12dB. C) -20dB. The overlapping ERP components have the ex act opposite scalp topography. So the estimated amplitude is smaller than 1 fo r high SNR data. It converges to 0.42 and 0.41 for component 1 and 2 for large respectively. Notice the huge bias for low SNR (especially -20dB) without regularization. But it conve rges to as small as 0.69 and 0.27 for component 1 and 2 for large respectively. 10 2 10 3 10 4 10 5 106 107 0 0.1 0.2 0.3 0.4 0.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 10 6 107 0.996 5 0.997 0.997 5 0.998 0.998 5 0.999 0.999 5 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 0.1 0.2 0.3 0.4 0.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 10 6 107 0.998 6 0.998 8 0.999 0.999 2 0.999 4 0.999 6 0.999 8 1 REG PARAMETER: LAMDA correlation coefficien t

PAGE 83

83 B C Figure 4-5 Continued 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 2.5 3 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 2.5 3 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 2 4 6 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.4 0.5 0.6 0.7 0.8 0.9 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 1 2 3 4 5 6 7 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.4 0.5 0.6 0.7 0.8 0.9 1 REG PARAMETER: LAMDA correlation coefficien t

PAGE 84

84 A Figure 4-6 Amplitude and scalp topography es timation II with regularization (constrained optimization) under 3 SNR conditions. A) 12dB. B) -12dB. C) -20dB. The overlapping ERP components have the sa me scalp topography. So the estimated amplitude is larger than 1 for high SNR da ta. It converges to 1.6 for both components for large Notice the huge bias for low SNR (especially -20dB) without regularization. It converges to 1.8 an d 1.7 for component 1 and 2 for large respectively. 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 10 6 107 0.999 7 0.999 8 0.999 8 0.999 9 0.999 9 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 10 6 107 0.999 9 0.999 9 0.999 9 0.999 9 1 1 1 1 1 REG PARAMETER: LAMDA correlation coefficien t

PAGE 85

85 B C Figure 4-6 Continued 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.95 0.96 0.97 0.98 0.99 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.985 0.99 0.995 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 1 2 3 4 5 6 7 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 1 2 3 4 5 6 7 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 REG PARAMETER: LAMDA correlation coefficien t

PAGE 86

86 A Figure 4-7 Amplitude and scalp topography estimation I with regularization (unconstrained optimization) under 3 SNR conditions. A) 12dB. B) -12dB. C) -20dB. The overlapping ERP components have the ex act opposite scalp topography. So the estimated amplitude is smaller than 1 fo r high SNR data. The estimated amplitude mean generally increases with increasing while the variance is not affected much. It becomes unstable for large 10 2 10 3 10 4 105 106 0 0.1 0.2 0.3 0.4 0.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 106 0.996 5 0.997 0.997 5 0.998 0.998 5 0.999 0.999 5 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 0.1 0.2 0.3 0.4 0.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 106 0.998 6 0.998 8 0.999 0.999 2 0.999 4 0.999 6 0.999 8 1 REG PARAMETER: LAMDA correlation coefficien t

PAGE 87

87 B C Figure 4-7 Continued 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficient 10 2 10 3 10 4 105 106 0 2 4 6 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.4 0.5 0.6 0.7 0.8 0.9 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 2 4 6 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.4 0.5 0.6 0.7 0.8 0.9 1 REG PARAMETER: LAMDA correlation coefficien t

PAGE 88

88 A Figure 4-8 Amplitude and scalp topography estimation II with regularization (unconstrained optimization) under 3 SNR conditions. A) 12dB. B) -12dB. C) -20dB. The overlapping ERP components have the sa me scalp topography. So the estimated amplitude is larger than 1 for high SN R data. The estimated amplitude mean generally increases with increasing while the variance is not affected much. It becomes unstable for large 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 106 0.999 7 0.999 8 0.999 8 0.999 9 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 106 0.999 9 0.999 9 0.999 9 0.999 9 1 1 REG PARAMETER: LAMDA correlation coefficien t

PAGE 89

89 B C Figure 4-8 Continued 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.95 0.96 0.97 0.98 0.99 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.985 0.99 0.995 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 2 4 6 8 10 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 2 4 6 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 REG PARAMETER: LAMDA correlation coefficien t

PAGE 90

90 CHAPTER 5 APPLICATIONS TO COGNITIVE ERP DATA In this chapter, we apply the spatiotemporal filtering method proposed in Chapter 3 to the single-trial ERP estimation problem in two different experiments. The first application is an oddball target detection task with different pictures as stimuli, where the difficulty of the task or saliency of the stimuli leads to decreased P300 amplitude. The second one is the habituation study where the subjects were repeatedly presen ted identical pictures and the amplitude of certain ERP components is expected to d ecrease rapidly with respect to trials. 5.1 Oddball Target Detection 5.1.1 Materials and Methods Because we were interested in single-trial, si ngle-subject analyses of amplitude and latency, we selected 4 participants that met a minimum signal-to-noise ratios based on their averaged ERPs, from a pilot study (n=8) on implicit cont ent processing during feat ure selection. They were right-handed according to the Edinburgh Handedness Questionnaire and all had normal or corrected vision. Stimuli consisted of pictures from the Inte rnational Affective Pict ure System, depicting adventure scenes, emotionally neutral social intera ctions, erotica, attack scenes, and mutilations. Their color content was manipulated such that they contained only shades of green or shades of red, and for each, color brightness was systemati cally manipulated to yield one bright and one dim version (Figure 5-1). All pi ctures were presented for 200 ms on the center of a 21-inch monitor, situated 1.5 m in front of the subjects. From this vi ewing distance the checkerboards subtended 4.0 deg. x 4.0 deg. of visual angle. A fixation cross was always present, even when no picture was presented on the screen. Target stimu li (p = 0.25) were defined for each experimental block (see below) by a combination of color a nd brightness.. All pictur es were presented in

PAGE 91

91 randomized order, with an inte r-stimulus-interval varying rand omly between 1000 to 1500 ms in 4 blocks of 120 trials each. One block lasted 7 min. on average. At the beginning of each block subjects were instructed to atte nd either to the bright/dark green or red pictures and to press the space bar of the computer keyboard when they dete cted a target. The target color and brightness were designated in counter-bal anced order. Furthermore, the responding hand was changed half way through the experiment, and the sequen ce of hand usage was counterbalanced across subjects. Subjects were also instructed to avoi d blinks and eye-movements and to maintain gaze onto the central fixation cross. Practice trials were provided for each subject for each condition to make sure that every subject had fully understood the task. EEG was recorded continuously from 257 el ectrodes using an Electrical Geodesics (EGI) EEG system and digitized at a rate of 250 Hz, using Cz as a recording reference. Impedances were kept below 50 k as recommended for the Elec trical Geodesics high inputimpedance amplifiers. A subset of EGI net electrodes lo cated at the outer canthi as well as above and below the right eye was used to determine ho rizontal and vertical Electrooculogram (EOG). All channels were preprocesse d on-line by means of 0.1 Hz high-pass and 100 Hz low-pass filtering. Epochs of 1000 ms (280 ms pre-, 720 ms pos t-stimulus) were obtained for each picture from the continuously recorded EEG, relative to picture onset. The mean voltage of a 120-msec segment preceding startle probe ons et was subtracted as the baselin e. In a first step, data were low-pass filtered at a frequency of 40 Hz (24 dB / octave) and then submitted to the procedure proposed by (Junghfer et al., 2000), which uses statis tical parameters to exclude channels and trials that are contaminated with artifacts. This procedure resulted in rejection of trials that were contaminated with artifacts (including ocular artifacts). Artifacts were also evaluated by visual

PAGE 92

92 inspection and respective trials were rejected. R ecording artifacts were fi rst detected using the recording reference (i.e. Cz). Subsequently, gl obal artifacts were detected using the average reference and distinct sensors fr om particular trials were rem oved interactively, based on the distribution of their mean amplit ude, standard deviation and maximum slope. Data at eliminated electrodes were replaced with a st atistically weighted spherical sp line interpolation from the full channel set. The mean number of approximated ch annels across conditions and subjects was 20. With respect to the spatial arrangement of th e approximated sensors, it was ensured that the rejected sensors were not located within one region of the scalp, as this would make interpolation for this area invalid. Spherical spline interp olation was used throughout both for approximation of sensors and illustration of voltage maps (Junghfer et al., 1997). Single epochs with excessive eye-movements and blinks or more than 30 channels containing artifacts in the time interval of interest were discarded. The vali dity of this procedure was further tested by visually inspecting the ver tical and horizontal EOG as computed from a subset of the electrodes that were part of the electrode ne t. Subsequently, data were arithmetically transformed to the average refe rence, which was used for all analyses. After artifact correction an average of 69 % of the trials were retained in the analyses. The present analysis highlighted the most reliable signal avai lable in this feature-based target identification task, which is the P300 component in response to a target stimulus (defin ed by a combination of color and brightness, irrespective of picture cont ent). Thus, all subseque nt analyses focused on amplitude and latency estimates for single tr ials belonging to the target condition. 5.1.2 Estimation Results The present study illustrates th e application of the method for a single late potential component. In reality, we do not know a priori how many ERP com ponents there are in a single-

PAGE 93

93 trial recording, nor do we know exactly when they occur. However, we may be able to estimate these values from single-trial EEG data in the data analysis session. This is a good time to mention one technical requirement of our latency estimation. The single-trial latency is estimated from the cost function in (3.9), which involves the inversion of the matrix TXX In reality, this matrix is usually ill-conditioned for dense-array EEG data (it will certainly be rank-deficient if there are any bad channels which were linearly interpolated from other channels.). This poses a computational problem in practice. Thus the solution in (3.9) somehow has to be regularized. Here, we adopt a simple approach and ad d a regularization term I (0 ) to the matrix TXX before taking the matrix invers ion operation. The regularization parameter acted as a smoother to the cost function in (3.9). Generally, the solution is rather irregular without regula rization, leading to too many local minima and spurious candidates for single-trial latencies due to large noise. With increasing the cost function becomes smoother. This is clearly seen in Fig. 5-2, which show s the cost function in (3.9) for four different for a particular trial from subject 2. With a smoot h cost function, we can avoid the dilemma of choosing the right latency from too many candidates. Now we have to select an a ppropriate value (or a meaningful range) for the regularization parameter A good value for is one that achieves a balan ce between two extremes: too few and too many local minima. The idea is this: for a particular we group all the candidates for single-trial latencies (time lags corresponding to local minima) together and perform 1D density estimation on these candidates. We count the number of modes (peaks) from the estimated probability density function (pdf). If this number is close to the average number of candidates for each trial, then the regularization parameter is at least internally consistent. Otherwise, it will contradict with itself and should not be used.

PAGE 94

94 We illustrate our point using the results from one subject. Fig. 5-3 shows the estimated pdf of the candidates for single-trial latency from 200ms up to 600ms after stimulus onset when the regularization parameter equals 510 We used the Parzen windo wing pdf estimator (Parzen, 1962) with a Gaussian kernel size of 4.2. The ke rnel size was selected according to Silvermans rule (Silverman, 1986), which is given by 0.21.06 hN, where N is the number of samples, and is the standard deviation of the data. The nu mber of peaks depends on the kernel size, but we found that a kernel size between 0.5h and 2h will give the same number of peaks in the estimated pdf for this data. We can see that the pdf consists of 4 modes (peaks) after 200ms of the stimulus onset. There are 418 local minima and 102 trials in total, so the average number of local minima for each trial is about 4.1 (very clos e to the number of peaks in estimated pdf). This indicates that =510 gives an internally consistent estimate for latency. We can repeat the above procedures for a wi de range of regularization parameters and compute the ratio of the number of peaks in esti mated pdf to the average number of local minima for each trial. For instance, the ratio was computed as around 4.75, 1.03, 0.96, 0.74 for the 4 regularization parameters in Fi g. 5-2 respectively. Clearly, th e first and last regularization parameter should not be used since they generate self-contradictory results. It is interesting to note that for a wide range of regularization parameters (from 510 to 010), the results are quite similar. This can also be seen from Fig.5-2, where both cost functions display 4 local minima and all time lags are near to their counterparts. For practical purposes, we can select any value from this range as a regularization parameter. We were primarily interested in the P300 comp onent, preferably the largest one. From the ensemble average, we know that the maximu m ERP occurred around 380ms after stimulus onset. In Fig. 5-3, the estimated pdf displays a m ode around 420ms. Thus, we searched around this

PAGE 95

95 latency and set the single-trial peak latency as th e one that was closest to it. The mode of latency is 360ms, 420ms and 400ms for the ot her three subjects, respectively. We should point out that since there is about 1 local minimum per mode, the search need not be around the true mode for latency (we do not know this anyway). The results would be almost the same as long as the estimated mode is not skewed to its two neighboring true modes. Figure 5-4 shows the scalp topographies for the four subjects plotted using EEGLAB (Delorme et al., 2004). As expected for a P300 topography, it has a large positive topography around the Pz area. To evaluate the single-trial estimation of the scalp topography, we compute the correlation between the single-trial scalp t opography in (3.15) and the overall normalized scalp topography (3.17). For comparison, we also compute the correlation between the singletrial scalp topography in (3.15) and the scalp topography obtained from ensemble average for each subject. We name these two correlations 1r and 2r respectively. Statistical inference based directly on the correla tion itself is difficult since its di stribution is complicated. A popular approach is to first apply the Fisher Z transformation to correlati on and then do inference on the transformed variable. The Fisher Z transform is given by (Fisher, 1915): 1 0.5ln 1r Z r (5-1) Z has a simpler distribution and it converges mo re quickly to a normal distribution. We can calculate the mean and confidence interval of Z based on the correlation, if we assume that the estimation error in (3-15) is a no rmal distribution. The statistics of the correlation can be easily obtained from the inverse transform of (5-1). The results are summarized in Table 5-1. We can see that there is a moderate amount of correlation between the single-tria l and overall scalp topography (the average mean correlation for 4 subjects is around 0.40) although the mean co rrelation is lower for su bject 3 at around 0.20.

PAGE 96

96 There is a small degradation in mean correlation when the overall scalp topography is computed from the ensemble average. This is expected sinc e the estimate in (3-17) is close to the ensemble averaged estimate. The correlation between these two estimates for the four subjects are: 0.80, 0.85, 0.89, 0.79 respectively. To evaluate the effectiveness of the single-trial amplitude estimation, we related our estimates to a behavioral measure of target iden tification: response time in target trials. Response time was selected because task difficulty was re latively low, and theref ore error rate did not show pronounced variability, with only limited numbers of misses (mean of 3.9 % across 4 participants) and false alarms (mean of 1.2 % across 4 participants). Thus, response time was used as a measure of target identification, with short response times i ndicating facilitated discrimination and long response times indicating di fficulties with identifica tion in a given trial. Using these measures, we were interested in the relationship between P300 amplitude and response time, expecting that trials in which participants found disc rimination relatively easy (short RT trials) should be associated with greater P300 amplitude, which also indicates successful encoding of the target features and prep aration for responding to a target that has been identified. There seems to be little relationship between the response time and estimated single-trial peak latency. The correlation coe fficients between thes e two for the four s ubjects are: 0.022, 0.248, 0.168 and 0.093 respectively. However, ther e were reliable negative correlations between the response time and estimated single-trial amp litude. Figure 5 shows the scatter plot of the response time versus the estimated amplitude for each single trial for the four subjects. To evaluate the statistical significan ce of the results, we performe d linear regression on the response time and estimated single-trial amplitude for the four subjects. The results are summarized in

PAGE 97

97 Table 5-2. the negative slope parameter estimated from linear regression is statistically significant under a signif icance level of 0.05 fo r all the four subject s, which supports our hypothesis that larger amplitude correspond to smaller response time, and vice versa. To compare our results with conventional methods, we calculated the average P300 amplitude at channel 100 for subject #2. This was simply the average single-trial amplitude times the 100-th entry of the scalp topog raphy in (3.17). It was found to be 22.1mV, compared with the 17.3mV from the ensemble average ERP. Taking in to account of the possible latency jitter of P300, the true amplitude could be only larger than 17.3mV. Therefore, we obtained an upper bound of 28% on the positive bias of our average P300 estimate in channel 100. The coefficient of variation, which is defined as the ratio of the standard deviation to the mean of a positive random variable, is used as a measure of disp ersion of the estimated amplitude and it was found to be around 0.60. This compares favorably wi th 0.79 obtained using the simple peak-picking method around its ensemble average peak at 400ms. Although the gain may seem small, we should keep in mind that this variation will incorp orate the estimation error as well as that of the underlying change in P300 amplitude itself, because there are systematic changes in P300 amplitude as suggested above. So the estimation va riance of our method is re duced by a factor of at least 1.7 from the peak-picking method. For inst ance, if one half of th e total variance of our method came from the underlying P300 amplitude, this roughly means that our method reduced the estimation variance by a factor of 2.5 (assu ming additive and uncorrelated estimation error). Of course, the comparison would be much more direct and informative if the P300 amplitude was expected to remain constant. All the above results were obtained using a fixed Gamma template with 11, 5 k If we change the template, specifi cally, the spread parameter the estimated amplitude will also

PAGE 98

98 change. However, we found that the amplitude estimation is only slightly affected by this change. For instance, the average estimated P 300 amplitude in channel 100 for subject #2 was around 20.5mV when 1 (this is too small for P300, ri se time 40ms) and was around 23.8mV when 8 (this is too large, rise time 320ms). Th ere is less than 8% change from the result (22.1mV) obtained with the original template with 5 This agrees with our earlier findings using simulated ERP data (Li et al., 2008). 5.1.3 Discussions As a straightforward test of the present method, we examined the relationship between target detection performance and feat ures of the P300 component e voked by the targets in an oddball task with rare targets varying in terms of thei r salience on a trial-by-tria l basis. In the present case, we replicated and extended a standard resu lt in target detection studies in the visual domain: When target identification is made diffi cult or saliency is reduced (e.g., by presenting many targets in succession, Gonsalvez and Po lich, 2002), P300 amplitude often decreases (Polich et al., 1997). This pattern has been interpreted as reflecti ng reduced resource allocation to a given target stimulus (Keil et al., 2007). Notably, previous work in this area has typically relied on averages across all trials of an experimental condition, or on block by condition averages across many trials (for a review, see Kok, 2001). The present result s suggest that the relationship between response time and P300 amplitude in feat ure-based attention task is of a continuous nature, rather than a conseque nce of a bimodal function separating easy and hard trials. The sensitivity of the method was sufficient to demonstrate this linear relationship on a single-subject level, which is often desirable in clinical studi es. In a similar manner, other research questions will benefit from the ab ility to examine hypotheses as to th e time course and distribution of single brain responses, in terms of their magnitude and latency.

PAGE 99

99 5.2 Habituation Study 5.2.1 Materials and Methods EEG data were recorded from subjects dur ing a passive pictureviewing experiment, consisting of 12 alternating phases: the habitu ation phase and mixed phase. Each phase has 30 trials. During the 30 trials of the habituation pha se, the same picture was repeatedly presented 30 times. During the mixed phase, the 30 pictures are all different. Each trial lasts 1600 ms, and there is 600 ms pre-stimulus and 1000 ms post-stimulus. The scalp electrodes were placed according to the 128-channel Geodesic Sensor Nets standards. All 128 channels were referred to channel Cz and were digitally sampled for analysis at 250Hz. A bandpass filter between 0.01Hz and 40H z was applied to all channels, which were then converted to average referenc e. To correct for vertical and hor izontal ocular artifacts, an eye movement artifact movement correction proced ure (Gratton et al., 1983) was applied to EEG recordings. 5.2.2 Estimation Results We assume that the entire ERP may be d ecomposed into several monophasic components with compact support. We will estimate their pa rameters (amplitude and latency) one by one, using the Gamma template as in the simula tion study. The present study illustrates the application of the method for a single late poten tial component, and the Gamma is not adapted. Its parameters are selected based on ne urophysiology plausibility and are set as 5, 6 k corresponding to a rise time of 96ms. In reality, we do not know a priori exactly how many components there are in a singletrial, nor do we know when they occur. However, we may be able to estimate these values from single-trial EEG data in the data analysis se ssion. Following the same procedures in Section 5.1.2, we identified 5 distinct peaks after 300ms of the stimulus onset. Assuming that the error in

PAGE 100

100 the latency estimation is equally biased and inde pendent from trial to trial and since there are also about 5 local minima for each trial, we conjecture that these peaks correspond to the latencies of 5 distinct components. These components, whic h may have different origins, are likely to compromise the Late Po sitive Potentials (LPP). Accordi ng to Codispoti et al.(2006), the grand-average of LPP is maximal around 400ms to 500ms after stimulus. We will concentrate on the component with a latency of 500ms to ex emplify the methodology. We search between 440ms and 560ms (which correspond to the two neighboring local minima) and set the component latency as the local minimum closest to 500ms. To avoid the influence of EEG outliers from unexpected artifacts, we reject those trials with 3 times or larger amplitude of the minimum one. This will eliminate 14 trials from the total of 360 trials (rejection rate: 4%). The same rejection criterion wa s applied to the other two subjects, leading to the rejection of 8 (2%) and 33 (9%) trials, respectively. Figure 5-6 shows the results of estimated scalp topography of the LPP component for 3 subjects. They are similar in the sense that all show large projections in the posterior area. The difference with subject 2 is that the scalp topography shifts its strength a b it to the occipital area. It may be that the pictures shown to the 3 subject s caused some emotional bi as. It is also possible that the SNR of the ERP data is too low to allo w for a stable estimate of the scalp projection across subjects (note that in habituation phase, the LPP amplitude decreases quickly with the trial index). Figure 5-7 shows the results of estimated amplitude of the LPP component for 3 subjects. Each point in Fig. 5-7 stands for the average am plitude over 6 trials with the same index in the same phase (habituation or mixed). It is clear that for the habituation phase, the amplitude diminishes rapidly with the trial index, while for the mixed phase, the amplitude does not show

PAGE 101

101 significant decay. To make the figure more intuitiv e, we also include the best fit (in the least square sense) to the estimated amplitude for both habituation and mixed phase. We fitted a straight line for the mixed phase, while an exponential curve was fitted to the estimated amplitude of the habituation phase. The fitted exponential curve for the habituation phase has a time constant of around 1.5 trials, which suggests that after 3 or 4 trials, the LPP amplitude decreases close to zero. We es timate the SNR for the mixed phase at around -4.1dB. Similar results were obtained with the ERP data from 2 ot her subjects as shown in Fig. 5-7 (B) and (C). The fitted exponential curves for the habituation phase for these 2 subjects has a time constant of around 1.5 and 2.0 trials, respectively. The SNR of the mixed phase for these 2 subjects are estimated to be around -7.6dB and -2.4dB respectively.

PAGE 102

102 Table 5-1. Correlation statistics for th e 4 subjects: Scalp topography estimation 1r 2r Subject # Sample size Mean Confidence interval (95%) Mean Confidence interval (95%) 1 98 0.520 [0.362, 0.649] 0.393 [0.215, 0.546] 2 85 0.367 [0.167, 0.538] 0.304 [0.097, 0.486] 3 74 0.198 [-0.027, 0.404] 0.200 [-0.025, 0.406] 4 65 0.446 [0.233, 0.619] 0.359 [0.133, 0.551] Table 5-2. Regression statistics for response time and estimated amplitude Subjec t # Sample size Correlation R square Slope estimate t statisticp value Confidence interval (95%) 1 98 0.440 0.194 -0.712 -4.80 <0.0001 [-1.007, -0.418] 2 85 0.539 0.291 -0.310 -5.84 <0.0001 [-0.416, -0.205] 3 74 0.263 0.069 -0.105 -2.31 0.012 [-0.197, -0.014] 4 65 0.351 0.123 -0.108 -2.97 0.002 [-0.181, -0.036]

PAGE 103

103 Figure 5-1 Pictures used in the experiment as stimuli

PAGE 104

104 Figure 5-2 Cost function in (3.9) versus time lag for different regularization parameters for subject #2. A) 610. B) 510 C) 010 D) 210. Regularization parameter that is too small led to ragged cost function and spurious latency estimates; Regularization parameter that is too larg e led to over-smoothed cost function and missed candidates for latency. A B C D 200 250 300 350 400 450 500 550 600 0.5 1 1.5 2 2.5 3 3.5 x 10 -5 time lag (ms) cost function 200 250 300 350 400 450 500 550 600 0 0.5 1 1.5 2 2.5 3 3.5 x 10-4 time lag (ms)cost function 200 250 300 350 400 450 500 550 600 0.4 0.5 0.6 0.7 0.8 0.9 time lag (ms) cost function 200 250 300 350 400 450 500 550 600 3.6 3.65 3.7 3.75 3.8 3.85 3.9 3.95 time lag (ms)cost function

PAGE 105

105 100 200 300 400 500 600 700 0 0.5 1 1.5 2 2.5 3 x 10-3 time lag(ms)pdf Figure 5-3 Estimated pdf of time lags corresponding to local minima of the cost function in (3-9) using the Parzen windowing pdf estimator with a Gaussian kernel size of 4.2. Regularization parameter 510

PAGE 106

106 A B C D Figure 5-4 Scalp topographies for the four subjects. A) Subject 1. B) Subject 2. C) Subject 3. D) Subject 4.

PAGE 107

107 Figure 5-5 Scatter plot of the response time vers us the estimated amplit ude for each single trial for the four subjects. A) Subject 1. B) Subjec t 2. C) Subject 3. D) Subject 4. Note that the estimated amplitude is with respect to the EEG data in all the channels as a whole. There appears to be a negative relati onship between the response time and the estimated amplitude. A B C D 300 350 400 450 500 550 600 650 -100 0 100 200 300 400 Response time (ms) estimated amplitude 250 300 350 400 450 500 550 600 650 -100 -50 0 50 100 150 200 Response time (ms)estimated amplitude 300 400 500 600 700 800 900 1000 -100 -50 0 50 100 150 200 250 Response time (ms) estimated amplitude 200 300 400 500 600 700 800 900 1000 -50 0 50 100 150 200 Response time (ms)estimated amplitude

PAGE 108

108 A B C Figure 5-6 Estimated scalp topography for mixe d and habituation phase. A) Subject 1. B) Subject 2. C) Subject 3.

PAGE 109

109 0 5 10 15 20 25 30 -40 -20 0 20 40 60 80 100 trial indexamplitude(mV) Habituation Mixed expotential fit linear fit A 0 5 10 15 20 25 30 -20 -10 0 10 20 30 40 50 60 70 80 trial indexamplitude(mV) Habituation Mixed expotential fit linear fit B 0 5 10 15 20 25 30 -20 0 20 40 60 80 100 120 trial indexamplitude(mV) Habituation Mixed expotential fit linear fit C Figure 5-7 Estimated amplitude for mixed and habitu ation phase. A) Subject 1. B) Subject 2. C) Subject 3. Note that the LPP amplitude decreases with trials.

PAGE 110

110 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH 6.1 Conclusions Traditional ERP analysis has relied on ensemble average over a large number of trials to deal with the typically low SNR environments in EEG data. To analyze ERP on a single event basis, we have introduced a new spatiotemporal filtering method for the problem of single-trial ERP estimation. Our method relies on explicit modeling of ERP components (not the full ERP waveform), and its output is limited to local descriptors (amplitude and latency) of these components. The reason that we model the ERP co mponents instead of the full ERP waveform is to exploit the localization of scalp projection for each sing le ERP component, which is impossible to do for the entire ERP. Indeed, note that the ensemble ERP in different channels usually have different morphology because there are multiple neural sources originating from different locations of the brain that give rise to different scalp pr ojections. Since one spatial filter can extract effectively only one scalp projection, in or der to utilize the spat ial information in a meaningful way, only a component based analysis is viable. C oncentrating only on latency and amplitude of each component togeth er with optimal spa tial filtering presents an alternative to deal with the negative SNR. Moreover, since th ese are in fact the feat ures of importance in cognitive studies, the methodology has the same descriptive power of traditional approaches. The proposed methodology can be seen as a generalization of Woodys filter (Woody 1967) in the spatial domain for latency estimation. It also obtains an explicit expression for amplitude estimation on a single-trial basis. By design, th e method is especially su itable to extract ERP features in the spontaneous EEG activity, in contrast to PCA and ICA which work best for reliable (large) signals. Another distinction is that, unlike most methods based on PCA and ICA, our method utilizes explicitly the timing informa tion, as well as the spatial information. The

PAGE 111

111 methodology as presented is based on least square s, but it can be furt her extended to robust estimation (Li et al. 2007) for better results. Using simulated ERP data, we have show n that although the mi smatch between the presumed and synthetic ERP components introd uces a bias for both latency and amplitude estimation, the bias for the latency is relatively small and the estimated amplitudes are still comparable across experimental conditions fo r ERP data with a SNR higher than -20dB. Furthermore, the mismatch of components has minimal influence on the estimation of scalp projection. These all compare favorably with some of the popular methods (Li et al., 2008). Despite its advantages over traditional met hods, there are still some issues with our spatiotemporal filtering method. First it is base d on the linear generativ e EEG model in (3.1). While this greatly simplifies the analysis, it may not be adequate to fully describe the complex information processing in the brain. One weak link of the method is that it requires an explicit template that is unknown a priori. Mismatch between the template and the true ERP component waveform brings both bias and larger variance to the estimation of the latency and amplitude that increases with decreasing SNR (Li, et al. 2008). It may be desirable to be able to adapt the template while estimating the model paramete rs. Another weakness of the method is the assumption of statistical unco rrelatedness among all the ERP components in deriving (3.14). With monophasic waveforms, this is equivalent to the condition that all the ERP components do not overlap in time (but overlap in space is allo wed), which is seldom satisfied in practice. Temporal overlap will bring bias to the amplitude estimation and poses a serious problem for the latency estimation, since it works effectively only for monophasic waveforms that are well separated in time. When there is heavy ove rlap among multiple components (e.g., P300 and possibly other unknown late componen ts), the peak latency estimation based on (3.10) may fail.

PAGE 112

112 Therefore, care must be taken not to over-int erpret the results of single-trial estimates. A crucial factor for amplitude estimation is a reasonably low SNR (>-20dB). This may not be satisfied for some ERP components under certain e xperimental conditions. Our ability to infer the template accurately, which are selected heur istically from real data, deteriorates with decreasing SNR. As a rule of thumb, we woul d recommend against the us e of the present method for data with SNR less than -15dB. 6.2 Future Research The use of a parametric template (Gamma func tion) provides the flexibility to change the shape and scale parameters continuously. However, this introduces undesirable bias when there is a mismatch between the template and true ERP component. Using a stochastic formulation, our method may be extended to a noisy template model and potentially the two nuisance Gamma parameters may be extracted from the data also for best fit. It is almost certain that activations of different ERP components overlap in time. If this is the case, the temporal overlap will introduce a bias to the estimation of single -trial scalp projection, because the derivation in (3.15) assumes the uncorrelatedness between the ERP component and all the other sources (includi ng the overlapping ERP compon ents) and unlike the background EEG, these overlapping components are coherent in all the trials. This bi as, together with the estimation variance due to finite-sample data, constitute the two main sources of error in the estimation of the scalp projection. Note that th is in turn will influence the estimation of the amplitude. In Chapter 4, we have proposed an explicit procedure to compensate for the overlapping effects for the amplitude and topography estimation. This assumes that the latency is (relatively) fixed and a fairly accurate knowledge of the shape of all the overlapping ERP components. When this is not the case (particular ly when we wish to find the latency change

PAGE 113

113 from trial to trial in the presence of com ponent overlap), we need to come up with new procedures to compensate for the overlapping issue. The current method considers the single-tri al amplitude of an ERP component as i.i.d. data. It may be advantageous to take into account th e dynamics of certain properties of the component with respect to the trial index. For instance, we expect that during the habituation phase, the amplitude of LPP components diminishes rapidly with the number of trials. Using regularization techniques, this a priori information may be incorporated into the proposed single-trial estimation method to provide more stable estimate for the amplitude. In the end, the evolution of the single-trial amplitude with trial index may be inferred with more resolution and more confidence.

PAGE 114

114 APPENDIX A PROOF OF VALIDITY OF THE PEAK LATENCY ESTIMATION IN (3-10) We justify the use of the time lag corresponding to the local minimum of ()J in (3-10) as the peak latency. Given the single-trial data matrix X the peak latency of an ERP component coincides with the local minimum of ()J if the following condi tions are satisfied: (1), the presumed component os and the actual component s have the same morphology; (2), ()0T io ns for 1,...,, iN and S (the signal and noi se are uncorrelated); (3), X is full rank. Proof: The optimal spatial filter is give n by (3-8). We plug it into (3-7) and get: 2 2()()()()()TTTT oooJ sCIsCCCCIs where, 1 TT CXXXX note that T CCC, so, 2 1()()()()()()()()()TTTTT oooooooJ sCsssssaRass where, T RXX is a positive definite matrix independent of the time lag With the constraint that ()()T ooconst ss the minimum of the cost function () J is achieved when ()T o ss achieves its maximum, since 1 TaRa is positive. This happens when coincides with the peak latency l of the actual ERP component s.

PAGE 115

115 APPENDIX B GAMMA FUNCTION AS AN APPROXIMATI ON FOR MACROSCOPIC ELECTRIC FIELD The macroscopic electrical field is created from spike trains by a nonlinear generator with a second-order linear component w ith real poles (Freeman 1975) Suppose that the transfer function of the second-order syst em with stable real poles a, b is: 1 () ()() Hs sasb where, without loss of generality: 0 ba Then the impulse response in the time domain is: 1 ()atbthtee ab This is also a monophasic wavefo rm with a sing le mode at 01 ln(/) tba ab The rising time depends on the relative magn itude of the two real poles. The impulse response can be expanded: () 1() 11 ()1 !n btabtbt nabt hteee ababn Thus we can see that the impulse response is a sum of infinite weighted Gamma functions. However, it is always possible to find a few dominant terms ar ound the mode, where, 0()ln(/) tabba If we knew the values for a, b we can choose the shape parameter K of the Gamma function as the largest term, i.e., the integer part of ln(/) ba A special case is when the system has two id entical poles. Then, the impulsive response is exactly modeled by a single Gamma function with K = 1, 1/ a This is also approximately true when the magnitude of one pole is much larger than the other one.

PAGE 116

116 APPENDIX C DERIVATION OF THE UPDATE RULE FO R THE CONSTRAINED OPTIMIZATION PROBLEM Using one Lagrange multiplier, we convert the constrained optimization problem in (4-1) to an unconstrained optimization problem. 2 22 0 22 ,argmin1T F aXasaaa Note that, 2 2 2 22 2 2T TTT F TTTT TTTT TTTTTr TrTrTr TrTrTr Tr XasXasXas XXXsaassa XXaXsssaa XXaXsssaa Setting the gradient of the Lagrangian function to 0 with respect to ,, a respectively, we find that the following set of equations holds: 2 0 2 222220 220 10T TTT ssaXsaaa aXsssaa a Solving for a, we have: 0 0 2 T T aXs ss aXs a aXs This is not a closed-form solution for the optimal values. However, it can be effectively used as a fixed point update to itera tively find the optimal values.

PAGE 117

117 APPENDIX D DERIVATION OF THE UPDATE RULE FOR THE UNCONSTRAINED OPTIMIZATION PROBLEM The unconstrained optimization problem is: 22 0 2 2argminT FaXasaaa First we note that: 2 000 222 2 2 000 22 0 22 22T TTT T aaaaaaaaa aaaaaaaa aaa Taking the derivative to 0, we get, 3 00 222220TT ssaXsaaaaaa Or equivalently, 0 2 3 0 2 TT Xsaa a ssaaa This is not a closed-form solution for a However, it can be effectively used as a fixed point update to iteratively find the optimal values.

PAGE 118

118 APPENDIX E MAP SOLUTION FOR TH E ADDITIVE MODEL With the assumptions indicated in Section 4.4, the posteriori probability can be rewritten as: 111 1(,...|...)(|,)()()(...)K oKKkkokoK kpppppaaaaaaaa Given the model, maximization of the pos terior probability is equivalent to: 1,... 1argmax(|,)()()oKK kokko kpppaaaa We assume a uniform (flat) a priori distribution for k Since oa is constrained to have a unit norm, its a priori distribution is a Di rac delta function: 21o a If we assume that ku is zero-mean i.i.d. Gaussian noise with the same covariance matrix 2dI across all the trials, maximization of the posterior probab ility can be further simplified: 1 12 22 ,...,... 1 1argmaxlog(|,)()argmaxlog1oKoKK K kokokkoo k kpp aaaaaaaa This can be converted to a c onstrained optimization problem: 2 2 1minK kko kJaa S.t. 21o a Setting the derivative to zero, a ne cessary condition for minimum is: T kok aa Plug this into the above cost function. We have:

PAGE 119

119 22 22 11 1 1 11 KK TT kokokook kk K T TTT kooook k K TT kook k KK TTT kkokko kkJ aaaaaaaa aIaaIaaa aIaaa aaaaaa The first term does not depend on oa, so MAP is equivalent to: argmaxoT ooaaAa S.t. 21o a where 11 KK T kkk kkAAaa The matrix A is symmetric, so it can be diagonalized. The maximum occurs when oa is the normalized eigenvector of the matrix A corresponding to the largest eigenvalue.

PAGE 120

120 APPENDIX F NORMALIZED ADDITIVE NOISE MODEL 1: MAP SOLUTION Following the same rationale in model 1, we attempt to derive the MAP solution for the model 2. The difference is in the covariance matrix of conditional probability: 2(|,)~0,kokkpNaaI So, 2 /2 21 (|,)exp2 2kokkko D D kp aaaa The MAP becomes: 1 1,... 1 2 22 ,... 1argmax(|,)()() argmaxlog1oK oKK kokko k K kkoo kppp a aaaa aaa where, we have used the prior distribution for k : () D kkp This is an improper prior (but still a prior for Bayesian inference). It is mainly motivated by analytical tractability. Similarly as in mode l 1, we can write the following constrained optimization problem: 2 2 1argminokK kko kaaa S.t. 21o a We first find a necessary condition for minimum: TT kkkok aaaa Let T kk k T kk aa B aa Note that kB is idempotent, i.e., kkk BBB Plug this into the above cost function. We have:

PAGE 121

121 2 2 1 1 1argmin argmin argmin argmaxo o o oK ko k K T oko k K T oko k T ooK a a a aIBa aIBa aBa aBa where, 11 T KK kk k T kk kk aa BB aa Again, the maximum occurs when oa is the normalized eigenvector of the matrix B corresponding to the largest eigenvalue.

PAGE 122

122 APPENDIX G NORMALIZED ADDITIVE NOISE MODEL 2: MAP SOLUTION Given the model: 2 k ok k a au a we attempt to find the MAP solution for the posteriori probability 1 1 2 22 2 1argmax(|...) argmaxlog(|)() argmaxlog1o o ooK K koo k K kkoo kp pp a a aaaa aaa aaaa The equivalent constraine d optimization problem is: 2 2 2 1argmaxoK kko k aaaa S.t. 21o a This can be solved in a straightforward way: 2 1 2 1 2 K kk k o K kk k aa a aa

PAGE 123

123 LIST OF REFERENCES Al-Nashi H. (1986). A maximum likelihood me thod for estimating EEG evoked potentials. IEEE Transactions on Biomedical Engineering, 33:1087-1095. Aunon J.I. and McGillem C.D. (1975). Techni ques for processing si ngle evoked potentials. Proceedings San Diego Biomedical Symposium, 211-218 Aunon J.I., McGillem C.D., and Childers D.G.. (1981). Signal processing in evoked potential research: averaging and modeling. CRC Critical Reviews in Biomedical Engineering, 4:323367. Basar E, Demiralp T, Schurmann M, Basar-Erglu C, Ademoglu A. (1999). Oscillatory brain dynamics, wavelet analysis, and cognition. Brain Lang;66:146-183. Bell A., Sejnowski T., (1995). An Informati on Approach to Blind Separation and Blind Deconvolution, Neural Computation., vol. 7: 1129-1159. Belouchrani A., Abed-Meraim K., Cardoso J-F, Moulines E., (1997). A Blind Source Separation Technique Based on Second-Order Statistics, IEEE Transaction on Signal Processing, vol.45, 434. Brazier. M.A.B. (1964). Evoked responses record ed from the depths of the human brain. Annals of New York Academy of Sciences, 112: 33-59. Bruin K.J., Kenemans J.L., Verbaten M.N., Van der Heijden A.H., (2000) Habituation: an eventrelated potential and dipole source analysis study. Intional Journal of Psychophysiol. 36, 199-209. Cardoso J.-F. (1998). Blind signal sepa ration: statistical principles, In Proceedings of the IEEE, special issue on blind iden tification and estimation, R.-W. Liu and L. Tong editors. Vol. 9(10), 2009-2025. Cardoso J.-F. (1999). High-Order Contrast s for Independent Component Analysis. Neural Computation, vol. 11, 157. Caspers, H., Speckmann, E.-J., & Lehmenkhler, A. (1980). Electrogenesis of cortical DC potentials. (eds. H. H. Kornhuber & L. Deecke), Progress in brain research: Vol. 54. Motivation, motor and sensory processes of the brain: Electrical pote ntials, behavior and clinical use 3-15. Amsterdam: Elsevier. Cerutti S., Bersani V., Carrara A ., and Liberati D. (1987). Analysis of visual evoked potentials through Wiener filtering applied to a small number of sweeps. Journal on Biomedical Engineering, 9:3-12.

PAGE 124

124 Chapman, R., McCrary, J., ( 1995). EP component identification and measurement by principal components analysis. Brain Cognition 27 (3), 288. (Erratum in: Brain Cogn. 28 (3) 342, 1995.) Choi S., Cichocki A., Park H. -M., and L ee S. -Y. (2005). Blind source separation and independent component analysis: A review. Neural Information Processing Letters and Review, vol. 6, no. 1, 1-57. Cichocki A. and Amari S. (2002). Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications. New York: Wiley. Ciganek Z. (1969). Variability of the huma n visual evoked potential: normative data. Electroencephalography and Clinical Neurophysiology, 27:35-42. Codispoti M., Ferrari V., Bradley M., (2006). Repetitive picture processing: autonomic and cortical correlates. Brain Research. 1068: 213-220. Coifman R.R., Wickerhauser MY. Wavelets (1996). adapted waveforms and denoising. Clinical Neurophysiology Supplement; 45:57-78. Coles MGH, Rugg MD. (1995). Event-rela ted brain potentials : an introduction. Electrophysiology of mind: event-rela ted brain potentials and cognition. (ed. Rugg MD, Coles MGH), Oxford University Press. Oxford, UK. 1. Comon P. (1994). Independent comp onent analysis, a new concept? Signal Processing, vol. 36, 287. Cuthbert, B.N., Schupp, H. T., Bradley, M.M., a nd Lang, P.J. (1996). Affective picture viewing: Task and stimulus effects on startle P3 and blink. Psychophysiology, 33. Daubechies. I. (1992). Ten Lectures on Wavelets. SIAM. Debener S, Makeig S, Delorme A, Engel AK, (2005). What is novel in the novelty oddball paradigm? Functional significance of the novelty P3 event-related potential as revealed by independent component analysis, Experimental Brai n Research: Cognitive Brain Research. 22:309-321. Delorme A, Makeig S, Fabre-Thorpe M, Sejnowski TJ, (2002). From single-trial EEG to brain area dynamics, Neurocomputing, 44-46: 1057-1064. Delorme, A., Makeig, S. (2003) EEG changes accompanying learning re gulation of the 12-Hz EEG activity. IEEE Transactions on Reha bilitation Engineering, 11(2), 133-136. Delorme, A., Sejnowski, T., Makeig, S. (2007) Improved rejection of artifacts from EEG data using high-order statistics and independent component analysis. Neuroimage, 34, 1443-1449

PAGE 125

125 Dien J. (1998). Addressing misallo cation of variance in principal components analysis of eventrelated potentials. Brain Topography;11(1):43. Dien J, Beal DJ, Berg P. (2005). Optimizing pr incipal components analysis of event-related potentials: Matrix type, f actor loading weighting, extraction, and rotations. Clinical Neurophysiology;116:1808. Doncarli C. and Goerig L. (1988). Adaptive sm oothing of evoked potentials: a new approach. Proceedings 10th Annual Confereence IEEE-EMBS, New Orleans, Lousiana, 1152-1153 Donchin E. (1966). A multivariate approach to the analysis of average evoked potentials. IEEE Transaction on Biomedical Engineering;13(3):131. Donchin E, Heffley EF. (1978). Multi variate analysis of event-rela ted potential da ta: a tutorial review. In: Otto DA, editor. Multidisciplinary perspectives in event-related brain potential research. Proceedings of the fourth international congress on event-related slow potentials of the brain (EPIC IV), Hendersonville, NC, Apr il 4, 1976. Washington, DC: The Office; 555. Donchin, E., Ritter, W., McCallum, C. (1978). Cognitive psychophysiology: The endogenous components of the ERP. Brain event-related potentials in man (ed. E. Callaway, P. Tueting, S. Koslow), Academic Press, New York. 349-441. Donchin, E. (1979). Event-relate d brain potentials: a tool in the study of human information processing. Evoked potentials a nd behavior (ed. H. Begleite r), Plenum, New York. 13-75. Donchin, E. (1981). Surprise! surprise? Psychophysiology, 18, 493-513. Donchin, E. and Coles, M. G. H. (1988a). Is the P300 component a ma nifestation of context updating? Behavioral and Brain Sciences, 11, 355-72. Donchin, E. and Coles, M. G. H. (1988b). On the conceptual foundations of cognitive psychophysiology. Behavioral and Brain Sciences, 11, 40617. Donchin, E., Spencer, K.M. and Dien, J. (1997) The varieties of de viant experience: ERP manifestations of deviance processors. In: G. J.M. van Boxtel and K. B.E. Bocker (Eds.), Brain and Behavior: Past, Present, and Future, Tilburg: Tilburg University Press,: 67-91. Donoho D.L. and Johnstone I.M. (1994). Idea l spatial adaptation by wavelet shrinkage. Biometrika, 81:425. Dornhege G., Blankertz B., Krauledat M., Losc h F., Curio G., Mller K.-R. (2006). Combined optimization of spatial and temporal filters for improving Brain-Computer Interfacing. IEEE Transactions on Biomedical Engineering, 53(11), 2274.

PAGE 126

126 Duffy, F.H., Jones, K., Bartels, P., Albert, M ., McAnulty, G.B. and Als, H. (1990). Quantified neurophysiology with mapping: Statistical infere nce, exploratory and confirmatory data analysis. Brain Topography,3(1): 3-12. Dyrholm M, Christoforou C., Parra L.C., (2007) Bilinear discriminant component analysis, Journal of Machine Learning Research, vol. 8, 1097-1111 Effern A, Lehnertz K, Fernandez G, Grunwal d T, David P, Elger CE. (2000). Single trial analysis of event related potentials: non-linear de-noising with wavelets. Clinical Neurophysiology; 11:2255. Fabiani, M., Gratton, G Karis, D., & Donchin, E. ( 1987). Definition, identification and reliability of measurement of the P300 component of the event-related brain potential. In P. K. Ackles, J. R. Jennings, & M. G. H. Coles (Eds.) Advances in psychophysiology, Vol. 2 (pp. 1-78). Greenwich CT : JAI Press. Farwell L.A., Martinerie J.M., Bashore T.R ., Rapp P.E., and Goddard P.H. (1993). Optimal digital filters for long-latency components of the event-relate d brain potential. Psychophysiology, 30:306-315. Fisher, R.A., (1915). Frequency di stribution of the values of the correlation coefficient in samples of an indefinitely large population. Biometrika, 10, 507-521. Freeman W., (1975). Mass activation in the nervous system. Academic Press, New York Gaillard AWK, Ritter WK. (1983). Tutorials in event-related potential research: endogenous components. Amsterdam: North-Holla nd Publishing Company. Gansler T. and Hansson M. (1991). Estimati on of event-related potentials with GSVD. Proceedings 13th Annual Conference. IEEE-EMBS, 423-424. Garoosi V. and Jansen B. (2000). Developmen t and evaluation of the piecewise prony method for evoked potential analysis. IEEE Transaction on Biomedical Engineering. 47(12): 15491554. Glaser EM, Ruchkin DS. (1976). Principles of neurobiol ogical signal analysis. New York: Academic Press. Gonsalvez CL, Polich J (2002) P300 amplitude is determined by target-to-target interval. Psychophysiology 39:388-396 Gratton, G., Coles, M.G.H., Donchin, E., (1983 ). A new method for off-line removal of ocular artifact. Electroencephalography Clin ical Neurophysiology. 55, 468484 Gratton G., Kramer A.F., Coles M.G.H., and Donc hin E. (1989). Simulation studies of latency measures of components of the event-related brain potential. Psychophysiology, 26:233-248.

PAGE 127

127 Hansson M. and Cedholt T. (1990). Estim ation of event related potentials. Proceedings 12th Annual Conference IEEE-EMBS, 901-902. Hansson M., Gansler T., and Salomonsson G. (1996). Estimation of single event-related potentials utilizing the prony method. IEEE Transactions on Biomedical Engineering, 43(10):973-981. Hendrickson AE, White PO. (1964). Promax: a qui ck method for rotation to oblique simple structure. Br J Stat Psychol;17:65. Hesse C.W. and James C.J., (2006). On Semi-Bli nd Source Separation usi ng Spatial Constraints with Applications in EEG Analysis, IEEE Transactions on Biomedical Engineering. 53(12): 2525-2534. Horn JL. (1965). A rationale and test for th e number of factors in factor-analysis. Psychometrika; 30(2):179. Huber P., (1981). Robust Statistics. New York: Wiley. Hyvrinen A., Oja E. (1997). A Fast Fixed-Point Algorithm for Independent Component Analysis, Neural Computation, vol. 9: 1483. Hyvaerinen A., (1999). Survey on independent component analysis. Neural Computation. Sur., vol. 2: 94. Hyvarinen A., Karhunen J. and Oja E. (2001). Independent Co mponent Analysis. New York: Wiley James, C. and Hesse, C. (2004). Independent co mponent analysis for biomedical signals. Physiological Measurement, 26, (1), 15-39. Jaskowski P. and Verleger R. (1999). Amplitudes and latencies of single-trial ERPs estimated by a maximum-likelihood method. IEEE Transactions on Biomedical Engineering, 46(8):987-993. Johnson, R., Jr. (1986). A triarc hic model of P300 amplitude. Psychophysiology, 23, 367 384. Johnson R. (1989). Developmental evidence for modality-dependent P300 generators: a normative study. Psychophysiology 26: 651-667. Jung T-P, Makeig S, Wester field M, Townsend J, Courchesne E, and Sejnowski TJ, (1999). Analyzing and Visualizing Singletrial Event-related Potentials. Advances in Neural Information Processing Systems, 11:118-124.

PAGE 128

128 Jung T-P, Makeig S, Lee T-W, McKeown M.J., Brown G., Bell, A.J. and Sejnowski TJ, (2000). Independent Component Analys is of Biomedical Signals, The 2nd Int'l Workshop on Indeppendent Component Anal ysis and Signal Separation, 633-644. Jung T-P, Makeig S, McKeown M.J., Bell, A.J. Lee T-W, and Sejnowski TJ, (2001). Imaging Brain Dynamics Using Indepe ndent Component Analysis, Proceedings of the IEEE, 89(7):1107-1122. Junghfer M, Elbert T, Leiderer P, Berg P, Rockstroh B (19 97) Mapping EEGpotentials on the surface of the brain: a strategy for uncovering cortical sources. Brain Topography 9:203-217 Junghfer M, Elbert T, Tucker DM Rockstroh B (2000) Statistical control of artifacts in dense array EEG/MEG studies. Psychophysiology 37:523-532 Kaiser HF. (1958). The Varimax criterion fo r analytic rotation in factor analysis. Psychometrika. 23:187. Kalman R.E. (1960). A new approach to linear filtering and prediction problems. Transactions ASME, Journal of Basic Engineering. 82D:35-45. Karjalainen P.A., Kaipio J.P., Koistinen A. S., and Karki T. (1996). Recursive Bayesian estimation of single trial evoked potentials. Proceedings 18th Annual Conference IEEEEMBS, Amsterdam Karjalainen PA. (1997). Regularization and Bayesian Method s for Evoked Potential Estimation. Ph.D. Thesis, Kuopio University Public ations C. Natural and Environmental Sciences 61 Karjalainen P.A., Kaipio J.P., Koistinen A. S., Vauhkonen M. (1999). Subspace regularization method for the single-trial estimation of evoked potentials. IEEE Transaction on Biomedical Engineering 46(7):849-60. Keil A, Bradley MM, Junghoefer M, Russmann T, Lowenthal W, Lang PJ (2007) Cross-modal Attention Capture by Affective Stimuli: Eviden ce from Event-Related Potentials. Cognitive, Affective, & Behavioral Neuroscience 7:18-24 Koch C., Poggio T., Torre V., (1983 ). Nonlinear interactions in a dendritic tree: localization, timing, and role in information processing. Proceedings National Academy of Science USA 80: 2799 Koles ZJ, Lazar MS, Zhou SZ. (1990). Spatial pa tterns underlying population differences in the background EEG. Brain Topography; 2(4):275-284. Koles ZJ, Lind JC, Flor-Henry P. (1994). Spatia l patterns in the bac kground EEG underlying mental disease in man. Electroenceph Clinical Neurophysiology, 91:319-328.

PAGE 129

129 Kok A., (2001) On the utility of P3 amp litude as a measure of processing capacity. Psychophysiology 38:557-577 Krieger S., Timmer J., Lis S., and Olbrich H. M. (1995). Some considerations on estimating event-related brain signals. Journal of Neural Trans Gen Sect, 99(1-3):103-129. Lang, P., Bradley, M. M., and Cuthbert, B. N. (1997). Motivated atten tion: Affect, activation, and action. In P. Lang, R. F. Simons, & M. Balaban (Eds.), Attention and orienting: Sensory and motivational processes. Hillsdale, NJ: Erlbaum. 97. Lang PJ, Bradley MM, Cuthbert BN (2005) Intern ational Affective Pict ure System: Technical Manual and Affective Ratings. In. Gainesville, FL: NIMH Center for the Study of Emotion and Attention Lange D.H. (1996). Variable si ngle-trial evoked potential esti mation via principal component identification. Proceedings 18th Annual Conference IEEE-EMBS, Amsterdam. Lee T. W., Girolami M., and Sejnowski T. J., (1999). Independent compone nt analysis using an extended infomax algorithm for mixed subGaussian and superGaussian sources. Neural Computation., vol. 11, 417. Lee T.W., Girolami M., Bell A. J., and Sejnowski T. J., (2000). A unifying information-theoretic framework for independent component analysis. Computing Math Application., vol. 39, 1 21. Lemm S., Blankertz B., Curio G., and Mller K.-R ., (2005). Spatio-spectral filters for improved classification of single trial EEG, IEEE Transactions on Biomedical Engineering, vol. 52, no. 9, 1541. Li R., Principe J., (2006). Blinking Artifact Removal in C ognitive EEG Data using ICA, International Conference of Engine ering in Medicine and Biology Society, 6273-6278. Li R., Principe J.C., Bradley M., Ferrari V., ( 2007) Robust single-trial ERP estimation based on spatiotemporal filtering. Proceedings IEEE EMBS Conference. 5206-5209 Li R., Principe J.C., Bradley M., Ferrari V., (2008) A spatiotemporal filtering methodology for single-trial ERP component estimation. IEEE Transactions on Biom edical Engineering, In Press Liberati D., Bertolini L., and Colombo D. C. (1991). Parametric method for the detection of interand intrasweep variability in VEP processing. Medical and Biological Engineering and Computing, 29:159-166,. Linden D. (2005). The P300: Where in the brai n is it produced and what does it tell us? Neuroscientist, 11(6): 563-576.

PAGE 130

130 Liu W., Pokharel P., Principe J., (2007). Correntropy: Properties a nd Applications in NonGaussian Signal Processing, IEEE Transactions on Signal Processing, Vol. 55, Issue 11 Luo A. and Sajda P. (2006) Learning discri mination trajectories in EEG sensor space: application to inferring task difficulty, Journal of Neural Engineering, 3 (1) L1-L6. Maccabee P.J., Pinkhasov E. I., and Cracco R. Q. (1983). Short latency evoked potentials to median nerve stimulation: Effect of low-frequency filter. Electroencephalography and Clinical Neurophysiology, 55:34-44. Madhavan G. P., Bruin H. de, and Upton A.R.M. (1984). Evok ed potential processing and pattern recognition. Proceedings 6th Annual Conference IEEE-EMBS, 699-702. Madhavan G. P., Bruin H. de, and Upton A. R. M. (1986) Improvements to adaptive noise cancellation. Proceedings 8th Annual Conference IEEE-EMBS, Fort Worth, TX, 482-486. Madhavan G. P. (1988). Comments on ad aptive filtering of evoked potentials. IEEE Transactions on Biomedical Engineering, 35:273-275. Madhavan G. P. (1992). Minimal repetition evok ed potentials by modified adaptive line enhancement. IEEE Transactions on Biomedical Engineering, 39(7):760-764. Makeig, S., Bell, A., Jung, T., Sejnowski, T ., (1996). Independent component analysis of electroencephalographic data. Advances in Neural Information Processing Systems, vol. 8. MIT Press, 145. Makeig S, Jung T-P, Bell AJ, and Sejnowski TJ, (1997). Blind Separation of Auditory Eventrelated Brain Responses into Independent Components. Proceedings of National Academy of Sciences, 94:10979-10984. Makeig S, Westerfield M, Jung T-P, Covington J, Townsend J, Se jnowski TJ, and Courchesne E, (1999). Functionally Independent Components of the Late Positive Event-Related Potential during Visual Spatial Attention," Journal of Neuroscience, 19: 2665-2680. Makeig S., Jung T.-P., Ghahremani D., and Se jnowski T. J. (2000). Independent component analysis of simulated ERP data. in Hum. High. Func. I: Adv. Meth., T. Nakada, Ed. Makeig S, Westerfield M, J ung T-P, Enghoff S, Townsend J, Courchesne E, Sejnowski TJ. (2002). Dynamic brain sources of visual evoked responses. Science, 295:690-694. Makeig S, Delorme A, Westerfield M, Jung TP, Townsend J, Courchesne E, Sejnowski TJ. (2004). Electroencephalographic brain dynamics following visual targets requiring manual responses, PLOS Biology, 2(6):747-762. Marple S. Lawrence Jr. (1987). Digital Spectral Analysis with Appli cations. Prentice Hall, Englewood Cliffs.

PAGE 131

131 McGillem C.D. and Aunon J.I. (1987). Analysis of event-related potentials, chapter 5, Elsevier Science Publisher. 131-169. Mocks J, Verleger R. (1991). Application of principal component analysis to event-related potentials. In Weitkunat R., editor. Multiv ariate methods in biosignal analysis: Digital Biosignal Processing. Amsterdam: Elsevier; 399. Mller-Gerking J., Pfurtscheller G., and Flyvbjerg H., (1999). Desi gning optimal spatial filters for single-trial EEG classifi cation in a movement task, Clinical Neurophysiology, 110: 787798. Naatanen, R. and Picton, T. W. (1987). The N 1 wave of the human electric and magnetic response to sound: a review and an an alysis of the component structure. Psychophysiology, 24, 375-425. Nishida S., Nakamura M., and Shibasaki H. (1993). Method for singl e-trial recording of somatosensory evoked potentials. Journal on Biomedical Engineering, 150:257-262. Parra, L.C., Alvino, C., Tang, A., Pearlmutter, B., Young, N., Osman, A., Sajda, P., (2002). Linear spatial integratio n for single-trial detec tion in encephalography. NeuroImage 17, 223 230 Parra, L.C., Sajda, P., (2003). Blind source separation via generalized eigenvalue decomposition. Journal of Machine Learning Research. 4, 1261. Parra, L., Spence, C., Gerson, A., Sajda, P., ( 2005). Recipes for the linea r analysis of EEG. Neuroimage, 28, 326-341. Parzen, E., (1962) On estimation of a probability density function and mode. The Annals of Mathematical Statistics, Vol. 33, No. 3. 1065-1076 Patterson R. D., Robinson K., Ho ldsworth J., McKeown D., Zha ng C., and Allerhand M. H., (1992) Complex sounds and auditory images, In Auditory Physiology and Perception, (Eds.) Y Cazals, L. Demany, K.Horner, Pergamon, Oxford, 429-446 Pham D. T., Mocks J., Kohler W., and Gasser T. (1987). Variable latenc ies of noisy signals: Estimation and testing in brain potential data. Biometrika, (74) 525. Picton TW, Bentin S, Berg P, Donchin E, Hillyard SA, Johnson R Jr, Miller GA, Ritter W, Ruchkin DS, Rugg MD, Taylor MJ. (2000). Guidelines for using human event-related potentials to study cogniti on: recording standards and publication criteria. Psychophysiology. 37(2):127-52

PAGE 132

132 Polich J, Alexander JE, Bauer LO, Kuperman S, Morzorati S, O'Connor SJ, Porjesz B, Rohrbaugh J, Begleiter H ( 1997) P300 topography of amplit ude/latency correlations. Brain Topogr 9:275-282 Polich J. (1999) P300 in clinical alications. Electroencephalography: basic principles, clinical alications and related fields (Eds. E. Niedermayer and F. Lopes de la Silva). Urban and Schwartzenberger, Baltimore-Munich. 1073-1091. Quian Quiroga R., Garcia H. (2003). Single-trial event-related potentials with wavelet denoising. Clinical Neurophysiology. 114: 376. Ramoser, H., Mueller-Gerking, J., Pfurtscheller, G., (2000). Optimal spa tial filtering of single trial EEG during imagined hand movement. IEEE Transaction on Rehabilitation Engineering. 8 (4), 441 446. Reilly, J., (1992). Applied Bioelectricity. Springer. New York Roberts S. and Everson E. (2001) Independent Component Analysis: Principles and Practice Cambridge: Cambridge University Press Ruchkin D.S. and Glaser E.M. (1978). Simple digital filters for examining CNV and P300 on a single trial basis. (ed. Otto D.A.) Multidisciplinary perspectives on event-related brain potential research, U.S. Government Printing Office, Washington DC. 579-581. Ruchkin D.S. (1987). Measurement of event-relate d potentials. In Human Event-Related Potentials, volume 3 of Handbook of Electroence phalography and Clinical Neurophysiology, Elsevier. 7-44. Samar VJ, Swartz KP, Raghuveer MR. (1995). Mu ltiresolution analysis of event related potentials by wavelet decomposition. Brain Cognition; 27:398-438. Sams, M., Alho, K., Ntnen, R. (1984). Shortterm habituation and dishabituation of the mismatch negativity of the ERP. Psychophysiology, 21, 434-441. Scherg, M. and Picton, T. W. (1991). Separation and identification of ev ent-related potential components by brain electric source analysis. Event-related brain research, EEG Sul. 42, (ed. C. H. Brunia, G. Mulder, and M. N. Verbaten), Elsevier, Amsterdam. 24-37. Schiff SJ, Aldrouby A, Unser M, Sato S. (1994). Fast wavelet transform of EEG. Electroenceph Clinical Neurophysiology; 91:442-455. Silva F.H. Lopes da. (1993). Event-related potentials: Methodology a nd quantification. Electroencephalography: Basic pr inciples, clinical applic ations and related fields, Williams & Wilkins,877-886.

PAGE 133

133 Silverman B.W. (1986), Density Estimation for Stat istics and Data Analysis. Chapman and Hall, London Spencer KM, Dien J, Donchin E. (1999). A comp onential analysis of th e ERP elicited by novel events using a dense electrode array. Psychophysiology;36:409. Spencer KM, Dien J, Donchin E. (2001). Spatiotemporal analysis of the late ERP responses to deviant stimuli. Psychophysiology; 38(2):343. Spreckelsen M. and Bromm B. (1988). Estimation of single-evoke d cerebral potentials by means of parametric modeling and filtering. IEEE Transactions on Biomedical Engineering, 35:691-700. Steeger G.H., Herrmann O., and Spreng M. (1983). Some improvements in the measurements of variable latency acoustically e voked potentials in human EEG. IEEE Transactions on Biomedical Engineering, 30:295-303. Sutton, S., Braren, M., Zubin, J., and John, E. R. (1965). Evoked potential correlates of stimulus uncertainty. Science, 150, 1187-8. Tang A., Pearlmutter B., Malaszenko N., Phung D ., Reeb B., (2002) Inde pendent components of magnetoencephalography: localization. Neural Computation. 14 (8), 1827 1858 Thakor N.V. (1987). Adaptive filtering of evoked potentials. IEEE Transactions on Biomedical Engineering, 34(1):6-12. Thakor N, Xin-rong G, Yi-Chun S, Hanley D. (1993). Multiresol ution wavelet analysis of evoked potentials. IEEE Transactions on Biomedical Engineering; 40:1085. Truccolo, W., Knuth, K.H., Shah, A., Schroeder, C., Bressler, S.L., Di ng, M. (2003) Estimation of single-trial multi-component ERPs: different ially Variable Compone nt Analysis (dVCA). Biological Cybernetics, 89, 426-438. Tsai AC, Liou M, Jung TP, Onton JA, Cheng PE, Huang CC, Duann JR, Makeig S. (2006). Mapping single-trial EEG records on the corti cal surface through a spatiotemporal modality. Neuroimage. 32(1):195-207. van Boxtel GJM. (1998). Comput ational and statistical methods for analyzing event-related potential data. Behavioral Research Methods Instrumental Computation. 30(1): 87. Vigario R., Jousmki V., Hmlinen M., Hari R., and Oja E., (1998). Independent component analysis for identification of artifacts in magnetoencephalographic recordings, Proceedings NIPS, Cambridge, MA, MIT Press, 229-235.

PAGE 134

134 Vigario R, Sarela J, Jousmaki V, Hamalain en M, Oja E. (2000). Independent component approach to the analysis of EEG and MEG recordings. IEEE Transaction on Biomedical Engineering. 47(5):589-593. Vila C.E. Da, Welch A.J., and Rylander H.G. III. (1986). Adaptive estimation of single evoked potentials. Proceedings 8th Annual Conference IEEE-EMBS, 406-409. Wan Eric A. and Nelson Alex T., (1996). Du al Kalman Filtering Methods for Nonlinear Prediction, Estimation, and Smoothing, in Advances in Neural Information Processing Systems 9. Widrow B. and Steams S. D. (1985). Adaptiv e Signal Processing. Englewood Cliffs, NJ: Prentice-Hall. Wood CC, McCarthy G. (1984). Principal compone nt analysis of event-related potentials: Simulation studies demonstrate misallo cation of variance across components. Electroencephalography Clin ical Neurophysiology. 59:249. Wood, C. C. (1987). Generators of event-related potentials. A textbook of clinical neurophysiology (ed. A. M. Halliday, S. R. Butler, and R. Paul), Wiley, New York. 535-567. Woody C.D., (1967) Characte rization of an adaptive filter for the analysis of variable latency neuroelectric signals. Medical and Biological Engineering and Computing, 5:539-553 Yu K. and McGillem C.D. (1983). Optimum filte rs for estimating evoked potential waveforms. IEEE Transactions on Biomedical Engineering, 30:730-737.

PAGE 135

135 BIOGRAPHICAL SKETCH Ruijiang Li was born in Shandong, China. He re ceived the B.S. degree in automation with emphasis on systems and control in 2004, from Zhejiang University, Hangzhou, China. Since 2004, he has been working toward his Ph.D. at the Electrical and Computer Engineering Department at the University of Florida, under th e supervision of Jose C. Principe. His current research interests include statistical signal proces sing, machine learning and their applications in biomedical engineering.