<%BANNER%>

New Approaches to Multiscale Signal Processing

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 E20101205_AAAAAG INGEST_TIME 2010-12-05T06:41:57Z PACKAGE UFE0017765_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 27067 DFID F20101205_AAAFSW ORIGIN DEPOSITOR PATH hu_j_Page_006.pro GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
98b8e9993d617cfb85e5b15059978d24
SHA-1
6f8d843e84e03b7c74c2e6dd3367c71f03bbd64d
36003 F20101205_AAAEWT hu_j_Page_094.pro
da82da9e6d9fa62ca0864a0e7278f033
1ce662909d38335771ace2d12fa32c4f71959c4b
59354 F20101205_AAAFTK hu_j_Page_060.pro
c8dc5a6a4ed4f6677f0a1b3715f24e32
183edf712b67b62072302eff16e575da51df93b7
5170 F20101205_AAAEXH hu_j_Page_044thm.jpg
90b61b9abc70474dad00eff2ec9b2062
b09fc4e4340bf7502573550fd3fa376535faa804
6105 F20101205_AAAEWU hu_j_Page_031thm.jpg
60705500fd358fbd03636dc21eb863bd
1edcc53fcfc31160a5cac0f79ec2ccc4ed8f9c25
31420 F20101205_AAAFTL hu_j_Page_063.pro
9488ebe8d2c90507caed738cd4235862
c5191ff6ebd63f75f540963cebe7b1ae92118c1a
58443 F20101205_AAAEXI hu_j_Page_039.pro
3431c088ed1b5e71b6b1111cd2682923
3a1ab560e6523fb4702e677ad2390289ca371c8c
35271 F20101205_AAAFSX hu_j_Page_018.pro
e9ed6a9495aa056c6725ede86079bb5f
b1d22d9d7d86858d49d0a493e7acecb708c8c1d7
5673 F20101205_AAAEWV hu_j_Page_001.QC.jpg
5c6c5f9453966a867865105f28040181
7bc38a4acab784d52fbbf9b081c80a5b964914b8
2291 F20101205_AAAFUA hu_j_Page_010.txt
e0dfc17ab6a831fc1bee4642bffb1b0c
94b0aea26382e85594cc84095a41fcd351369afc
34458 F20101205_AAAFTM hu_j_Page_066.pro
4d0170d2881e1f5a95f7f1b778a7c957
3dcae085ed48f319ccbe3679a50ae30c5b36e401
51472 F20101205_AAAEXJ hu_j_Page_009.pro
e239bb5cf372db1c00f86186a28da13f
6e01f448cb9244dc9648130c660c8eca67eb9dd4
60667 F20101205_AAAFSY hu_j_Page_021.pro
47f6bd104f31e8a33dfd892abb4869bb
7473f1ad8384a7966eebdf4077c4656d923d6bc4
4663 F20101205_AAAEWW hu_j_Page_087thm.jpg
5a3f40ab2cf7292e53a69a9f5c0a6773
3dd141bda253f8f2164b500f2d5648b513688b60
2487 F20101205_AAAFUB hu_j_Page_011.txt
4af7779974641da2eaa19f51ed13e003
f6ec952e3b654a5ca00afb2e74fe9d646ee65d4a
61145 F20101205_AAAFTN hu_j_Page_071.pro
d1128b5029b8414d986233152cc5ac67
94ba249f518b828acfc319d6eaa962c7d77ac1a5
26766 F20101205_AAAEXK hu_j_Page_039.QC.jpg
ce2d83b385dd26d4ca23783591a93660
45e05e789d698125759d2b1bb342598392548dd4
53875 F20101205_AAAFSZ hu_j_Page_029.pro
4ef6eb070ae85530dfd35d39db82c9d6
d9a241e43be3ad0e83443f86a3e64e5481fc34a7
1716 F20101205_AAAEWX hu_j_Page_052.txt
859ef3e2dcf8a12139d4df98760e685a
fc45af66ecabdda4a7e2c87371ff078c6ce11e8b
1700 F20101205_AAAFUC hu_j_Page_012.txt
33dc9e9a2a47003f6a6e645f4529d487
dd160d7df2d23157babd08ec0ec1c13abdcf4aa2
57142 F20101205_AAAFTO hu_j_Page_074.pro
18be0f54c3d666afb6c0d24d70b123e4
c68c17928bcea449a666bd183cd76a41dfb4de59
4280 F20101205_AAAEXL hu_j_Page_032thm.jpg
741617cc07f59fee2da652e330cac4ef
0a92ee43208799f30718b1647cc1a3a88122d076
492 F20101205_AAAEWY hu_j_Page_032.txt
ba9290d038907d356893b0886b73160e
93108efabd0c7fe9b5eed080f542c3cbde8ee782
1400 F20101205_AAAFUD hu_j_Page_014.txt
cbf578f8af6c20e0da85e48cd82d168c
680c7ed6f9b3a90aae3624527644a614f4ed9508
1051975 F20101205_AAAEYA hu_j_Page_083.jp2
c54d2927295756a3c08f57a336531ea1
02b1cfad5432059a63a4906b1ee1418bded3b9d2
60343 F20101205_AAAFTP hu_j_Page_076.pro
31845fcc54fca7eb4f3dd11e370943ea
752217312b4100245921d82f043d388db9286ac3
1496 F20101205_AAAEXM hu_j_Page_079.txt
b3ae7a82c482d4b3bbfa213e49bb961e
a7808ee9cb12bc7ba930dcea642da131466c5e00
62865 F20101205_AAAEWZ hu_j_Page_012.jpg
e47d228c04db90b7a43d0d5a6a93ed37
a1979a97a96415409038ca00d99648928cfff034
1438 F20101205_AAAFUE hu_j_Page_018.txt
7c98a88b522a09557ffab7d5039f2284
45df0a507904e61f47eb8c11ae674e254884aaf6
93085 F20101205_AAAEYB hu_j_Page_061.jpg
18358c2075a942ce113816dd0f1a9fcb
9d21a9a690e4c1dce3351535f2200e0ba52433f9
30082 F20101205_AAAFTQ hu_j_Page_081.pro
67891733d655c372ce62dc30377237b4
fe304c94df2d43b646a90c676dc572a8f582fd90
44608 F20101205_AAAEXN hu_j_Page_043.pro
547e4d7be34ed46990e70dc0f1d19a31
4a2d7d3db0909a69fe517797b5d704e06dfbcb34
2141 F20101205_AAAFUF hu_j_Page_022.txt
fea018161afb960c2588b1f2b94bc2be
f12995abc90f1bd93d7f94f57e734429e253730e
1911 F20101205_AAAEYC hu_j_Page_019.txt
27c6453f90e47077525aa2db9262a81c
2f6218e4baf16667ed77f13ab06bd86a146fad1b
53127 F20101205_AAAFTR hu_j_Page_089.pro
4f4cdf653d3097d5acc24c9c2d39ec7a
5633953d7b0f39a52d495a9af7877478ba02059c
58812 F20101205_AAAEXO hu_j_Page_006.jpg
9ac32e2076b04b71fcd0354200957a35
9c9687758162f7f0f91fdf3f986eee6cf5fc54fd
2381 F20101205_AAAFUG hu_j_Page_024.txt
3c9128d7d4c1d97284c78972aff544f0
3ed29cd15e96276fdc9dc7d4d4b0a716f602ed01
523788 F20101205_AAAEYD hu_j_Page_048.jp2
554e22acea288677964e1a7df6b4ad57
413f9b1bd6e970fa58e073e2d729dfd5ec0dbba7
54911 F20101205_AAAFTS hu_j_Page_091.pro
1f5c71c1e1f59a44b5e5c776e7a6e5a5
6614e2020227d40ce2bff8d6d138d0b435a22bb5
5490 F20101205_AAAEXP hu_j_Page_066thm.jpg
d996eec864f97806727b76ff243cfc8c
b785e22a9bdb66fa6e9328937fee6cc57580c5df
1696 F20101205_AAAFUH hu_j_Page_034.txt
8b70c30c8531448947f0d84b58203156
6ae26ea4956fd6c50d7b19b4d55da362022f2c0c
25271604 F20101205_AAAEYE hu_j_Page_086.tif
4193334f2d7421c17c333b60b87979c1
24d8c5cb416e9e2dfad8ee21b44217c83e6896c1
34342 F20101205_AAAFTT hu_j_Page_095.pro
5b8eca5d212d9bfceb587a5b7c064bb2
48104d19dbfe1ea468f3c234afbe8fb833734cd3
64052 F20101205_AAAEXQ hu_j_Page_034.jpg
e86b0834573e3e802b96cea77dd86442
a8e0ad197e8c821045b866be289102381fe4465a
2313 F20101205_AAAFUI hu_j_Page_036.txt
9d9c62a8c40a1f6b8f12babde6cfd4ad
4c41fd04f9f72af5159011bf6338f47400bba19f
2389 F20101205_AAAEYF hu_j_Page_106.txt
960fa71a1ab014d4ca3aea48103ebfaa
b257fab0d9066abed738d0cd014a702fad38f0e4
54672 F20101205_AAAFTU hu_j_Page_098.pro
9626d8df30c005461d87e7c7638c0840
1e63235f7dc765baa7e8dc9fcfc6f85a6addb877
2308 F20101205_AAAEXR hu_j_Page_082.txt
eacb7bec30af985df1864a5853c034fa
6322052887cc6753cc4f1f58db05f2f5b33c7002
2356 F20101205_AAAFUJ hu_j_Page_039.txt
eea43dd6b4b5cfb8d0582a8a4d814bdf
db0c24a0aecfa9ccbf526887614a685e6b28f9bb
5600 F20101205_AAAEYG hu_j_Page_092thm.jpg
12ad2c0eafa8e7e749df050effc01432
dc707ad0b016c056e38f8f12e9e1a700ba93b109
28351 F20101205_AAAFTV hu_j_Page_100.pro
5ad1a38376a59a7ddc62b8231ea75659
7aedf839292507efb8fd5f1878df484c21739567
94540 F20101205_AAAEXS hu_j_Page_077.jpg
edeee4dc9705c669bf0114de4f988357
44d02805486b5a7fbd9d332df349cd5bd94b575c
1582 F20101205_AAAFUK hu_j_Page_042.txt
ebe360862fcfe9c867bc0c0d05f8ef4b
92b2e00469c419d766d037840ff88a3a1b66f63e
1051971 F20101205_AAAEYH hu_j_Page_077.jp2
dc83205ddc3ea3f0385d2b7c384e4dab
6f27c4e9145b14e0999f80b6bdf776ffa8981e22
35962 F20101205_AAAFTW hu_j_Page_101.pro
b8f0c3f394647b31a8036c86fc879581
ca06c3f70c86f4aa3ca64d7503d3f65b491aa78d
1051945 F20101205_AAAEXT hu_j_Page_035.jp2
c0c553c0cb098bf1dd3676ff77a1bff1
09e49656921d54c7a4b67f6dedc5cd3f87f44cec
1817 F20101205_AAAFUL hu_j_Page_044.txt
6c896bb3a4a36869ea067e5852ec69c8
259303c05a52ed4cde2c19c14d8d5e663424956c
2059 F20101205_AAAEYI hu_j_Page_013.txt
3df288f4bf4a3a8c58b9a71124ffa8df
66c87476abbd2f42e673ebd05d3de4295257a78f
30296 F20101205_AAAFTX hu_j_Page_103.pro
76c1cee03c9f09908e55834a5026756e
7228f77e05408827d0ad10583ad460e50f4e380c
2305 F20101205_AAAEXU hu_j_Page_108.txt
3452ba926dee15180df5239b6a718ca4
39ece21100ac32118bf1ef11283c213ddbdfe6de
1076 F20101205_AAAFVA hu_j_Page_002.QC.jpg
7a8645558280fdc1361d239542121eae
39a798a36ad0de1715b7805ffcd45a0c28b674b6
1841 F20101205_AAAFUM hu_j_Page_055.txt
5552f34bb3b1ea24274927f909ab1736
c83f69d0a0ddee45ea87847f5b7efda61c22890c
45142 F20101205_AAAEYJ hu_j_Page_013.pro
62430275395e94b1bf676a1a52268bdd
dbfbe29ed57b93d9e0e376703748f4c3a0eef66b
67734 F20101205_AAAEXV hu_j_Page_072.jpg
bf376da80a771e800f58529903390229
069322bbde0c267a433b5a53ddd8ca354c6c4eb0
5646 F20101205_AAAFVB hu_j_Page_007thm.jpg
e5e7fcd95120ba02da6ef57d63b88090
586e5cff833a548b9d572c88cfa52a2eda39c5ad
2095 F20101205_AAAFUN hu_j_Page_056.txt
47e1cae169ce8db79dda0377041abeba
d9b4edf2ac976bf3104ba30ab2115d299226b252
48039 F20101205_AAAEYK hu_j_Page_097.jpg
e3c723131492cfe966766901e96f9130
a204c65726a03e66ede7ab6f9f123d7a0c5699f9
1619 F20101205_AAAFTY hu_j_Page_004.txt
191a6d3360e77ecfec924d960ade6077
c0bbc39c22668f7ebf5027833a6ea77899f9e7ac
F20101205_AAAEXW hu_j_Page_027.tif
2a4f9c73254d25a69f6b55d3276a1851
dcd6f20b26653e9259571e112014ba4c137c8c1c
20913 F20101205_AAAFVC hu_j_Page_008.QC.jpg
a4d936fd4e1e8c5b0da26d51e94c3ae5
bf1bf3840df9bbd2b60204811a062ff79b46f769
369 F20101205_AAAFUO hu_j_Page_057.txt
ca11ab10223e925a71a931ba3be394da
956b2211c500d6d01cb663f59add0a9db193461d
55404 F20101205_AAAEYL hu_j_Page_015.pro
93119138ba7f80bb3de8ca3f24773687
38b5f9fe4fe8d698b3a5b6ebe9b554842a419f15
1218 F20101205_AAAFTZ hu_j_Page_006.txt
2499936b2ba67d56bc18b2dac1094ff2
d9c46f71a48ee2862225edc54074a6ab9b35b0f3
1867 F20101205_AAAEXX hu_j_Page_115thm.jpg
ec50f5bf113b56342c864623fbb313c1
69a943709ad5a476cfe4e5c255a2c8a7f752b562
5198 F20101205_AAAFVD hu_j_Page_009thm.jpg
241d19d89e8e218bac50ff18283887ba
a415143564792f6d63716cfe58979b71d0ed9d94
97568 F20101205_AAAEZA hu_j_Page_110.jpg
eb770b25b65d147544509a93a9a02816
c4513ac8a5b94b93b358e769783f4a8ccc9c3a38
874 F20101205_AAAFUP hu_j_Page_058.txt
4ed68e4a908c11656afde582b66ce83a
df04472bcd57f8fbe434815d29773b86eb6d789d
2399 F20101205_AAAEYM hu_j_Page_060.txt
b676de6e022bf14e680e0ebbfaf16073
c6482ff0954de3938d1b81a1b9655ea525a5d309
52469 F20101205_AAAEXY hu_j_Page_049.jpg
3370c1d6fefa9eb8f44c053fa71659fd
00af2550d65f2be6569e3b53cea1615e4378b36c
18250 F20101205_AAAFVE hu_j_Page_014.QC.jpg
ea4e2a9113ab7c97c61c9524a25a1e93
0ecbe1923d549202c6f9eccc8aa581c6aacd50fa
1272 F20101205_AAAEZB hu_j_Page_059thm.jpg
cf2cb94f610a68e5ae8cdc991c14bfaf
46b2ab48dbd7f98cae96128a614dd0f5da2691df
2108 F20101205_AAAFUQ hu_j_Page_067.txt
23db198937b8e7e2296796c0348d16d1
700e7ed403eb2ab366933eb6a5f67a4d51c6d2fe
1051977 F20101205_AAAEYN hu_j_Page_064.jp2
ffaf9510c2c2906172ea322e338f8b93
1838cfdd69a53140a114737e9aea7eef6da1e165
1051974 F20101205_AAAEXZ hu_j_Page_078.jp2
d81dfa3a1041279cdc12958da72f7cb3
23085cc3ba16d333c504ae45ed01c1b562eb87d2
18719 F20101205_AAAEZC hu_j_Page_012.QC.jpg
d40d3166680ac72dca589038c0efdd7e
c4e2e7f2967b2f601ff28854acfb6c8ed842e5e3
377 F20101205_AAAFUR hu_j_Page_075.txt
f02410c65f37f3542466ee5f75273d8c
0a80f1dc8d4e27379d7c9678d6f60350d837c434
2207 F20101205_AAAEYO hu_j_Page_091.txt
45d71797d0fd87780af5e7afb7ee5e03
ab4d3716186e18b887938d8ab72e117ed1ed38f0
25717 F20101205_AAAFVF hu_j_Page_015.QC.jpg
ddc8155d0953b4fdad92577806f2c44d
e7e68921701a237dee465258715cb0e6945ba43d
1053954 F20101205_AAAEZD hu_j_Page_001.tif
ecaaa7af525c53aa67bc38d60d6e4c15
70b92fae77e2a75c57047f31f72802d785d23307
2396 F20101205_AAAFUS hu_j_Page_077.txt
6d71413940340dd5d9a8a7b448fac950
e248b8698d0a695ed91c7df904f1562c86836c8b
21273 F20101205_AAAEYP hu_j_Page_096.QC.jpg
a64bcc4ad7181f83beb75b797ac61eaf
0bacfb57abc20c9bc05733db52c581bbaf185c0c
26129 F20101205_AAAFVG hu_j_Page_016.QC.jpg
1a8727af6fba1d5720ff0631cb5cbf98
c13c2910df1f82c709dc3a7278461c50b72f7dbc
26485 F20101205_AAAEZE hu_j_Page_079.pro
ab4a19c1a163ca3fa40f71dc936bd6fa
089f4e33d275877ef4c139d48bd32539929196a8
2333 F20101205_AAAFUT hu_j_Page_078.txt
18f60236046f73f83b4c1f9213cd9842
d204f73a6297f234e27cfcf0610fcf7f33df12f0
24268 F20101205_AAAEYQ hu_j_Page_092.QC.jpg
57331e6898d1b07c1f657ed57c4dbca8
f1218a287ed0b85f11e0cd81dd131a829fb503f1
5416 F20101205_AAAFVH hu_j_Page_018thm.jpg
63fae37de542285f93d45e554d2235c6
7a7d322ad57f9621c37dbb8015bf7e5b4414847b
61108 F20101205_AAAEZF hu_j_Page_062.pro
10bda61c1abe53ec7b8f723df4754c35
05e3f278cdf8d639142ad1681f3e6f3568dea9e4
2210 F20101205_AAAFUU hu_j_Page_089.txt
95994990d096b9e637db9706224e2502
bcf721fe3f4d01700b20fdf085db0957e936940d
1051984 F20101205_AAAEYR hu_j_Page_082.jp2
c9892306c0c4ecb63931bf83f4dc6338
8dd4b73dceece087795ae925dab96250e6aee7cf
23056 F20101205_AAAFVI hu_j_Page_019.QC.jpg
a59d941ce4527b2ca228c86297b98cdc
b062b74422b5219c7aaff07fdae986122b739bd3
740 F20101205_AAAEZG hu_j_Page_097.txt
78515ac09b5734ffee4b5b004e9f8443
78095f6f10d1f526c21910d9e717a8ed3e925c12
1513 F20101205_AAAFUV hu_j_Page_094.txt
756e532fd7de73324762002607ddb6e7
267414e20c2b96e1abf81e85540af5d807ad8df3
6823 F20101205_AAAEYS hu_j_Page_026.QC.jpg
5d3dbb12f4be92f22795962b8b459267
ec628971cdd47ab0a9315aa6e944bee5963a0f80
25967 F20101205_AAAFVJ hu_j_Page_020.QC.jpg
1f15f919aed39af6b92cfbe1986a84cd
9a026134c918a3845d3be697288fc810f4f5e065
51202 F20101205_AAAEZH hu_j_Page_022.pro
7fcd95e4af49b3ac82585f38c4571e0d
08857f74fe89432051e4bff0a724d90a74cfa2a7
1535 F20101205_AAAFUW hu_j_Page_095.txt
bc54b5a5996924aa5a40d93f5f4b7058
66932172acd0ce0f317ce18055af91d1f530206b
1051982 F20101205_AAAEYT hu_j_Page_008.jp2
d9daed8dfa0a7ebfd5b1c82efa407b5a
1ebdea80ba39b640b4aa0fef647079d5a33b79ce
26122 F20101205_AAAFVK hu_j_Page_021.QC.jpg
f8a0c7ae82d435f446976533127eb310
cd1c9d9784a6bfe6fc9e08d2615a9ed3a7bcbe15
61328 F20101205_AAAEZI hu_j_Page_081.jpg
d8549b07b1caff4e4043d1545b94e2f8
4d97bc4d8fd99b59d1e541b270489e8b48f2a7ca
2397 F20101205_AAAFUX hu_j_Page_099.txt
ed537249dfcd03876bce8aa4fb32f031
effbfc2eda1c1c5ebeb69a616f84009bc80b3c19
56522 F20101205_AAAEYU hu_j_Page_031.pro
fa094a319d8a71e1f3f1d2da7f51bb3f
37f0adab8179b73f978e9c0f456ff8e5c7dcbb79
5753 F20101205_AAAFVL hu_j_Page_021thm.jpg
6e22344b415a8b6e5b9932372e366a1d
3cf13176ba1172f6c434da38535b327278a04b5d
23653 F20101205_AAAEZJ hu_j_Page_105.QC.jpg
1ea056ec0b875168bd3aa12b2babcf62
c3319d0d6543d310f58a3ab97ef8e69169542b00
2706 F20101205_AAAFUY hu_j_Page_113.txt
5665b1cf75a2ff969fa9be5281b6c8fc
b560fb5d8fa87d93a438305163440007bc93036d
89443 F20101205_AAAEYV hu_j_Page_016.jpg
b171eeff14ddecebd123bc0bf7dfaede
452c37ca3cf343ccd7478413fa006761e881565e
15453 F20101205_AAAFWA hu_j_Page_053.QC.jpg
b269e7162b04a9e38bf28ba996d3b7f0
d2ae8491eedfa310b2d13bd16605ed72703d84e0
24338 F20101205_AAAFVM hu_j_Page_023.QC.jpg
d37e88c69cd9da81d0460ae40a9afaf6
6a92ab3edeb1b8fcac110de834e18a710d1c11b5
F20101205_AAAEZK hu_j_Page_107.tif
872f2ffefefb2a0b564deddc67fd6657
3e39d460f6e3618ff88917b59372e32891b4808b
2355 F20101205_AAAEYW hu_j_Page_061.txt
6b7dbd51626c44db503075ef6bccb559
0d3d426696c6b133477e849a914ca5da595db2d2
23747 F20101205_AAAFWB hu_j_Page_056.QC.jpg
ea94502820c61c4157327d1d45796040
4d384287f4d3118d4277a513fd1b47f34731a98e
5930 F20101205_AAAFVN hu_j_Page_027thm.jpg
1a9263bff386bfab802edff0fd417ff4
e239b6e05941a95acd1be19fb5c7e4fd2c54c0ce
15783 F20101205_AAAEZL hu_j_Page_088.jpg
4ec8adf7c82d6d1e55b388ba4c195aa5
81098bfd5c65daf0ff19b9aa22888097f0f1fffc
1328 F20101205_AAAFUZ hu_j_Page_001thm.jpg
607dbef615171d646ee5e67922a2a7ff
898e7f9255736bb38ca44c3da41f01c6429f95a2
F20101205_AAAEYX hu_j_Page_047.tif
6ad0d0d6410fd4accfac1daf4008e555
c7732b28c34883efdc2497e1b2892b4945ffcde5
3179 F20101205_AAAFWC hu_j_Page_058thm.jpg
286210296dbfba30e24d73fcdb10836f
dd86432175b14d3097a2faa9e29d9dfc4327bc9a
24137 F20101205_AAAFVO hu_j_Page_028.QC.jpg
ccd9825fc83c4a0338407e7ec2ce69c0
3c6b853713b95d38d4780793b2f3ee59e3f7740c
1254 F20101205_AAAEZM hu_j_Page_072.txt
5da17ee27dec32c7f2de2e4f70c329a8
6510f4af4d46cc867e4d83eb3409c399a2a2120d
22024 F20101205_AAAEYY hu_j_Page_043.QC.jpg
f6c735113a097c837bb9219b171ae14e
116b62c1c74d7b5ab9a343c9428b3352db94e90a
6088 F20101205_AAAFWD hu_j_Page_060thm.jpg
eb89df667036a31d4ff7910896828432
97184b4861c51a07f4b0cc30483351e6c8a4e99f
5491 F20101205_AAAFVP hu_j_Page_028thm.jpg
87173a044271bd5fb69553bba0da3af0
60e69e148f0bcab73d2aa2a1fe5dc190a3256c31
353308 F20101205_AAAEZN hu_j_Page_058.jp2
d7783759177b40933944c1edc182ee23
33ffbc324ae9f6fb20885bdc68e0d3b67c23c61b
28182 F20101205_AAAEYZ hu_j_Page_071.QC.jpg
691e541795f3a350ac1feacfb66ca3dc
d37c845530c33e8f1ca32c3c5c6a4a4ea2862d9a
6303 F20101205_AAAFWE hu_j_Page_061thm.jpg
6d2c4443e3529791465271ee6b05b488
7af7f65608a4733009853a84ec8a76173ef1ed8f
26795 F20101205_AAAFVQ hu_j_Page_031.QC.jpg
0710ce5ecaae5566a48c21ae98a65e96
523e5c87298fdc951a7e943292ca05abc63b657e
24994 F20101205_AAAEZO hu_j_Page_029.QC.jpg
d638dd3a9dcefb0d90af536a4a8ff6fd
7045e6015d9254e8c3d8d33d29f0a4b3d52628e7
19073 F20101205_AAAFWF hu_j_Page_063.QC.jpg
9326ae93750a1629bf9535fdf3be9940
eb3b3c922a682ef1d428e1104ff39800ed9bbbe6
19539 F20101205_AAAFVR hu_j_Page_033.QC.jpg
8aad52c320ac916f8115e5167209919b
72c57fb615fff4d8c9b4919a5d8c78e24a2e4e7b
5126 F20101205_AAAEZP hu_j_Page_090thm.jpg
92f25fa570ec95835fad611dadb9848d
0d9d63dab6ee881afdc234321741d7fa132001be
20845 F20101205_AAAFWG hu_j_Page_066.QC.jpg
3529848a9da04e3ad3e1770ebceb1f78
e84c5fa52b692c9a18cda3566a20e3003c37af42
5053 F20101205_AAAFVS hu_j_Page_034thm.jpg
b5b8b1861dc1b48a5fe56a70dfcd3a9f
c7d844e8f35d941be5de26af1e16dbddb4a51f0f
6161 F20101205_AAAFWH hu_j_Page_069thm.jpg
f9d42a7f54185b62a3a2d422d498ead7
5bb46999c0570f76773cfc0f41080262dee2f2e6
65466 F20101205_AAAEZQ hu_j_Page_045.jpg
5f5958888217340fbfa9f2a55ca676d7
798a5ea1d24484e495b5b94596b4ca3e745fc206
29672 F20101205_AAAFVT hu_j_Page_038.QC.jpg
9d2ad90a5358353858485feeb4f97bec
af850ef786e8bded7a3e22908abe8a43e5ca2aff
6255 F20101205_AAAFWI hu_j_Page_071thm.jpg
5cee2a3e545f1d50f89ac78bc7754240
4b11c31276292631dc1c44c97ff8ba39dd6669bd
F20101205_AAAEZR hu_j_Page_079.tif
5d1d6d06924882bac09a73f87c364e7e
e7d1f1b35edaccedafc6f0803ca611677487fc61
16328 F20101205_AAAFVU hu_j_Page_042.QC.jpg
f75522793995b6faaaa88c1278cd5b54
8afd0d7dcef02d9536216a99d7c1408946ce75f9
5136 F20101205_AAAFWJ hu_j_Page_072thm.jpg
a767ee6fbd15bb64857395969939eb7e
d450da0a0622af21638f394e6dea24c3ac00dfc3
1051983 F20101205_AAAEZS hu_j_Page_006.jp2
a4193b689092fc93b9c6bf792dc6053c
eaca569c5f1d2bb81746d6837646a93b270bfb40
3925 F20101205_AAAFVV hu_j_Page_048thm.jpg
ad13ec0aeb6dbf4231be130e7ef49d98
fe9f7219635a611ec3072a41c27455344c27cf14
5143 F20101205_AAAFWK hu_j_Page_073thm.jpg
dabdb8763265779e80e4504f886a3a5e
b11143d43649e2fd16a9d5b388b6fa90fb0be5cd
671650 F20101205_AAAEZT hu_j_Page_079.jp2
24795017846cfe4ff8affdf598f97468
415ff15ee76a8cef5bb6434f5af90211502a0018
15283 F20101205_AAAFVW hu_j_Page_049.QC.jpg
1629ea622a191e4e0b0d5740e4457669
bbe54617998779dd8e1b14fc0349c69c18986168
28216 F20101205_AAAFWL hu_j_Page_076.QC.jpg
45d0c4fd73ee98efbb9d7e6e292f6ac6
70b09ac283f3ed81610bbc3c4352cd8f4c2bd9a4
3983 F20101205_AAAEZU hu_j_Page_040thm.jpg
620e37b152354d6eaf730e48fa6677b5
0511e2613e4d9a28b2c980e00e1a7733e3aa82b2
4809 F20101205_AAAFVX hu_j_Page_050thm.jpg
209c7999dfb09a7ee42ad670d7305b8a
f73cac990f78e9660f1e3f386b96b8fd1c4c4067
6045 F20101205_AAAFXA hu_j_Page_109thm.jpg
34e8ad63ca95e036468731565ba97d14
40a6e9eb36d8db8757168ecbc7db9a959d9953a9
4394 F20101205_AAAFWM hu_j_Page_081thm.jpg
c80281165d53b7457cd653327ae45d48
1e35715296d88e44ef98207aec857bb51a31ebae
60593 F20101205_AAAEZV hu_j_Page_099.pro
e0ccf776fa41c480687d887438974e9d
8060702229a47ccc552c2f127537f64a44ca6de0
10567 F20101205_AAAFVY hu_j_Page_051.QC.jpg
030d4f14c4f53ddb083a0080a9855920
3569ccdbdcae3a896ffefc75c6405032440b1009
24708 F20101205_AAAFXB hu_j_Page_111.QC.jpg
d2d2a3fee8468278161cc1c2b4e1627d
2984eb0d85e1d27d3e64d8b4a2460b12cf13b432
27509 F20101205_AAAFWN hu_j_Page_082.QC.jpg
6e00abb1b4bc5d8ce047387ffe580761
a1bd86821a97cdc1d2be122886cce79ad722fd99
6208 F20101205_AAAEZW hu_j_Page_077thm.jpg
705d9561845d14c0a6d34a5d364ce39f
f52b23d0bc7f93aefddf9487cc265818aacf93f0
3340 F20101205_AAAFVZ hu_j_Page_051thm.jpg
862a6cfc8afb3b5e73ec3a04851cadca
61fc942a1d3752fc8a016840fe648785160e56a8
5854 F20101205_AAAFXC hu_j_Page_111thm.jpg
de7ce3e60b6e2c1e4c4d0a2e4e3e499a
038feac6ed263ed195b191ed14dfb477d40b0922
28103 F20101205_AAAFWO hu_j_Page_083.QC.jpg
3aee24cf4310a65773f3b61ab1a1fd51
af56560371283cf1dd4afcb20521809212c72910
1037639 F20101205_AAAEZX hu_j_Page_043.jp2
bd9f0c57b9df2bcd31a55fd929a8b271
01434d61c060362ee1e523a7198105dea47d8e6b
25145 F20101205_AAAFXD hu_j_Page_112.QC.jpg
6118056b9ef2c8b3809a8a30bacbb78c
d6a5130e9588f817e6c385564e4e83e954e5d895
6197 F20101205_AAAFWP hu_j_Page_083thm.jpg
a38026566090d06e656a0a77906c00b5
b963e3b8d0788121879ad842d0f27de3446a0a93
2500 F20101205_AAAEZY hu_j_Page_110.txt
4c28ebb6feda27eb059e33afc64364a0
92fa3f2f03715e337cdbd24800c6f4b5d73bc3ae
131147 F20101205_AAAFXE UFE0017765_00001.mets FULL
aa60e60b0d292bf1c002f1837a3d6193
3ba7d1c43791ac35a93e172ac249659fb10f267f
4580 F20101205_AAAFWQ hu_j_Page_085thm.jpg
12568d88e3e195fda3ee30a43192c324
0267fb3fcd855246dc41cdf004f29812166df5ca
19785 F20101205_AAAEZZ hu_j_Page_001.jpg
196b0479a2bc13fad3ce95738251ff12
4727605e1c4ea425dcb4b69d9dedb9e3d5df66ae
27297 F20101205_AAAFWR hu_j_Page_086.QC.jpg
dc8c38e94d4cd90e5504d051f94d4541
f35bc1368e5609257b8682043c6c4792457fa95d
5028 F20101205_AAAFWS hu_j_Page_088.QC.jpg
775aa53536bd633aa80f199bf7a5e372
afbec18d8bf4315423c8ada86f318aa823a8ccd5
4557 F20101205_AAAFWT hu_j_Page_094thm.jpg
71a1cefbd797651046420956b76e9916
ab0be11c38fa25cef228576e6b961437abd53288
5971 F20101205_AAAFAA hu_j_Page_110thm.jpg
5fbbc582c7b815a0fe0c6449f6ce32a2
f498bb225c365e63226445d9f09b00285e5058cd
17550 F20101205_AAAFWU hu_j_Page_095.QC.jpg
fcda931d443ae4716abe640f5832ff3b
062d12423efe156a41fed45afa43cec156dc78c9
29981 F20101205_AAAFAB hu_j_Page_072.pro
e3f81ba468f0aac710719655d84cc8ac
b1dc27b9fb5cac4e98fa28d9d17bb98918e46e05
4297 F20101205_AAAFWV hu_j_Page_097thm.jpg
701b4238a09812d13e306356a3ea79c7
62e57f89d211da961981ade5e463e9441e06e6a9
1992998 F20101205_AAAFAC hu_j.pdf
0e0ced08cfac78b7b803e2cf95d551c0
d8744612ca2d72bed522cc89bfe2b39a4f4ae929
25891 F20101205_AAAFWW hu_j_Page_098.QC.jpg
315286fe5431f349cf24ba6b221bfe98
a0b23346d88d0425d35ea63bea165e1af90fa5f3
2393 F20101205_AAAFAD hu_j_Page_007.txt
f3979c1069e9443539b3879e7370665c
5d552d30334d41e024e9cb80bf0feee2ff634e0b
5828 F20101205_AAAFWX hu_j_Page_098thm.jpg
560b5c8e26aa58e42295129b4018aab9
3391270a934898eb4da539be8dfe5bbb88f12b7f
F20101205_AAAFAE hu_j_Page_017.jp2
8a726d7d68847485b26685a7acf59b25
7d4d7e2ad969609261b23ff8765ef03a108b6e0a
4142 F20101205_AAAFWY hu_j_Page_102thm.jpg
bcada386aee55127e12c6ef189896554
5852cc0cc051c46815f6c145eb7d4b40f4cf2728
26062 F20101205_AAAFWZ hu_j_Page_107.QC.jpg
fb06621afff6e9d5f8f2967ab12ddac2
ef947572288913df9b3b23d3a6a5488152158692
41796 F20101205_AAAFAF hu_j_Page_068.pro
ef0022b5c0d812905554c8fa17fa9c75
2d1c03eae6cf290c2ed4e4cd523181dcc055dd77
F20101205_AAAFAG hu_j_Page_087.tif
4a52e48ec743ace6e994c9d5f6906b5f
f8e3465d0aa9da2b1377ce1185979e3b08d64282
677642 F20101205_AAAFAH hu_j_Page_049.jp2
8e1c2564082c2a2427ae3dce106c2982
ce481fbd898d6da9e42e9b3981a6325090c3b1e1
89377 F20101205_AAAFAI hu_j_Page_039.jpg
0a4703bc5dd878f3b431081e01b85f14
be44f449350cc0761d074db671cadb7d4b8bf1af
25770 F20101205_AAAFAJ hu_j_Page_007.QC.jpg
383e481e8a8789415fc6537406aa7f4b
9827d8c2e8660dd62cf2230129a29db53467c213
902869 F20101205_AAAFAK hu_j_Page_050.jp2
7dc1cd5b2c14246b09a88eae761e5055
b3b9aa36342b36f686d4a21207c30ae7002608d0
5929 F20101205_AAAFBA hu_j_Page_067thm.jpg
964f05198cc5cf46c394e3299f9689e4
4ceafe2941eae8e036d8541663e453ce14441d31
4530 F20101205_AAAFAL hu_j_Page_047thm.jpg
c4a5193d4390f09f7a3c1369aaf341c2
8a03431068f1d06d2af836b1478f7fe5557f8520
F20101205_AAAFBB hu_j_Page_019.tif
35bb2119b923ea0d0ff1cdf3bf19d8ee
35bbbd0e68ebdcd1a9e8368ed31abd75ae1a207f
35442 F20101205_AAAFAM hu_j_Page_014.pro
4870c5377e828ab57a5a9d1940987a6c
68a64c9a60dfdba7eea3128782f009cfd340b6a6
27833 F20101205_AAAFBC hu_j_Page_069.QC.jpg
ca2fdc39d556bcb6c04ceb7c3ee25f3c
4c5be902beca1927a6b302630d50aba0580d8379
970631 F20101205_AAAFAN hu_j_Page_090.jp2
3beb2956811e8d69d70f93965bd02944
e2f0c5e902d613ecce04ccdc61590696a598c841
6188 F20101205_AAAFBD hu_j_Page_025thm.jpg
0eda5a1e7ce6fbef8edf49646ead57f9
7a15cc9291c949549463056d4958f17680b52bce
77303 F20101205_AAAFAO hu_j_Page_022.jpg
6c5c9bc4e09906c26039a69b0ac889de
a2dd6064f8b62a59de7f0628b4466f6793d0d808
2227 F20101205_AAAFBE hu_j_Page_016.txt
f6b5f84f49cbc21deff65fe3453ae317
fc972a93200609c9a16153bfbf25e3f1d32b2550
1593 F20101205_AAAFAP hu_j_Page_037.txt
e96523b4997140e0afaaef192c875919
d21c1d89872db809386f020bf50a46e6c82a88d9
59382 F20101205_AAAFBF hu_j_Page_024.pro
cc22ed22ae1c3b3c4e4093cc16e2291e
56e001efb97a8912ca66691a9aed75da52fa5542
62338 F20101205_AAAFAQ hu_j_Page_110.pro
389629e5184df5b397b35dc741a77a6e
78088624f8d31fdb0662b7ea25bbb8cd0f3f3e40
22151 F20101205_AAAFAR hu_j_Page_055.QC.jpg
11a19a889ce8adf872d66aea8380ad0e
38ac7cf232481e75df61fffee0cc61ae8d30f45d
1051925 F20101205_AAAFBG hu_j_Page_068.jp2
894e0fd4f54fcd54eb959cd4b56391a0
601400e934605449089d34236e50d43865316b96
25667 F20101205_AAAFAS hu_j_Page_035.QC.jpg
131209e9e21332695454d9091d8f6217
68ed54c596cb5b9fbff30f71066ea1ae6c8090c2
2427 F20101205_AAAFBH hu_j_Page_107.txt
a600c7748c875e061d96b23536e83a02
b8a4b97413b7c3b341b689cf1c575f965c8a2c76
F20101205_AAAFAT hu_j_Page_083.tif
6b629069a9559594138bd2c34f7dd54e
7a5555addfa74a74ca89d4d1ffc8f836956d7574
1000788 F20101205_AAAFBI hu_j_Page_093.jp2
084380fa71dfeae8481ff5d58f5a80dd
194c10d30a565b3b8b5f04fd3fbcab8ce00c7f4d
28031 F20101205_AAAFAU hu_j_Page_060.QC.jpg
b77bfbd064f9feac2c9fc197c327ecc3
10310d431cd86eb2f5bb8e8c3c28f290ea9cf21d
4945 F20101205_AAAFBJ hu_j_Page_037thm.jpg
23d5bf87b1347b1d72742dc422619bb2
dab4e42fc589d17c4c79fbc11761e5a1f80060c7
1051952 F20101205_AAAFAV hu_j_Page_099.jp2
f080848560a63370496ac80204a039e9
41ae0b750c5d56a84cb76d780f9d853dc8c292a0
618 F20101205_AAAFBK hu_j_Page_002.pro
943c2ad9e74b6a51840ef32f8afefc75
e3db317029b8cdeff8e77f27bc613b3f72f3047e
38718 F20101205_AAAFAW hu_j_Page_045.pro
57b34666d8cadbec9ab508ece87378f3
b5b5d91ced9c2e65035cdb98851b1c8a2a8e27dc
F20101205_AAAFBL hu_j_Page_088.tif
37ab8a39ad63fd68a2c4535a9a9c3257
86919686322a287ba938e6b93787977c671a1f5e
F20101205_AAAFAX hu_j_Page_115.tif
66c4f105f4ad80e3afc0d28a44051603
e3e9543cf48038620b499965431052f414053990
5872 F20101205_AAAFCA hu_j_Page_035thm.jpg
46e9b0d2ac6598d94b0e7fd37c2d3ff5
81a51819d5014b90bcbb429f025d7f956a53cb9c
4480 F20101205_AAAFBM hu_j_Page_049thm.jpg
5ccd52395f532e5978c7ea8ce14d2bc9
d76823c776d0e91f63e3baa55bde8a90f9cc5781
959699 F20101205_AAAFAY hu_j_Page_096.jp2
607d5d8dd0c81368cd1d01a74b9efd70
a33fbfab3d316cc3b4e77894d2bdab9cc8fc270a
2191 F20101205_AAAFCB hu_j_Page_029.txt
cb7dc6b6af2f8cd8232cd1aad3587f18
ad46be51cf2da377ce149630223649fc16f7f29f
67393 F20101205_AAAFBN hu_j_Page_113.pro
c96e369bd8111eca473d9d63b66bfaa2
abe828b982ddfd10595eeb1df4d71b49e99c78a5
62811 F20101205_AAAFAZ hu_j_Page_011.pro
a2cdf346f6f4b6d4c846ce00bae57c32
41b4a436f7fa682dd22bcd74055653ce8cef5f00
F20101205_AAAFCC hu_j_Page_080.jp2
5fcecd8103f4fbe17662fe9ca1182eba
7b805981358fb88119c65269c7c3f9108fe0251b
42100 F20101205_AAAFBO hu_j_Page_080.pro
bd7b0b4d340c4f3fea9a3d35aa1d429f
0b880fac22e42af45aa48da024ac46d98220f58e
922407 F20101205_AAAFCD hu_j_Page_063.jp2
86002e053837b6071355c15420de106c
40489e70ba068f68927344ddca6d9de7d2d86753
F20101205_AAAFBP hu_j_Page_057.tif
70fb7e454930d35a83f770b91e3cc62b
30669024a91bfc64c50da98d7c425bfc30ed0019
410 F20101205_AAAFCE hu_j_Page_001.txt
8005e9edf5ae8dbff51a49a31df904ab
b24da48db61d04029bf6aabd518418f7ea4b66a7
15790 F20101205_AAAFCF hu_j_Page_040.QC.jpg
dffff843679a62a20a45e2d9e04181b8
719438892217e43bc101b3e68e4ef60e1cd0f5d0
900380 F20101205_AAAFBQ hu_j_Page_100.jp2
8eb680b4c0184221ebb1fc745fbbef97
4be5859d36b725146cce168158425d6585fc4abc
85814 F20101205_AAAFCG hu_j_Page_030.jpg
b4bca82debac89668b0be342692f4cb5
0cbbc5169d5dd47f8502652edd1ce1b016b4a8c2
20014 F20101205_AAAFBR hu_j_Page_072.QC.jpg
6d6fcbae36afcd52babee77462614adf
ce08ec07c2b3f65984ed238955b91c24ca506e2f
59621 F20101205_AAAFBS hu_j_Page_069.pro
15596f4a59d2d75e104da5e45e169f3b
c53050486b4a545dc58ad42097ea5c80fd4ac0ab
3339 F20101205_AAAFCH hu_j_Page_006thm.jpg
5095060c5ed4bdcfced6d97b72e05453
1858a083a0d35ac17f1043eb5797debb489748b6
66879 F20101205_AAAFBT hu_j_Page_052.jpg
3d24e58b69bb8aa2ab5c9f932b686e9d
255e0cff3a10808e195017932b089fdd03bf599b
93429 F20101205_AAAFCI hu_j_Page_069.jpg
3b52e4010d97698406df51a0b7dbe2fd
f4839377842c890fe689bec800e341a2d45dfbd8
36116 F20101205_AAAFBU hu_j_Page_019.pro
0dec4537578c06cfa51a2d02198e8f1d
150b5e22c4de0386f52ef8dddcb9e67478772412
94562 F20101205_AAAFCJ hu_j_Page_099.jpg
a76ea6ca013dabf9fa929276d07a7111
e110fe00646d5071e7762e256c00f1f2e2808450
30438 F20101205_AAAFBV hu_j_Page_049.pro
5b1464507201abe24ae13350131450d1
f966c140a796aa68dc028648228468bbe0f718d6
4693 F20101205_AAAFCK hu_j_Page_008thm.jpg
9a8329cb08f6139849e76d8105d041f4
3112dbae7450e7de40a3d742ebdd3a392a12d9ad
6029 F20101205_AAAFBW hu_j_Page_080thm.jpg
e3ac18e850473deb794adfdf92906159
7daf3fb6ef8be34b6388caaac3a12323b3d37b52
F20101205_AAAFDA hu_j_Page_065.tif
7479850329c98a82b37388df855967df
b13d7f3a4af195c524be53d98d997d189fd5b55e
2346 F20101205_AAAFCL hu_j_Page_069.txt
b1af1fe8b5c57ec03468ddb5075ebdbf
ad2d385874c78357dcca5048956b0b390d56c573
718179 F20101205_AAAFBX hu_j_Page_042.jp2
ecce1c1b346d6ee9b21b5aa07a64ce10
1506250cef3f890f234cd5a535d1fb4a509ab76c
40759 F20101205_AAAFDB hu_j_Page_057.jp2
adab2a39e16bb5d2a02005f9ad251dc7
0b2fcaf99d38028bc605c7b5971ac6b185d45c54
16001 F20101205_AAAFCM hu_j_Page_032.QC.jpg
e513cd93de3696134525783e55a38b53
f8df0c5b3e152bf382082868321057f7594c4f15
1816 F20101205_AAAFBY hu_j_Page_088thm.jpg
cc92e9330cece47b48c048ee447d1eb4
67a951d573a69d792992449d4c42b48a51680328
F20101205_AAAFDC hu_j_Page_105.tif
d51b9fbe4a08f323b13d0a5e9fb407b3
56a370d88712a55b6164c73f3a053a335a4ea62b
81587 F20101205_AAAFCN hu_j_Page_023.jpg
17123a34da2c9532eaaaf3b654a79c4f
b26f4bf20de0cf434c4681a679e4c14a6d1d78fd
5355 F20101205_AAAFBZ hu_j_Page_065thm.jpg
c4c5b310754b9cb594d75efba0956749
4990351865d4bd8dd02345213fda1aef13c79d47
39471 F20101205_AAAFDD hu_j_Page_004.pro
c4d6d8d6c0b4759578a7f0046a76bb07
2c9c2180121f126bd813d0cb4e78fcbb8505684c
55847 F20101205_AAAFCO hu_j_Page_105.pro
fa18dde63d0d4a02f4da55b9a01bda37
beeacd05fb31bed714949b00a47d402727b3fce7
6086 F20101205_AAAFDE hu_j_Page_036thm.jpg
a8b2cbc7d335f50c5debdcf67d73c1b0
278a37286ad6f448d56058390824338a27131571
122689 F20101205_AAAFCP hu_j_Page_109.jp2
d5462841063e0b5a8a028d4e06f45205
93cbff040ff9809e432e4b4236db982fa76d780d
40873 F20101205_AAAFDF hu_j_Page_044.pro
cbdb89deb945a0c4dcc29632bc1c5c74
8d31620f7210b08e8dfdd3aadcedd5dfd84ce62e
961466 F20101205_AAAFCQ hu_j_Page_013.jp2
893e1fbb1e190b63c9e23fc0b403db57
20975d902e7475b9a8034ad8abb13149f68c1430
F20101205_AAAFDG hu_j_Page_044.jpg
70a4d420531f41ba64965866205f1bde
28f6d4ecd3f1194009d2135c0ab15f4140d2fb2a
2249 F20101205_AAAFCR hu_j_Page_105.txt
bbde44bd1c34fb78dbb4963a3beb2375
3c1fbe01da740601ef348b160c27dcec87caf764
59279 F20101205_AAAFDH hu_j_Page_064.pro
aa313776f7091e847e6d8c14fdfce500
94c546011c8a71c8e2d5a45bd0040ad6443cbfb3
F20101205_AAAFCS hu_j_Page_104.tif
f058b07b92d4b61cb71fb3bcd539caa2
c8f1cd336e35eff8e98f9f9437315f2896b7c5e0
17805 F20101205_AAAFCT hu_j_Page_081.QC.jpg
f61fe7082e56059fbbcfb127b2359863
9e3a55c111e10d2a1652ef0898251e44c1c360a2
105 F20101205_AAAFDI hu_j_Page_003.txt
be581df2f662f4eebf73734cf5834399
81d1cc0e83bbb5f1efab784eb27caa46b261bb39
27227 F20101205_AAAFCU hu_j_Page_070.QC.jpg
0beeca8e4db375b5cd17c3d3819ffc71
cfd96a38ee72fe072912343a076ea74af53b95ae
1051968 F20101205_AAAFDJ hu_j_Page_089.jp2
feeeccea2f0fce0bba2983ebf995b3c6
043e26cfbade8d21917002fb922e61c4b3ab5359
84110 F20101205_AAAFCV hu_j_Page_089.jpg
9940668aca6d1c3951fbe8ebd0df22aa
3e843cd8b5d427ca316e1bf090b35abf7e016ea1
F20101205_AAAFDK hu_j_Page_075.tif
aae382088801db5095a4c586d718fdfe
d61b5927a26d8667f18cad20dbf84eed9ab2e9a7
5498 F20101205_AAAFCW hu_j_Page_056thm.jpg
559be042c89f57539f6c1fa720bcd0cc
2623f46b6c05d7b24e9d1d77b5a3e02f9f7a2854
4225 F20101205_AAAFDL hu_j_Page_059.QC.jpg
3184ff0e997773ffe75ba70eced00f86
8f42126741ffcd59a05d5e3bd1ea641fb2f3c0b0
9371 F20101205_AAAFCX hu_j_Page_051.pro
a943c8d5453b47dc52610a254868d013
f8e10ced49456345fed635ffcd4d8b14509aa087
5379 F20101205_AAAFEA hu_j_Page_054thm.jpg
ccad448c9d22fbac5d3f35f604516120
6a0bdb3a1155f35eca3925117cd98dfbb153cd3c
81829 F20101205_AAAFDM hu_j_Page_056.jpg
38e6e355c9d7566f56de742854722260
3c841ce3e76958ca560cc1199b5c15dce05c6864
2321 F20101205_AAAFCY hu_j_Page_112.txt
5d6d68b4cd3fa60fb11461a437fba834
e8a31a8593c8a02c024072618f49008a6953715c
6005 F20101205_AAAFEB hu_j_Page_011thm.jpg
725f5de96a12947b08453c181d5e0add
ab86d6a75ce76382b60e4c0041787d5727c3d17f
80442 F20101205_AAAFDN hu_j_Page_080.jpg
7386dee45caa389e9ce427acbd06573b
8e1555b67ac08c63d7aad3893079dd51601d373c
F20101205_AAAFCZ hu_j_Page_007.jp2
eee401de88d492b92fccac326eda063e
3f3a5cebbb8d4e00cc26b4750243ac1aba769e17
F20101205_AAAFEC hu_j_Page_092.tif
643092ef437bc7ca4d502df0e3082ead
0e82fbce364840dd439c4f6f8acbda69be8313c3
F20101205_AAAFDO hu_j_Page_068.tif
3c45c0155b7e9ae95aebdb7e94192bec
f4ddd0c82a3ef81b4d4e11988846e4f9552062b0
7545 F20101205_AAAFED hu_j_Page_075.pro
301bb94f236a44675795600c7a2be3eb
9cde14da99ffcbb4b1d19240502f83c5920ff5df
5092 F20101205_AAAFDP hu_j_Page_063thm.jpg
87dbe522a4e1c644fdc1fb7d98c31deb
68b5d50a865cf5fa0d042a690ab519d026b56b04
118677 F20101205_AAAFEE hu_j_Page_020.jp2
cc59c33076fc57d2beb66f8d2e47d8f7
9a2bedd27d808e44474a5e1571fb0b626331c42f
41360 F20101205_AAAFDQ hu_j_Page_048.jpg
29a3b61c7b38f01ee96400caf9c46336
0790aea1d25b266fec742f8c3705d31195f1cfba
F20101205_AAAFEF hu_j_Page_090.tif
b9f37573d5ab46b2bf53553783d5b554
7112b850586d1b7c9092f89b03421a87b2c4d352
21996 F20101205_AAAFDR hu_j_Page_041.QC.jpg
347928d0f210258c2c39f26e76de89bb
cd30f277b26ba5e04a90af00789f6ed723a6a665
57913 F20101205_AAAFEG hu_j_Page_007.pro
067a98357352c1eab9d0ab0ff19b006b
057a5b1e50e24b2654a2a7868ef79f74d30cb220
302 F20101205_AAAFDS hu_j_Page_088.txt
9f5da03fdfaaf085eb432f9e6d80c966
cb9606008b3b5b9fb9324a83518fa984a739039b
27351 F20101205_AAAFEH hu_j_Page_011.QC.jpg
6569e287df35874fb0968dc99daa8b25
bda3b652b2e1435ff1ba81a9c9208f0f2aca2a02
F20101205_AAAFDT hu_j_Page_110.tif
111f882ad91299aa110a61812dfa658b
2fd6d3770cad9d1f639b68f2445316d633b26de4
107642 F20101205_AAAFEI hu_j_Page_017.jpg
6edf5cd65ec4f4b15fa50d7f2a87d2a9
18140c686e74cbe4daa8ec6b0f1e4f4d5300eb27
26169 F20101205_AAAFDU hu_j_Page_065.pro
ca5d276e8ccd8cb065743a9408102220
bae8e49cca7f15e23101321b5d3b4a561f876eed
F20101205_AAAFDV hu_j_Page_063.tif
8654ebadff0ae9948ff3fd175120ab6c
2de0e8e37ece1f56951cb76ac452c5ab874a083c
2409 F20101205_AAAFEJ hu_j_Page_071.txt
524cf9d172b0a9c0d4dbe964c267de12
7386d2fb580c9c35b571adaa0718f92d4ac8fe26
53144 F20101205_AAAFDW hu_j_Page_028.pro
5198a003493f7abdcd34e0e83ed8e3f7
482b9b2d7a08e9bc48bc2200d5a163dda682bce4
25847 F20101205_AAAFEK hu_j_Page_091.QC.jpg
d50e8604cb625d69f8618707c6ef3f4d
98a4f8958e276848e630abaf798495580f079ec5
F20101205_AAAFDX hu_j_Page_022.tif
12c3cd41aec71385e457aa8ec4624bb9
8a7cc11f75462abc75e9a8a424321434ab02b626
F20101205_AAAFFA hu_j_Page_113.tif
e22705c4208bf8fb0ee7ca0392a29a9b
6a54f663ff34ada03960943b78d532e5edc1e04f
F20101205_AAAFEL hu_j_Page_008.tif
93f3cbaf2f2c4368175d9ec7967a7d07
e0caa001675a2ca329f8e4dcddffabda2bc6ee6c
89851 F20101205_AAAFDY hu_j_Page_036.jpg
225ba6445266aedc5b2c0d57126a0604
7138eb19419d0f91a02a7821c114c8925e9ed2ce
858610 F20101205_AAAFFB hu_j_Page_066.jp2
5b3f46edc6d2e397be45fc3b54a7b604
f635868dc1d75980386562c84e3a335a30151ffb
2328 F20101205_AAAFEM hu_j_Page_064.txt
6e0cb6416125bc51db4a42bf1e000df2
4b4fb9e703a37e8e009491a357f44cc046d358c7
F20101205_AAAFDZ hu_j_Page_044.tif
9f4dda3a0c34e173755c6f74a75aa8e4
b75a59226ea5a5a45f4d73c306584687892c4597
1312 F20101205_AAAFFC hu_j_Page_063.txt
824b14a22a4f06330e62623616a2ebd2
41bf3a64c6a2bd8f326c776f87b7c848bfd16940
F20101205_AAAFEN hu_j_Page_017.tif
597dd2a26db26ab01c84f037d606e504
6cc0cabede9b90788bdfcc4dbc6356582854c54a
19746 F20101205_AAAFFD hu_j_Page_034.QC.jpg
49e33e55beccfd8d88811ab4fd589ff0
664ee40f1e4e4d3743c8f52e7c3bb2c662d27e93
1734 F20101205_AAAFEO hu_j_Page_085.txt
dbb9fa296442afd7db6449b3bc0dfb5f
34d072b6b5db9d86a5f7941947993b1efa7090f4
18867 F20101205_AAAFFE hu_j_Page_084.QC.jpg
a935e2a454a71b9d46d6be6aa95a5028
021da07037a7a026aebf237e20b8eb37bfb57f38
F20101205_AAAFEP hu_j_Page_054.tif
c58ce4f7b349a723bb22e33d86cd8140
83c1e1cf03f98cc6b7f3054f4f92846643783ddb
51859 F20101205_AAAFFF hu_j_Page_053.jpg
f1152c148256df197da39b210c4a43c0
f5a27301a2d4050ed2581674a441e501673d99e3
66081 F20101205_AAAFEQ hu_j_Page_066.jpg
4479290a2646f77b6d67e6e4abe37c3a
4d2de9f74816b9bc2569edc2861bb34e845c530c
1051954 F20101205_AAAFFG hu_j_Page_069.jp2
4d47ca28bc6bb5161a3606cc875a0f77
7e5ad2f8aa8613089973ec4790be048f5bc3ebb4
4738 F20101205_AAAFER hu_j_Page_012thm.jpg
f6b303ef3450f8f1790690597062397a
0975031e05f0cb94275ba28df258779d695a7740
F20101205_AAAFFH hu_j_Page_102.tif
a29dc7822f17c9f132df1b870e13b8f0
3ea6886f7000fd0f0f02c223d8f3bcb29b274fb8
54876 F20101205_AAAFES hu_j_Page_030.pro
ac2dcbe3f41cc799849a94770297f5e2
cc3f6ed74a35ec03294f0cc3c3ab8fd3210f66c2
F20101205_AAAFFI hu_j_Page_091.tif
8b94eb3a128fe2e1a0c8263a167e11cc
e7aaa86c22540bd0980f2544ac136aac5890ecf4
88183 F20101205_AAAFET hu_j_Page_091.jpg
f1a5dc759248c9279affb6a278c6f87d
1c40c7c30e45522d93adc3412576c4b139c76119
2044 F20101205_AAAFFJ hu_j_Page_041.txt
315b9188f8ed6c55e01fa5f01ca85f7d
795a8eb0fb0b3a6676fde3994be9d58fc72e8a3e
4933 F20101205_AAAFEU hu_j_Page_045thm.jpg
914ded693a5935e7e85e03a2209f9b94
dc7dda81743f952593cffcde204a00e7186a7b36
71843 F20101205_AAAFEV hu_j_Page_041.jpg
07e83585450d13c3681c234974c62ceb
344d0954592edfd1435b9fa480352fb1fc925332
4996 F20101205_AAAFFK hu_j_Page_079thm.jpg
abe55abba0660b57e28d66d5bb1b6951
76eb7f39b655036c1652f2fc8136b67dcb89048d
4209 F20101205_AAAFEW hu_j_Page_004thm.jpg
f546d55b05c67a37c705537a3ab24fc4
981295ee2ccb49700ad31a8f8e1b96b7e939df1f
1051961 F20101205_AAAFGA hu_j_Page_076.jp2
55a3ead740ce84e8d1c0bfc441f1d3ce
7df6fe235b0941575d63bb09b42d1b2c78473870
2185 F20101205_AAAFFL hu_j_Page_028.txt
ae119e4fa6b48abd1331c9bebdbcbefc
bce91be0bfdf5441e0c70cf18347b25da14d0e54
2204 F20101205_AAAFEX hu_j_Page_009.txt
1d2ea8bc28241d0bcddda9de885a43b8
04c7c45db490c624f54f31f8fcdd377e5cc8fdd3
73641 F20101205_AAAFGB hu_j_Page_055.jpg
885d613f0491ee4e9addca673e8f3af7
fb077a2f3418f692c41f4c86074a88bab3f5f18e
2208 F20101205_AAAFFM hu_j_Page_015.txt
7d24b79c34a297a6c1bb6d080a8b0912
566182ff0598599e32070896235a29364228dd06
987733 F20101205_AAAFEY hu_j_Page_055.jp2
373d3563a4a523e3965a2c310e113d88
44671df8a9e0d33dc007847e2e3d69607dc94bf3
1011010 F20101205_AAAFGC hu_j_Page_037.jp2
12bbc7f3699b0fb51dc714763b5a59b8
e54bef15b818baae8d87a7ba6330de5361c21ae2
F20101205_AAAFFN hu_j_Page_093.tif
ae2e135c445afa8fe1d93cb655a67ef4
01d4dde5f52c1da62dbdaaef6815f0c189a4d029
839382 F20101205_AAAFEZ hu_j_Page_094.jp2
d7f5c8eb28912ad753be18d73b3e00c7
bd673120efd52ae81b961673373176063d9c89e7
5922 F20101205_AAAFGD hu_j_Page_068thm.jpg
3caa20e56d4083e4bae0e2bdd83fbc27
aad328a5d62c41ce504206efbe7eae1ddf217316
F20101205_AAAFFO hu_j_Page_026.tif
ddb1d1217b477a2a410a5c4c46233146
cdc66ae112de9111e98194ee3df2cb01f50496cf
80293 F20101205_AAAFGE hu_j_Page_009.jpg
a788aa6d6773417a6be532ae4661c083
04877cb4599a4fd66730a5b9280670bff140690f
76 F20101205_AAAFFP hu_j_Page_002.txt
b8ba83cad2cbadfbbbca6324359cda28
1ed2910aa448cc33adba1c2637cf4e8d1672aacc
20257 F20101205_AAAFGF hu_j_Page_052.QC.jpg
51e84487515c7505789d14c394cdd916
d8704b7281d224d4d0b3843f5fd966098b0801f9
F20101205_AAAFFQ hu_j_Page_028.tif
238cb4dadae679ef91c019046e392f9f
2fe98ceeaaa6011c4a17cdc4042457de98e9d711
77269 F20101205_AAAFGG hu_j_Page_018.jpg
f6499c67de1dfda6eb0981b2e1b86743
b3f98e7648ad81a7e9e9355bc5e37a668ce59caf
31855 F20101205_AAAFFR hu_j_Page_042.pro
ba2f2ea836b4001135f486d3379dbcb2
3a4d36d95d74089ff0e9efb7139f2a6f7b3390b1
F20101205_AAAFGH hu_j_Page_007.tif
74e3d5a28376df4917c746289b52c4dd
92f0846217b0116434cf361f63c8d497eb4fb2ea
F20101205_AAAFFS hu_j_Page_032.tif
2bae3410e655a6159b9abf67dfd4d23b
b8a56556499ea09949d489bd0502d5492ecdea5b
4246 F20101205_AAAFGI hu_j_Page_042thm.jpg
5b564ae832157d5da8634313fa2f8fde
e84f4df8d60a596854dc9513bbccd3584d134a21
6127 F20101205_AAAFFT hu_j_Page_019thm.jpg
178854e5e47aeeb3b80d1070d56b110c
7e39ac4ef8860a68be87004fa74c9f9fc188b744
F20101205_AAAFGJ hu_j_Page_041.tif
554ecc34127b0d6d370ca7cf06847a07
dfb1692718d20bce48ce296c44990da97b1d2ebd
54171 F20101205_AAAFFU hu_j_Page_023.pro
95c2516497bab6b2339bea91baf4b20b
5e1ce134dfb16b27cda394cf5b19391bbf86fd1b
27310 F20101205_AAAFGK hu_j_Page_027.QC.jpg
1ae10eff60eb00f29bfa69dc6de91774
a8c2433949df64bf252030d71a9fcb1417e34554
31366 F20101205_AAAFFV hu_j_Page_085.pro
608fa85a982684a05904159128273c48
372a810d66a1682a4a1900cbe3688a1fe17f8042
67937 F20101205_AAAFFW hu_j_Page_100.jpg
42c57c4c42ca4b6c0ffdfff5e9168b56
756a6ab273eef01afccb64ab085efe8983d1471a
119927 F20101205_AAAFGL hu_j_Page_108.jp2
240074d05c8afecce26494fdf8287da7
b22851e97c465a454277fe9434e1d1796adba591
19733 F20101205_AAAFFX hu_j_Page_065.QC.jpg
5c36ea1cc55e53c2f61373f77bf6eea9
31ba39895bc888046898336dcbbfe2d9c9a2c215
2292 F20101205_AAAFHA hu_j_Page_020.txt
b247e2c033a5926db340b3e90f6a1f06
0c0ad84e50871320e0e1ef4fb9e01f004e65b3a2
30360 F20101205_AAAFGM hu_j_Page_051.jpg
6c5c8267badc3e1a13c9622a40d63b97
f8a09b82ed3136337c49943feec9861869d404ff
1213 F20101205_AAAFFY hu_j_Page_053.txt
ffb3b027521bafdbc9df021edeeb4bda
43a3489271ba27f4485b77ca87629f3a7cb64a1a
1660 F20101205_AAAFHB hu_j_Page_045.txt
9b111ef1402cc965b37c9564893deea1
b62e881f61888b7caea81e6b430107d462aeb3a6
F20101205_AAAFFZ hu_j_Page_010.tif
02133b0788f608007649925e79812c15
e9762c8883a48918e9b02635598fcef4056961c2
1885 F20101205_AAAFHC hu_j_Page_090.txt
184b47dd9c3dd2f10a66e1fdd421c0ee
4f21a751e59dc74fadf930c37f617cc47af5e199
74914 F20101205_AAAFGN hu_j_Page_093.jpg
4726c57b837be177267b7e1797b258c9
197ea2ecdd7765acfa1568745911887b209ad651
F20101205_AAAFHD hu_j_Page_108thm.jpg
a3dff506d2f67b3b2d636c6666c5d295
8cc160694a7fecf6f1215edc8d151ec92e9ea569
5688 F20101205_AAAFGO hu_j_Page_029thm.jpg
300d9985fe81e72c03afbccb330c9d69
aa57f9e0f53745c850a50b6b39de5d22d248a905
F20101205_AAAFHE hu_j_Page_101.tif
c2976c0243b362ff9ae6f5f51373e116
bf459011226ef943c4c91afadd5866f9933e2b2f
112724 F20101205_AAAFGP hu_j_Page_104.jp2
87472e1aab040126033bf64fbe5f8af4
0a82dde6bc26ea20b007c09970224d3992098734
60881 F20101205_AAAFHF hu_j_Page_070.pro
a7090afb69899b0d0116e6c8b233fcca
be5d1ad3bd4a1d01f3a887b2c34d6e183faa7765
2724 F20101205_AAAFGQ hu_j_Page_005.txt
77045915a767c97c2629864c4025d6ee
bba7dd72151726208bfda677080b698be126814f
F20101205_AAAFHG hu_j_Page_112.tif
9c183663530ee3d39432b9fa753f45a4
df27eaaae5ed7938ada66e2d5ab46935658692ee
2188 F20101205_AAAFGR hu_j_Page_023.txt
7172d60a01d02ec79cc29950f95c64e2
d69283c41b760aae32b123efb65ac782de6b1255
57662 F20101205_AAAFHH hu_j_Page_020.pro
b6de8a203c25b5253fe542c63263178e
d9fd0a83e2ce7c07ec823949fd4a6334d59f486a
F20101205_AAAFGS hu_j_Page_096.tif
0aa8d92993326e1212b8b8747703d44f
3256a4e3eb9da5dbc219126ff970c241d3082ece
25017 F20101205_AAAFHI hu_j_Page_104.QC.jpg
e99c47e054cb922e61cf18640f2cec67
7cfefbe1776e9873e87cdfe6220ee19a075435f3
83976 F20101205_AAAFGT hu_j_Page_067.jpg
81c41a3f7c6b095bac8fa5b5821eb7ab
3b80aecd600bca45e7833ddf926e76d7bbbc7266
30401 F20101205_AAAFHJ hu_j_Page_058.jpg
0444511bd36ac30cb063b80d17d049f1
e69d780bb93aa2fbf9174fa51b26b7cf972233d8
19391 F20101205_AAAFGU hu_j_Page_004.QC.jpg
c53ba90023cd50c66ae5aa17bc421ab9
051823cd23b6f544e007991100de5c2c7b875d2f
5281 F20101205_AAAFHK hu_j_Page_005thm.jpg
23297401e6f96d721bfcad678d31488c
77e4fcbfd168c4e3978ce6a30bf2b6862e3e6ee7
60056 F20101205_AAAFGV hu_j_Page_109.pro
84730bd385b978c549bf3694eb5411e3
b554708cf32b9aa2b9288d800eb0b97e64215b7b
58406 F20101205_AAAFHL hu_j_Page_078.pro
8926a69c4c5f7a7c8930d7f3a6d94080
be96547ea95ca558bf30811de4ac8533b80c5613
18699 F20101205_AAAFGW hu_j_Page_094.QC.jpg
1516fbaa65846425f120b67f4ae78de6
b9507a72f375eedfe5bcbccd004b9afdfb5eb637
4853 F20101205_AAAFIA hu_j_Page_033thm.jpg
ce45277d463fedd1dbfa3400864bb926
a8998d295448b75f8453d2b33706c6647e550c90
F20101205_AAAFGX hu_j_Page_099.tif
e8894003d9e978f80f3a892909641798
25ab47b877e5ab3ad1b172addf00794345d8e749
5786 F20101205_AAAFIB hu_j_Page_015thm.jpg
59b3e90b2565727e8d4be39f67d577a0
939e3b59344cd85230f9b2399a131d649d420a03
F20101205_AAAFHM hu_j_Page_098.tif
3912e213a5465394a4de67d7ad74d560
6e040c3a6b0fd2ed7ad0c76bac62fc4ff6845818
F20101205_AAAFGY hu_j_Page_081.tif
c1cbb4b5fd3b525319793f5e99a2da45
041c47efe507643796b8409f8da603692b85e480
59756 F20101205_AAAFIC hu_j_Page_061.pro
e8dcc7306d623f8c2500ac4735800190
856e97388c6e2153b9dcdb288b0fcf3766b397d1
8146 F20101205_AAAFHN hu_j_Page_114.QC.jpg
d3f8c0ae2d252d5a8ec8d790baeeeaaf
310673bd51cfe125891ea1bff1d664b52719d226
68312 F20101205_AAAFGZ hu_j_Page_050.jpg
6863ce0861f19d448a354b2edf7df394
598d1c062f261a329dfa96349e9a998af08b9363
2323 F20101205_AAAFID hu_j_Page_027.txt
1b1ed5eb243dc55c71ead69c81b22eab
575a2d37e6dca377a9bd553dc3c96c89abfa982a
21832 F20101205_AAAFHO hu_j_Page_101.QC.jpg
165ca436748b70d9e013039ccfd1141c
c1b1ff6f0833c8dbca6798e572f978ebaa019099
4643 F20101205_AAAFIE hu_j_Page_084thm.jpg
45ecc562f4854f9b4d3183085eca3bbd
5e7d3df8329ba4ec742bc39c770a349723ec8606
5739 F20101205_AAAFHP hu_j_Page_089thm.jpg
aea2699e7c26a4ff7ede0c1b3efdfc33
aa198a676b0ac5d1d68c1719c12a7fc515efa794
F20101205_AAAFIF hu_j_Page_094.tif
b51efdc056e81fad0df9aaf799df7375
654947ac67697a4c2d665857ac22020a3f0baf20
92877 F20101205_AAAFHQ hu_j_Page_107.jpg
5ec95adbdfa6484383288649ad2ab1a3
9f3b117e78d59d5b33ca86037096faf71ac2d49b
55440 F20101205_AAAFIG hu_j_Page_092.pro
068e64ed1d6eb82ed6d062bdd4d5dfda
3bd6702d1c4933db2394de0fba76666629d9b161
3906 F20101205_AAAFHR hu_j_Page_053thm.jpg
951af680de1fa5529c996f2611f26cee
cd3528ceb6f2cf44bbe8097143857d8ebd1ca6c7
2267 F20101205_AAAFIH hu_j_Page_030.txt
5ba204fa26d1156a91198050ec23e980
9ac44a432e22ab6f8fae834f0a4dfc520019b1e0
650 F20101205_AAAFHS hu_j_Page_115.txt
0e847ca996f51865373eae87a4c5bd5c
c15e3c3a9568cfb696c1ded74407c477cbd2ce5b
26104 F20101205_AAAFII hu_j_Page_053.pro
35bed6802f55f83397488e0a47473728
820191d23c6a46ec86bd81fe323b96b46a7b4969
9956 F20101205_AAAFHT hu_j_Page_058.QC.jpg
4d0242f3c62400eeef50a6f128235d18
0cdcb266b484c6f493da83de30716f70d630e1f4
F20101205_AAAFIJ hu_j_Page_011.tif
762b8084919023080786a5f6bcf359eb
8ff18005e32c76145e6527b8ae67d3f33e1e4a59
25275 F20101205_AAAFHU hu_j_Page_073.pro
ddcd9b40ebcef023c8c424a838449fc0
99af0d0e4d6524e326ccf10e64c4755b29ed13cf
21472 F20101205_AAAFIK hu_j_Page_013.QC.jpg
bc6459029134a0b366808cfcd5b48601
a5b018eb144fde8615caa77954f9b0bf5f1b2f33
6037 F20101205_AAAFHV hu_j_Page_076thm.jpg
50770c26f9c3cd08ff5c0e50dd1ac32a
fb1ef20b0da01b3d39009024f3b43f8f5618e68b
391 F20101205_AAAFIL hu_j_Page_002thm.jpg
835a739b64f92c6eb9252d78dca49f45
bfcb69a47092a97a71d6618147233f363e70c6ad
84073 F20101205_AAAFHW hu_j_Page_004.jp2
d959d28edf24ac3c5a9ef0832c1205fc
c07f2fc60453b450a01dbf8f41d81db58448fbf1
56872 F20101205_AAAFIM hu_j_Page_016.pro
f7c1b993d3a8f3555add9711d7ec8860
9f6a5657e6397a0abf7461cec3353d2d65a86582
26857 F20101205_AAAFHX hu_j_Page_036.QC.jpg
883dbfda4dc7ba9bc22cc8719fb73695
d6967adb5884141a634148e4f0b5078667d91a9f
125697 F20101205_AAAFJA hu_j_Page_107.jp2
6caba09aad00294e79f0d8d2bdad623e
d4baa5e05430007afaba570b630738412e7594c0
57186 F20101205_AAAFHY hu_j_Page_036.pro
973b7f0549e90ce7d4fc3e55fea08107
ddc0ac71045605f8387b08f64c734153ec3b5429
25379 F20101205_AAAFJB hu_j_Page_089.QC.jpg
78db83004e6f0b84235cd10a0f04165f
069e86c30e491eca4256f9c6b0e72861601a2c5a
1573 F20101205_AAAFIN hu_j_Page_033.txt
51c8f7c1201aa7496eb177f235cd2cf3
3f18a089c0cef360ced050089a1cd16afb364111
1051985 F20101205_AAAFHZ hu_j_Page_031.jp2
b0465424b387631c4f21755c0fc7cf1b
a2ea97bb878459cc65ec97005c1cbce87d0cd5f6
22792 F20101205_AAAFJC hu_j_Page_093.QC.jpg
c69084bf43e0b56176b23d8f95d63c42
571a2fea3e0f3c7e9f7976d27e916ade62d65cb7
6272 F20101205_AAAFIO hu_j_Page_100thm.jpg
dec57fdc2a4270c68fa421f309a76741
a8cb453d1cb276a981d395176b1ff61e33b7d06a
F20101205_AAAFJD hu_j_Page_091.jp2
79a4e7a7b46386d1cd188e47742e57d8
1ca347654396e4f25d90b0efadeaaa01d0e2ed1d
763 F20101205_AAAFIP hu_j_Page_087.txt
d5cb98f4a0e396d21ce3d117f1526477
62f8fa4fba27964359316acbb0065e0e5a478017
5825 F20101205_AAAFJE hu_j_Page_112thm.jpg
003a1d7b566cd4e86eb45ea0b9b820fb
ee0a15ee6c1c5c4a70635427e51d6a9a0ed3b752
F20101205_AAAFIQ hu_j_Page_023.tif
4c9c45794080adfef57a797e1505b803
698db2a5fa4d4ca26e1f3cbd084b3f3099a89601
4941 F20101205_AAAFJF hu_j_Page_101thm.jpg
23e0743a3afcd30a9c617e7b8d6c2fbf
7713db774def18b35a193dfa51aade653d954ccf
84140 F20101205_AAAFIR hu_j_Page_029.jpg
4207172707254f3c32ccedb4a170afaa
947b7fac9805a1d1a0c6db37aa0c7c5f0e2ff938
56292 F20101205_AAAFJG hu_j_Page_010.pro
33e0ab7de072bbb426f0ab7ac330ad83
9f5854090699d68bb15cdf1bdbb5cac175e9163e
F20101205_AAAFIS hu_j_Page_059.tif
4400363bdd193835f5f8652deffba432
96bf19676c7fadfc9f077904223277f42ac69b80
5588 F20101205_AAAFJH hu_j_Page_096thm.jpg
a23a42a478777832a24db911d5342981
d13ea9db696027d3dfe4d399312112d14b477fd6
1164 F20101205_AAAFIT hu_j_Page_003.pro
b0deacef6f4683631222c92e0953f010
c4c8c3d9b33ffdfcb97579fbaf13374ec303448c
8997 F20101205_AAAFJI hu_j_Page_057.QC.jpg
4bd1a131c3efc8bc897d7dd8f4a85afc
c188647c9a468fbf4a185b09116483bdc0a0fcf8
48215 F20101205_AAAFIU hu_j_Page_032.jpg
81012e3c1485975335665e0fc31d60c3
2cc9a4cc9d9cdd5f03df880a50ba8426bad0cb5c
28667 F20101205_AAAFJJ hu_j_Page_062.QC.jpg
c8d86ff8350ff9cf9f849eb4f458a47e
be20c930e2255e5c980c400fb66e08ded9820d4f
1877 F20101205_AAAFIV hu_j_Page_096.txt
3c29ce304b5bc9faf09600d55ec81670
182e74e63985b2efd563bc1ae6818b2c58adc40c
5407 F20101205_AAAFJK hu_j_Page_074thm.jpg
37eaa297b0cd5205d91bbe5cb59ab74d
9c83c83ee0c13f7c183f06060ecfba93904e509f
1051957 F20101205_AAAFIW hu_j_Page_092.jp2
b6b7409cf3edf2874387eb28d38c12d0
e7fbdb2ce68de0ebb1411dc85653ef5fd58d687c
F20101205_AAAFJL hu_j_Page_042.tif
c897aa66c391d0e7e825e31d6a528dcb
75bf447d82816dac8a11e7345ca4bc5d44e1ee66
47846 F20101205_AAAFIX hu_j_Page_087.jpg
aca88d272865eaca2db29ea04549acf9
60000c834fd41373ab00cce7f7dd2293ff78836c
74286 F20101205_AAAFKA hu_j_Page_043.jpg
627e80d25515ebf0a4e012cd34d656ab
d2efbf5f31a933abbf9db2b3661cb634c0bf0dd3
F20101205_AAAFJM hu_j_Page_070.txt
a5dfa73446b05769110f67eff179999e
276e9fe5c967d394bebc60f95ce8fcfe058b4374
1413 F20101205_AAAFIY hu_j_Page_081.txt
6aed43f8504512f8edbd444df052895b
63540a4722e2e8d8f8d6a66c20a10552b0f45b7f
2309 F20101205_AAAFKB hu_j_Page_111.txt
d5d07477dff946007399109da12cec10
e1f8faad0935e5c0159c8baad704841630570ed8
20737 F20101205_AAAFJN hu_j_Page_044.QC.jpg
464cdadc3e6e7adbd95c54b44dc57282
de606ece5cfa808831f296c55287fcd5c2c713e0
95345 F20101205_AAAFIZ hu_j_Page_076.jpg
81054bd8f5c8a2d683783d947afab42e
2de0ae229eee4d91b5552489c219cd4d02446c89
5888 F20101205_AAAFKC hu_j_Page_010thm.jpg
727c80fed25e00e187bd7825ad509269
a892024a9f2371b55f2853cbf77dd6abd688a2ea
92839 F20101205_AAAFKD hu_j_Page_082.jpg
13328edb1c0074f7caee7ef81b4547db
89614ab80126a66d65aded0662b43ba6dd33eb9d
79265 F20101205_AAAFJO hu_j_Page_068.jpg
0c5bd7b993d06b4c64a0b29a9e82033f
2c0763c2e94ef975dd5ac2e949adc9d3ff7e38f3
64238 F20101205_AAAFKE hu_j_Page_065.jpg
0a4a79535a5e01162421da3f0e030ca1
9ba9980807200b889518417fe48d1005e31d1683
95584 F20101205_AAAFJP hu_j_Page_011.jpg
dd65dfa603b3a3f905b0f4cd6458b00c
305fb6a4342c68de28c02d0249ebf4f5e24d7ce4
43287 F20101205_AAAFKF hu_j_Page_090.pro
c01398da605e0e37577c52f16a95e97e
3703ade508d0d5ec2f8504686b35824719c878e0
627799 F20101205_AAAFJQ hu_j_Page_097.jp2
58a32fc8aa3209e758672731f906e56f
2bbaeae41e918ffe230181aa9eb9b16954b3f0b4
1044 F20101205_AAAFKG hu_j_Page_073.txt
2117aaf25bcba094cd59531ce5927660
cb2f53a9cc7a6cf6ebc8ed3559177fe8b05a4588
27726 F20101205_AAAFJR hu_j_Page_113.QC.jpg
01141de35b146b7987342c7effa72691
a66579ce5f3034349f1e822efbbc46face8c1e59
F20101205_AAAFKH hu_j_Page_076.tif
7c9ad9144213e70ba6281d43184efa96
97a109ce2a50114fb897decae929a7886705deb5
60285 F20101205_AAAFJS hu_j_Page_083.pro
28b9684d655aaffaef5e3b0125808d0e
a49b928ede365286fccbbb29c4b26caed3a45f98
1051956 F20101205_AAAFKI hu_j_Page_010.jp2
01cb94e6e4e08ee05e3f912e15a3ca02
da71c48e8aa0f881eb92d5e0cf781f7d71c14605
867533 F20101205_AAAFJT hu_j_Page_034.jp2
387a344eab701fe429aac97b1aef1bdf
536e6480de5e1557141d4bc3765721c344481c7e
17650 F20101205_AAAFKJ hu_j_Page_102.pro
3ea1eb9886255430dc72cd39432601e5
2a0ca1d880be36bfb9b69ae9e9e2dce8ee9bdf06
F20101205_AAAFJU hu_j_Page_085.tif
a9cb81e081e48f4da2e7f9b621eed13d
5fd23b072b0d6aa61b4a94060c4f2915b316a314
6089 F20101205_AAAFKK hu_j_Page_082thm.jpg
42d923859cee736487d70c6f62b15762
6bfe91cbc02ca2792dbce4f48c6d93252ad4e2ca
F20101205_AAAFJV hu_j_Page_012.tif
46c9902660e34f7929212a558eb844a0
bff846615c185a3ac71e4f07777fa69e5d300a94
F20101205_AAAFKL hu_j_Page_005.tif
bb5c36e987a7fdfab5da3f561a1bce91
8000fb6e070706914b5ffd9a87195314ed1c30aa
129696 F20101205_AAAFJW hu_j_Page_110.jp2
ebedfd49733563f888bb9bcda17b0387
a6514efbae645610939437f19b94e54378e35f99
470 F20101205_AAAFLA hu_j_Page_026.txt
35cc98c970d40ffef1205e46962eb499
3c4adc53273bb6a74a7568261eb0ba5a66311493
32412 F20101205_AAAFKM hu_j_Page_047.pro
d865a941d4c2887edeea72bc4b19c4e7
20c9bb1617a9bb39d09cb93408d103d2596742f5
1730 F20101205_AAAFJX hu_j_Page_080.txt
a8b437f11b0a3ced3ef6a982975acf63
154f02b90e035aa332dede72b51e96a408e5b67c
97893 F20101205_AAAFLB hu_j_Page_025.jpg
55a941a47bb6fa4e6f1113bc63777ef6
fc51dd5d44b36f6d25fbbdbf5c2b601e5d6ec164
6270 F20101205_AAAFKN hu_j_Page_062thm.jpg
3e6cb2ee7921281406e4b90483f14c2f
25325fd4389da669f8766ea438851a1a8ee109a1
15487 F20101205_AAAFJY hu_j_Page_103.QC.jpg
c5f9d3c588effef77abaaf24ecaa34ce
5695567c24e2024a2f30983619036778c67aae7a
57306 F20101205_AAAFLC hu_j_Page_111.pro
cc5a33aaf1fa0d0f71b09dbf60beffcc
ec0008a0e51d8ae4fc20cbcb74bbb6a7c70a014e
F20101205_AAAFKO hu_j_Page_095.tif
a3acc3f849d4de142a2060a7780ea54e
2d70222c5339b3fdd34b95da238b1d91d045ae9f
1986 F20101205_AAAFJZ hu_j_Page_008.txt
23436efca78716049e730af4ca8546b9
87c9fd9fa40d687bd4913612ac78a3d96293873b
535 F20101205_AAAFLD hu_j_Page_003thm.jpg
907dd3cc99e1769255574568b26513b4
f3b4e14aa131a79b2cfe84ac759d5382c5549029
F20101205_AAAFLE hu_j_Page_111.tif
4865296d05b79a0b62012277b63f40d0
c199d20e45162eea312151b16f82afb86386db1c
120873 F20101205_AAAFKP hu_j_Page_021.jp2
3b651e81932b1d405c4cd8b2eb643cb0
084ae85c00951ee6372c19676e9ea0243dcfeea0
79150 F20101205_AAAFLF hu_j_Page_019.jpg
ea92501138eaca2c864fda6a2bf11065
90513c36b5c225d80de42271091f7705740ffab8
38930 F20101205_AAAFKQ hu_j_Page_051.jp2
cd3b025653544ce1f6a961b26b6d3c56
a8e56e174d5defc9c8f0dbd2d06323e52579409d
136252 F20101205_AAAFLG hu_j_Page_113.jp2
960b23cf14192f5d31eafb00cb32b2f2
0c13ea20736a7f49ae00299478f3acb7978b7551
F20101205_AAAFKR hu_j_Page_024.tif
d4b1bea4c3c6a551c15a130d24755842
931b0fc85c3f776d7296a2e3999bd7e7f7d85fb0
58773 F20101205_AAAFLH hu_j_Page_082.pro
76e3811151271cb6422af61689f06836
d1150f8355c33076e833eea6b82fe017fc5cafb3
2327 F20101205_AAAFKS hu_j_Page_031.txt
e484f622c1af12bc94d0aa0d40bb923f
c89031fbc056f71ebdc4d75062f042bbec02772e
5405 F20101205_AAAFLI hu_j_Page_104thm.jpg
f7dc245c6723c94c57b0e6f2ce987763
75225dbf61a85b242e7aac27d32a05ea7688abe6
3112 F20101205_AAAFKT hu_j_Page_057thm.jpg
022e0bf84cb84a125534a56ee1d457f1
32c53696902f94a02d97e8d1b59c5672292db473
92461 F20101205_AAAFLJ hu_j_Page_064.jpg
91fbc60a2915d2a59f2fca431296dcf3
0c0456c0de8cea275741dd41238e5f8b06d6b503
59431 F20101205_AAAFKU hu_j_Page_084.jpg
18fe0e0e88c1a5443dd88be33a2d3ea1
9ba6ec0bfbeee838408b0e83e7de0ff10130b8fc
664 F20101205_AAAFLK hu_j_Page_051.txt
de92adc1dccc6e4b4596f3273284b56f
f00df38c9c342e68604fe88c9574059e77fe3dde
15907 F20101205_AAAFKV hu_j_Page_006.QC.jpg
8c3eeb829236949e97d0161e4059b2fa
d82dfe23a858312591da32f6637210777c4bb3d4
59533 F20101205_AAAFKW hu_j_Page_106.pro
3475c54ee2a83e19e581c9cf829389e3
8f919d2dc6ca211a87c73d351074a724b9300b4d
8423 F20101205_AAAFLL hu_j_Page_115.QC.jpg
46b14c782d148dac01aaee4f790f0ce1
8298f074aa555b7251a0ea02079f6127038386ba
21470 F20101205_AAAFKX hu_j_Page_018.QC.jpg
9e3a9a7bf076a929fe3355bb71b3364c
328ce12d642065bbffb854d03291d97d14c41f2f
3644 F20101205_AAAFMA hu_j_Page_103thm.jpg
fc2b13e1ff0cac42f20a0f385925e3d7
886e7348126fb3e802f128e997481b03a66fac3f
5634 F20101205_AAAFLM hu_j_Page_105thm.jpg
0a8f491e859e17fb19ed356f0685f9e7
6b887602911e2d7a77374f219c9bfe2a0695a018
61770 F20101205_AAAFKY hu_j_Page_025.pro
b6092c1ed9e1126f6f51728a4f4c67f1
847d44e8e0186dc2141dbd27d92715b868be70e4
F20101205_AAAFMB hu_j_Page_039.tif
cf72034154babec5334d78201fd905c8
d07cce79d69370f1c5281b50c48da286797e92c6
22242 F20101205_AAAFLN hu_j_Page_100.QC.jpg
ea89c794d86f591467f563778a789943
ffa6a8400bf261325fc2ad4ab5572b3d9d31fa23
5264 F20101205_AAAFKZ hu_j_Page_055thm.jpg
f16563da11dd0f6beaace8903271ec10
4e1acb8328995622a7369587b36736878ac4f30d
F20101205_AAAFMC hu_j_Page_109.tif
dc151eb70f1dc038dde017bc3874cf23
22dfa441bb10c5cdc0b4d5f3ccd2586e9f3ee723
88432 F20101205_AAAFLO hu_j_Page_020.jpg
32eade3cbe0897d7efd729220df4b7fe
19f1c29beb080636ee12537293e4c67c5d857582
75098 F20101205_AAAFMD hu_j_Page_037.jpg
23a1d98265c05b3a3e298db245bd165a
eee033a2255cff12ca3764aa245f9bf852b87112
1224 F20101205_AAAFLP hu_j_Page_040.txt
ebcec236ac51146763ad90d73a71244f
f333eb1902ea5891191d811dbec616bc50c10921
1757 F20101205_AAAFME hu_j_Page_068.txt
de01e8732c8c6c6579250956381b38a1
aa7260e332fa964ea94921726a796434ec2c243a
2365 F20101205_AAAFMF hu_j_Page_083.txt
e4eb7f277c2b7e18258da6f74b4c0881
e79eec3b7dba2f3e182ab9ceed7ae74e9edbeea8
16724 F20101205_AAAFLQ hu_j_Page_097.pro
c0efd26d818fba9d3e965032a645962f
ed64b30355e436914b4aab521832d53789fa6eee
F20101205_AAAFMG hu_j_Page_114.tif
c9334d08da090ccd72adb8d33ac55a95
4373aeebaca397ce805e7237316266d152dede5c
23698 F20101205_AAAFLR hu_j_Page_009.QC.jpg
e61e984b06635220b5d1ed6b424618c2
18d5ddef4113750ef391880c458dc63dd28f84bf
47437 F20101205_AAAFMH hu_j_Page_008.pro
eceb76598227e13f4442f00c3467117b
20c780b4ef97eb459ed6d21b99c11f0bd8a58b8b
5486 F20101205_AAAFLS hu_j_Page_088.pro
e00e65d69d6a063610b65f0dc8fb7166
768089f8ec8b1844c8b1b9b2e1775007e04948e3
2170 F20101205_AAAFMI hu_j_Page_104.txt
aa604ce870138b22dffd54cd8aeb4d4c
c4dc960aa77949b97273a63d4a7447c84cf462dd
847486 F20101205_AAAFLT hu_j_Page_012.jp2
95faaafea31ea32e9d7144cb625a8f2c
1c5cc40d287ada60429e92634408871dba8410e1
27825 F20101205_AAAFMJ hu_j_Page_061.QC.jpg
5208ac21642b94308eaab35937a2af03
7b306c6df0afd94e25f4d61ee4c411d00db3ef1a
2520 F20101205_AAAFLU hu_j_Page_038.txt
42437b9248574b1fb14d52f29c672f5e
1a03c80bbc8d1442b3747aaaba320a6bbdfa39a2
89050 F20101205_AAAFMK hu_j_Page_021.jpg
ddd2cbfa90d17965781653329c02f3c4
c1f53522319f8a746ec05fcaa2188fb263be07c5
42809 F20101205_AAAFLV hu_j_Page_093.pro
f9406baa033507aba2277d08a6c0c000
16dac9721c86112212ad653803a32b61ba65cee0
505613 F20101205_AAAFML hu_j_Page_032.jp2
7e19f0a0822d7961105de9ec1cd790e0
a469c2fd68393fd546ac1259159e28b4310e789c
6411 F20101205_AAAFLW hu_j_Page_038thm.jpg
45ea473588628cd976d60949c94493a2
66658c6fe8229d1023d2a0a70337ecc32c2593b1
29474 F20101205_AAAFNA hu_j_Page_084.pro
cc74bf719f4e1c6c42d5a4c82c18d730
35f3ec665f633d0c5322b7d175a552c3100f3872
49398 F20101205_AAAFMM hu_j_Page_102.jp2
4847687ec7c9f0fa565f686f5c716571
251e311fe34bd744d0bdddcc2f3357e4d8afe3e0
25244 F20101205_AAAFLX hu_j_Page_067.QC.jpg
6a080ef21e79734e17a9e35452e5cd8b
b65ee6506ca4827d127e1be92647a15612c0a392
266 F20101205_AAAFNB hu_j_Page_059.txt
0ccbdde78294ca6937934f555cfe029e
b425a78f174d7a7d7683920ce522819a7b330b44
F20101205_AAAFMN hu_j_Page_058.tif
24e9d0ec90284713370b55cde998ddc2
0b77b4ed5fe5ad38dce7e35b714777a1b62d25ae
5616 F20101205_AAAFLY hu_j_Page_020thm.jpg
94fa23df05fd721fcabdf8ca4a78efac
4066fe57db6e48b0e3705ed309c162fb07a62b2e
6046 F20101205_AAAFNC hu_j_Page_091thm.jpg
d3e33e51a137e8799e268a36d093afcf
b80d4c2389bf9bb7933a1a8bd6420ae4e9fa5b08
2278 F20101205_AAAFMO hu_j_Page_086.txt
33672d091e6ab9cb31abe851aee3d8ce
8766413a6f4c1aae48dac92fcc29fe562175f1fe
1207 F20101205_AAAFLZ hu_j_Page_103.txt
2ceda5879daaf88f0169d62bbab80a66
2035e55d947faf663e7e0082ce84619cd5dbe9ff
17239 F20101205_AAAFND hu_j_Page_047.QC.jpg
e53a12460b40752b81daf638d2a2fdf4
bada0820f1084d51ef73123a7410a49e88492191
2180 F20101205_AAAFMP hu_j_Page_035.txt
bd7bb455e955b2f6f4632f8295485a52
b08d8008bbe79ae801df34efab0b1a12a43d07b5
6471 F20101205_AAAFNE hu_j_Page_059.pro
c188506b4e94edcefb99238a72330d01
5ba062d00ffa0ebfa92d230eabf0798777db3432
22719 F20101205_AAAFMQ hu_j_Page_022.QC.jpg
f7dc39f6196e75fcc061c425fa7edae6
e2327f516c5268171712cffc49a18cb67633377c
5831 F20101205_AAAFNF hu_j_Page_016thm.jpg
cc0b9039d89704dc2e09ed927721e4f8
b5d1a0aebbaf528d6b14fe7d1a0c16d60c5e1aec
668868 F20101205_AAAFNG hu_j_Page_103.jp2
f93f0f7f3c9f28b88480a4ca60eb1eb5
d92b82dcbeb177ac36a58f8857f764ed5a741571
53114 F20101205_AAAFMR hu_j_Page_104.pro
788f944abda37ac066b40407d58cba50
3471f4591db2bd812ae758f6b7e848c3d817ca11
4870 F20101205_AAAFNH hu_j_Page_052thm.jpg
a20d50b551b4b17c75f5c1cfb384d059
e9427f5630c211a7b6d9c805860a695fcf9e6c7d
F20101205_AAAFMS hu_j_Page_061.tif
446810be117122c3b97e402c3e43b37e
0587bad5eaf3efbf83b9ee698d7c51fedaf5acd2
F20101205_AAAFNI hu_j_Page_029.jp2
a8aab84605c99292196c3ae92eb466f7
a4513433b75779846560741c0b498ec8fb026932
1872 F20101205_AAAFMT hu_j_Page_054.txt
82ae9f8ae97a11d2c716091384f00cab
0bc52175e6793c5edd7f24bc7f7d7689c59b4016
28186 F20101205_AAAFNJ hu_j_Page_077.QC.jpg
4321db500cd30325fd6b89f0b1095050
8417f57576e2c042eaf6de6f3c0e1f472e4ab595
66242 F20101205_AAAFMU hu_j_Page_063.jpg
63e05a9d1cf6d4a4503bff35c3359d64
f625028118899baeb9949e45e1de362c6465f2c5
6168 F20101205_AAAFNK hu_j_Page_113thm.jpg
2b9f797f3c4dada9d4fbff9d70bb9a71
9724af2ad7782c8331c5895b7f90a954cf98d9e6
F20101205_AAAFMV hu_j_Page_025.tif
a5d545e041d89385d517926eabc9d46e
304fde93c678c2cab9832cff38feb08bae891b6d
110319 F20101205_AAAFNL hu_j_Page_023.jp2
4b91957eced27b4f9387261b492da4ba
1b1826f878748667bd532711caf87cdaa3ea2177
1313 F20101205_AAAFMW hu_j_Page_084.txt
d60359e21f0886452746925a8b317cd0
e178b9f3ee5103873e1921c722c0ef00bcb168a9
F20101205_AAAFNM hu_j_Page_021.tif
d4f50cd17dfaf4f5aa3c2e0fae60b8a1
c1d998bb196c4a949913c0b2963052be2cc42297
2433 F20101205_AAAFMX hu_j_Page_076.txt
ede5bf99035f5707170497ca24ef04e8
5297be8b53f22afdef3c4826d5e783bee2c007c4
18348 F20101205_AAAFOA hu_j_Page_085.QC.jpg
5cf2c61a8f2ef8f080fb3bf61ca59b6a
23dcee62e090130c9b41e79568ab9c4304bdc4eb
25903 F20101205_AAAFNN hu_j_Page_030.QC.jpg
5e07f51bf99b0f8005167491d0f4945c
774e5a7582b67f776e1b3532d99aeb06e2e9540b
F20101205_AAAFMY hu_j_Page_037.tif
a1270884bb7926d8540d0cbef177e3ac
a439c965f0eb06da14d58953d4017a91bac9a0bc
96676 F20101205_AAAFOB hu_j_Page_062.jpg
8f2cf5d02a2b85f3629ee1c13aeaf9c2
8a744d1eb71dfb00cdf613b35b265af1b58b624b
16531 F20101205_AAAFNO hu_j_Page_114.pro
d71a1a138db21d16e8e17afea2410d29
5bb58c1bce27a68fbf50a518846d17f4dd9835dc
87049 F20101205_AAAFMZ hu_j_Page_098.jpg
a6d140cf7e5be3587f46f7a958739ee5
b7ceff765505440cbb52b5f296849905f6439625
895263 F20101205_AAAFOC hu_j_Page_044.jp2
3aacf4cb5faa89a7a00d0fd12c6a4681
cb54bf9961e2df5ad41e13f9be787a3248585c20
F20101205_AAAFNP hu_j_Page_049.tif
5a9bcdbf724e29740ead4119e1cc4490
56ad0851cacb22a3d9809380426c4d07b9c2d65d
1873 F20101205_AAAFOD hu_j_Page_043.txt
bf8a9d0effe5ec2ccc4f1e249f94ac1b
233788e510d5ba169deec62d820c377974e2783d
834690 F20101205_AAAESB hu_j_Page_072.jp2
c9c6866a4f84264d2294513423b928a8
2260cd35c5c297184e9d412ff95df554885478d4
6093 F20101205_AAAFNQ hu_j_Page_078thm.jpg
3ccfbcb8ae34727f8f1902d0e573ba3a
58f026f63d12b5c621556c559af6c8dff9558600
1051981 F20101205_AAAFOE hu_j_Page_025.jp2
77660272c1c14aff0f10a3f308ea5184
f256f7e0a4dcdb2c3017248716f04f02d36b931e
5874 F20101205_AAAESC hu_j_Page_003.jp2
c24f7d526d19347d0b2cbbc626f43340
ee6dbf837e744aba2dae14e6236b942fee896c3f
8066 F20101205_AAAFNR hu_j_Page_017thm.jpg
b9c0457a365f7c0f8385e27a53f1e1f4
2eb702f374f1089c4b4f666006c249727a54db27
24591 F20101205_AAAFOF hu_j_Page_080.QC.jpg
f6b37e93686429140347d39b1478c52e
5c1eeaed0ca4b7a210e89ebbfea45bfc359a10d4
24704 F20101205_AAAESD hu_j_Page_108.QC.jpg
5178ac9c96e9ebf6889fdead80accb14
26390864d180b02db8a8592176378e73584c272b
54533 F20101205_AAAFOG hu_j_Page_040.jpg
81798e7f2a02073156913ce40c2bc113
1c73fcf6595f91d206ef3feba3144e6b13fcb0ef
39995 F20101205_AAAESE hu_j_Page_050.pro
c05a1f80cf958fb94be15642448ff1e9
f7b369fec4034c220f62da743694bd18251d5679
812977 F20101205_AAAFNS hu_j_Page_081.jp2
bd9cc3fd2b136d9c361a5a578290aab9
59f5ddfc2c212c0b2d8ec47ef07a187e0f13d90c
1886 F20101205_AAAFOH hu_j_Page_050.txt
883b46f69d913bc627d9d4de371d5e14
919c90b004480b0c66669d5ce711390f1a7ab339
F20101205_AAAFNT hu_j_Page_074.tif
c21bd921bb3ad3631d0e7c2827b7e587
252805a5fa6a8a3d587372f4e905dd499228663e
2410 F20101205_AAAFOI hu_j_Page_109.txt
bcb40f9b71d328f925af0cea15a6a8fe
03a9a72304d2d3f8aec8f73f48605718e9aed705
1237 F20101205_AAAESF hu_j_Page_003.QC.jpg
9d4e7467216ca546b66de6d42d65a6a4
006a71aea7450b2dff4d873d19bcb032612bd58d
27013 F20101205_AAAFNU hu_j_Page_024.QC.jpg
643ac8881b142e66d53f05a14911ab99
dd41e1fa94dcc56ca324748d68405bcd62bd77c4
27669 F20101205_AAAFOJ hu_j_Page_078.QC.jpg
6d8982c4d86c5decaaac1f04a3520c45
9319eeb5c22c4b41292893f29ebd9c3b59d83f11
33511 F20101205_AAAESG hu_j_Page_033.pro
ee538e0fdbd50c1c156a94078423ab1a
2ffa39174dd6e6d84e51c48830159be0fad1821a
6085 F20101205_AAAFNV hu_j_Page_039thm.jpg
938b61d12074080db001bee71b3c5aa9
2eb155e3c710fd35df02b4a5a8ebc1da6448903a
28220 F20101205_AAAFOK hu_j_Page_057.jpg
6460852393190843454d70827e1154e6
c047258594dd5dadb16b2b3be623d16315a2b0a8
60833 F20101205_AAAESH hu_j_Page_077.pro
5a50fe16cc3b0f760eb030db55e52a72
ee91c3df57016017402fc0f274a0ed90d7197353
24882 F20101205_AAAFNW hu_j_Page_106.QC.jpg
b48b3624fb2f5daf4e13c4a3251a6799
83f253ccec65a7e22515a4abf1e71d29a036a9db
32460 F20101205_AAAFOL hu_j_Page_017.QC.jpg
78366cbd17521b0792747d4eabb1d671
8dc322933b83091bc65bc3efd223546fe1bec41e
15505 F20101205_AAAESI hu_j_Page_046.QC.jpg
8608ea011b29fe9cc07d153346c12834
4fbde97ea62f618de9d95cb9949a980f16ce1b7c
72657 F20101205_AAAFNX hu_j_Page_013.jpg
7afd3506a997f4bdd70eacab85830ba8
9d01a8936d45ffa8a314b2f4a635db6676c9fca0
F20101205_AAAFPA hu_j_Page_061.jp2
aa512f827bb91a8e75cfcc40a13f03f8
28ddc5147dbaf5ab791eeec4097263965857bc9a
34950 F20101205_AAAFOM hu_j_Page_114.jp2
eca68555d4a1e52b9cdf6a3efde643bc
53589ed3b5b887cde1172b12e73bd30e6ab8ead8
F20101205_AAAESJ hu_j_Page_034.tif
fa27e681d676f811b16db67a0ab95210
dace22462008857201e6429774c7b63344c0724a
639323 F20101205_AAAFNY hu_j_Page_087.jp2
d05769b576911d04f401b9b99bf0d609
f5aeea4ecfd57df40b10cdf660be4cc83e472948
2415 F20101205_AAAFPB hu_j_Page_025.txt
2f0274ab1257f936e8d90b1d8458e720
06ee2eec784a9f924280d5cdd82c6dc2b809e4fc
43253 F20101205_AAAFON hu_j_Page_041.pro
4a52229f31bbfb9220e9817af80cb27f
d6d6e72b8932164500e2e22ec9e9104fe9e24cfe
4546 F20101205_AAAESK hu_j_Page_002.jp2
2ecf932de958b933637859240a89e82c
d5e3243350217c449e5a78cc55688aac98ccbf07
37591 F20101205_AAAFNZ hu_j_Page_012.pro
035cb8bcf12ede916edfe91824c09949
ccdc2bbbe066a959ba666e5befafc0e325fde6d3
169508 F20101205_AAAFPC UFE0017765_00001.xml
c2557907045551c1c259bc0c7d923a1b
f0348cc50edaabb21fb46cf16603af73db103145
7152 F20101205_AAAFOO hu_j_Page_001.pro
fd4785c8589eb48e775cae3d7cbccd64
c298fa10d60ca70e04cc8da0b7d7717d6ca23cd5
25687 F20101205_AAAESL hu_j_Page_005.QC.jpg
fa0a33a5b59b984eb1e3f88dc2b1dd76
f6f9301e972536fc3479f9684b1f3e6ad71893dd
57295 F20101205_AAAETA hu_j_Page_108.pro
2c123939d903eacf531fa04e4f12707c
cfcbad3776993c27e363c0030d0f953b58507b0e
895 F20101205_AAAFOP hu_j_Page_102.txt
bb021ce284f9cfbfd3eb1223e5beebc8
e97a0d3fe45fc106a361749c53fe748440aea64f
5373 F20101205_AAAESM hu_j_Page_022thm.jpg
5bd32cc6edd3d6bdd86f0f2b0de66b06
0647236603af03cd3e6376e5f7142524cd7c6aa7
F20101205_AAAETB hu_j_Page_030.tif
7193e4f839226d6f76036dae603d6e9f
e0dea7dfd6615d58430eeda8aaeded200e691475
68245 F20101205_AAAFOQ hu_j_Page_096.jpg
8bf8d14d09fd974929a7298235d964cf
ad75cfa737e96a99c341259a7a72d7263fcf09f1
51912 F20101205_AAAESN hu_j_Page_067.pro
f96c523e66c7917af7f667221c2f23db
0aa21deffe3d745b88957cc0eef58f87e65c57a6
3616 F20101205_AAAFPF hu_j_Page_003.jpg
a2ffb7b9018830dc1ed8bcd719aec6b2
52c3b984a9cbcaa14d82e5f9dec007fb7d16d6eb
14555 F20101205_AAAETC hu_j_Page_102.QC.jpg
dbf662ad8830b6bddbdc75a209b3e164
5a6505ae6d51b095db395052af593074433ae92d
23760 F20101205_AAAFOR hu_j_Page_068.QC.jpg
2ffe8345f8fa96852b7d279cba4e5d69
a1bbdcb51530b3d1c7cbcdb39cd6b5ee8a497e0e
45695 F20101205_AAAESO hu_j_Page_054.pro
eb01cd4b87145961aa077d4d7e05a3a4
9cbe136dc801d16cd06014c1f2887c6fcceb75b3
63810 F20101205_AAAFPG hu_j_Page_004.jpg
e62e01a2df210eb8abc278dcde4eaffb
89c7bca599e0e1b4314d417b7f3dde85a869f305
1589 F20101205_AAAETD hu_j_Page_101.txt
56b8bbb5d8f699eef614928983cbe125
7292a5f1efd06b868c56dcfa5ec3c120232b6405
97792 F20101205_AAAFOS hu_j_Page_071.jpg
36cb6e6a1cb41a3380754f4117c518d0
f3a5f93e6e9c38b47818ff3f0a528c1bf5461d59
11752 F20101205_AAAESP hu_j_Page_026.pro
808b8eb505b85ef235e97a0b61ef90a4
e6c42ebd2e02b68b1e80e47f4741ab5ecc0a3a9f
97625 F20101205_AAAFPH hu_j_Page_005.jpg
da3c03f438a0d312c2f4e891de5c0c4b
6d82d1e7275b8ed3fb0344d6767a012aa2a16bda
56904 F20101205_AAAETE hu_j_Page_086.pro
30c82f9dffc4259ff014b2560535a034
c70c2559143180cc25aa790ec4238d2255e18731
5333 F20101205_AAAESQ hu_j_Page_041thm.jpg
40dc4d838022b289d52ecdf2002c87c7
1ea2a1260ccae52bed67ceb643174198e2d8320a
70239 F20101205_AAAFPI hu_j_Page_008.jpg
70fbc9ff83359bd180867fff34d3b842
991e15605998ac8005940fd80880e87e567b0a5a
28856 F20101205_AAAETF hu_j_Page_025.QC.jpg
8349be01bb1296aa2a9c0634a30b281f
ffa02d691febbc085a02eb8c56b9196e5239400a
57361 F20101205_AAAFOT hu_j_Page_112.pro
4a44d8b20e499672317e59bd4b67983c
7951c39383fa6c5351917d3d90d89b1a1429bf97
90860 F20101205_AAAESR hu_j_Page_086.jpg
ec1b2e8d5f09fcbcfcc8bafbe4d9b17e
14298ce3f1923e67b560719d0149b85575d0a870
88822 F20101205_AAAFPJ hu_j_Page_010.jpg
437c7959fa2093c9f02e2324e238af4e
79b822035a3a640153534794273f089dceefcf80
43499 F20101205_AAAETG hu_j_Page_096.pro
f95ab80c8319d89609e0f8ecaf016002
ef9fe6e4ecbdfd97211cb962c7891f6d11d43fef
6153 F20101205_AAAFOU hu_j_Page_107thm.jpg
a8624e94b99d830cf8ebaef35f8fb572
d9c4b420d0c5f06bc9f3247099eb0aa12bd9c2c3
1051939 F20101205_AAAESS hu_j_Page_086.jp2
75f71fa4a1614f840d17cb8ad59c2dbb
34c214fcbc0f75937912fdfefeef38d1cd73136f
86588 F20101205_AAAFPK hu_j_Page_015.jpg
a03d161d29de2853be79e8c1e6d0c6f5
d1dfe21e1f4b3cef7370afa4a9b71c99db6c0f8c
36556 F20101205_AAAETH hu_j_Page_115.jp2
eb02f9d8b94b940452d61b7d56ba90b9
621e752b854639a65325fce3a26de14d989e7f88
875539 F20101205_AAAFOV hu_j_Page_045.jp2
4a1ce6bcc8166904659593201785664e
428f82a35140fe3261461056af7de84a39796bdf
45694 F20101205_AAAEST hu_j_Page_075.jpg
5802537732b0055746bc604d229feb7a
a81628b74b0f05268b657d71cdfe88840308495a
90277 F20101205_AAAFPL hu_j_Page_024.jpg
97a99ed2f6900f32f1c9b2f150bfb859
66b57c6f04fc83b5abd53d980f3accee33d07beb
62493 F20101205_AAAETI hu_j_Page_094.jpg
4ff5573e16214c737a4b943eab578b0e
af4afb874b5b0e9483334a739c2eaa9d87b25752
988592 F20101205_AAAFOW hu_j_Page_033.jp2
7c4bd72c3b059a0ae8234d5575b7edac
3dfd1b21cdd04da242c3bca39285327e1ecc3ccf
51028 F20101205_AAAESU hu_j_Page_056.pro
f761f787758b8cee4984bbe2629455e7
a8db33f3b4a03ab86b84d0c47c5fffb2e294d7ca
59402 F20101205_AAAFQA hu_j_Page_095.jpg
e349fabef4032bab4e170541f5aced03
2b8038be22561f5c4d6aa541cb94ef122642b006
21944 F20101205_AAAFPM hu_j_Page_026.jpg
be8edf81aef308c281dce0ee85bbb84c
9cfa84e8ef753f6956505ba5e8b92b36d587f2e1
88715 F20101205_AAAETJ hu_j_Page_007.jpg
a2390624fc22462c1e9010da00c8dbfa
c959052c4a7ad78779414b1103c40649fbdb6798
677 F20101205_AAAFOX hu_j_Page_114.txt
52b225552ebdf99d05ec723654b8cd10
781c998148ee1c0ac05fdd66e94caae8413fcf01
25114 F20101205_AAAESV hu_j_Page_109.QC.jpg
902df85e3ba7ff47dc08b97ffe8fbf11
f9d89e32f8bd8faefd286fd3aaca1177a5af7264
74310 F20101205_AAAFQB hu_j_Page_101.jpg
153011594db145386db8ad57c49d8248
7b3f1ab2f3cd95eb3540031dffb3a8d3f854513b
83182 F20101205_AAAFPN hu_j_Page_028.jpg
3988ac2efbb584e53f976468d6c2bdaa
4712b767fc88e982cc8599fb43918b362fa7a76b
14125 F20101205_AAAETK hu_j_Page_075.QC.jpg
c3c0aefe47aa22b37d30eabaa156581c
1dbed06619c95103210dfb0c7e074e07ec8e26fb
15439 F20101205_AAAFOY hu_j_Page_115.pro
9a77b1275494240279c504d925415cae
02337630e43f5012ceaf940bdc61ffdc978f2584
59730 F20101205_AAAESW hu_j_Page_014.jpg
46a777114facf22de0dcf6377b58394a
a786ee480e51e83de445e6369ef71428ec740c07
46698 F20101205_AAAFQC hu_j_Page_102.jpg
bdce6d9829045e263b778ef99d610878
fee2d550aeab495d723a81ee2c53e2fc84b95db6
87498 F20101205_AAAFPO hu_j_Page_031.jpg
9688fb8ec4e5a5b07b1236526efd224d
225435fbe26c39e14c4ec04d99b54d10f7d58c99
5193 F20101205_AAAETL hu_j_Page_043thm.jpg
f00035ab51782748e30310eba183a533
ceff8c07fe71835af5ea5ec844864b27ff7fd683
F20101205_AAAFOZ hu_j_Page_098.txt
6195072ce6a7bcf33c7c900bbd2b9613
26e494ae891c270d19322cd68a36612b26c002b5
13838 F20101205_AAAESX hu_j_Page_048.QC.jpg
f63a277ac1a8aa9343f3ec9799cfd3d0
ae45619fb755f92fe9e2d61a1a38e3e53de3d693
50861 F20101205_AAAFQD hu_j_Page_103.jpg
8532f9ad068d46eb562a41f75881250f
3933ded6ebde3e3aca87e369e60f21767b00b525
6286 F20101205_AAAEUA hu_j_Page_064thm.jpg
5b12725347408e7d55102aeb8952c88c
de8cc667e4a61207f33ccee8d1e6b7533f4429ac
63773 F20101205_AAAFPP hu_j_Page_033.jpg
e3d538068fc23ffcbf1deac5dbdfb8cc
86a8418a10440a10dfe10643b2a44ff7ce2e5058
812313 F20101205_AAAETM hu_j_Page_084.jp2
307eb3abeda68983930c771e44149474
be762977dd20d7757ab55b5c6be9742007f4b21e
83173 F20101205_AAAFQE hu_j_Page_104.jpg
b3e97497690d90aa432167a1582a9e47
2f1a54bfdd5241436346cc17768348a4eab2b0fb
5963 F20101205_AAAEUB hu_j_Page_106thm.jpg
c4e18b9637b6a72d98f485012784d4aa
5e2c63280fc8a1dc015912669197b5063d3f74f7
99235 F20101205_AAAFPQ hu_j_Page_038.jpg
869b5bac6fcaf7c0b153bedf426279b6
2204b7297f1804c4c384e5e96c6d9136055c4dcd
17997 F20101205_AAAETN hu_j_Page_079.QC.jpg
a306dcd2726694df36c835e3824fa89f
6bd7c803b98a228001a7174ba7851863034b913d
F20101205_AAAESY hu_j_Page_050.tif
ebeeeadd3424f70ab3d5abca16712921
61f37a99a674895ee4cca36e7c77a3335f5bea4d
85843 F20101205_AAAFQF hu_j_Page_105.jpg
06a6d5a2b2cfe8d8ad31f28d9b213e04
f23e06dd37f7f0b300b44a7d119da8b80df549e1
F20101205_AAAEUC hu_j_Page_073.tif
638e7fe17e08c02145ec8415590c6867
745c0680428c3440e0ecf01abf07b65e7489d0a8
52292 F20101205_AAAFPR hu_j_Page_042.jpg
ab1a75f3def5ce54c8296d4d92d825ba
f5eafa87b900813c1e6400643430af87b222cbaf
20657 F20101205_AAAETO hu_j_Page_050.QC.jpg
e9e8867e96f01184e942311cd75925e1
4ec05eeb358e8fddc304c076d670221b8da0e117
21422 F20101205_AAAESZ hu_j_Page_090.QC.jpg
7af71355908ea41eb9db938ca0e29a45
915933e38996e2e5133957a110d190cecd90f135
89457 F20101205_AAAFQG hu_j_Page_106.jpg
841cbdf930de3bcdd26a33da5d43dc87
d70a7562acd7b03e253ac75555431d77b1ac4415
57753 F20101205_AAAEUD hu_j_Page_047.jpg
44ec206df40b78e2723124c9868e39f1
d1e86dacc4cab52d2a62f8ef865ed9fa47fb9085
49613 F20101205_AAAFPS hu_j_Page_046.jpg
17a9dd20b9c352dcbcb5778a18a0535e
e51eccb120071cf89451e62056884f588fd1e3e6
38492 F20101205_AAAETP hu_j_Page_034.pro
49f7a98366ffca142dc8feb47ccac7f6
12ff55efbbeb6e353c05f1caa1e24282d47b9752
91092 F20101205_AAAFQH hu_j_Page_109.jpg
122a88aaf6644a264e7b691adb9d056c
dda20d39b0fdb10aa2f0da255bbdbba8085c6e38
27518 F20101205_AAAEUE hu_j_Page_064.QC.jpg
9f7115d8d22bc9c503136cc924fb2fea
f54c525daa788e2534d2e1dd420ff9b13bc1b22b
75651 F20101205_AAAFPT hu_j_Page_054.jpg
c0d96a402e913983e2fcc54867ded62d
a195303c9d58728fc5937257e44351f4ffa33374
14010 F20101205_AAAETQ hu_j_Page_097.QC.jpg
3b4ce12aed3d1bf13589c774dafe2bc7
3ed968d2727d2c6008a01975b1b62952a58e0832
74057 F20101205_AAAEUF hu_j_Page_090.jpg
255373caa9b815ba8f32c2b977157a8f
23330997c117002c046bc7591cbdb050708f246e
58803 F20101205_AAAETR hu_j_Page_079.jpg
9aa85580ce63105ae35487976622adc7
d43070e797548acd0859a8a70604acc26162fe3a
91623 F20101205_AAAFQI hu_j_Page_111.jpg
3abb837bef47bfe2285949256e8825e2
b9a4f4cbd32c20191799f3e1e7dfb56c5ef86bf9
1898 F20101205_AAAEUG hu_j_Page_114thm.jpg
69437b6c5ea3fb1b51fbb04a46a0931f
bea674294e15be0f4c92c2e59f1d72cd7ec0bca5
12601 F20101205_AAAFPU hu_j_Page_059.jpg
524682d0cf398987fa86a2f25cd52e4f
d61653dddfe0f30bca50b2e6121a6af48a09332d
1240 F20101205_AAAETS hu_j_Page_065.txt
3db2eea3d353c1ec32ee2e14dcf1ef09
b80c764d97690b2c25029b0255dc2be757e9993f
89626 F20101205_AAAFQJ hu_j_Page_112.jpg
7ca6e16238fffc312ab12d4decb6939c
390b629da94a6ea5ebafcd954e55d5035480ba82
5783 F20101205_AAAEUH hu_j_Page_070thm.jpg
15d326d70d16d931b16e3a9269e3b705
549a9cc43416ad8c1ba0a5f4b136e280cc25ad48
94220 F20101205_AAAFPV hu_j_Page_060.jpg
1fd09979c30c9eb77f39a18958ac8138
9755a673fc488c1832260d115d3cb4e2fc0d61d7
F20101205_AAAETT hu_j_Page_053.tif
7742fca72b4d794e891781f69d43c795
3eea30bb1168d75b65a204ba43d23f2caf1764c5
100091 F20101205_AAAFQK hu_j_Page_113.jpg
f9336550cf8f80cc4518cd628a0cf290
49f418c21965da045f36fcef0fc81700e5d3dca3
5043 F20101205_AAAEUI hu_j_Page_013thm.jpg
b547fc0bc108ba8951bb39c7604adf53
85c57b90e49b8d7ef47985749734ea160d223576
95311 F20101205_AAAFPW hu_j_Page_070.jpg
f70c1b90bbe9407e9557f2538a770e71
77dff2f61678be67717e5af20c9496e9baff3ab4
4251 F20101205_AAAETU hu_j_Page_046thm.jpg
23d4f2df18d4c77ed9bbefc89724afd4
6e551983250aef46833629e36fc6bfa1d01610d9
28433 F20101205_AAAFQL hu_j_Page_114.jpg
513051de80fe346d098e950b831f8a52
41bffb8e82463bbae7cb8083705a2bd783a03be7
F20101205_AAAEUJ hu_j_Page_021.txt
ee1da7cf4baa8aca6e8978d893c816fb
a94da4f840d3b9a23330c16a02fb225229511e20
67096 F20101205_AAAFPX hu_j_Page_073.jpg
cb69abf49a5262a25ebaf0ead3e9d8a6
c1057a07b29c32a22b4f65ca3446f24cd2437366
F20101205_AAAETV hu_j_Page_046.tif
185fe1d29f91e9225fcc95428a477fec
673fc5450c5d60848f060d92c1a3a9e7fdff6cfd
670318 F20101205_AAAFRA hu_j_Page_046.jp2
bcdf8185c882bd53ed37180d8e5ea69b
43bf91aff753bc2948e1dfd877a91f1c00510ac1
21178 F20101205_AAAFQM hu_j_Page_001.jp2
840ab22aa8848db87d7d18ba9cdd54bc
36b7c32d9b8275c5e95d0793c05dc1aaa9bac2e6
23117 F20101205_AAAEUK hu_j_Page_054.QC.jpg
e759a09e9b857ad4a5ee8b486c4953c2
900c23d1696ad49fa5d5e6b7e10388502db61f6f
93412 F20101205_AAAFPY hu_j_Page_078.jpg
cb63b13dca99b9428399d2b7ac448605
2374ff21c6f0b59e88f446086970c71b13c33cfe
F20101205_AAAETW hu_j_Page_077.tif
7a07737dba4d0c9b4be8d897da7ed113
588570a1ee3243a79208afbe34b5463bc2d66cbb
759279 F20101205_AAAFRB hu_j_Page_047.jp2
975651da0f5f766eab9b0845f3cbdc6c
bb1737fd1c515be85c7afabed2d0d2286ca62dfd
130881 F20101205_AAAFQN hu_j_Page_011.jp2
4fa8d494819025d11aa541c0e2e8bff3
33c8ab0e84c5590b6d08517a4070415b5f1a5606
F20101205_AAAEUL hu_j_Page_082.tif
faaa22e8da28e8bbcaa45f40ac7db59d
0a16a0858a692d520e3ed606ebe1aff51cc3dd11
84923 F20101205_AAAFPZ hu_j_Page_092.jpg
54cb5ed54808c64d67895b332e1b11b6
0bcbc2a87a4c276b440fa1cbeba867d3b830e9f0
17102 F20101205_AAAETX hu_j_Page_087.pro
909873d456a5503a3067f553e2ff7132
e73f12cdad5a144e98f7c63d76a33b1151c9d6f2
871754 F20101205_AAAFRC hu_j_Page_052.jp2
643a4d4547786f7d9cc7fc9c97ef5084
fd7945e68aac761006c6b301c8ebdc2a6c608a12
115919 F20101205_AAAFQO hu_j_Page_015.jp2
3360b2a98f4197e6009c8c047694690b
e6920923bae6c0da5621260dcecd435606db7e7c
1051972 F20101205_AAAEUM hu_j_Page_071.jp2
8faa0cc8c349f1991da8355f96cb254c
484a491ddbec485007883cf787e2fa961eab8fa0
87847 F20101205_AAAETY hu_j_Page_108.jpg
15809deeef657fcd15d305c1a79abe1a
62567c05f0a132e5853d6424287d627a38d03d84
1040159 F20101205_AAAFRD hu_j_Page_054.jp2
ace06c62dce289bbca863a04b8d8b3f2
8e8ca449c6a026d1d42df5b32b0a2e91ffdc404b
F20101205_AAAEVA hu_j_Page_108.tif
4797d77affee97c62a36384ba5054005
ec5abe4c3e7772243e2d28eebe4661cba49ee1b8
97272 F20101205_AAAFQP hu_j_Page_018.jp2
84f0211695226aaeb95c6b2b835fbf57
f99a1b22e195720746333528ee279f6575ad73ea
38347 F20101205_AAAEUN hu_j_Page_037.pro
55da8a38de522ccad5145259ae07bf0e
93e2c329ffc9a451f29ccfebd59e59813f1b1a0e
F20101205_AAAFRE hu_j_Page_056.jp2
7b73cbdb6164021f9c53a336a2af9abb
aded19dfe3fefded7e36bd1a6070c70b5ab12b59
92654 F20101205_AAAEVB hu_j_Page_027.jpg
996a156567092556dbd9edb91b3753c6
0f5a3d42328e2cf6608235026c68c4dbcabe3d72
1051938 F20101205_AAAFQQ hu_j_Page_019.jp2
1551bc2deaf775fa7a73f8dbbcac6c1c
bd3d695176e2154a08be84dc0128ce2f42d5c23d
F20101205_AAAEUO hu_j_Page_031.tif
e5b0c0633af67386d6b2f740a4eaa0ef
195d711066d3172d1aaf0ffe8faedfc06f63aece
6170 F20101205_AAAETZ hu_j_Page_099thm.jpg
f99d7c4f0f44020a90e4c6f3907500a7
592ab14445b157a31de1d3ec7aef9543ee7eb2dc
1051967 F20101205_AAAFRF hu_j_Page_060.jp2
e04f5c79bfa9cb53e3eec07e8d716a72
942b3ce2a0878b636753c7c58f8ad686760b02ea
18908 F20101205_AAAEVC hu_j_Page_045.QC.jpg
b5ff073c67282d6c99f653f17e432221
bb692de981dfc88352fd9008efb225ce0937b374
105315 F20101205_AAAFQR hu_j_Page_022.jp2
22a3261bd9af81329aeb943f6ae19b81
944cbfe841c184966bc9e3449efbaee18e04a330
F20101205_AAAEUP hu_j_Page_062.txt
f708daf3bac12a965861cf947b9dc13b
fef08d10aab6735dd8b6d42be28b70585b4eb365
F20101205_AAAFRG hu_j_Page_062.jp2
1c1bd673e5af15763ca839b09d737b0f
98e796e0fb6dd8b0086445f14802d7d0b69de087
5361 F20101205_AAAEVD hu_j_Page_093thm.jpg
8cc8a07c473dc94f6ab61e8d1046832f
7ab9cb747e74df431fb24b85595fa094c99ac15f
124426 F20101205_AAAFQS hu_j_Page_024.jp2
77ee582a260a52f2ab08e2e09dd03b1c
1826fd1751b129871debf5f4810585aa3a3d6e36
149716 F20101205_AAAEUQ hu_j_Page_059.jp2
022f3b81ea3163578a5732791a181bdd
8846bbb4daef0841618dbe0d4dda88d82e184ed4
F20101205_AAAFRH hu_j_Page_067.jp2
c0c4bdff289b6a0b1637abfcceea2794
d0cc4d0392d44d6a765bdb817e7a961d743a20ae
4606 F20101205_AAAEVE hu_j_Page_014thm.jpg
904d27037f13908a0add2078c9c6177b
c04c038b1bb03d30e388561ec2bd853fdbfd8466
276195 F20101205_AAAFQT hu_j_Page_026.jp2
9e08749cb167c1d2d36cab6e950ec6a2
158825197d0d6f987ae8b301244d2de2a9ace681
1422 F20101205_AAAEUR hu_j_Page_100.txt
9c89c7c8b6582c0a50cc0b8681520399
50985c5c375d5e2fb1caaf7b79365220f9d1a7a3
F20101205_AAAFRI hu_j_Page_070.jp2
3fb7de4b3e7898efa1715b30201839f7
7fbe190df68b0a0f127021ae6a118eef8760ce75
20190 F20101205_AAAEVF hu_j_Page_037.QC.jpg
99184a82056492aabf41ede5777847ab
13afc718ccdb71ab12eaf4200f863a3f6e2bd0cf
1051926 F20101205_AAAFQU hu_j_Page_027.jp2
8b4173883cd45cb18862536b32118a95
0a69deba7801ec5e1191ee87652b7987926d1a0b
1051946 F20101205_AAAEUS hu_j_Page_074.jp2
0a5a86cc0c5b202c4bea03743654c2a2
8744e90254d5f2ee7cf8254cb82f2d15dde120b9
862649 F20101205_AAAFRJ hu_j_Page_073.jp2
183c4b7c9c93c2a781c107305e70257b
cfc3a0119dc7cf96bf403aa6b497b9377814d5d8
F20101205_AAAEVG hu_j_Page_045.tif
d6e5854ffb3d31674f87e3648db14696
5c80e30520bcdf2f2e06508817e0d1008c149abf
F20101205_AAAEUT hu_j_Page_048.tif
54b2944c1215464c273ed4fc75f62fa9
dfb013f0fccace9456b21d97b2f8bf36b0ddd2bc
520817 F20101205_AAAFRK hu_j_Page_075.jp2
4c0d6210175e316926fd201d986273d2
4a146adf3fb6c337c40b8dff6d49c4589c865893
56597 F20101205_AAAEVH hu_j_Page_027.pro
60933f0581b6641b51857284bd00ce97
53a5795fd6510b80efd5891a712d3c4436661eac
F20101205_AAAFQV hu_j_Page_028.jp2
9359d45a713c3e42f141773a5ffbd3b1
4ae9eae89f7d7928b5cd4b182e14a9d0b2e9abd5
71784 F20101205_AAAEUU hu_j_Page_053.jp2
f326d0eac8e0e762dd0c689ad5a00816
6e2772a8e274f21ff1b70b74d9086292c6579210
782219 F20101205_AAAFRL hu_j_Page_085.jp2
327295757935481760b8be649339054f
c4ab30acbead917e85867f9c81e09e7c4d9e2e84
1396 F20101205_AAAEVI hu_j_Page_017.txt
197f341893f49e39339bf3eeef4fab77
8f0e7995460fb5d17e37aea46c097d2086c77879
1051964 F20101205_AAAFQW hu_j_Page_030.jp2
2fde8b2078503ee7cfcfd573316dd1a4
c7fe12c1ba0857a8ecfe2b6a4455ee4d4a149468
F20101205_AAAEUV hu_j_Page_002.tif
e88b1e2ebabc2ce250bd31eb5f374248
288679663c723fd23bda483d9129922e2f6a0a95
F20101205_AAAFSA hu_j_Page_033.tif
5b2d2203b2e7b4df8f5e09f22e4a923f
1c8c1c3837a00699ee36e18ddf0d9356d029a947
789693 F20101205_AAAFRM hu_j_Page_095.jp2
1c5e9e995665a66388931ee1e45869d2
ebfc0122f53bea90d57689152e6c0592070e9af3
15243 F20101205_AAAEVJ hu_j_Page_087.QC.jpg
619a7fdc19134b7f5aa92926e9e413b7
42aa87987cc0067f0603a7d8c6c0ee0f48dbca94
1051922 F20101205_AAAFQX hu_j_Page_036.jp2
cb179cf94e2e007d82ea97441269ff42
e262cb82cb6485973ef5f8dc0a4f93882d2d55e8
831 F20101205_AAAEUW hu_j_Page_048.txt
62809cd71b659cf036a731c2df74955a
bb41fe6c4a9925a92e0f0e18ea1fd1d046f37c8d
F20101205_AAAFSB hu_j_Page_035.tif
80cf78388eb76000e7d299eb1af3d970
83ccab1128cf4672dd650d27aa393482fa0af238
1051986 F20101205_AAAFRN hu_j_Page_098.jp2
afb82d4b230c2d0362b7fe6d0f6d0171
af7d18f67fe5028205d7eefb083127df95a8d278
34088 F20101205_AAAEVK hu_j_Page_017.pro
dbfb9d3789698a2d70313cf28bb0cc51
c76824d0b87ee412a5de5ab6e9459dcedc844bfc
F20101205_AAAFQY hu_j_Page_039.jp2
6ea83fa7ca05ab4f5183c4c87c94df20
7cdb9d3bf410929ca1b2b86eeab81d8c6418667e
113773 F20101205_AAAEUX hu_j_Page_105.jp2
88d267c7154019bb12565e28dd2ac03e
bf332179aa5c5553286e7abd33dfe276c049ff8c
F20101205_AAAFSC hu_j_Page_036.tif
77bbb60bebb6c2c56cadc5cbf93cc1b4
9f71cf8a2841c36972bbfa055d92dbae8652afe0
F20101205_AAAFRO hu_j_Page_101.jp2
aa41e1a2f847911a4dbf474786e44a95
90283f2ed70892fb9d5f07a02e24459837102166
1805 F20101205_AAAEVL hu_j_Page_026thm.jpg
f151c3d1410246862b1d5eab73b109bd
49b1d75eab565937b8f8838e943e73246b9151fc
992253 F20101205_AAAFQZ hu_j_Page_041.jp2
c7c3984abf00d616ffcd893f3e5c2c99
ff1dca6f7eb099da1359856dcdc0c2af57996b7b
F20101205_AAAEUY hu_j_Page_092.txt
62d81ea71d17f220a860835112d92d12
e8d986d91e4179423acf317706f8baa59e4aac6e
F20101205_AAAFSD hu_j_Page_038.tif
e3cc2e531763cc299804278d64a4ec67
bb83c24c173c66798f43053d420c734b7ce9b86b
66733 F20101205_AAAEWA hu_j_Page_040.jp2
7ba0129a97c2bd876730182cc836f693
725f6810987a016f64c757da910120ce55f32ede
120265 F20101205_AAAFRP hu_j_Page_106.jp2
0b14f8603eb4354b844b75bc69957ca9
74edb978a6b8f32177edf868463be1049ad0dd93
F20101205_AAAEVM hu_j_Page_005.jp2
5f3459f5127abb3c3be9a06711e4d42d
b0b50481ffc6590b2bfe83534ba3d23d7c720928
1643 F20101205_AAAEUZ hu_j_Page_066.txt
15c25891ec6aaf3922f91b048e5d8602
2b0dbba1fea8f731cb94d738ba7807f80b63f386
F20101205_AAAFSE hu_j_Page_040.tif
693206b0da6a3f2a4872178d8e2e93c8
012b630808a1542f3f64ea2288737a361ce84c1a
1349 F20101205_AAAEWB hu_j_Page_049.txt
9a425ea01a09770c0942539a88d0b8ad
2d7d32d9ce6c850d00429b43c9861dda4f1f4644
121334 F20101205_AAAFRQ hu_j_Page_111.jp2
d9baf8f5ff88b61f773d627d051e9182
5e85982c0e1f5de304575d28295d9f4f80da85ce
4012 F20101205_AAAEVN hu_j_Page_075thm.jpg
015a0c77f3f2ac36abb007587aa50bbc
3ae8c80af365c51862a0a2a2b8b8aee67a2bf218
F20101205_AAAFSF hu_j_Page_055.tif
30cec41f9700ab521796966f284c4eca
c30eda60967af4612efe107a97409cfbe225a29d
F20101205_AAAEWC hu_j_Page_038.jp2
bd766d02058117aca045bbd922830b07
76d41a0231911f0e2830cf7f566ef25c14ba81ab
120916 F20101205_AAAFRR hu_j_Page_112.jp2
3e2f5d516f42a37d5f71669d4abc26cc
ec60672c80c7f325d638d48e75907e9f677b2a51
25838 F20101205_AAAEVO hu_j_Page_110.QC.jpg
1a551b197e1b84e4aa543b701f3f7aa3
f2cd88b960520d1d883670cdace6e0c709ef6fa3
F20101205_AAAFSG hu_j_Page_060.tif
cf3adff796638991cbe5ca16667ce0f4
f80c19bafdeb9335d19399a9d6c60d9660919078
26299 F20101205_AAAEWD hu_j_Page_010.QC.jpg
c9d2ef664151902fba3504fc14a4eea1
993747bf979d60e32fa9a62a97d30389b1d1f002
F20101205_AAAFRS hu_j_Page_003.tif
619d76a2477dd3bb128ff592eb4246f6
c4dc5ff2fbb25c6f799ae005177ad9f309850245
5567 F20101205_AAAEVP hu_j_Page_023thm.jpg
01a792f89f73bfd8d1a5357ab58e4b8f
3f9c7a572528fe4c01967836a0633c40417cd2e0
F20101205_AAAFSH hu_j_Page_062.tif
9a24bbfb6b74f3ae0c051b4eba133ef5
15442547d4bbe94204c30d7fbde278ad2041366e
157545 F20101205_AAAEWE hu_j_Page_088.jp2
898c04a0d7a0c01566e82389447f674c
127e1593a71346ec1dadaed40d631a6bb1a349fe
F20101205_AAAFRT hu_j_Page_004.tif
7291eef7562506a7f5b476559f2b78f8
2d806955573df6c7022dfbd71c608a7b9c4e5828
F20101205_AAAEVQ hu_j_Page_056.tif
a3bbca8faa1ca9a465d23999eda2e53a
51f0e9a1fe9ab88cb6540a9c175d6bf857af2efc
F20101205_AAAFSI hu_j_Page_066.tif
f68c833f67a0cfa0cd80a56a7bcf860e
c7916336e54370f6c632c9119372f1af615532db
28223 F20101205_AAAEWF hu_j_Page_115.jpg
2bd2542293067f86820115e59e970182
671cd80da8c6af5bdb6b53c7955769b7425695b6
F20101205_AAAFRU hu_j_Page_006.tif
7639a5411644802aade8400081686822
9e3c520539a2b4966815d28fd662503694bbb9bc
110506 F20101205_AAAEVR hu_j_Page_009.jp2
d508b4d93384e44771d491a938fd108c
b1f2bc702267b3f495760002ff922720942eb6f1
F20101205_AAAFSJ hu_j_Page_067.tif
b20dfed8a6ae6b19995eba78f694bb19
210989b990d712231d79ed30335d1ec107c92f84
785865 F20101205_AAAEWG hu_j_Page_014.jp2
bbb3003d8b73a2bb1483547924b8527c
1696122883d8cc5df066f8b27f4ade8ff3ddf258
F20101205_AAAFRV hu_j_Page_013.tif
67665ec4cb5bf8d79097eb6e225be9c1
5ac1023a799adae7a185d0d431f92957ce37a4bf
F20101205_AAAEVS hu_j_Page_086thm.jpg
81fe3c067b8245d85ff4f0f6b3b23b20
9d47adfb2ae825adc01c9e18f049323044cce634
F20101205_AAAFSK hu_j_Page_069.tif
244edf2817424ca89d9a083d1a65f440
df146e02eac2c2c6a0e4ef609ae30c4c8104a72d
F20101205_AAAEWH hu_j_Page_051.tif
0dc053890e9d8acc1b7f426e1b345a0e
e5f7e00fbc23b1ca7c3f6bc017d364e1f588a739
787025 F20101205_AAAEVT hu_j_Page_065.jp2
6ba0ed5cd7303f9d8de906bee5bda741
f18b2b07a8b498423e2201d3dbe811cf3449903c
F20101205_AAAFSL hu_j_Page_070.tif
c7461705523c1a94283fa92d4207aa35
b92ccd5635abd90568002c4ec6cda0f764869ea4
1051928 F20101205_AAAEWI hu_j_Page_016.jp2
b65604d7536631d2a5b7a21395ec7024
639c73b6b99623f09e406dee099eeccef3301c01
F20101205_AAAFRW hu_j_Page_015.tif
dffa221569ed714aa860f7e80ede392b
1f9a9da6f175260a14adcf07c07d180661494e97
58493 F20101205_AAAEVU hu_j_Page_085.jpg
fa187ad7d074a0496834136adaa11143
f68e1c86ca81966a702297c29e3bf0df2adebabd
F20101205_AAAFSM hu_j_Page_071.tif
3b63e028923c1565ceada2c6860fc5d7
1082d814d37af6a4443454eb59a9828833bb6b0e
1523 F20101205_AAAEWJ hu_j_Page_047.txt
c9d6821da158c74256abd2a8feeae571
94df8aefe0c7df2f7fd2287f06c6ce961dafeec2
F20101205_AAAFRX hu_j_Page_016.tif
251d05355c23bc414953b6bf5a8d9089
f0aee9e82f3ec8cc54dfa88a189ed0cd435942a9
5918 F20101205_AAAEVV hu_j_Page_024thm.jpg
89760c9abc55384e93f6da6b86a50183
5a4dbfa01a45b0e3dfc4d95741cdffaf87805b90
12196 F20101205_AAAFTA hu_j_Page_032.pro
6e1929416f2409a4567a2e9142f2152d
1383fdc2c2da7b4cafb7a8d697f74ff9b8ea7b68
F20101205_AAAFSN hu_j_Page_078.tif
69305ed6f9dd1eb37dd770b0de87e149
7f95c4ac91be6d3738696424e59e3bd8d8845922
94813 F20101205_AAAEWK hu_j_Page_083.jpg
aa692069e0785eca8ca488edd2655f58
5cf5134dfc954bcbbb859dae277e4fae0243b959
F20101205_AAAFRY hu_j_Page_020.tif
3fe85ab6777df6f08d5795504f22a1b0
1492670a3b7666d0c74a8d3c6aabe0fcb2d794a1
F20101205_AAAEVW hu_j_Page_009.tif
c18bd4c4674f870f6be4d3698d82097f
a1fba58205eadece0b3c576ce47812ea7428a13a
54231 F20101205_AAAFTB hu_j_Page_035.pro
e5c85b0a2b8313c71ae14fc29625365b
fb63a2ad7a70f761179ccafeab914e2804862a3b
F20101205_AAAFSO hu_j_Page_080.tif
7e8f28140d4c4412a6c9466a1e87d0d5
6c4c19cf763f15f10c824e08877c2da36bd8829f
F20101205_AAAEWL hu_j_Page_052.tif
abbfb57105883fad8a46c3bcac9a1a85
e33d438a424f71de44a8b62d86361e7daf21c670
F20101205_AAAFRZ hu_j_Page_029.tif
04739d60f794dcb9aa7e0299710c8473
09244baabfe1d9bd84a1d5b9374b2ef4c76f36a5
F20101205_AAAEVX hu_j_Page_064.tif
efd86868395236c0b783734954c1deb8
ca6df8c4ffa4520e80d67075081a2a47f283f8c5
64313 F20101205_AAAFTC hu_j_Page_038.pro
d5f5e1eb6647fc9aa0c2f6d5bd6eb12e
5644a88711b26d7e898b47a50411bf6e8e08085f
1937 F20101205_AAAEXA hu_j_Page_093.txt
4b0cd12c1e96853830f25fd12814eae4
f24173bafc18c9f4dc6278859d1687263dcdf31e
F20101205_AAAFSP hu_j_Page_084.tif
5c3a9dbec86ee3ab73e1d51e37acb15f
fd9a68d23a936bae9abf757370b34fb06d38d7ba
F20101205_AAAEWM hu_j_Page_018.tif
f27b995d121ff9a258fe6aefd212b280
62f7aef100ffd6b442f752121c8e723f995ff108
1025 F20101205_AAAEVY hu_j_Page_046.txt
05448c1b70b81146952affc3cb22fa3f
ce1324c9180e19690f7c8a10e72ac5e779b61b5a
27319 F20101205_AAAFTD hu_j_Page_040.pro
2b7e7f32fc748e07b21c4d3a7c537e0e
bb7781d37dac89526751164bbea30717721e01c5
25042 F20101205_AAAEXB hu_j_Page_074.QC.jpg
5035e1cd20121fd1acd5d541db1f4562
aa88f11e871e86a6e3a70ffdb8a07e42aef7cb9d
F20101205_AAAFSQ hu_j_Page_089.tif
90bfc8ad92deb14491c0215d7e346307
e2067f9bde2507a321e080fa0a66728284f07bd7
19825 F20101205_AAAEWN hu_j_Page_073.QC.jpg
d0a66ebbb43456dfe2b16a03449097dc
a8d3c7083acd5cdf87b1b0604fb27a0c9ff1bbf6
85674 F20101205_AAAEVZ hu_j_Page_035.jpg
e04697afa86ca1ab80c443375486fb06
70eba4ea1737aa690099fb566f197a3234ee04ce
19281 F20101205_AAAFTE hu_j_Page_046.pro
93ed97d91cb3604cd98a7f9f7bc1b51f
3e922f787e7d449c9b1ec3ce091eab0fe387f533
F20101205_AAAFSR hu_j_Page_097.tif
6ab71c34ec180796895b9529cad96edb
9ef7abc6796993cc3e06de0c25f6ebb1188e3402
5626 F20101205_AAAEWO hu_j_Page_030thm.jpg
c644cd13a7e72cc1f6e1f02678c9fbd8
e433ce7f9aee699827123899492c731d3e2a537a
16334 F20101205_AAAFTF hu_j_Page_048.pro
ca8555b20959ef1f55afff2d1aa5174e
120db3a67289c14208c81fdeee47dc9bf39254ce
3144 F20101205_AAAEXC hu_j_Page_002.jpg
44c97872b094eb54b6f00a17f23a3a4c
b85dabb043ade467ac6a78cdf25e99e20b130132
F20101205_AAAFSS hu_j_Page_100.tif
d5ddefbd7498466e429405b760339993
a51c664361f994148b1d5d6a05ea6b6e41204709
F20101205_AAAEWP hu_j_Page_043.tif
16274678110ee229238e90c71045a034
652b34ec54515131c8ea9ffdc1d32fe49c10427c
39745 F20101205_AAAFTG hu_j_Page_052.pro
d50603ff7c27884ea3fb364e25990052
8f0e7bd2a2b20744307bb898359a94448c499084
28123 F20101205_AAAEXD hu_j_Page_099.QC.jpg
5c9a4e600c5ff1e68d0c0e8b7e9ec513
e252e41b6b6888637c7dda4d9b8c0cc7a83229e2
F20101205_AAAFST hu_j_Page_103.tif
f0d83a38b46945517688e54183fc2d70
af129c65ca615480eca778cb315fd57de5f804df
F20101205_AAAEWQ hu_j_Page_072.tif
5f3b45b268ebad46f9a5249293754c6f
c994b11a6a444869afd58be760ac4f5cbc201d83
43610 F20101205_AAAFTH hu_j_Page_055.pro
6de1b402f9c028d7caa3bb5f8b169d6a
6060ec8c88f1597ea02f31459005f98025d0fb4e
60487 F20101205_AAAEXE hu_j_Page_107.pro
1d9a1886b672aadd32d295e965df362d
85804646f4def66ae13d0482e823cd2f54b08897
F20101205_AAAFSU hu_j_Page_106.tif
ff270720dc5fb9e8cf53cc56ff6f914a
4d472d6aa7e47fb06689e7db20fda5cc866d6fcb
4604 F20101205_AAAEWR hu_j_Page_095thm.jpg
8b62a185285efa91eb560140fd836f32
4fee2328c9c525bb5789f6f0ae7e5619053ca6e9
5638 F20101205_AAAFTI hu_j_Page_057.pro
5aef181588b63da0dc549769521f46ff
62b6582476f279498437cfc20e7430031ab0673e
90894 F20101205_AAAEXF hu_j_Page_074.jpg
d677538ce4efd060db0326438ea27647
b2b2982c4bb35f2fc285a23a8c4fad1afca5b61d
59018 F20101205_AAAFSV hu_j_Page_005.pro
de75f996d95d74d48de96d712a303a6c
014de233fa81c5cc9123a234ff4e1fd82380cde0
2310 F20101205_AAAEWS hu_j_Page_074.txt
a46745050185c1b9c331e9c3e76a380d
de19ce3b794f3c86c8122edebbc46803d18295c9
13171 F20101205_AAAFTJ hu_j_Page_058.pro
c8b57ec05b541171c88e51889b4e73e3
f12156b8b0d631c12ec149a5dd61d3800ba40f4b
F20101205_AAAEXG hu_j_Page_014.tif
b77dda726a47efccbe201278fd95bf24
4d795cf26028934122014a7b1d74da038cd827bc



PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

Firstofall,Iwouldliketothankmyadvisor,ProfessorJianboGao,forhisinvaluableguidanceandsupport.Iwasluckytoworkwithsomeonewithsomanyoriginalideasandsuchasharpmind.Withouthishelp,myPh.D.degreeandthisdissertationwouldhavebeenimpossible.DuringmyyearsattheUniversityofFlorida,Iwasmovedbyhismagnicentpersonality.AfamousChineseadagesays,\Onedayasmyadvisor,entirelifeasmyfather."ProfessorGaohighlyqualies.Ialsowanttothankmycommitteemembers(ProfessorsSu-ShingChen,YuguangFang,JoseC.Principe,andKeithD.White),fortheirinterestinmyworkandvaluablesuggestions.Mythanksofcoursealsogotoallthefaculty,staandmyfellowstudentsintheDepartmentofElectricalandComputerEngineering.IextendparticularthankstomyocematesJaeminLee,YiZheng,andUngsikKim,fortheirhelpfuldiscussions.Iexpressmyappreciationtoallofmyfriendswhooeredencouragement.IspeciallythankQianZhan,XingyanFan,LingyuGuandhiswife(JingZhang),YanminXu,YanLiu,andYanaLiang.TheygavemealotofhelpduringmyearlyyearsintheUniversityofFlorida.Theytaughtmetocookandmademylifemorecolorful.Finally,Ioweagreatdebtofthankstomyparents,grandparents,andmyboyfriend,JingAi,fortheirpillar-likesupportandtheirunwaveringbeliefandcondenceinme|withoutthisIdonotthinkIwouldhavemadeitthisfar! 4

PAGE 5

page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 1.1OverviewofDynamicalSystems ........................ 11 1.2ExamplesofMultiscalePhenomena ...................... 15 1.3BriefIntroductiontoChaos .......................... 20 1.4BriefIntroductiontoFractalGeometry .................... 21 1.5ImportanceofConnectingChaosandRandomFractalTheories ...... 23 1.6ImportanceoftheConceptofScale ...................... 24 1.7StructureofthisDissertation ......................... 24 2GENERALIZATIONOFCHAOS:POWER-LAWSENSITIVITYTOINITIALCONDITIONS(PSIC) ................................ 27 2.1DynamicalTestforChaos ........................... 27 2.2GeneralComputationalFrameworkforPower-lawSensitivitytoInitialConditions(PSIC) ............................... 33 2.3CharacterizingEdgeofChaosbyPower-lawSensitivityofInitialConditions(PSIC) ...................................... 36 3CHARACTERIZINGRANDOMFRACTALSBYPOWER-LAWSENSITIVITYTOINITIALCONDITIONS(PSIC) ........................ 41 3.1Characterizing1=fProcessesbyPower-lawSensitivitytoInitialConditions(PSIC) ...................................... 41 3.1.1CharacterizingFractionalBrownianMotion(fBm)ProcessesbyPower-lawSensitivitytoInitialConditions(PSIC) .......... 43 3.1.2CharacterizingON/OFFProcesseswithParetodistributedONandOFFPeriodsbyPower-lawSensitivitytoInitialConditions(PSIC) 50 3.2CharacterizingLevyProcessesbyPower-lawSensitivitytoInitialConditions(PSIC) ...................................... 54 4STUDYOFINTERNETDYNAMICSBYPOWER-LAWSENSITIVITYTOINITIALCONDITIONS(PSIC) ........................... 60 4.1StudyofTransportDynamicsbyNetworkSimulation ............ 61 4.2ComplicatedDynamicsofInternetTransportProtocols ........... 69 5

PAGE 6

............... 76 5.1SeaClutterData ................................ 78 5.2NonChaoticBehaviorofSeaClutter ..................... 79 5.3TargetDetectionwithinSeaClutterbySeparatingScales .......... 81 5.4TargetDetectionwithinSeaClutterbyPower-lawSensitivitytoInitialConditions(PSIC) ............................... 86 6MULTISCALEANALYSISBYSCALE-DEPENDENTLYAPUNOVEXPONENT(SDLE) ........................................ 89 6.1BasicTheory .................................. 89 6.2ClassicationofComplexMotions ....................... 92 6.2.1Chaos,NoisyChaos,andNoise-inducedChaos ............ 92 6.2.2ProcessesDenedbyPower-lawSensitivitytoInitialConditions(PSIC) .................................. 95 6.2.3ComplexMotionswithMultipleScalingBehaviors .......... 96 6.3CharacterizingHiddenFrequencies ...................... 99 7CONCLUSIONS ................................... 104 REFERENCES ....................................... 105 BIOGRAPHICALSKETCH ................................ 115 6

PAGE 7

Figure page 1-1Exampleofatrajectoryinthephasespace ..................... 14 1-2Giantoceanwave(tsunami). ............................. 17 1-3Exampleofseaclutterdata. ............................. 18 1-4ExampleofHeartratevariabilitydataforanormalsubject. ........... 19 2-1Time-dependentexponent(k)vs.evolutiontimekcurvesforLorenzsystem. 32 2-2Time-dependentexponent(k)vs.evolutiontimekcurvesfortheMackey-Glasssystem. ........................................ 33 2-3Time-dependentexponent(t)vs.tcurvesfortimeseriesgeneratedfromlogisticmap. .......................................... 37 2-4Time-dependentexponent(t)vs.tcurvesfortimeseriesgeneratedfromHenonmap. .......................................... 40 3-1WhitenoiseandBrownianmotion ......................... 46 3-2SeveralfBmprocesseswithdierentH. ....................... 48 3-3Time-dependentexponent(k)vs.lnkcurvesforfractionalBrownianmotion. 51 3-4ExampleofON/OFFprocesses. ........................... 51 3-5Time-dependentexponent(k)vs.lnkcurvesforON/OFFprocesseswithParetodistributedONandOFFperiods. .......................... 53 3-6ExamplesforBrownianmotionsandLevymotions. ................ 57 3-7Time-dependentexponent(k)vs.lnkcurvesforLevyprocesses. ........ 58 4-1Exampleofquasi-periodiccongestionwindowsizeW(i)timeseries. ....... 63 4-2ExamplesofchaoticT(i)timeseries. ........................ 65 4-3Complementarycumulativedistributionfunctions(CCDFs)forthetwochaoticT(i)timeseriesofFigs. 4-2 (a,c). .......................... 66 4-4ThetimeseriesT(i)extractedfromaW(i)timeseriescollectedusingnet100instrumentsoverORNL-LSUconnection. ...................... 68 4-5TimeseriesforthecongestionwindowsizecwndforORNL-GaTechconnection. 72 4-6Time-dependentexponent(k)vs.kcurvesforcwnddatacorrespondingtoFig. 4-5 ........................................ 73 7

PAGE 8

4-5 ........................................ 75 5-1Collectionofseaclutterdata ............................ 79 5-2Typicalseaclutteramplitudedata. ......................... 79 5-3Twoshortsegmentsoftheamplitudeseaclutterdataseverelyaectedbyclipping. 80 5-4Examplesofthetime-dependentexponent(k)vs.kcurvesfortheseaclutterdata. .......................................... 81 5-5VariationsoftheLyapunovexponentestimatedbyconventionalmethodsvs.the14rangebinsforthe10HHmeasurements. .................. 84 5-6Variationsofthescale-dependentexponentcorrespondingtolargescalevs.therangebinsforthe10HHmeasurements. ...................... 85 5-7Examplesofthe(k)vs.lnkcurvesfortheseaclutterdata. ........... 87 5-8Variationoftheparameterwiththe14rangebins. ............... 87 5-9FrequenciesofthebinswithouttargetsandthebinswithprimarytargetsfortheHHdatasets. ................................... 88 6-1Scale-dependentLyapunovexponent()curvesfortheLorenzsystemandlogisticmap. .......................................... 93 6-2Scale-dependentLyapunovexponent()curvefortheMackey-Glasssystem. .. 94 6-3ThefunctionsF(x)andG(x). ............................ 97 6-4Scale-dependentLyapunovexponent()forthemodeldescribedbyEq. 6{9 . 97 6-5Power-spectraldensity(PSD)fortimeseriesgeneratedfromLorenzsystem. .. 100 6-6Lorenzattractor. ................................... 101 6-7Hiddenfrequencyphenomenonoflaserintensitydata. ............... 102 8

PAGE 9

Complexsystemsusuallycomprisemultiplesubsystemswithhighlynonlineardeterministicandstochasticcharacteristics,andareregulatedhierarchically.Thesesystemsgeneratesignalswithcomplexcharacteristicssuchassensitivedependenceonsmalldisturbances,longmemory,extremevariations,andnonstationarity.Chaostheoryandrandomfractaltheoryaretwoofthemostimportanttheoriesdevelopedforanalyzingcomplexsignals.However,theyhaveentirelydierentfoundations,onebeingdeterministicandtheotherbeingrandom.Tosynergisticallyusethesetwotheories,wedevelopedanewtheoreticalframework,power-lawsensitivitytoinitialconditions(PSIC),toencompassbothchaosandrandomfractaltheoriesasspecialcases.Toshowthepowerofthisframework,weappliedittostudytwochallengingandimportantproblems:Internetdynamicsandseaclutterradarreturns.WeshowedthatPSICcanreadilycharacterizethecomplicatedInternetdynamicsduetointerplayofnonlinearAIMD(additiveincreaseandmultiplicativedecrease)operationofTCPandstochasticuserbehavior,androbustlydetectlowobservabletargetsfromseaclutterradarreturnswithhighaccuracy. Wealsodevelopedanewmultiscalecomplexitymeasure,scale-dependentLyapunovexponent(SDLE),thatcanbecomputedfromshortnoisydata.Themeasurereadilyclassiedvarioustypesofcomplexmotions,andsimultaneouslycharacterizedbehaviorsofcomplexsignalsonawiderangeofscales,includingcomplexirregularbehaviorsonsmallscales,andorderlybehaviors,suchasoscillatorymotions,onlargescales. 9

PAGE 10

Adynamicalsystemisonethatevolvesintime.Dynamicalsystemscanbestochastic(inwhichcasetheyevolveaccordingtosomerandomprocesssuchasthetossofacoin)ordeterministic(inwhichcasethefutureisuniquelydeterminedbythepastaccordingtosomeruleormathematicalformula).Whetherthesysteminquestionsettlesdowntoequilibrium,keepsrepeatingincycles,ordoessomethingmorecomplicated,dynamicsarewhatweusetoanalyzethebehaviorofsystems. Complexdynamicalsystemsusuallycomprisemultiplesubsystemswithhighlynonlineardeterministicandstochasticcharacteristics,andareregulatedhierarchically.Thesesystemsgeneratesignalswithcomplexcharacteristicssuchassensitivedependenceonsmalldisturbances,longmemory,extremevariations,andnonstationarity[ 1 ].Astockmarket,forexample,isstronglyinuencedbymulti-layereddecisionsmadebymarketmakers,aswellasinuencedbyinteractionsofheterogeneoustraders,includingintradaytraders,short-periodtraders,andlong-periodtraders,andthusgivesrisetohighlyirregularstockprices.TheInternet,asanotherexample,hasbeendesignedinafundamentallydecentralizedfashionandconsistsofacomplexwebofserversandroutersthatcannotbeeectivelycontrolledoranalyzedbytraditionaltoolsofqueuingtheoryorcontroltheory,andgiverisetohighlyburstyandmultiscaletracwithextremelyhighvariance,aswellascomplexdynamicswithbothdeterministicandstochasticcomponents.Similarly,biologicalsystems,beingheterogeneous,massivelydistributed,andhighlycomplicated,oftengeneratenonstationaryandmultiscalesignals.Withtherapidaccumulationofcomplexdatainlifesciences,systemsbiology,nano-sciences,informationsystems,andphysicalsciences,ithasbecomeincreasinglyimportanttobeabletoanalyzemultiscaleandnonstationarydata. Multiscalesignalsbehavedierentlydependinguponwhichscalethedataarelookedat.Inordertofullycharacterizesuchcomplexsignals,theconceptofscalehastobe 10

PAGE 11

Thisdissertationaimstobuildaneectivearsenalformultiscalesignalprocessingbysynergisticallyintegratingapproachesbasedonchaostheoryandrandomfractaltheory,andgoingbeyond,tocomplementconventionalapproachessuchasspectralanalysisandmachinelearningtechniques.Tomakesuchanintegrationpossible,twoimportanteortshavebeenmade:Power-lawsensitivitytoinitialconditions(PSIC):Wedevelopedanewtheoreticalframeworkforsignalprocessing,tocreateacommonfoundationforchaostheoryandrandomfractaltheory,sothattheycanbebetterintegrated.Scale-dependentLyapunovexponent(SDLE):Wedevelopedanexcellentmultiscalemeasure,whichisavariantofthenitesizeLyapunovexponent(FSLE).Weproposedahighlyecientalgorithmforcalculatingit,andshowedthatitcanreadilyclassifydierenttypesofmotions,includingtrulylow-dimensionalchaos,noisychaos,noise-inducedchaos,random1=fand-stableLevyprocesses,andcomplexmotionswithchaoticbehavioronsmallscalesbutdiusivebehavioronlargescales.Themeasurecanaptlycharacterizecomplexbehaviorsofrealworldmultiscalesignalsonawiderangeofscales. Intherestofthischapter,weshallrstpresentanoverviewofdynamicssystems,especiallyweshallpointoutwhynonlinearsystemsaremuchhardertoanalyzethanlinearones.Wethengiveafewexamplesofmultiscalephenomenatoshowthedicultiesandexcitementofmultiscalesignalprocessing.Afterthat,weshallbrieyintroducesomebackgroundknowledgeaboutchaosandfractalgeometry.Thenwediscusstheimportanceofconnectingchaostheoryandrandomfractaltheoryforcharacterizingthebehaviorsofmultiscalesignalsonawiderangeofscales.Wealsoemphasizetheimportanceofexplicitlyincorporatingtheconceptofscaleindevisingmeasuresforcharacterizingmultiscalesignals.Finally,weoutlinethescopeofthepresentstudy. 11

PAGE 12

dt2+bdx dt+kx=0(1{1) isanordinarydierentialequation,becauseitinvolvesonlyordinaryderivativesdx=dtandd2x=dt2.Thatis,thereisonlyoneindependentvariable,thetimet.Incontrast,theheatequation@u @t=@2u @x2 Averygeneralframeworkforordinarydierentialequationsisprovidedbythesystem _x1=f1(x1;;xn)..._xn=fn(x1;;xn):(1{2) Heretheoverdotsdenotedierentiationwithrespecttot.Thus_xi=dxi=dt.Thevariablesx1;;xnmightrepresentconcentrationsofchemicalsinareactor,populationsofdierentspeciesinanecosystem,orthepositionsandvelocitiesoftheplanetsinthesolarsystem.Thefunctionsf1;;fnaredeterminedbytheproblemathand. Forexample,thedampedoscillator(Eq. 1{1 )canberewrittenintheformof(Eq. 1{2 ),thankstothefollowingtrick:weintroducenewvariablesx1=xandx2=_x. 12

PAGE 13

1{2 )is_x1=x2_x2=b mx2k mx1: Forexample,theswingingofapendulumisgovernedbytheequationx+g Lsinx=0; Lsinx1: Itturnsoutthatthependulumequationcanbesolvedanalytically,intermsofellipticfunctions.Butthereoughttobeaneasierway.Afterall,themotionofthependulumissimple:atlowenergy,itswingsbackandforth,andathighenergyitwhirlsoverthetop.Thereshouldbesomewayofextractingthisinformationfromthesystemdirectly|usinggeometricmethods. Hereistheroughidea.Supposewehappentoknowasolutiontothependulumsystem,foraparticularinitialcondition.Thissolutionwouldbeapairoffunctionsx1(t)andx2(t),representingthepositionandvelocityofthependulum.Ifweconstructan 13

PAGE 14

Exampleofatrajectoryinthephasespace abstractspacewithcoordinates(x1;x2),thenthesolution(x1(t);x2(t))correspondstoapointmovingalongacurveinthisspace,asshownFig. 1-1 .Thiscurveiscalledatrajectory,andthespaceiscalledthephasespaceforthesystem.Thephasespaceiscompletelylledwithtrajectories,sinceeachpointcanserveasaninitialcondition. Ourgoalistorunthisconstructioninreverse:giventhesystem,wewanttodrawthetrajectories,andtherebyextractinformationaboutthesolutions.Inmanycases,geometricreasoningwillallowustodrawthetrajectorieswithoutactuallysolvingthesystem! Thephasespaceforthegeneralsystem(Eq. 1{2 )isthespacewithcoordinatesx1;;xn.Becausethisspaceisn-dimensional,wewillrefertoEq. 1{2 asann-dimensionalsystemorannth-ordersystem.Thusnrepresentsthedimensionofthephasespace. Aswehavementionedearlier,mostnonlinearsystemsareimpossibletosolveanalytically.Whyarenonlinearsystemssomuchhardertoanalyzethanlinearones?Theessentialdierenceisthatlinearsystemscanbebrokendownintoparts.Theneachpartcanbesolvedseparatelyandnallyrecombinedtogettheanswer.Thisideaallowsa 14

PAGE 15

Butmanythingsinnaturedonotactthisway.Wheneverpartsofasysteminterfere,orcooperate,orcompete,therearenonlinearinteractionsgoingon.Mostofeverydaylifeisnonlinear,andtheprincipleofsuperpositionfailsspectacularly.Ifyoulistentoyourtwofavoritesongsatthesametime,youwillnotgetdoublethepleasure!Withintherealmofphysics,nonlinearityisvitaltotheoperationofalaser,theformationofturbulenceinauid,andthesuperconductivityofJosephsonjunctions. 15

PAGE 16

Also,ithasbeenobservedthatthefailureofasingleroutermaytriggerroutinginstability,whichmaybesevereenoughtoinstigatearouteapstorm.Furthermore,packetsmaybedeliveredoutoforderorevengetdropped,andpacketreorderingisnotapathologicalnetworkbehavior.AsthenextgenerationInternetapplicationssuchasremoteinstrumentcontrolandcomputationalsteeringarebeingdeveloped,anotherfacetofcomplexmultiscalebehaviorisbeginningtosurfaceintermsoftransportdynamics.Thenetworkingrequirementsforthesenextgenerationapplicationsbelongto(atleast)twobroadclassesinvolvingvastlydisparatetimescales:Highbandwidths,typicallymultiplesof10Gbps,tosupportbulkdatatransfers,Stablebandwidths,typicallyatmuchlowerbandwidthssuchas10to100Mbps,tosupportinteractive,steeringandcontroloperations. 1-2 .Tobequantitative,inFig. 1-3 ,two0.1sdurationseacluttersignals,sampledwithafrequencyof1KHz,areplottedinFig. 1-3 (a,b),a2sdurationsignalisplottedinFig. 1-3 (c),andanevenlongersignal(about130s)isplottedinFig. 1-3 (d).Itisclearthatthesignalisnotpurelyrandom,sincethewaveformcanbefairlysmoothonshorttimescales(Fig. 1-3 (a)).However,thesignalishighlynonstationary,sincethefrequencyofthesignal(Fig. 1-3 (a,b))aswellastherandomnessofthesignal(Fig. 1-3 (c,d))changeovertimedrastically.OnethuscanperceivethatnaiveFourieranalysisordeterministicchaoticanalysisofseacluttermay 16

PAGE 17

Giantoceanwave(tsunami).Supposeoureldofobservationincludesthewavetipoflengthscaleofafewmeters,itisthenclearthatthecomplexityofseaclutterismainlyduetomassivereectionofradarpulsesfromawavyandeventurbulentoceansurface. notbeveryuseful.FromFig. 1-3 (e),whereX(m)tisthenon-overlappingrunningmeanofXoverblocksizem,andXistheseaclutteramplitudedata,itcanbefurtherconcludedthatneitherautoregressive(AR)models(Asanexample,wegivethedenitionfortherst-orderARmodel,whichisgivenbyxn+1=axn+n,wheretheconstantcoecientasatises06=jaj<1andnisawhitenoisewithzeromean.)nortextbookfractalmodelscandescribethedata.ThisisbecauseARmodelingrequiresexponentiallydecayingautocorrelation(whichamountstoVar(X(m)t)m1,oraHurstparameterof1/2.SeelaterchaptersforthedenitionoftheHurstparameter),whilefractalmodelingrequiresthevariationbetweenVar(X(m)t)andmtofollowapower-law.However,neitherbehaviorisobservedinFig. 1-3 (e).Indeed,albeitextensiveworkhasbeendoneonseaclutter,thenatureofseaclutterisstillpoorlyunderstood.Asaresult,theimportantproblemof 17

PAGE 18

Exampleofseaclutterdata.(a,b)two0.1sdurationsignal;(c)a2sdurationseacluttersignal;(d)theentireseacluttersignal(ofabout130s);and(e)log2[m2Var(X(m))]vs.log2m,whereX(m)=fX(m)t:X(m)t=(Xtmm+1++Xtm)=m;t=1;2;gisthenon-overlappingrunningmeanofX=fXt:t=1;2;goverblocksizem,andXistheseaclutteramplitudedata.TobetterseethevariationofVar(X(m)t)withm,Var(X(m)t)ismultipliedbym2.Whentheautocorrelationofthedatadecaysexponentiallyfast(suchasmodeledbyanautoregressive(AR)process),Var(X(m)t)m1.HereVar(X(m)t)decaysmuchfaster.Afractalprocesswouldhavem2Var(X(m)t)m.However,thisisnotthecase.Therefore,neitherARmodelingnoridealtextbookfractaltheorycanbereadilyappliedhere. targetdetectionwithinseaclutterremainsatremendouschallenge.Weshallreturntoseaclutterlater. 18

PAGE 19

ExampleofHeartratevariabilitydataforanormalsubject.(a)theentiresignal,(b,c)thesegmentsofsignalsindicatedasAandBin(a);(d,e)powerspectraldensityforthesignalsshownin(b,c). maintainedconstantandnoperturbinginuencescanbeidentied.IthasbeenobservedthatHRVisrelatedtovariouscardiovasculardisorders.Therefore,analysisofHRVisveryimportantinmedicine.However,thistaskisverydicult,sinceHRVdataarehighlycomplicated.AnexampleisshowninFig. 1-4 ,foranormalyoungsubject.Evidently,thesignalishighlynonstationaryandmultiscaled,appearingoscillatoryforsomeperiodoftime(Figs. 1-4 (b,d)),andthenvaryingasapower-lawforanotherperiodoftime(Figs. 1-4 (c,e)).Thelatterisanexampleoftheso-called1=fprocesses,whichwillbediscussedindepthinlaterchapters.Whilethemultiscalenatureofsuchsignalscannotbefullycharacterizedbyexistingmethods,thenonstationarityofthedataisevenmoretroublesome,sinceitrequiresthedatatobeproperlysegmentedbeforefurtheranalysis 19

PAGE 20

Atthecenterofchaostheoryistheconceptofsensitivedependenceoninitialconditions:averyminordisturbanceininitialconditionsleadstoentirelydierentoutcomes.AnoftenusedmetaphorillustratingthispointisthatasunnyweatherinNewYorkcouldbereplacedbyarainyonesometimeinthefutureafterabutteryapsitswingsinBoston.Suchafeaturecontrastssharplywiththetraditionalview,largelybasedonourexperiencewithlinearsystems,thatsmalldisturbances(orcauses)canonlygenerateproportionaleects,andthatinorderforthedegreeofrandomnesstoincrease,thenumberofdegreesoffreedomhastobeinnite. Nodenitionofthetermchaosisuniversallyacceptedyet,butalmosteveryonewouldagreeonthethreeingredientsusedinthefollowingworkingdenition: Chaosisaperiodiclong-termbehaviorinadeterministicsystemthatexhibitssensitivedependenceoninitialconditions.\Aperiodiclong-termbehavior"meansthattherearetrajectorieswhichdonotsettledowntoxedpoints,periodicorbits,orquasi-periodicorbitsast!1.Forpracticalreasons,weshouldrequirethatsuchtrajectoriesarenottoorare.Forinstance,wecould 20

PAGE 21

Chaosisalsocommonlycalledastrangeattractor.Thetermattractorisalsodiculttodeneinarigorousway.Wewantadenitionthatisbroadenoughtoincludeallthenaturalcandidates,butrestrictiveenoughtoexcludetheimposters.Thereisstilldisagreementaboutwhattheexactdenitionshouldbe. Looselyspeaking,anattractorisasettowhichallneighboringtrajectoriesconverge.Stablexedpointsandstablelimitcyclesareexamples.Moreprecisely,wedeneanattractortobeaclosedsetAwiththefollowingproperties: 1. 2. 3. Nowwecandeneastrangeattractortobeanattractorthatexhibitssensitivedependenceoninitialconditions.Strangeattractorswereoriginallycalledstrangebecausetheyareoftenfractalsets.Nowadaysthisgeometricpropertyisregardedaslessimportantthanthedynamicalpropertyofsensitivitydependenceoninitialconditions.Thetermschaoticattractorandfractalattractorareusedwhenonewishestoemphasizeoneortheotherofthoseaspects. 21

PAGE 22

Fornow,weshallbesatisedwithanintuitivedenitionofafractal:asetthatshowsirregularbutself-similarfeaturesonmanyorallscales.Self-similaritymeansapartofanobjectissimilartootherpartsortothewhole.Thatis,ifweviewanirregularobjectwithamicroscope,whetherweenlargetheobjectby10times,orby100times,orevenby1000times,wealwaysndsimilarobjects.Tounderstandthisbetter,letusimaginewewereobservingapatchofwhiteclouddriftingawayinthesky.Oureyeswereratherpassive:wewerestaringmoreorlessatthesamedirection.Afterawhile,thepartofthecloudwesawdriftedaway,andwewereviewingadierentpartofthecloud.Nevertheless,ourfeelingremainsmoreorlessthesame. Mathematically,fractalischaracterizedbyapower-lawrelation,whichtranslatestoalinearrelationinlog-logscale.Fornow,letusagainresorttoimagination|wewerewalkingdownawildandveryjaggedmountaintrailorcoastline.Wewouldliketoknowthedistancecoveredbyourroute.Supposeourrulerhasalengthof|whichcouldbeourstep-size,anddierenthikersmayhavedierentstep-sizes|apersonridingahorsehasahugestep-size,whileagroupofpeoplewithalittlechildmusthaveatinystep-size.Thelengthofourrouteis whereN()isthenumberofintervalsneededtocoverourroute.ItismostremarkablethattypicallyN()scaleswithinapower-lawmanner, withDbeinganon-integer,1
PAGE 23

Nowletusexaminetheconsequenceof1
PAGE 24

24

PAGE 25

2 ,werstintroducethenewconceptofPSIC.ThenweextendtheconceptofPSICtohigh-dimensionalcase.WeshallpresentageneralcomputationalframeworkforassessingPSICfromrealworlddata.Wealsoapplytheproposedcomputationalproceduretostudynoise-freeandnoisylogisticandHenonmapsattheedgeofchaos.weshowthatwhennoiseisabsent,PSICishardtodetectfromascalartimeseries.However,whenthereisdynamicnoise,motionsaroundtheedgeofchaos,beitsimplyregularortrulychaoticwhenthereisnonoise,allcollapseontothePSICattractor.Hence,dynamicnoisemakesPSICobservable.InChapter 3 ,wedemonstratethatthenewframeworkofPSICcanreadilycharacterizerandomfractalprocesses,including1=fprocesseswithlong-range-correlationsandLevyprocesses.FromourstudyinChapters 2 and 3 ,itisclearthattheconceptofPSICnotjustbridgesstandardchaostheoryandrandomfractaltheory,butinfactprovidesamoregeneralframeworktoencompassboththeoriesasspecialcases.ToillustratethepoweroftheframeworkofPSIC,inChapters 4 and 5 ,weapplyittostudytwoimportantbutchallengingproblems:Internetdynamicsandseaclutterradarreturns,respectively.Bothareoutstandingexamplesofcomplexdynamicalsystemswithbothnonlinearityandrandomness.ThenonlinearityofInternetdynamicscomesfromAIMD(additiveincreaseandmultiplicativedecrease)operationofTCP(TransmissionControlprotocol),whilethestochasticcomponentcomesfromtherandomuserbehavior.Seaclutterisanonlineardynamicalprocesswithstochasticfactorsduetointerferenceofvariouswindandswellwavesandtolocalatmosphericturbulence.WeshallshowthatthenewtheoreticalframeworkofPSICcaneectivelycharacterizethecomplicatedInternetdynamicsaswellastorobustlydetectlowobservabletargetsfromseaclutterradarreturnswithhighaccuracy.InChapter 6 ,weshallintroducethebasictheoryofanexcellentmultiscalemeasure|thescale-dependentLyapunovexponent(SDLE),anddevelopaneectivealgorithmtocomputethemeasure.To 25

PAGE 26

7 26

PAGE 27

Theformalismofnonextensivestatisticalmechanics(NESM)[ 2 ]hasfoundnumerousapplicationstothestudyofsystemswithlong-range-interactions[ 3 { 6 ]andmultifractalbehavior[ 7 8 ],andfullydevelopedturbulence[ 8 { 10 ],amongmanyothers.Inordertocharacterizeatypeofmotionwhosecomplexityisneitherregularnorfullychaotic/random,recentlytheconceptofexponentialsensitivitytoinitialconditions(ESIC)ofdeterministicchaoshasbeengeneralizedtopower-lawsensitivitytoinitialconditions(PSIC)[ 7 11 ].Mathematically,theformulationofPSICcloselyparallelsthatofNESM.PSIChasbeenappliedtothestudyofdeterministic1-Dlogisticlikemapsand2-DHenonmapattheedgeofchaos[ 7 11 { 20 ],yieldingconsiderablenewinsights.Inthischapter,werstbrieydescribesomebackgroundknowledgeaboutchaos.Inparticular,wewilldiscussadirectdynamicaltestfordeterministicchaosdevelopedbyGaoandZheng[ 21 22 ].Thismethodoersamorestringentcriterionfordetectinglow-dimensionalchaos,andcansimultaneouslymonitormotionsinphasespaceatdierentscales.ThenweintroducethemoregeneralizedconceptofPSIC,andextendthenewconcepttohigh-dimensionalcase.Specically,wepresentageneralcomputationalframeworkforassessingPSICinatimeseries.Wealsoapplytheproposedcomputationalproceduretostudytwomodelsystems:noise-freeandnoisylogisticandHenonmapsattheedgeofchaos. 27

PAGE 28

where1ispositiveandcalledthelargestLyapunovexponent.Duetotheboundednessoftheattractorandtheexponentialdivergencebetweennearbytrajectories,astrangeattractortypicallyisafractal,characterizedbyasimpleandelegantscalinglawasdenedbyEq. 1{4 .Anon-integralfractaldimensionofanattractorcontrastssharplywiththeinteger-valuedtopologicaldimension(whichis0fornitenumberofisolatedpoints,1forasmoothcurve,2forasmoothsurface,andsoon). Conventionally,ithasbeenassumedthatatimeserieswithanestimatedpositivelargestLyapunovexponentandanon-integralfractaldimensionischaotic.However,ithasbeenfoundthatthisassumptionmaynotbeasucientindicationofdeterministicchaos.Onecounter-exampleistheso-called1=fprocesses.Suchprocesseshavespectraldensity where0
PAGE 29

23 { 27 ],estimationofthestrengthofmeasurementnoiseinexperimentaldata[ 28 29 ],pathologicaltremors[ 30 ],shear-thickeningsurfactantsolutions[ 31 ],diluteshearedaqueoussolutions[ 32 ],andserratedplasticows[ 33 ]. Nowletusbrieydescribethedirectdynamicaltestfordeterministicchaos[ 21 22 ].Givenascalartimeseries,x(1);x(2);:::;x(N),(assuming,forconvenience,thattheyhavebeennormalizedtotheunitinterval[0,1]),onerstconstructsvectorsofthefollowingformusingthetimedelayembeddingtechnique[ 34 { 36 ]: withmbeingtheembeddingdimensionandLthedelaytime.Forexample,whenm=3andL=4,wehaveV1=[x(1);x(5);x(9)],V2=[x(2);x(6);x(10)],andsoon.Fortheanalysisofpurelychaoticsignals,mandLhastobechosenproperly.Thisistheissueofoptimalembedding(see[ 21 22 37 ]andreferencestherein).Afterthescalartimeseriesisproperlyembedded,onethencomputesthetimedependentexponent(k)curves (k)=lnkVi+kVj+kk kViVjk(2{4) withrkViVjkr+r,whererandrareprescribedsmalldistances.Theanglebracketsdenoteensembleaveragesofallpossiblepairsof(Vi;Vj).Theintegerk,calledtheevolutiontime,correspondstotimekt,wheretisthesamplingtime.Notethatgeometrically(r;r+r)denesashell,andashellcapturesthenotionofscale.Thecomputationistypicallycarriedoutforasequenceofshells.ComparingEq. 2{4 withEq. 2{1 ,onenoticesthatkVi+kVj+kkplaystheroleofd(t),whilekViVjkplaystheroleofd(0).Figure 2-1 (a)showsthe(k)vs.kcurvesforthechaoticLorenzsystemdrivenby 29

PAGE 30

dt=16(xy)+D1(t)dy dt=xz+45:92xy+D2(t)dz dt=xy4z+D3(t)(2{5) with=0,=ij(tt0),i;j=1;2;3.NotethatD2isthevarianceoftheGaussiannoiseterms,andD=0describesthecleanLorenzsystem.WeobservefromFig. 2-1 (a)thatthe(k)curvesarecomposedofthreeparts.Thesecurvesarelinearlyincreasingfor0kka.Theyarestilllinearlyincreasingforkakkp,butwithaslightlydierentslope.Theyareatforkkp.NotethattheslopeofthesecondlinearlyincreasingpartgivesanestimateofthelargestpositiveLyapunovexponent[ 21 22 ].kaisrelatedtothetimescaleforapairofnearbypoints(Vi;Vj)toevolvetotheunstablemanifoldofViorVj.Itisontheorderoftheembeddingwindowlength,(m1)L.kpisthepredictiontimescale.Itislongerforthe(k)curvesthatcorrespondtosmallershells.Thedierencebetweentheslopesoftherstandsecondlinearlyincreasingpartsiscausedbythediscrepancybetweenthedirectiondenedbythepairofpoints(Vi;Vj)andtheunstablemanifoldofViorVj.ThisfeaturewasrstobservedbySatoetal.[ 38 ]andwasusedbythemtoimprovetheestimationoftheLyapunovexponent.Therstlinearlyincreasingpartcanbemadesmallerorcanevenbeeliminatedbyadjustingtheembeddingparameterssuchasbyusingalargervalueform.Notethatthesecondlinearlyincreasingpartsofthe(k)curvescollapsetogethertoformanenvelope.Itisthisveryfeaturethatformsthedirectdynamicaltestfordeterministicchaos[ 21 22 ].Thisisbecausethe(k)curvesfornoisydata,suchaswhitenoiseor1=fprocesses,areonlycomposedoftwoparts,anincreasing(butnotlinear)partfork(m1)Landaatpart[ 21 22 ].Furthermore,dierent(k)curvesfornoisydataseparatefromeachother;henceanenvelopeisnotdened.Wealsonote,mostotheralgorithmsforestimatingthelargestLyapunovexponentisequivalenttocompute(k)forr
PAGE 31

Onecanexpectthatthebehaviorofthe(k)curvesforanoisychaoticsystemliesinbetweenthatofthe(k)curvesforacleanchaoticsystemandthatofthe(k)curvesforwhitenoiseorfor1=fprocesses.Thisisindeedso.AnexampleisshowninFig. 2-1 (b)forthenoisyLorenzsystemwithD=4.WeobservefromFig. 2-1 (b)thatthe(k)curvescorrespondingtodierentshellsnowseparate.Therefore,anenvelopeisnolongerdened.Thisseparationislargerbetweenthe(k)curvescorrespondingtosmallershells,indicatingthattheeectofnoiseonthesmall-scaledynamicsisstrongerthanthatonthelarge-scaledynamics.Alsonotethatka+kpisnowontheorderoftheembeddingwindowlength,andisalmostthesameforallthe(k)curves.Withstrongernoise(D>4),the(k)curveswillbemorelikethoseforwhitenoise[ 21 22 ]. Finally,weconsidertheMackey-Glassdelaydierentialsystem[ 39 ].Whena=0:2;b=0:1;c=10;=30,ithastwopositiveLyapunovexponents,withthelargestLyapunovexponentcloseto0.007[ 21 ]. 1+x(t+)cbx(t)(2{6) HavingtwopositiveLyapunovexponentswhilethevalueofthelargestLyapunovexponentofthesystemisnotmuchgreaterthan0,onemightbeconcernedthatitmaybediculttodealwiththissystem.Thisisnotthecase.Infact,thissystemcanbeanalyzedasstraightforwardlyasotherdynamicalsystemsincludingtheLogisticmap,HenonmapandtheRosslersystem.Anexampleofthe(k)vs.kcurvesforthechaoticMackey-Glass 31

PAGE 32

Time-dependentexponent(k)vs.evolutiontimekcurvesfor(a)cleanand(b)noisyLorenzsystem.Sixcurves,frombottomtotop,correspondtoshells(2(i+1)=2;2i=2)withi=8,9,10,11,12,and13.Thesamplingtimeforthesystemis0.03sec,andembeddingparametersarem=4,L=3.5000pointsareusedinthecomputation. 32

PAGE 33

Time-dependentexponent(k)vs.evolutiontimekcurvesfortheMackey-Glasssystem.Ninecurves,frombottomtotop,correspondtoshells(2(i+1)=2;2i=2)withi=5;6;,and13.Thecomputationwasdonewithm=5;L=1,and5000pointssampledwithatimeintervalof6. delaydierentialsystemisshowninFig. 2-2 ,wherewehavefollowed[ 21 ]andusedm=5;L=1,and5000pointssampledwithatimeintervalof6.Clearly,weobservethatthereexistsawelldenedcommonenvelopetothe(k)curves.Actually,theslopeoftheenvelopeestimatesthelargestLyapunovexponent.Thisexampleillustratesthatwhenoneworksintheframeworkofthe(k)curvesdevelopedbyGaoandZheng[ 21 22 ],onedoesnotneedtobeveryconcernedaboutnon-uniformgrowthrateinhigh-dimensionalsystems. 2{7 ,wherex(0)istheinnitesimaldiscrepancyintheinitialcondition,x(t)isthediscrepancyattimet>0. x(0)(2{7) 33

PAGE 34

Tsallisandco-workers[ 7 11 ]havegeneralizedEq. 2{9 to whereqiscalledtheentropicindex,andqisinterpretedtobeequaltoKq,thegeneralizationoftheKolmogorov-Sinaientropy.Eq. 2{10 denesthePSICinthe1-Dcase.Obviously,PSICreducestoESICwhenq!1.ThesolutiontoEq. 2{10 is Whentislargeandq6=1,(t)increaseswithtasapower-law, whereC=[(1q)q]1=(1q).ForEq. 2{12 todeneanunstablemotionwithq>0,wemusthaveq1.Laterweshallmapdierenttypesofmotionstodierentrangesofq. ToapplyPSICtotheanalysisoftimeseriesdata,onecanrstconstructaphasespacebyconstructingembeddingvectorsViasdenedbyEq. 2{3 .Eq. 2{11 canthenbegeneralizedtohigh-dimensionalcase[ 40 ], jjV(0)jj=1+(1q)(1)qt1=(1q)(2{13) wherejjV(0)jjistheinnitesimaldiscrepancybetweentwoorbitsattime0,jjV(t)jjisthedistancebetweenthetwoorbitsattimet>0,qistheentropicindex,and(1)qistherstq-Lyapunovexponent,correspondingtothepower-lawincreaseoftherstprincipleaxisofaninnitesimalballinthephasespace.(1)qmaynotbeequaltoKq.Thisisunderstoodbyrecallingthatforchaoticsystems,theKolmogorov-Sinaientropyisthesum 34

PAGE 35

2{13 againgivesapower-lawincreaseof(t)witht.Notethatundertheaboveframework,themotionmaynotbelikefullydevelopedchaos,thusnotergodic[ 41 ]. WenowconsiderthegeneralcomputationalframeworkforPSIC.Givenanitetimeseries,theconditionofjjV(0)jj!0cannotbesatised.Inordertondthelawgoverningthedivergenceofnearbyorbits,onethushastoexaminehowtheneighboringpoints,(Vi;Vj),inthephasespace,evolvewithtime,byformingsuitableensembleaverages.Noticethatif(Vi1;Vj1)and(Vi2;Vj2)aretwopairsofnearbypoints,whenjjVi1Vj1jjjjVi2Vj2jj1,thentheseparationssuchasjjVi1+tVj1+tjjandjjVi2+tVj2+tjjcannotbesimplyaveragedtoprovideestimatesforqand(1)q.Infact,itwouldbemostconvenienttoconsiderensembleaveragesofpairsofpoints(Vi;Vj)thatallfallwithinaverythinshell,r1jjViVjjjr2,wherer1andr2areclose.Theseargumentssuggestthatthetime-dependentexponentcurvesdenedbyEq. 2{4 provideanaturalframeworktoassessPSICfromatimeseries.Thisisindeedso.Infact,wehave ln(t)(t)=lnkVi+tVj+tk kViVjk(2{14) Now,bythediscussionsinSec. 2.1 ,itisclearthatPSICisageneralizationofESIC:aslongasthe(k)curvesfromdierentshellsformalinearenvelope,thenq=1andthemotionischaotic.Thenextquestionis:DoesPSICalsoincluderandomfractalsasspecialcases?Theanswerisyes.Itwillbegiveninthenextchapter.There,wewillalsogainabetterunderstandingofthemeaningof(1)q.TherestofthischaptershowsexamplesofcharacterizingtheedgeofchaosbythenewframeworkofPSIC.Specically,westudytimeseriesgeneratedfromnoise-freeandnoisylogisticandHenonmaps. 35

PAGE 36

26 ],andphysiologicalnoiseinbiologicalsystems[ 30 ]),westudyboththedeterministicandthenoisylogisticmap: whereaisthebifurcationparameterandnisawhiteGaussiannoisewithmeanzeroandvariance1.Theparametercharacterizesthestrengthofnoise.Forthecleansystem(=0),theedgeofchaosoccursattheaccumulationpoint,a1=3:569945672.Weshallstudythreeparametervalues,a1=a10:001,a1,anda2=a1+0:001.Whennoiseisabsent,a1correspondstoaperiodicmotionwithperiod25,whilea2correspondstoatrulychaoticmotion.Weshallonlystudytransient-freetimeseries.InFigs. 2-3 (a-c),wehaveplottedthe(t)vs.tcurvesforparametervaluesa1,a1,anda2,respectively.WeobservefromFig. 2-3 (a)thatthevariationof(t)withtisperiodic(withperiod16,whichishalfoftheperiodofthemotion)whenthemotionisperiodic.Thisisagenericfeatureofthe(t)curvesfordiscreteperiodicattractors,whentheradiusoftheshellislargerthanthesmallestdistancebetweentwopointsontheattractor(whenaperiodicattractoriscontinuous,(t)canbearbitrarilycloseto0).IthasbeenfoundbyTsallisandco-workers[ 11 ]thatattheedgeofchaosforthelogisticmap,(t)isgivenbyEq. 2{11 withq0:2445.Surprisingly,wedonotobservesuchadivergenceinFig. 2-3 (b).Infact,ifoneplots(t)vs.lnt,oneonlyobservesacurvethatincreasesveryslowly(similartothatshowninFig. 2-4 (a)).ThemoreinterestingpatternistheonethatisshowninFig. 2-3 (c),whereweobservealinearlyincreasingly(t)vs.tcurve.Infact,showninFig. 2-3 (c)aretwosuchcurves,correspondingtotwodierentshells.Veryinterestingly,thetwocurvescollapsetoformacommonenvelopeinthelinearlyincreasingpartofthe 36

PAGE 37

Time-dependentexponent(t)vs.tcurvesfortimeseriesgeneratedfromthenoise-freelogisticmapwith(a)a1=a10:001,wherethemotionisperiodicwithperiod25,(b)a1=3:569945672,and(c)a2=a1+0:001,wherethemotionischaotic.Plottedin(d-f)are(t)vs.lntcurvesforthenoisylogisticmapwith=0:001.Verysimilarresultswereobtainedwhen=0:0001.Shownin(c-f)areactuallytwocurves,correspondingtotwodierentshells.104pointswereusedinthecomputation,withembeddingparametersm=4;L=1.However,solongasm>1,theresultsarelargelyindependentofembedding.Whenm=1,the(t)curvesarenotsmooth,andtheestimated1=(1q)valueismuchsmallerthanthetheoreticalvalue. curve.TheslopeoftheenvelopegivesagoodestimateofthelargestpositiveLyapunovexponent.Thisisagenericfeatureofchaos[ 21 22 ],asexplainedinSec. 2.1 .Sincethechaosstudiedhereisclosetotheedgeofchaos,thecurvesshowninFig. 2-3 (c)arelesssmooththanthosereportedearlier[ 21 { 23 26 ]. WhycannotthetheoreticalpredictionofPSICattheedgeofchaosforthelogisticmapbeobservedfromacleantimeseries?Inarecentlypublishedveryinterestingand 37

PAGE 38

42 ]suggeststhatdynamicnoisemaybeoffundamentalimportancetotheTsallisNESM.MaydynamicnoiseplaysimilarlysignicantroleforPSIC?Asisshownbelow,theanswerisyes.InFigs. 2-3 (d-f),wehaveshownthe(t)vs.lntcurvesforthethreeparametersconsidered,withnoisestrength1=0:001.Infact,shownineachgurearetwocurves,correspondingtotwodierentshells.Theyparallelwitheachother.Theslopesofthosecurvesareabout1.20,closetothetheoreticalvalueof1=(10:2445)1:32.WhileitisverysatisfactorytoobservePSICattheedgeofchaos,itismorethrillingtoobservethecollapseofregularaswellaschaoticmotionsontothePSICattractoraroundtheedgeofchaos.ThissigniesthestablenessofPSICwhenthereisdynamicnoise.ItisimportanttoemphasizethattheresultsshowninFigs. 2-3 (d-f)arelargelyindependentofthenoisestrength,solongasnoiseisneithertooweaknortoostrong.Forexample,verysimilarresultshavebeenobservedwith2=1=10=0:0001. BeforewemoveontodiscusstheHenonmapneartheedgeofchaos,letuscommentonthedierencebetweentheESICandthePSIC.WhentheESICisthecase,the(t)vs.tcurvesarestraightforTatTp,whereTpisapredictiontimescale,andTaisatimescalefortheinitialseparationtoevolvetothemostunstabledirection.When(t)vs.tcurvescorrespondingtodierentshellsareplottedtogether,theycollapsetogetherforaconsiderablerangeoft.WhenthePSICisthecase,the(t)vs.lntcurvesarefairlystraight,andthe(t)vs.lntcurvescorrespondingtodierentshellsseparateandparallelwitheachother.Thisfeatureisthefundamentalreasonthatensembleaveragesaremostconvenientlyformedbyrequiringneighboringpairsofpointstoallfallwithinathinshell.Forexample,iftwodistinctthinshells,describedbyr1jjXiXjjjr2andr2jjViVjjjr3,arejoinedtogethertoformasinglethickershell,describedbyr1jjViVjjjr3,andoneaveragestheseparationbetweenXiandXjbeforetakingthelogarithm,thentheresultingslopebetween(t)andlnt,assumedtobestilllinear,willhavetobesmallerthanthatestimatedfromthetwothinshellsseparately.Atthispointitisalsoworthnotingthatthelinearlyincreasingpartofthe(t)vs.lntcurves 38

PAGE 39

2{13 whendealingwithexperimentaltimeseries.Infact,wehavefoundthatifwedoso,theestimated1=(1q)valuefromFigs.1(d-f)increasestoabout1.27,muchclosertothetheoreticalvalueof1.32. NextletusconsidertheHenonmap,wherexandyarewhiteGaussiannoise,andareuncorrelatedwitheachother.Theparametermeasuresthestrengthofthenoise. Tirnakli[ 20 ]studiedthedeterministicversionofthismapattheedgeofchaos,forparametervaluesac=1:40115518;b=0;ac=1:39966671;b=0:001;andac=1:38637288;b=0:01.Wehavestudiedboththenoise-freeandnoisymapforparametervalueslistedabove,andfoundverysimilarresultstothosepresentedinFig. 2-3 .InFigs. 2-4 (a,b),wehaveplotted(t)vs.lntcurvesforthenoise-freeandnoisymapforac=1:38637288;b=0:01.Ascanbeobservedclearly,Fig. 2-4 (b)isverysimilartoFigs. 2-3 (d-f),whiletheslopeforthecurvesinFig. 2-4 (a)ismuchsmallerthanthatinFig. 2-4 (b).Again,wehaveobserved(butnotshownbygureshere,sincetheyareverysimilartoFigs. 2-3 (d-f)andFig. 2-4 (b))thatdynamicnoisemakestheregularandchaoticmotionstocollapseontothePSICattractorforparametersaroundthosedeningtheedgeofchaos,andthatsuchtransitionsarelargelyindependentofthenoisestrength. Tosummarize,wehavedescribedaneasilyimplementableprocedureforcomputationallyexaminingPSICfromatimeseries.Bystudyingtwomodelsystems:noise-freeandnoisylogisticandHenonmapsneartheedgeofchaos,wehavefoundthatwhenthereisno 39

PAGE 40

Time-dependentexponent(t)vs.tcurvesfortimeseriesgeneratedfrom(a)thenoise-freeHenonmapwithac=1:38637288;b=0:01.Plottedin(b)aretwo(t)vs.lntcurves(correspondingtotwodierentshells)forthenoisymapwith=0:001.Similarresultswereobtainedwith=0:0001.104pointswereusedinthecomputation,withembeddingparametersm=4;L=1. noise,thePSICattractorcannotbeobservedfromascalartimeseries.However,whendynamicnoiseispresent,motionsaroundtheedgeofchaos,beitsimplyregularortrulychaoticwhenthereisnonoise,allcollapseontothePSICattractor.WhiletheseexaminationssignifytheubiquityofPSIC,theyalsohighlighttheimportanceofdynamicnoise.Theexistenceofthelatterisperhapstheveryreasonthattrulychaotictimeseriescanseldombeobserved. 40

PAGE 41

InChapter 2 ,wehavediscussedthatPSICisageneralizationofESIC:aslongasthe(t)curvesfromdierentshellsformalinearenvelope,thenq=1andthemotionischaotic.WehavealsoshownthatedgeofchaoscanbewellcharacterizedbyPSIC.Inthischapter,weshallstudytwomajortypesofrandomfractals:1=fprocesseswithlong-range-correlationsandLevyprocesses.WeshallshowbothanalyticallyandthroughnumericalsimulationsthatbothtypesofprocessescanbereadilycharacterizedbyPSIC. for>0,0
PAGE 42

3{3 canbesimplyderivedfromEq. 3{4 ;wehavelisteditasaseparateequationforconvenienceoffuturereference.Ifweconsideronlysecond-orderstatistics,orifaprocessisGaussian,Eq. 3{2 toEq. 3{4 canbeusedinsteadofEq. 3{1 todeneaself-similarprocess. Averyusefulwayofdescribingaself-similarprocessisbyitsspectralrepresentation.Strictlyspeaking,theFouriertransformofX(t)isundened,duetothenonstationarynatureofX(t).Onecan,however,considerX(t)inaniteinterval,say,0
PAGE 43

with ProcesseswithpowerspectraldensitiesasdescribedbyEq. 3{7 arecalled1=fprocesses.Typically,thepower-lawrelationshipsthatdenetheseprocessesextendoverseveraldecadesoffrequency.Suchprocesseshavebeenfoundinnumerousareasofscienceandengineering.Someoftheolderliteraturesonthissubjectcanbefound,forexample,inPress[ 45 ],Bak[ 46 ],andWornell[ 47 ].Someofthemorerecentlydiscovered1=fprocessesareintracengineering[ 48 { 50 ],DNAsequence[ 51 { 53 ],humancognition[ 54 ],ambiguousvisualperception[ 55 56 ],coordination[ 57 ],posture[ 58 ],dynamicimages[ 59 60 ],andthedistributionofprimenumbers[ 61 ],amongmanyothers.Itisfurtherobservedthatprinciplecomponentanalysisofsuchprocessesleadstopower-lawdecayingeigenvaluespectrum[ 62 ]. Twoimportantprototypicalmodelsfor1=fprocessesarethefractionalBrownianmotion(fBm)processesandtheON/OFFintermittencywithpower-lawdistributedONandOFFperiods.Below,weapplytheconceptofPSICtobothtypesofprocesses. 1. AllBrownianpathsstartattheorigin:B(0)=0. 2. For0
PAGE 44

WemayinferfromtheabovedenitionthattheprobabilitydistributionfunctionofBisgivenby: 2sdu(3{9) Thisfunctionalsosatisesthescalingproperty: Inotherwords,B(t)and1=2B(t)havethesamedistribution.Thus,weseefromEq. 3{1 thatBmisaself-similarprocesswithHurstparameter1/2. SupposewehavemeasuredB(t1);B(t2);t2t1>0.WhatcanwesayaboutB(s);t1
PAGE 45

WenotethataBrownianpathisnotadierentiablefunctionoftime.Heuristically,thiscanbeunderstoodinthisway:considerthevariable,B(t+s)B(t),withvariances.Itsstandarddeviation,whichisameasureofitsorderofmagnitude,isp AlthoughB(t)isalmostsurelynotdierentiableint,symbolicallyonestilloftenwrites wherew(t)isstationarywhiteGaussiannoise,andextendstheaboveequationtot<0throughtheconvention,Zt0Z0t 3{12 Toseehow,letuspartition[0;t]intonequallyspacedintervals,t=t=n.Sincew(t)areGaussianrandomvariableswithzeromeanandunitvariance,inorderforB(t)tohavevariancet,weshouldhave Thatis,thecoecientis(t)1=2insteadoft,asonemighthaveguessed.Typically,t1;hence,(t)1=2t.Thus,ifoneincorrectlyusestinsteadof(t)1=2,oneisseverelyunderestimatingthestrengthofthenoise. 45

PAGE 46

Whitenoise(a)andBrownianmotion(b).Axesarearbitrary. Whenthetimeisgenuinelydiscrete,ortheunitsoftimecanbearbitrary,onecantakettobe1unit,andEq. 3{13 becomes Eq. 3{14 providesperhapsthesimplestmethodofgeneratingasampleofBm.AnexampleisshowninFig. 3-1 Eq. 3{14 isalsoknownasarandomwalkprocess.Amoresophisticatedrandomwalk(orjump)processisgivenbysummingupaninnitenumberofjumpfunctions:Ji(t)=Ai(tti),where(t)isaunit-stepfunction andAi;tiarerandomvariableswithGaussianandPoissondistributions,respectively. 46

PAGE 47

whereWisaGaussianrandomvariableofzeromeanandunitvariance,andHistheHurstparameter.WhenH=1=2,theprocessreducestothestandardBrownianmotion(Bm).ItistrivialtoverifythatthisprocesssatisesthedeningEq. 3{1 foraself-similarstochasticprocess. FractionalBrownianmotionBH(t)isaGaussianprocesswithmean0,stationaryincrements,variance andcovariance: 2ns2H+t2Hjstj2Ho(3{18) whereHistheHurstparameter.DuetoitsGaussiannature,accordingtoEq. 3{2 toEq. 3{4 ,theabovethreepropertiescompletelydetermineitsself-similarcharacter.Fig. 3-2 showsseveralfBmprocesseswithdierentH. Roughly,thedistributionforanfBmprocessis[ 63 ]: (H+1=2)Zt(t)H1=2dB()(3{19) where(t)isthegammafunction:(t)=Z10yt1eydy Inotherwords,thisprocessisinvariantfortransformationsconservingthesimilarityvariablex=tH. 47

PAGE 48

SeveralfBmprocesseswithdierentH. TheintegraldenedbyEq. 3{19 diverges.Themoreprecisedenitionis (H+1=2)ZtK(t)dB()(3{21) wherethekernelK(t)isgivenby[ 63 ] TheincrementprocessoffBm,Xi=BH((i+1)t)BH(it);i1,wheretcanbeconsideredasamplingtime,iscalledfractionalGaussiannoise(fGn)process.ItisazeromeanstationaryGaussiantimeseries.NotingthatE(XiXi+k)=Ef[BH((i+1)t)BH(it)][BH((i+1+k)t)BH((i+k)t)]g

PAGE 49

3{18 ,oneobtainstheautocovariancefunction(k)forthefGnprocess: 2n(k+1)2H2k2H+jk1j2Ho;k0(3{23) Noticethattheexpressionisindependentoft.Therefore,withoutlossofgenerality,onecouldtaket=1.Inparticular,wehave(1)=1 222H2 (i) WhenH=1=2,(k)=0fork6=0.Thisisthewell-knownpropertyofwhiteGaussiannoise. (ii) When00. Properties(ii)and(iii)areoftentermedanti-persistentandpersistentcorrelations[ 43 ],respectively. Next,weconsiderthebehaviorof(k)whenkislarge.Notingthatwhenjxj1,(1+x)1+x+(1) 2x2 hence, whenH6=1=2.WhenH=1=2,(k)=0fork1,theXi'saresimplywhitenoise. Nowwearereadytocharacterize1=fprocessesintheframeworkofPSIC.Recallingthatthedeningpropertyfora1=fprocessisthatitsvarianceincreaseswithtast2H. 49

PAGE 50

Therefore, Expressingtintermsof,wehave ComparingwiththedeningequationofPSIC(Eq. 2{10 ),wendthat 2(3{29) Noticing0
PAGE 51

Time-dependentexponent(k)vs.lnkcurvesforfractionalBrownianmotion. ON/OFFsourceswithN=3,X1(t),X2(t),X3(t),andtheirsummationS3(t). 51

PAGE 52

Thisexpressionisoftencalledthecomplementarycumulativedistributionfunction(CCDF)ofaheavy-taileddistribution.Infact,itismorepopularforexpressingaheavy-taileddistribution,sinceitemphasizesthetailofthedistribution.Itiseasytoprovethatwhen<2,thevarianceandallhigherthan2nd-ordermomentsdonotexist.Furthermore,when1,themeanalsodiverges. Whenthepower-lawrelationextendstotheentirerangeoftheallowablex,wehavetheParetodistribution: x;xb>0;2>0(3{31) whereandbarecalledtheshapeandthelocationparameters,respectively.Itcanbeproven[ 64 ]thatwhen12,theHurstparameteroftheON/OFFprocesseswithParetodistributedONandOFFperiodsisgivenas OurpurposehereistocheckwhethertheON/OFFprocesseswithParetodistributedONandOFFperiodscanbecharacterizedbyPSIC.InFig. 3-5 wehaveshownafewexamplesfor=1:2;1:6,and2.0.NoticingEq. 3{32 ,weseethattheslopesofthestraightlinesagaincorrectlyestimatetheHparameter.Alsonotethatwhen0<<1,1
PAGE 53

Time-dependentexponent(k)vs.lnkcurvesforON/OFFprocesseswithParetodistributedONandOFFperiods. Beforeendingthissection,wewouldliketoemphasizethatthepossibilityofmisinterpreting1=fprocessesbeingdeterministicchaosneveroccursifoneworksundertheframeworkofPSIC.Underthisframework,onemonitorstheevolutionoftwonearbytrajectories.Iftwonearbytrajectoriesdivergeexponentiallyfast,thenthetimeseriesischaotic;ifthedivergenceincreasesinapower-lawmanner,thenthetrajectoriesbelongto1=fprocesses.Theseideascanbeeasilyexpressedprecisely.LetjjViVjjjbetheinitialseparationbetweentwotrajectories.Thisseparationisassumedtobenotlargerthansomeprescribedsmalldistancer.Aftertimet,thedistancebetweenthetwotrajectorieswillbejjVi+tVj+tjj.Fortruechaoticsystems,jjVi+tVj+tjj/jjViVjjjet

PAGE 54

65 { 70 ].TherearetwotypesofLevyprocesses[ 71 ].OneisLevyights,whicharerandomprocessesconsistingofmanyindependentsteps,eachstepbeingcharacterizedbyastablelaw,andconsumingaunittimeregardlessofitslength.TheotherisLevywalks,whereeachsteptakestimeproportionaltoitslength.ALevywalkcanbeviewedassampledfromaLevyightwithauniformspeed.TheincrementprocessofaLevywalk,obtainedbydierencingtheLevywalk,isverysimilartoanON/OFFtrainwithpower-lawdistributedONandOFFperiods.Therefore,inthefollowing,weshallnotbefurtherconcernedaboutit.WeshallfocusonLevyights.NotethatLevyights,havingindependentsteps,arememorylessprocessescharacterizedbyH=1=2,irrespectiveofthevalueoftheexponentcharacterizingthestablelaws[ 71 ].Inotherwords,methodssuchasdetrendeductuationanalysis(DFA)[ 72 ]cannotbeusedtoestimatetheparameter. NowwedeneLevyightsmoreprecisely.A(standard)symmetric-stableLevyprocessfL(t);t0gisastochasticprocesswiththefollowingproperties[ 73 ]: 1. 2. 3. 54

PAGE 55

whereY1;Y2;areindependentrandomvariables,eachhavingthesamedistributionasY.FromEq. 3{33 ,onethenndsthatnVarY=n2=VarY.Forthedistributiontobevalid,0<2.When=2,thedistributionisGaussian,andhence,thecorrespondingLevyightisjustthestandardBrownianmotion.When0<<2,thedistributionisheavy-tailed,P[Xx]x;x!1,andhasinnitevariance.Furthermore,when0<1,themeanisalsoinnite. Atthispoint,itisworthspendingafewparagraphstoexplainhowtosimulatean-stableLevyprocess.Thekeyistosimulateitsincrementprocess,whichfollowsanstabledistribution.TherstbreakthroughforthesimulationofstabledistributionswasmadebyKanter[ 74 ],whogaveadirectmethodforsimulatingastabledistributionwith<1and=1.TheapproachwaslatergeneralizedtothegeneralcasebyChambersetal.[ 75 ].WerstdescribethealgorithmforconstructingastandardstablerandomvariableXS(;;1). (cos0cos)1=ncos[0+(1)] hasaS(;;1)distribution. Forsymmetricstabledistributionwith=0,theaboveexpressioncanbegreatlysimplied.TosimulateanarbitrarystabledistributionS(;;;),wecansimplytake 55

PAGE 56

whereZisgivenbyEq. 3{34 WehavesimulatedanumberofsymmetricstableLevymotionswithdierent.Figure 3-6 showstwoexamplesofLevymotionswith=1:5and1.ThedierencebetweenthestandardBrownianmotionsandLevymotionsliesinthatBrownianmotionsappeareverywheresimilarly,whileLevymotionsarecomprisedofdenseclustersconnectedbyoccasionallongjumps:thesmallertheis,themorefrequentlythelongjumpsappear. LetusnowpauseforawhileandthinkwhereLevyights-likeesotericprocessescouldoccur.Onesituationcouldbethis:amosquitoheadtoagiantspidernetandgotstuck;itstruggledforawhile,andluckily,withthehelpofagustofwind,escaped.Whenthemosquitowasstruggling,thestepsitcouldtakewouldbetiny;butthestepleadingtoitsfortunateescapemustbehuge.Asanotherexample,letusconsider(American)footballgames.Formostofthetimeduringagame,theoenseanddefensewouldbefairlybalanced,andtheoenseteammayonlybeabletoproceedforafewyards.Butduringanattackthatleadstoatouchdown,theherogettingthetouchdownoften\ies"tensofyards|somewhathehasescapedthedefense.Whilethesetwosimplesituationsareonlymeantasanalogies,rememberingthemcouldbehelpfulwhenreadingresearchpaperssearchingforLevystatisticsinvarioustypesofproblemssuchasanimalforaging.Atthispoint,wealsowishtomentionthatLevy-ights-likepatternshavebeenusedasatypeofscreensaverforLinuxoperatingsystems. ThesymmetricstableLevyightis1=self-similar.Thatis,forc>0,theprocessesfL(ct);t0gandfc1=L(t);t0ghavethesamenite-dimensionaldistributions.BythisargumentaswellasEq. 3{33 ,itisclearthatthelengthofthe 56

PAGE 57

ExamplesforBrownianmotionsandLevymotions.1-Dand2-DBrownianmotions(a,b)vs.Levymotionswith=1:5and1for(c,d)and(e,f),respectively. 57

PAGE 58

Time-dependentexponent(k)vs.lnkcurvesforLevyprocesses. motioninatimespanoft,L(t),isgivenbythefollowingscaling: L(t)/t1=(3{36) ThiscontrastswiththescalingforfractionalBrownianmotion: BH(t)/tH(3{37) Wethusidentifythat1=playstheroleofH.Therefore, Noticing0<2,fromEq. 3{38 ,wehave1q<1. 58

PAGE 59

3-7 .Theslopesofthestraightlinescorrectlyestimatethevaluesofusedinthesimulations. 59

PAGE 60

Timeseriesanalysisisanimportantexerciseinscienceandengineering.Oneofthemostimportantissuesintimeseriesanalysisistodeterminewhetherthedataunderinvestigationisregular,deterministicallychaotic,orsimplyrandom.Alongthisline,alotofworkhasbeendoneinthepasttwodecades.Abitsurprisingly,quiteoftentheexponentialsensitivitytoinitialconditions(ESIC)intherigorousmathematicalsensecannotbeobservedinexperimentaldata.Whilecurrentconsensusistoattributethisfacttothenoiseinthedata,weshallshowthatthemoregeneralconceptofpower-lawsensitivitytoinitialconditions(PSIC)providesaninterestingframeworkfortimeseriesanalysis.WeillustratethisbystudyingthecomplicateddynamicsofInternettransportprotocols. TheInternetisoneofthemostcomplicatedsystemsthatmanhasevermade.Inrecentyears,twotypesoffascinatingmulti-scalebehaviorshavebeenfoundintheInternet.Oneisthetemporal-domainfractalandmultifractalpropertiesofnetworktrac(see[ 76 77 ]andreferencestherein),whichtypicallyrepresentsaggregationofindividualhoststreams.Theotheristhespatial-domainscale-freetopologyoftheInternet(see[ 78 ]andreferencestherein).Also,ithasbeenobservedthatthefailureofasingleroutermaytriggerroutinginstability,whichmaybesevereenoughtoinstigatearouteapstorm[ 79 ].Furthermore,packetsmaybedeliveredoutoforderorevengetdropped,andpacketreorderingisnotapathologicalnetworkbehavior[ 80 ].AsthenextgenerationInternetapplicationssuchasremoteinstrumentcontrolandcomputationalsteeringarebeingdeveloped,itisbecomingincreasinglyimportanttounderstandthetransportdynamicsinordertosustaintheneededcontrolchannelsoverwide-areaconnections.Typically,thetransportdynamicsovertheInternetconnectionsarearesultofthenonlineardynamicsofthewidely-deployedTransmissionControlProtocol(TCP)interactingwiththeInternettrac,whichisstochasticandoftenself-similar.Thisleadsonetonaturallyexpectthe 60

PAGE 61

81 ].Iftransportdynamicsareindeedchaotic,thensteady-stateanalysisandstudyofconvergencetoasteady-statewillnotbethesolefocusinpractice,particularlyforremotecontrolandcomputationalsteeringtasks.Rather,oneshouldalsofocusonthe\transient"non-convergenttransportdynamics.Althoughrecentlyalotofresearchhasbeencarriedout[ 82 { 88 ]tobetterunderstandthisissue,alotoffundamentalproblemsremaintobeanswered.Forexample,theobservationofchaosin[ 84 ]cannotberepeated,whichleadstothesuspicionthatthechaoticmotionsreportedin[ 84 ]maycorrespondtoperiodicmotionswithverylongperiods[ 89 ].Fundamentally,weshallshow,bydevelopinga1-Ddiscretemap,thatthenetworkscenarioswithonlyafewcompetingTCPowsstudiedin[ 84 ]cannotgeneratechaoticdynamics.However,withotherparametersettings,chaosispossible. Inordertoexaminethetemporalevolutionofadynamicalsystem,oneoftenmeasuresascalartimeseriesataxedpointinthestatespace.Takensembeddingtheorem[ 34 { 36 ]ensuresthatthecollectivebehaviorofthedynamicsdescribedbythedynamicalvariablescoupledtothemeasuredonecanbeconvenientlystudiedbythemeasuredscalartimeseries.ToillustratethequalitativeaspectsofTCPdynamicsbyonlymeasuringascalartimeseries,itismosteectivetoconsiderasinglebottlenecklink,as 61

PAGE 62

84 ]consistingoflonglivedTCPowsonasinglelink.Thissetuphasfourparameters,thelinkspeedCinMbps,thelinkpropagationdelaydinms,thebuersizeBinunitofpacketsof1000bytes,andthenumberofcompetingTCPowsN.WehavefoundthatNactsasacriticalbifurcationparameter.Forexample,whenC=0:1,d=10ms,B=10andNissmall,wehaveobservedonlyperiodicandquasi-periodicmotions(Figs. 4-1 (a-d)).WhenNislarge,suchasN=17,chaosisobserved(seeFig. 4-2 (a,b)).WethusconcludethatoneroutetochaosinTCPdynamicsisthewell-knownquasi-periodicroute.Inordertoshowmorediversieddynamicalbehaviorsuchasquasi-periodicmotionswithmorethantwoincommensurate(i.e.,independent)frequencies,below,however,weshallvaryalltheparameters.Inparticular,weshallshowthatwhenNissmall,chaoscannothappen. ForeachoftheNcompetingTCPows,onecanrecorditscongestionwindowsizedata.Letusonlyrecordoneofthem,anddenoteitbyW(i).Anexampleofquasi-periodicW(i)isshowninFig. 4-1 (a),whereC=0:1Mbps,delayd=10ms,B=10,N=3.Itspowerspectraldensity(PSD)isshowninFig. 4-1 (b).Weobservemanydiscretesharppeaks.Notethatwhenonlyaround105pointsareusedtocomputethePSD,onedoesnotobservemanypeaksinthespectrum.Ratheroneobservesabroadspectrum,andhence,istemptedtointerprettheW(i)timeseriestobechaotic.Therefore,theW(i)timeseriesisnotideallysuitedforthestudyof(quasi-)periodicmotionswithlongperiods.Thismotivatesustodeneanewtimeseries,T(i),whichisthetimeintervalbetweentheonsetoftwosuccessiveexponentialbacko(i.e.,multiplicativedecrease)ofTCP,asindicatedinFig. 4-1 (a).Thestartofanexponentialbackoindicatestriggeringofalossepisodeorheavycongestion.Hence,theT(i)isrelatedtothewell-knownroundtriptime(RTT).Infact,itcanbeconsideredtobemoreorlessequivalenttothetimeintervalbetweentwosuccessivelossbursts.TheT(i)timeseriesfortheW(i)ofFig. 4-1 (a)isshowninFig. 4-1 (c),togetherwithits 62

PAGE 63

Exampleofquasi-periodiccongestionwindowsizeW(i)timeseriessampledbyanequaltimeintervalof10ms(a)anditspowerspectraldensity(PSD)(b).TheparametersusedareC=0:1Mbps,delayd=10ms,B=10packets,whereapacketisofsize1000bytes,andthenumberofTCPstreamsN=3.(c)TheT(i)timeseriescorrespondingto(a)and(d)itsPSD.(e)AnotherT(i)timeseriescorrespondingtoC=0:5Mbps,d=10ms,B=10,N=3,and(f)itsPSD. PSDinFig. 4-1 (d).WeobservethatthisT(i)timeseriesisasimpleperiodicsequence,andhence,theW(i)timeseriesisquasi-periodic.However,T(i)timeseriescanstillbequasi-periodic.AnexampleisshowninFig. 4-1 (e),withC=0:5Mbps,delayd=10ms,B=10packets,N=3,togetherwithitsPSDisFig. 4-1 (f).NotethattherearestillmanydiscretesharppeaksinFig. 4-1 (f).Toappreciatehowmanyindependentfrequencies 63

PAGE 64

4-1 (e),wehavefurtherextractedtwonewtimeseries,T(1)(i),whichistheintervaltimeseriesbetweenthesuccessivelocalmaximaoftheT(i)timeseries,andT(2)(i),whichistheintervaltimeseriesbetweenthesuccessivelocalmaximaoftheT(1)(i)timeseries.WehavefoundthattheT(2)(i)issimplyperiodic.Hence,weconcludethattheW(i)timeseriesisquasi-periodicwithfourindependentfrequencies. BeforeweproceedtodiscussthechaoticTCPdynamics,wenotethatsimpleperiodicorquasi-periodicW(i)timeserieswithlessormorethanfourindependentfrequenciescanallbeobserved.Infact,wehavefoundthe\chaotic"congestionwindowsizedatastudiedin[ 84 ]tobequasi-periodicwithtwoindependentfrequencies.Furthermore,theprobabilitydistributionsfortheconstant,periodic,andquasi-periodicT(i)timeseriesareonlycomposedofafewdiscretepeaks.Thissuggeststhattheobservedquasi-periodicT(i)timeseriesconstitutesadiscretetorus. LetusnowturntothediscussionofchaoticTCPdynamics.ShowninFigs. 4-2 (a,c)aretwoirregularT(i)timeseries.TheparametersC,d,andBusedforFig. 4-2 (a)arethesameasthoseforFigs. 4-1 (a-d).NotingfromFig. 4-2 thatT(i)oftencanbeontheorderof104to105,hence,anottoolongW(i)timeseriesisevenmoreill-suitedforthestudyoftheunderlyingirregulardynamics. ToshowthattheT(i)timeseriesinFigs. 4-2 (a,c)isindeedchaotic,weemploythedirectdynamicaltestfordeterministicchaosdevelopedbyGaoandZheng[ 21 22 ].The(k)curvesfortheT(i)timeseriesofFigs. 4-2 (a,c)areshowninFigs. 4-2 (b,d),wherethefourcurves,frombottomtotop,correspondtoshellsofsizes(2(i+1)=2;2i=2),i=14;15;16;17.wehavesimplychosenm=6andL=1.VerysimilarcurveshavebeenobtainedforotherchoicesofmandL.WeobserveaverywelldenedlinearcommonenvelopeatthelowerleftcornerofFigs. 4-2 (b,d).TheexistenceofacommonenvelopeguaranteesthatarobustpositiveLyapunovexponentwillbeobtainednomatterwhichshellisusedinthecomputation.Hence,thetwoT(i)timeseriesareindeedchaotic. 64

PAGE 65

TwoexamplesofchaoticT(i)timeseriesobtainedwithparameters(a)C=0:1,delayd=10ms,B=10,andthenumberofTCPstreamsN=17and(c)C=0:1,d=10ms,B=12,andN=19.The(k)curvesfor(a)and(c)areshownin(b)and(d),respectively. WehaveremarkedthattheprobabilitydistributionsfortheT(i)timeseriesofregularmotionsarecomprisedofafewdiscretepeaks.WhatarethedistributionsforthechaoticT(i)timeseriessuchasshowninFigs. 4-2 (a,c)?Theyarepower-law-like,P(Tt)t,foralmosttwoordersofmagnitudeint,asshowninFigs. 4-3 (a,b).Theexponentisnotauniversalquantity,however. Wenowdevelopasimple1-DmaptodescribetheoperationofTCP.TCPreliesontwomechanismstosetitstransmissionrate:owcontrolandcongestioncontrol. 65

PAGE 66

Complementarycumulativedistributionfunctions(CCDFs)forthetwochaoticT(i)timeseriesofFigs. 4-2 (a,c). Flowcontrolensuresthatthesendersendsnomorethanthesizeofthereceiver'slastadvertisedow-controlwindowbasedonitsavailablebuersizewhilecongestioncontrolensuresthatthesenderdoesnotunfairlyoverrunthenetwork'savailablebandwidth.TCPimplementsthesemechanismsviaaow-controlwindow(fwnd)advertisedbythereceivertothesenderandacongestion-controlwindow(cwnd)adaptedbasedoninferringthestateofthenetwork.Specically,TCPcalculatesaneectivewindow(ewnd),whereewnd=min(fwnd;cwnd),andthensendsdataatarateewnd=RTT,whereRTTistheroundtriptimeoftheconnection.RecentlyRaoandChua[ 85 ]developedananalyticmodeldescribingTCPdynamics.ByassumingfwndbeingxedandtheTCP'sslowstartonlycontributingtotransientbehavior,RaoandChuahavedevelopedthew-updatemapM:[1;Wmax]![1;Wmax], 66

PAGE 67

Theabovemap(Eq. 4{1 )canbesimplied[ 90 ], Sincefwndisnotassumedtobeaconstant,themodiedmapismoregeneralthantheoriginalRaoandChua'smap.ToapplytheabovemaptothestudyofaTCPowcompetingwithN1otherTCPows,onemaylumptheeectoftheN1otherTCPowsasabackgroundtrac.Hence,Wmaxisdeterminedbycwndandthebackgroundtrac,andtypicallyisacomplicatedfunctionoftime,noticingthattheavailablebandwidthofabottlenecklinkcanvarywithtimeconsiderablyduetothedynamicnatureofbackgroundtrac.WehavecarriedoutsimulationstudiesofEq. 4{2 byassumingWmaxtobeperiodicandquasi-periodic,andhaveindeedobservedchaos. WearenowreadytounderstandwhychaoscannotoccurwhenthenumberofcompetingTCPowsissmall,suchas2.Toseethis,letusassumethatwiandwi+wiaswellaswi+1andwi+1+wi+1areallsmallerthanWmax.Itistheneasytoseethatjwi+1jjwij.Thatis,smalldisturbancedecays.ThismeansduringtheadditiveincreasephaseofTCP,nearbytrajectoriescontractinsteadofdiverge.Hence,thedynamicsarestable.InorderforthedynamicstobeunstablesothattheLyapunovexponentispositive,thetransitionfromtheadditiveincreasephasetothemultiplicativedecreasephasehastobefast.ThiscanbetrueonlywhenthenumberofcompetingTCPstreamsislarge.Wethusconcludethatthenetworkscenariosconsideredin[ 84 ]cannotgeneratechaoticTCPdynamics. HowrelevantisoursimulationresulttothedynamicsoftherealInternet?AretheirregularT(i)timeseriesshowninFigs. 4-2 (a,c)anartifactofTahoeversionofTCP 67

PAGE 68

ThetimeseriesT(i)extractedfromaW(i)timeseriescollectedusingnet100instrumentsoverORNL-LSUconnection.(a)thetimeseriesT(i),(b)thecomplementarycumulativedistributionfunction(CCDF)fortheT(i)timeseries. weusedormoregenerallyofthens-2simulator?Tondananswer,wecollectedW(i)measurementsbetweenORNLandLouisianaStateUniversityatmillisecondresolutionusingthenet100instrumentsandcomputedT(i)asshowninFig. 4-4 (a),whoseproleisqualitativelyquitesimilartoFig. 4-2 (a).Infact,italsofollowsa(truncated)power-law,asshowninFig. 4-4 (b).Theexponentforthepower-lawpartisevensmallerthanthetimeseriesofFig. 4-2 .Thetruncationinthepower-lawcouldbeduetotheshortnessoftheT(i)timeseries.Aswehavepointedout,theT(i)timeseriesisrelatedtoRTT.CottrellandBullothavebeenexperimentingwithmanyadvancedversionsofTCP,andalsoobservedRTTtimeseriesverysimilartothese.WealsohaveobservedfromRTTandlossdatameasuredongeographicallydispersedpathsontheInternet(witharesolutionof1sec)byresearchersattheSanDiegoSupercomputerCenterthattheprobabilitydistributionsforthetimeintervalbetweensuccessivelossburststypicallyfollowapower-law-likedistribution,withtheexponentalsosmallerthan1.Thus,wehavegoodreasontobelievethattheobservedirregularT(i)timeseriesisnotan 68

PAGE 69

91 ],butvaryconsiderablyindurationsdeterminedbyspecicapplications.Inthenextsection,wewillanalyzetheactualdynamicsoftheInternetmeasurements. 92 ]withwhichTCPcompetesforbandwidthandrouterbuers.Roughlyspeaking,theinteractionbetweenthetwoisintermsoftheadditionstoTCPcongestionwindow-size,denotedbycwnd,inresponsetoacknowledgments,andmultiplicativedecreasesinresponsetoinferredlosses(ignoringtheinitialslow-startphase).Anyprotocol,howeversimpleitsdynamicsare,isgenerallyexpectedtoexhibitapparentlycomplicateddynamicsduetoitsinteractionwiththestochasticnetworktrac.TCPinparticularisshown(albeitinsimulation)toexhibitchaoticorchaos-likebehaviorevenwhenthecompetingtracismuchsimplersuchasasinglecompetingTCP[ 84 ]orUserDatagramProtocol(UDP)stream[ 85 ].Thedicultyistounderstandthedynamicswhenboththephenomenaareineect,andequallyimportantlytounderstandtheirimpactonactualInternetstreams.Historically,themethodsfromstandardchaosandstochastictheorieshavebeenunabletooermuchnewperspectiveonthetransportdynamics. 69

PAGE 70

21 22 37 ].OurmajorpurposeistoelucidatehowthedeterministicandstochasticcomponentsoftransportdynamicsinteractwitheachotherontheInternet.ItisofconsiderableinteresttonotethatrecentlytherehavebeenseveralimportantworksonTCPdynamicswiththepurposeofimprovingcongestioncontrol[ 93 { 96 ].Infact,therehasbeenconsiderableeortindevelopingnewversionsand/oralternativestoTCPsothatthenetworkdynamicscanbemorestable.Conventionalwaysofanalyzingthenetworkdynamicsareunabletoreadilydeterminewhethernewermethodsresultinstabletransportdynamicsorhelpindesigningsuchmethods.Ouranalysisshowstwoimportantfeaturesinthisdirection: (a) Randomnessisanintegralpartofthetransportdynamicsandmustbeexplicitlyhandled.Inparticular,thestepsizesutilizedforadjustingthecongestionparametersmustbesuitablyvaried(e.g.usingRobbins-Monroconditions)toensuretheeventualconvergence[ 97 ].ThisisnotthecaseforTCPwhichutilizesxedstepsizes. (b) Thechaoticdynamicsofthetransportprotocolsdohaveasignicantimpactonpracticaltransfers,andtheprotocoldesignmustbecognizantofitseects.Inparticular,itmightbeworthwhiletoinvestigateprotocolsthatdonotcontaindominantchaoticregimes,particularlyforremotecontrolandsteeringapplications. Thetraditionaltransportprotocolsarenotdesignedtoexplicitlyaddresstheabovetwoissues,butjustiablysosincetheiroriginalpurposeisdatatransportratherthannercontrolofdynamics. Intherestofthissection,weshallrstbrieydescribethecwnddatastudiedhere.Thenweshallbrieyexplaintheanalysisprocedureanddescribetypicalresultsoftheanalysis. WehavecollectedanumberofcwndtracesusingsingleandtwocompetingTCPstreams.ThisdatawascollectedontwodierentconnectionsfromOakRidgeNationalLaboratory(ORNL)toGeorgiaInstituteofTechnology(GaTech)andtoLouisianaState 70

PAGE 71

4-5 (a-d)segmentsoffourdatasetsforORNL-GaTechconnection.Powerspectralanalysisofthesedatadoesnotshowanydominantpeaks,andhence,thedynamicsarenotsimplyoscillatory.SinceourdatawasmeasuredontheInternetwith\live"backgroundtrac,itisapparentlymorecomplicatedandrealisticthanthetracesobtainedbynetworksimulation. Nextletusanalyzecwnddata,x(i);i=1;;n,usingtheconceptsoftimedependentexponent(k)curves[ 21 22 37 ].Ashasbeendiscussedearlier,werstneedtoemploytheembeddingtheorem[ 34 { 36 98 ]toconstructvectorsoftheform:Vi=[x(i);x(i+L);:::;x(i+(m1)L)],wheremistheembeddingdimensionandLthedelaytime.Theembeddingtheorem[ 34 { 36 ]statesthatwhentheembeddingdimensionmislargerthantwicetheboxcountingdimensionoftheattractor,thenthedynamicsoftheoriginalsystemcanbestudiedfromasinglescalartimeseries.Wenotethattheembeddingdimensionusedin[ 84 ]isonlytwo,whichhastobeconsiderednotlargeenough(thismaycallforacloserexaminationoftheirconclusions). Beforewemoveon,wepointoutafewinterestingfeaturesofthe(k)curves: (i) Fornoise,onlyforkuptotheembeddingwindowsize(m1)Lwill(k)increase.Thus,whenever(k)increasesforamuchlargerrangeofk,itisanindicationofnon-trivialdeterministicstructureinthedata. (ii) Forperiodicsignals,(k)isessentiallyzeroforanyk. 71

PAGE 72

TimeseriesforthecongestionwindowsizecwndforORNL-GaTechconnection. (iii) Forquasi-periodicsignals,(k)isperiodicwithanamplitudetypicallysmallerthan0.1,hence,forpracticalpurposes,(k)canbeconsideredverycloseto0. (iv) Whenachaoticsignaliscorruptedbynoise,thenthe(k)curvesbreakthemselvesawayfromthecommonenvelope.Thestrongerthenoiseis,themorethe(k)curvesbreakawaytilltheenvelopeisnotdenedatall.Thisfeaturehasactuallybeenusedtoestimatetheamountofbothmeasurementanddynamicnoise[ 99 ]. Nowwearereadytocomputeandunderstandthe(k)curvesforcwndtraces.Thesetof(k)curves,correspondingtoFig. 4-5 areplottedinFigs. 4-6 .Inthecomputations,3104pointsareused,andm=10;L=1.Thesevencurves,fromthebottomto 72

PAGE 73

Time-dependentexponent(k)vs.kcurvesforcwnddatacorrespondingtoFig. 4-5 .Inthecomputations,3104pointsareused,andm=10;L=1.Curvesfromthebottomtotopcorrespondtoshellswithsizes(2(i+1)=2;2i=2),i=2;3;;8. top,correspondtoshellsofsizes(2(i+1)=2;2i=2),i=2;3;;8.Wemakethefollowingobservations: (i) Thedynamicsarecomplicatedandcannotbedescribedaseitherperiodicorquasi-periodicmotions,since(k)ismuchlargerthan0. (ii) Thedynamicscannotbecharacterizedaspuredeterministicchaos,sinceinnocasecanweobserveawell-denedlinearenvelope.Thustherandomcomponentofthedynamicsduetocompetingnetworktracisevidentandcannotsimplybeignored. 73

PAGE 74

Thedataisnotsimplynoisy,sinceotherwiseweshouldhaveobservedthat(k)isalmostatwhenk>(m1)L.Thus,thedeterministiccomponentofdynamicswhichisduetothetransportprotocolplaysanintegralroleandmustbecarefullystudied.Thefeatures(ii)and(iii)indicatethattheInternettransportdynamicscontainsbothchaoticandstochasticcomponents. (iv) Thereareconsiderabledierencesbetweenthedatawithonly1TCPsourceandwith2competingTCPsources.Inthelattercase,the(k)curvessharplyrisewhenkjustexceedstheembeddingwindowsize,(m1)L.Ontheotherhand,(k)forFig. 4-6 (a)withonlyoneTCPsourceincreasesmuchslowerwhenkjustexceeds(m1)L.Alsoimportantisthatthe(k)curvesinFigs. 4-6 (c,d)aremuchsmootherthanthoseinFigs. 4-6 (a,b).Hence,wecansaythatthedeterministiccomponentofthedynamicsismorevisiblewhentherearemorethan1competingTCPsources(alongthelinesof[ 84 ]). Sincetheincreasingpartofthe(k)curvesarenotverylinear,letusnextexamineifthecwndtracescanbebettercharacterizedbythemoregeneralizedconceptofpower-lawsensitivitytoinitialconditions(PSIC).Figure 4-7 showsthe(k)vs.lnkcurvesforORNL-GaTechconnection.Veryinterestingly,wehavenowindeedobservedabunchofbetterdenedlinearlines,especiallyforsmallscales.Inparticular,thetransportdynamicswithmorethan1competingTCPsourcesarebetterdescribedbytheconceptofPSIC. Tosummarize,byanalyzinganumberofhighqualitycongestionwindow-sizedatameasuredontheInternet,wehavefoundthatthetransportdynamicsarebestdescribedbytheconceptofPSIC,especiallyforsmallscales.Itisinterestingtofurtherexaminehowonemightsuppressthestochasticityofthenetworkbyexecutingmorecontrolsonthenetworkwhenmakingmeasurements,suchasusingalargenumberofcompetingTCPsourcestogetherwithaconstantUDPow.OuranalysismotivatesthatbothchaoticandstochasticaspectsbepaidacloseattentiontoindesigningInternetprotocolsthatarerequiredtoprovidethedesiredandtractabledynamics. 74

PAGE 75

Time-dependentexponent(k)vs.lnkcurvesforcwnddatacorrespondingtoFig. 4-5 .Inthecomputations,3104pointsareused,andm=10;L=1.Curvesfromthebottomtotopcorrespondtoshellswithsizes(2(i+1)=2;2i=2),i=2;3;;8. 75

PAGE 76

Understandingthenatureofseaclutteriscrucialtothesuccessfulmodelingofseaclutteraswellastofacilitatetargetdetectionwithinseaclutter.Tothisend,animportantquestiontoaskiswhetherseaclutterisstochasticordeterministic.Sincethecomplicatedseacluttersignalsarefunctionsofcomplex(sometimesturbulent)wavemotionsontheseasurface,whilewavemotionsontheseasurfaceclearlyhavetheirowndynamicalfeaturesthatarenotreadilydescribedbysimplestatisticalfeatures,itisveryappealingtounderstandseaclutterbyconsideringsomeoftheirdynamicalfeatures.Inthepastdecade,Haykinetal.havecarriedoutanalysisofsomeseaclutterdatausingchaostheory[ 100 101 ],andconcludedthatseaclutterwasgeneratedbyanunderlyingchaoticprocess.Recently,theirconclusionhasbeenquestionedbyanumberofresearchers[ 102 { 108 ].Inparticular,Unsworthetal.[ 105 106 ]havedemonstratedthatthetwomaininvariantsusedbyHaykinetal.[ 100 101 ],namelythe\maximumlikelihoodofthecorrelationdimensionestimate"andthe\falsenearestneighbors"areproblematicintheanalysisofmeasuredseaclutterdata,sincebothinvariantsmayinterpretstochasticprocessesaschaos.Theyhavealsotriedanimprovedmethod,whichisbasedonthecorrelationintegralofGrassbergerandProcaccia[ 109 ]andhasbeenfoundeectiveindistinguishingstochasticprocessesfromchaos.Still,noevidenceofdeterminismorchaoshasbeenfoundinseaclutterdata. Toreconcileevergrowingevidenceofstochasticityinseaclutterwiththeirchaoshypothesis,recently,Haykinetal.[ 110 ]havesuggestedthatthenon-chaoticfeatureofseacluttercouldbeduetomanytypesofnoisesourcesinthedata.Totestthispossibility,McDonaldandDamini[ 111 ]havetriedaseriesoflow-passlterstoremovenoise;butagaintheyhavefailedtondanychaoticfeatures.Furthermore,theyhavefoundthatthecommonlyusedchaoticinvariantmeasuresofcorrelationdimensionandLyapunovexponent,computedbyconventionalways,producesimilarresultsformeasuredsea 76

PAGE 77

Whiletheserecentstudieshighlysuggestthatseaclutterisunlikelytobetrulychaotic,anumberoffundamentalquestionsarestillunknown.Forexample,mostofthestudiesin[ 102 { 106 ]areconductedbycomparingmeasuredseaclutterdatawithsimulatedstochasticprocesses.Wecanask:canthenon-chaoticnatureofseaclutterbedirectlydemonstratedwithoutresortingtosimulatedstochasticprocesses?Recognizingthatsimplelow-passlteringdoesnotcorrespondtoanydenitescalesinphasespace,canwedesignamoreeectivemethodtoseparatescalesinphasespaceandtotestwhetherseacluttercanbedecomposedassignalsplusnoise?Finally,willstudiesalongthislinebeofanyhelpfortargetdetectionwithinseaclutter? Inthischapter,weemploythedirectdynamicaltestfordeterministicchaosdiscussedinSec. 2.1 toanalyze280seaclutterdatameasuredundervariousseaandweatherconditions.Themethodoersamorestringentcriterionfordetectinglow-dimensionalchaos,andcansimultaneouslymonitormotionsinphasespaceatdierentscales.However,nochaoticfeatureisobservedfromanyofthesedataatallthedierentscalesexamined.Butinterestingly,weshowthatscale-dependentexponentcorrespondingtolargescaleappearstobeusefulfordistinguishingseaclutterdatawithandwithouttargets.Thissuggeststhatseacluttermaycontaininterestingdynamicfeaturesandthatthescale-dependentexponentmaybeanimportantparameterfortargetdetectionwithinseaclutter.Furthermore,wendthatseacluttercanbeconvenientlycharacterizedbythenewconceptofpower-lawsensitivitytoinitialconditions(PSIC).Weshowthatforthepurposeofdetectingtargetswithinseaclutter,PSICoersamoreeectiveframework. Below,weshallrstbrieydescribetheseaclutterdata.Thenwestudytheseaclutterdatabyemployingthedirectdynamicaltestforlow-dimensionalchaosdevelopedbyGaoandZheng[ 21 22 ].Andweshowthatthescale-dependentexponentcorrespondingtolargescalecanbeusedtoeectivelydetectlowobservabletargetswithin 77

PAGE 78

5-1 .Thedistancebetweentwoadjacentrangebinswas15m.Oneorafewrangebins(say,Bi1,BiandBi+1)hitatarget,whichwasasphericalblockofstyrofoamofdiameter1m,wrappedwithwiremesh.Thelocationsofthethreetargetswerespeciedbytheirazimuthalangleanddistancetotheradar.Theywere(128degree,2660m),(130degree,5525m),and(170degree,2655m),respectively.Therangebinwherethetargetisstrongestislabeledastheprimarytargetbin.Duetodriftofthetarget,binsadjacenttotheprimarytargetbinmayalsohitthetarget.Theyarecalledsecondarytargetbins.Foreachrangebin,therewere217complexnumbers,sampledwithafrequencyof1000Hz.Amplitudedataoftwopolarizations,HH(horizontaltransmission,horizontalreception)andVV(verticaltransmission,verticalreception)wereanalyzed. Fig. 5-2 showstwoexamplesofthetypicalseaclutteramplitudedatawithoutandwithtarget.However,carefulexaminationoftheamplitudedataindicatesthat4datasetsareseverelyaectedbyclipping.ThiscanbereadilyobservedfromFigs. 5-3 (a,b).We 78

PAGE 79

Collectionofseaclutterdata Figure5-2. Typicalseaclutteramplitudedata(a)withoutand(b)withtarget. discardthose4datasets,andanalyzetheremaining10measurements,whichcontain280seacluttertimeseries. 21 22 ],toanalyzeseaclutterdata.Themethodoersamorestringentcriterionforlow-dimensionalchaos,andcansimultaneouslymonitormotionsinphasespaceatdierentscales.Themethodhasfoundnumerousapplicationsinthe 79

PAGE 80

Twoshortsegmentsoftheamplitudeseaclutterdataseverelyaectedbyclipping. studyoftheeectsofnoiseondynamicalsystems[ 23 24 26 27 ],estimationofthestrengthofmeasurementnoiseinexperimentaldata[ 28 29 ],pathologicaltremors[ 30 ],shear-thickeningsurfactantsolutions[ 31 ],diluteshearedaqueoussolutions[ 32 ],andserratedplasticows[ 33 ].Inparticular,thismethodwasusedbyGaoetal.[ 107 108 ]toanalyzeonesinglesetofseaclutterdata.Whilechaoswasnotobservedfromthatdataset,nodenitegeneralconclusioncouldbereached,duetolackoflargeamountofdataatthattime.Herewesystematicallystudy280seaclutterdatameasuredundervariousseaandweatherconditions,andexaminewhetheranychaoticfeaturescanbefoundfromtheseseaclutterdata. TheexplicitincorporationofscalesintheGaoandZheng'stest[ 21 22 ]enablesustosimultaneouslymonitormotionsinphasespaceatdierentscales.Wehavesystematicallyanalyzed280amplitudeseacluttertimeseriesmeasuredundervariousseaandweatherconditions.However,nochaoticfeaturehasbeenobservedfromanyofthesedataatallthescalesexamined.Typicalexamplesofthe(k)vs.kcurvesfortheseaclutteramplitudedatawithouttargetsareshowninFigs. 5-4 (a,c,e)andthecurvesforthedatawithtargetsshowninFigs. 5-4 (b,d,f),respectively.Wehavesimplychosenm=6and 80

PAGE 81

Examplesofthetime-dependentexponent(k)vs.kcurvesfortheseaclutterdata(a,c,e)withoutand(b,d,f)withthetarget.Sixcurves,frombottomtotop,correspondtoshells(2(i+1)=2;2i=2)withi=13,14,15,16,17,and18.Thesamplingtimefortheseaclutterdatais1msec,andembeddingparametersarem=6,L=1.217datapointsareusedinthecomputation. 5-4 aregenericamongallthe280seaclutterdataanalyzedhere.Hence,wehavetoconcludethatnoneoftheseaclutterdataischaotic. 81

PAGE 82

112 ],log-normal[ 113 ],K[ 114 115 ],andcompound-Gaussiandistributions[ 116 ],aswellasusingchaostheory[ 100 { 104 107 108 110 111 ],wavelets[ 117 ],neuralnetworks[ 118 119 ],wavelet-neuralnetcombinedapproaches[ 120 121 ],andtheconceptoffractaldimension[ 122 ]andfractalerror[ 123 124 ].However,nosimplemethodhasbeenfoundtorobustlydetectlowobservableobjectswithinseaclutter[ 125 ]. Inthissection,weexaminewhethertheLyapunovexponentestimatedbyconventionalmethodsandthescale-dependentexponentmaybehelpfulfortargetdetectionwithinseaclutter. WerstcheckwhethertheLyapunovexponentestimatedbyconventionalmethodscanbeusedfordistinguishingseaclutterdatawithandwithouttargets.Aspointedoutearlier,theconventionalwayofestimatingtheLyapunovexponentistocompute(k)=kt,where(k)isdenedasinEq. 2{4 ,subjecttotheconditionsthatkXiXjk
PAGE 83

ThemodiedapproachforestimatingtheLyapunovexponentusestheleast-squaresttotherstfewsamplesofthe(k)curvewherethe(k)curveincreaseslinearlyorquasi-linearly.Thisisdoneforallthe280seacluttertimeseries.TobetterappreciatethevariationoftheLyapunovexponentamongthe14rangebinsoftheseaclutterdata,wehaverstsubtractedtheparameterofeachbinbytheminimumofvaluesforthatmeasurement,andthennormalizedtheobtainedvaluesbyitsmaximum.Thevariationsoftheparameterswiththe14rangebinsforthe10HHmeasurementsareshowninFigs. 5-5 (a)-(j),respectively,whereopencirclesdenotetherangebinswiththetarget,andasteroidsdenotethebinswithoutthetarget.Theprimarytargetbinisexplicitlyindicatedbyanarrow.Similarresultshavebeenobtainedforthe10VVmeasurements.WhileFigs. 5-5 (a)and(h)indicatethattheprimarytargetbincanbeseparatedfrombinswithoutthetarget,ingeneral,wehavetoconcludethattheLyapunovexponentestimatedbymethodsequivalentorsimilartoconventionalmeanscannotbeusedfordistinguishingseaclutterdatawithandwithouttargets. Nextletusexaminewhetherthescale-dependentexponentcorrespondingtolargescalemaybeusefulfordetectingtargetswithinseaclutterdata.Byscale-dependentexponent,wemeanthatifweusetheleast-squaresttothelinearlyorquasi-linearlyincreasingpartofthe(k)curvesofdierentshells,theslopesofthoselinesdependonwhichshellsareusedforcomputingthe(k)curves.Inotherwords,ifweplottheexponentestimatedthiswayagainstthesizeoftheshell,thentheexponentvarieswiththeshellsizeorthescale.Thescale-dependentexponenthasbeenusedinthestudyofnoise-inducedchaosinanopticallyinjectedsemiconductorlasermodeltoexaminehownoiseaectsdierentscalesofdynamicsystems[ 26 ].Wenowfocusontheexponentcorrespondingtolargeshellorscale.Figures 5-6 (a)-(j)showthevariationsofthescale-dependentexponentcorrespondingtolargescalewiththe14rangebinsforthe 83

PAGE 84

VariationsoftheLyapunovexponentestimatedbyconventionalmethodsvs.the14rangebinsforthe10HHmeasurements.Opencirclesdenotetherangebinswithtarget,while*denotethebinswithouttarget.Theprimarytargetbinisexplicitlyindicatedbyanarrow. same10HHmeasurementsasstudiedinFig. 5-5 .Weobservethattheprimarytargetbincanbeeasilyseparatedfromtherangebinswithoutthetarget,sincethescale-dependentexponentfortheprimarytargetbinismuchlargerthanthoseforthebinswithoutthetarget. Itisthusclearthatthescale-dependentexponentcorrespondingtolargescaleisveryusefulfordistinguishingseaclutterdatawithandwithouttargets.Thissuggeststhatseacluttermaycontaininterestingdynamicfeaturesandthatthescale-dependentexponentmaybeanimportantparameterfortargetdetectionwithinseaclutter.This 84

PAGE 85

Variationsofthescale-dependentexponentcorrespondingtolargescalevs.therangebinsforthe10HHmeasurements.Opencirclesdenotetherangebinswithtarget,while*denotethebinswithouttarget.Theprimarytargetbinisexplicitlyindicatedbyanarrow. exampleclearlysigniestheimportanceofincorporatingtheconceptofscaleinameasure.Infact,theconceptofscaleisonlyincorporatedinthetimedependentexponent(k)curves[ 21 22 ]inastaticmanner.InChapter 6 ,weshallseethatwhenameasuredynamicallyincorporatestheconceptofscale,itwillbecomemuchmorepowerful. Notethatonedicultyofusingthescale-dependentexponentfortargetdetectionwithinseaclutteristhatwemayneedtochooseasuitablescaleforestimatingtheexponent.Thismakesthemethodnoteasytouse,anditishardtomakethemethodautomatic. 85

PAGE 86

5-7 (a)and(b),respectively,wherethecurvesdenotedbyasteroidsarefordatawithoutthetarget,whilethecurvesdenotedbyopencirclesarefordatawiththetarget.Weobservethatthecurvesarefairlylinearfortherstafewsamples.Alsonoticethattheslopesofthecurvesforthedatawiththetargetaremuchlargerthanthoseforthedatawithoutthetarget.Forconvenience,wedenotetheslopeofthecurvebytheparameter.Tobetterappreciatethevariationoftheparameterwiththerangebins,wenormalizeofeachbinbythemaximalvalueofthe14rangebinswithinthesinglemeasurement.Fig. 5-8 showsthevariationoftheparameterwiththe14rangebins.Itisclearthattheparametercanbeusedtodistinguishseaclutterdatawithandwithouttarget.Interestingly,thefeatureshowninFig. 5-8 isgenericallytrueforallthemeasurements.Itisworthpointingoutthatusuallythe(k)vs.lnkcurvesfordierentscalesarealmostparallel,thustheestimatedparameterisrelativelylessdependentonthescale.Thisisaverynicefeatureofthismethod. Letusexamineifarobustdetectorfordetectingtargetswithinseacluttercanbedevelopedbasedontheparameter.Wehavesystematicallystudied280timeseriesoftheseaclutterdatameasuredundervariousseaandweatherconditions.Tobetterappreciatethedetectionperformance,wehaverstonlyfocusedonbinswithprimarytargets,butomittedthosewithsecondarytargets,sincesometimesitishardtodeterminewhetherabinwithsecondarytargetreallyhitsatargetornot.Afteromittingtherangebindatawithsecondarytargets,thefrequenciesfortheparameterunderthetwohypotheses(thebinswithouttargetsandthosewithprimarytargets)forHHdatasets 86

PAGE 87

Examplesofthe(k)vs.lnkcurvesforrangebins(a)withoutand(b)withtarget.Opencirclesdenotetherangebinswithtarget,while*denotethebinswithouttarget. Figure5-8. Variationoftheparameterwiththe14rangebins.Opencirclesdenotetherangebinswithtarget,while*denotethebinswithouttarget. areshowninFig. 5-9 .WeobservethatthefrequenciescompletelyseparatefortheHHdatasets.Thismeansthedetectionaccuracycanbe100%. 87

PAGE 88

FrequenciesofthebinswithouttargetsandthebinswithprimarytargetsfortheHHdatasets. 88

PAGE 89

InChapter 1 ,wehaveemphasizedtheimportanceofdevelopingscale-dependentmeasurestosimultaneouslycharacterizebehaviorsofcomplexmultiscaledsignalsonawiderangeofscales.Inthischapter,weshalldevelopaneectivealgorithmtocomputeanexcellentmultiscalemeasure,thescale-dependentLyapunovexponent(SDLE),andshowthattheSDLEcanreadilyclassifyvarioustypesofcomplexmotions,includingtrulylow-dimensionalchaos,noisychaos,noise-inducedchaos,processesdenedbypower-lawsensitivitytoinitialconditions(PSIC),andcomplexmotionswithchaoticbehavioronsmallscalesbutdiusivebehavioronlargescales.Finally,weshalldiscusshowtheSDLEcanhelpdetecthiddenfrequenciesinlargescaleorderlymotions. 126 ].ThealgorithmforcalculatingtheFSLEisverysimilartotheWolfetal'salgorithm[ 127 ].Itcomputestheaverager-foldtimebymonitoringthedivergencebetweenareferencetrajectoryandaperturbedtrajectory.Todoso,itneedstodene\nearestneighbors",aswellasneedstoperform,fromtimetotime,arenormalizationwhenthedistancebetweenthereferenceandtheperturbedtrajectorybecomestoolarge.Suchaprocedurerequiresverylongtimeseries,andtherefore,isnotpractical.Tofacilitatederivationofafastalgorithmthatworksonshortdata,aswellastoeasediscussionofcontinuousbutnon-dierentiablestochasticprocesses,wecastthedenitionoftheSDLEasfollows. Consideranensembleoftrajectories.Denotetheinitialseparationbetweentwonearbytrajectoriesby0,andtheiraverageseparationattimetandt+tbytandt+t,respectively.Beingdenedinanaveragesense,tandt+tcanbereadilycomputedfromanyprocesses,eveniftheyarenon-dierentiable.Nextweexaminetherelationbetweent

PAGE 90

where(t)istheSDLE.Itisgivenby Givenatimeseriesdata,thesmallesttpossibleisthesamplingtime. ThedenitionoftheSDLEsuggestsasimpleensembleaveragebasedschemetocomputeit.Astraightforwardwaywouldbetondallthepairsofvectorsinthephasespacewiththeirdistanceapproximately,andthencalculatetheiraveragedistanceafteratimet.Thersthalfofthisdescriptionamountstointroducingashell(indexedask), whereVi;Vjarereconstructedvectors,k(theradiusoftheshell)andk(thewidthoftheshell)arearbitrarilychosensmalldistances.Suchashellmaybeconsideredasadierentialelementthatwouldfacilitatecomputationofconditionalprobability.Toexpeditecomputation,itisadvantageoustointroduceasequenceofshells,k=1;2;3;.Notethatthiscomputationalprocedureissimilartothatforcomputingtheso-calledtime-dependentexponent(TDE)curves[ 21 22 37 ]. Withalltheseshells,wecanthenmonitortheevolutionofallofthepairsofvectors(Vi;Vj)withinashellandtakeaverage.Wheneachshellisverythin,byassumingthattheorderofaveragingandtakinglogarithminEq. 6{2 canbeinterchanged,wehave wheretandtareintegersinunitofthesamplingtime,andtheanglebracketsdenoteaveragewithinashell.NotethatcontributionstotheSDLEataspecicscalefromdierentshellscanbecombined,withtheweightforeachshellbeingdeterminedbythe 90

PAGE 91

Intheaboveformulation,itisimplicitlyassumedthattheinitialseparation,kViVjk,alignswiththemostunstabledirectioninstantly.Forhigh-dimensionalsystems,thisisnottrue,especiallywhenthegrowthrateisnon-uniformand/ortheeigenvectorsoftheJacobianarenon-normal.Fortunately,theproblemisnotasseriousasonemightbeconcerned,sinceourshellsarenotinnitesimal.WhencomputingtheTDEcurves[ 21 22 37 ],wehavefoundthatwhendicultiesarise,itisoftensucienttointroduceanadditionalcondition, whenndingpairsofvectorswithineachshell.SuchaschemealsoworkswellwhencomputingtheSDLE.Thismeansthat,aftertakingatimecomparabletotheembeddingwindow(m1)L,itwouldbesafetoassumethattheinitialseparationhasevolvedtothemostunstabledirectionofthemotion. Beforeproceedingon,wewishtoemphasizethemajordierencebetweenouralgorithmandthestandardmethodforcalculatingtheFSLE.Aswehavepointedout,tocomputetheFSLE,twotrajectories,oneasreference,anotherasperturbed,havetobedened.Thisrequireshugeamountsofdata.Ouralgorithmavoidsthisbyemployingtwocriticaloperationstofullyutilizeinformationaboutthetimeevolutionofthedata:(i)Thereferenceandtheperturbedtrajectoriesarereplacedbytimeevolutionofallpairsofvectorssatisfyingtheinequality( 6{5 )andfallingwithinashell,and(ii)introductionofasequenceofshellsensuresthatthenumberofpairsofvectorswithintheshellsislargewhiletheensembleaveragewithineachshelliswelldened.LetthenumberofpointsneededtocomputetheFSLEbystandardmethodsbeN.Thesetwooperationsimplythatthemethoddescribedhereonlyneedsaboutp 91

PAGE 92

2{5 .Figure 6-1 (a)showsvecurves,forthecasesofD=0;1;2;3;4.Thecomputationsaredonewith10000pointsandm=4;L=2.Weobserveafewinterestingfeatures: Nextweconsidernoise-inducedchaos.Toillustratetheidea,wefollow[ 23 ]andstudythenoisylogisticmap 92

PAGE 93

Scale-dependentLyapunovexponent()curvesfor(a)thecleanandthenoisyLorenzsystem,and(b)thenoise-inducedchaosinthelogisticmap.Curvesfromdierentshellsaredesignatedbydierentsymbols. whereisthebifurcationparameter,andPnisaGaussianrandomvariablewithzeromeanandstandarddeviation.In[ 23 ],wereportedthatat=3:74and=0:002,noise-inducedchaosoccurs,andthoughtthatitmaybediculttodistinguishnoise-inducedchaosfromcleanchaos.InFig. 6-1 (b),wehaveplottedthe(t)forthisparticularnoise-inducedchaos.Thecomputationwasdonewithm=4;L=1and10000points.WeobservethatFig. 6-1 (b)isverysimilartothecurvesofnoisychaosplottedinFig. 6-1 (a).Hence,noise-inducedchaosissimilartonoisychaos,butdierentfromcleanchaos. Atthispoint,itisworthmakingtwocomments:(i)Onverysmallscales,theeectofmeasurementnoiseissimilartothatofdynamicnoise.(ii)The()curvesshowninFig. 6-1 arebasedonafairlysmallshell.ThecurvescomputedbasedonlargershellscollapseontherightpartofthecurvesshowninFig. 6-1 .Becauseofthis,forchaoticsystems,oneorafewsmallshellswouldbesucient.Ifonewishestoknowthebehaviorofoneversmallerscales,onehastouselongerandlongertimeseries. 93

PAGE 94

Scale-dependentLyapunovexponent()curvefortheMackey-Glasssystem.Thecomputationwasdonewithm=5;L=1,and5000pointssampledwithatimeintervalof6. Finally,weconsidertheMackey-Glassdelaydierentialsystem[ 39 ],denedbyEq. 2{6 .ThesystemhastwopositiveLyapunovexponents,withthelargestLyapunovexponentcloseto0.007[ 21 ].HavingtwopositiveLyapunovexponentswhilethevalueofthelargestLyapunovexponentofthesystemisnotmuchgreaterthan0,onemightbeconcernedthatitmaybediculttocomputetheSDLEofthesystem.Thisisnotthecase.Infact,thissystemcanbeanalyzedasstraightforwardlyasotherdynamicalsystemsincludingtheHenonmapandtheRosslersystem.Anexampleofthe()curveisshowninFig. 6-2 ,wherewehavefollowed[ 21 ]andusedm=5;L=1,and5000pointssampledwithatimeintervalof6.Clearly,weobserveawelldenedplateau,withitsvaluecloseto0.007.ThisexampleillustratesthatwhencomputingtheSDLE,onedoesnotneedtobeveryconcernedaboutnon-uniformgrowthrateinhigh-dimensionalsystems. Uptillnow,wehavefocusedonthepositiveportionof(t).Itturnsoutthatwhentislarge,()becomesoscillatory,withmeanabout0.Denotethecorrespondingscalesby 94

PAGE 95

3 ,wehaveseenthatpower-lawsensitivitytoinitialconditions(PSIC)providesacommonfoundationforchaostheoryandrandomfractaltheory.Here,weexaminewhethertheSDLEisabletocharacterizeprocessesdenedbyPSIC.Pleasingly,thisisindeedso.Below,wederiveasimpleequationrelatingtheqandqofPSICtotheSDLE. First,werecallthatPSICisdenedbyt=limjjV(0)jj!0jjV(t)jj jjV(0)jj=1+(1q)(1)qt1=(1q) 6{2 ,wendthat Whent!0,1+(1q)qt(1q)qt.SimplifyingEq. 6{7 ,weobtain Wenowconsiderthreecases: 95

PAGE 96

3{28 )and(1)q=H(Eq. 3{29 ).Therefore,(t)=H1 3{38 )and(1)q=1=(Eq. 3{39 ).Therefore,(t)=1 128 ]andstudythefollowingmap, where[xn]denotestheintegerpartofxn,tisanoiseuniformlydistributedintheinterval[1;1],isaparameterquantifyingthestrengthofnoise,andF(y)isgivenby ThemapF(y)isshowninFig. 6-3 asthedashedlines.ItgivesachaoticdynamicswithapositiveLyapunovexponentln(2+).Ontheotherhand,theterm[xn]introducesarandomwalkonintegergrids. Itturnsoutthissystemisveryeasytoanalyze.When=0:4,withonly5000pointsandm=2;L=1,wecanresolveboththechaoticbehavioronverysmallscales,andthenormaldiusivebehavior(withslope2)onlargescales.SeeFig. 6-4 (a). Wenowaskaquestion:Givenasmalldataset,whichtypeofbehavior,thechaoticorthediusive,isresolvedrst?Toanswerthis,wehavetriedtocomputetheSDLEwithonly500points.TheresultisshowninFig. 6-4 (b).Itisinterestingtoobservethatthechaoticbehaviorcanbewellresolvedbyonlyafewhundredpoints.However,the 96

PAGE 97

ThefunctionF(x)(Eq. 6{10 )for=0:4isshownasthedashedlines.ThefunctionG(x)(Eq. 6{11 )isanapproximationofF(x),obtainedusing40intervalsofslope0.Inthecaseofnoise-inducedchaosdiscussedinthepaper,G(x)isobtainedfromF(x)using104intervalsofslope0.9. Scale-dependentLyapunovexponent()forthemodeldescribedbyEq. 6{9 .(a)5000pointswereused;forthenoisycase,=0:001.(b)500pointswereused. 97

PAGE 98

Wehavealsostudiedthenoisymap.TheresultingSDLEfor=0:001isshowninFig. 6-4 (a),assquares.Wehaveagainused5000points.WhilethebehavioroftheSDLEsuggestsnoisydynamics,with5000points,wearenotabletowellresolvetherelation()ln.Thisindicatesthatforthenoisymap,onverysmallscales,thedimensionisveryhigh. Map( 6{9 )canbemodiedtogiverisetoaninterestingsystemwithnoise-inducedchaos.ThiscanbedonebyreplacingthefunctionF(y)inmap( 6{9 )byG(y)toobtainthefollowingmap[ 128 ], wheretisanoiseuniformlydistributedintheinterval[1;1],isaparameterquantifyingthestrengthofnoise,andG(y)isapiecewiselinearfunctionwhichapproximatesF(y)ofEq. 6{10 .AnexampleofG(y)isshowninFig. 6-3 .Inournumericalsimulations,wehavefollowedCencinietal.[ 128 ]andused104intervalsofslope0.9toobtainG(y).WithsuchachoiceofG(y),intheabsenceofnoise,thetimeevolutiondescribedbythemap( 6{11 )isnonchaotic,sincethelargestLyapunovexponentln(0:9)isnegative.Withappropriatenoiselevel(e.g.,=104or103),theSDLEforthesystembecomesindistinguishabletothenoisySDLEshowninFig. 6-4 forthemap( 6{9 ).Havingadiusiveregimeonlargescales,thisisamorecomplicatednoise-inducedchaosthantheonewehavefoundfromthelogisticmap. Beforeproceedingon,wemakeacommentonthecomputationoftheSDLEfromdeterministicsystemswithnegativelargestLyapunovexponents,suchasthemap( 6{11 )withoutnoise.Atransient-freetimeseriesfromsuchsystemsisaconstanttimeseries.Therefore,thereisnoneedtocomputetheSDLEorothermetrics.Whenthetimeseries 98

PAGE 99

2{5 .InFigs. 6-5 (a-c),wehaveshownthepower-spectraldensity(PSD)ofthex;y;zcomponentsofthesystem.WeobservethatthePSDofx(t)andy(t)aresimplybroad.However,thePSDofz(t)showsupaverysharpspectralpeak.Recallthatgeometrically,theLorenzattractorconsistsoftwoscrolls(seeFig. 6-6 ).ThesharpspectralpeakinthePSDofz(t)oftheLorenzsystemisduetothecircularmotionsalongeitherofthescrolls. TheaboveexampleillustratesthatifthedynamicsofasystemcontainsahiddenfrequencythatcannotberevealedbytheFouriertransformofameasuredvariable(say,x(t)),theninordertorevealthehiddenfrequency,onehastoembedx(t)toasuitablephasespace.Thisideahasledtothedevelopmentoftwointerestingmethodsforidentifyinghiddenfrequencies.OnemethodisproposedbyOrtega[ 129 130 ],bycomputingthetemporalevolutionofdensitymeasuresinthereconstructedphasespace.AnotherisproposedbyChernetal.[ 131 ],bytakingsingularvaluedecompositionoflocalneighbors.TheOrtega'smethodhasbeenappliedtoanexperimentaltimeseriesrecorded 99

PAGE 100

Power-spectraldensity(PSD)for(a)x(t),(b)y(t),(c)z(t),(d)(x)1(t),(e)(y)1(t),and(f)(z)1(t). fromafar-infraredlaserinachaoticstate.Thelaserdatasetcanbedownloadedfromthelink 6-7 (a)showsthelaserdataset.ThePSDofthedataisshowninFig. 6-7 (b),whereoneobservesasharppeakaround1.7MHz.Fig. 6-7 (c)showsthePSDofthedensitytimeseries,whereonenotesanadditionalspectralpeakaround37kHz.Thispeakisduetotheenvelopeofchaoticpulsations,whichisdiscernablefromFig. 6-7 (a). 100

PAGE 101

Lorenzattractor. ToappreciatethestrengthoftheadditionalspectralpeakinFig. 6-7 (c),wehavetonotethattheunitsofthePSDinbothFigs. 6-7 (b,c)arearbitrary|thelargestpeakistreatedas1unit,aswasdonebyOrtega.Infact,theactualenergyoftheoriginallaserintensitytimeseriesismuchgreaterthanthatofthedensitytimeseries.Therefore,theadditionalspectralpeakof37kHzin(c)isreallyquiteweak.Inotherwords,althoughtheOrtega'smethodisabletorevealhiddenfrequencies,itisnotveryeective.Infact,Chernetal.havepointedoutthattheeectivenessoftheirmethodaswellastheOrtega'smethodtoexperimentaldataanalysisremainsuncertain. Oneofthemajordicultiesoftheabovetwomethodsisconceptual|bothmethodsarebasedonverysmallscaleneighbors,whilethephenomenonitselfislargescale.Recognizingthis,onecanreadilyunderstandthattheconceptoflimitingscale,developedinSec. 6.2.1 ,providesanexcellentsolutiontotheproblem.Inotherwords,onecansimplytaketheFouriertransformofthetemporalevolutionofthelimitingscale,andexpectadditionalwell-denedspectralpeaks,ifthedynamicscontainhiddenfrequencies.To 101

PAGE 102

Hiddenfrequencyphenomenonoflaserintensitydata.(a)laserintensitydata,(b)thepowerspectraldensity(PSD)fortheoriginallaserintensitytimeseries,(c)thePSDforthedensitytimeseriesofthereconstructedphasespacefromthelaserdata,and(d)thePSDforthelimitingscaleofthelaserdata.In(c)and(d),theembeddingparametersarem=4;L=1. 102

PAGE 103

6-5 (d-f)thePSDofthelimitingscalesofx(t);y(t)andz(t)oftheLorenzsystem.Weobserveverywell-denedspectralpeaksinallthreecases.Infact,nowthezcomponentnolongerplaysamorespecialrolethanthexandycomponents.Themethodissimilarlyamazinglyeectiveinidentifyingthehiddenfrequencyfromthelaserdata,asshowninFig. 6-7 (d):wenownotonlyobserveanadditionalspectralpeakaround37KHzin(d),butalsothatthispeakisevenmoredominantthanthepeakaround1.7MHz.Thereasonfortheexchangeofthisdominanceisduetofairlylargesamplingtime|although80nsissmall,itisonlyabletosampleabout8pointsineachoscillation.Whentheembeddingwindow,(m1)L,becomesgreaterthan4,theNyquistsamplingtheoremisviolatedinthereconstructedphasespace,thenthispeakmayevendiminish. Itisimportanttonotethatthelimitingscaleisnotaectedmuchbyeithermeasurementnoiseordynamicnoise.Therefore,onecanexpectthatthehiddenfrequenciesrevealedbylimitingscaleswillnotbeaectedmuchbynoiseeither. 103

PAGE 104

Inthiswork,wehavedevelopedanewtheoreticalframework,power-lawsensitivitytoinitialconditions(PSIC),toprovidechaostheoryandrandomfractaltheoryacommonfoundation.Forpracticaluses,wehavedevelopedaneasilyimplementableprocedureforcomputationallyexaminingPSICfromascalartimeseries.Wehavefoundthatdierenttypeofmotions,suchaschaos,edgeofchaos,1=fprocesseswithlong-range-correlationsandLevyprocesses,arereadilycharacterizedbytheframeworkofPSICindierentrangeofq.Specically,trulychaoticmotionsarecharacterizedbyq=1;edgeofchaosisusuallycharacterizedbyaqvaluelargerthan0butsmallerthan1;and1=fprocessesandLevyprocessesarecharacterizedbyqintherangeof
PAGE 105

[1] P.Fairley,\Theunrulypowergrid,"IEEESpectrum,vol.41,pp.22-27,2004. [2] C.Tsallis,\PossiblegeneralizationofBoltzmann-Gibbsstatistics,"JStatPhys,vol.52,pp.479-487,1988. [3] A.R.PlastinoandA.Plastino,\Stellarpolytropesandtsallisentropy,"Phys.Lett.A,vol.174,pp.384-386,1993. [4] A.Lavagno,G.Kaniadakis,M.Rego-Monteiro,P.Quarati,andC.Tsallis,\Non-extensivethermostatisticalapproachofthepeculiarvelocityfunctionofgalaxyclusters,"Astrophys.Lett.Commun.,vol.35,pp.449-455,1998. [5] M.AntoniandS.Ruo,\ClusteringandrelaxationinHamiltonianlong-rangedynamics,"Phys.Rev.E,vol.52,pp.2361-2374,1995. [6] V.Latora,A.Rapisarda,andC.Tsallis,\Non-Gaussianequilibriuminalong-rangeHamiltoniansystem,"Phys.Rev.E,vol.64,pp.056134,2001. [7] M.L.LyraandC.Tsallis,\Nonextensivityandmultifractalityinlow-dimensionaldissipativesystems,"Phys.Rev.Lett.,vol.80,pp.53-56,1998. [8] T.ArimitsuandN.Arimitsu,\Tsallisstatisticsandfullydevelopedturbulence,"J.Phys.A,vol.33,pp.L235-L241,2000. [9] C.Beck,\Applicationofgeneralizedthermostatisticstofullydevelopedturbulence,"PhysicaA,vol.277,pp.115-123,2000. [10] C.Beck,G.S.Lewis,andH.L.Swinney,\MeasuringnonextensitivityparametersinaturbulentCouette-Taylorow,"Phys.Rev.E,vol.63,pp.035303,2001. [11] C.Tsallis,A.R.Plastino,andW.M.Zheng,\Power-lawsensitivitytoinitialconditions-Newentropicrepresentation,"Chaos,Solitons,&Fractals,vol.8,pp.885-891,1997. [12] U.M.S.Costa,M.L.Lyra,A.R.Plastino,andC.Tsallis,\Power-lawsensitivitytoinitialconditionswithinalogisticlikefamilyofmaps:Fractalityandnonextensivity,"Phys.Rev.E,vol.56,245-250,1997. [13] V.Latora,M.Baranger,A.Rapisarda,andC.Tsallis,\Therateofentropyincreaseattheedgeofchaos,"Phys.Lett.A,vol.273,pp.97-103,2000. [14] E.P.Borges,C.Tsallis,G.F.J.Ananos,andP.M.C.deOliveira,Phys.Rev.Lett.,vol.89,pp.254103,2002. 105

PAGE 106

U.Tirnakli,C.Tsallis,andM.L.Lyra,\Circular-likemaps:sensitivitytotheinitialconditions,multifractalityandnonextensivity,"Eur.Phys.J.B,vol.11,pp.309-315,1999. [16] U.Tirnakli,\Asymmetricunimodalmaps:Someresultsfromq-generalizedbitcumulants,"Phys.Rev.E,vol.62,pp.7857-7860,2000. [17] U.Tirnakli,G.F.J.Ananos,andC.Tsallis,\GeneralizationoftheKolmogorov-Sinaientropy:logistic-likeandgeneralizedcosinemapsatthechaosthreshold,"Phys.Lett.A,vol.289,pp.51-58,2001. [18] U.Tirnakli,\Dissipativemapsatthechaosthreshold:numericalresultsforthesingle-sitemap,"PhysicaA,vol.305,pp.119-123,2001. [19] U.Tirnakli,C.Tsallis,andM.L.Lyra,\Asymmetricunimodalmapsattheedgeofchaos,"Phys.Rev.E,vol.65,pp.036207,2002. [20] U.Tirnakli,\Two-dimensionalmapsattheedgeofchaos:NumericalresultsfortheHenonmap,"Phys.Rev.E,vol.66,pp.066212,2002. [21] J.B.GaoandZ.M.Zheng,\Directdynamicaltestfordeterministicchaosandoptimalembeddingofachaotictimeseries,"Phys.Rev.E,vol.49,pp.3807-3814,1994. [22] J.B.GaoandZ.M.Zheng,\Directdynamicaltestfordeterministicchaos," [23] J.B.Gao,C.C.Chen,S.K.Hwang,andJ.M.Liu,\Noise-inducedchaos,"Int.J.Mod.Phys.B,vol.13,pp.3283-3305,1999. [24] J.B.Gao,S.K.Hwang,andJ.M.Liu,\Whencannoiseinducechaos?"Phys.Rev.Lett.,vol.82,pp.1132-1135,1999. [25] J.B.Gao,S.K.Hwang,andJ.M.Liu,\Eectsofintrinsicspontaneous-emissionnoiseonthenonlineardynamicsofanopticallyinjectedsemiconductorlaser,"Phys.Rev.A,vol.59,pp.1582-1585,1999. [26] S.K.Hwang,J.B.Gao,andJ.M.Liu,\Noise-inducedchaosinanopticallyinjectedsemiconductorlaser,"Phys.Rev.E,vol.61,pp.5162-5170,2000. [27] J.B.Gao,W.W.Tung,andN.Rao,\NoiseInducedHopfBifurcation-typeSequenceandTransitiontoChaosintheLorenzEquations,"Phys.Rev.Lett.,vol.89,pp.254101,2002. [28] J.Hu,J.B.Gao,andK.D.White,\Estimatingmeasurementnoiseinatimeseriesbyexploitingnonstationarity,"Chaos,Solitons,&Fractals,vol.22,pp.807-819,2004. 106

PAGE 107

C.J.Cellucci,A.M.Albano,P.E.Rapp,R.A.Pittenger,R.C.Josiassen,\Detectingnoiseinatimeseries,"Chaos,vol.7,pp.414-422,1997. [30] J.B.GaoandW.W.Tung,\Pathologicaltremorsasdiusionalprocesses,"BiologicalCybernetics,vol.86,pp.263-270,2002. [31] R.BandyopadhyayandA.K.Sood,\Chaoticdynamicsinshear-thickeningsurfactantsolutions,"Europhys.Lett.,vol.56,pp.447-453,2001. [32] R.Bandyopadhyay,G.Basappa,andA.K.Sood,\ObservationofchaoticdynamicsindiluteshearedaqueoussolutionsofCTAT,"Phys.Rev.Lett.,vol.84,pp.2022-2025,2000. [33] S.Venkadesan,M.C.Valsakumar,K.P.N.Murthy,andS.Rajasekar,\Evidenceforchaosinanexperimentaltimeseriesfromserratedplasticow,"Phys.Rev.E,vol.54,pp.611-616,1996. [34] N.H.Packard,J.P.Crutcheld,J.D.Farmer,andR.S.Shaw,\Geometryfromatimeseries,"Phys.Rev.Lett.,vol.45,pp.712-716,1980. [35] F.Takens,inDynamicalsystemsandturbulence,LectureNotesinMathematics,vol.898,pp.366,1981,editedbyD.A.RandandL.S.Young,Springer-Verlag,Berlin. [36] T.Sauer,J.A.Yorke,andM.Casdagli,\Embedology,"J.Stat.Phys.,vol.65,pp.579-616,1991. [37] J.B.GaoandZ.M.Zheng,\Localexponentialdivergenceplotandoptimalembeddingofachaotictimeseries,"Phys.Lett.A,vol.181,pp.153-158,1993. [38] S.Sato,M.Sano,andY.Sawada,\PracticalmethodsofmeasuringthegeneralizeddimensionandthelargestLyapunovexponentinhighdimensionalchaoticsystems,"Prog.Theor.Phys.,vol.77,pp.1-5,1987. [39] M.C.MackeyandL.Glass,\Oscillationandchaosinphysiologicalcontrolsystems,"Science,vol.197,pp.287-288,1977. [40] J.B.Gao,W.W.Tung,Y.H.Cao,J.Hu,andY.Qi,\Power-lawsensitivitytoinitialconditionsinatimeserieswithapplicationstoepilepticseizuredetection,"PhysicaA,vol.353,pp.613-624,2005. [41] J.P.EckmannandD.Ruelle,\Ergodictheoryofchaosandstrangeattractors,"Rev.Mod.Phys.,vol.57,pp.617-656,1985. [42] C.Beck,\DynamicalFoundationsofNonextensiveStatisticalMechanics,"Phys.Rev.Lett.,vol.87,pp.180601,2001. [43] B.B.Mandelbrot,TheFractalGeometryofNature,SanFrancisco,Freeman,1982. 107

PAGE 108

D.Saupe,\Algorithmsforrandomfractals",in:H.Peitgen,D.Saupe(Eds.),TheScienceofFractalImages,Springer-Verlag,Berlin,1988,pp.71-113. [45] W.H.Press,\Flickernoisesinastronomyandelsewhere",Commentson Astrophysics,vol.7.pp.103-119,1978. [46] P.Bak,HowNatureWorks:theScienceofSelf-OrganizedCriticality,Copernicus,1996. [47] G.M.Wornell,Signalprocessingwithfractals:awavelet-basedapproach,PrenticeHall,1996. [48] W.E.Leland,M.S.Taqqu,W.Willinger,andD.V.Wilson,\Ontheself-similarnatureofEthernettrac(extendedversion)",IEEE/ACMTrans.onNetworking,vol.2,1-15,1994. [49] J.Beran,R.Sherman,M.S.Taqqu,andW.Willinger,\Long-range-dependenceinvariable-bit-ratevideotrac",IEEETrans.onCommun.,vol.43,pp.1566-1579,1995. [50] V.PaxsonandS.Floyd,\WideAreaTrac|ThefailureofPoissonmodeling,"IEEE/ACMTrans.onNetworking,vol.3,pp.226-244,1995. [51] W.LiandK.Kaneko,\Long-RangeCorrelationandPartial1/fSpectruminaNon-CodingDNASequence",Europhys.Lett.,vol.17,pp.655-660,1992. [52] R.Voss,\Evolutionoflong-rangefractalcorrelationsand1/fnoiseinDNAbasesequences",Phys.Rev.Lett.,vol.68,pp.3805-3808,1992. [53] C.K.Peng,S.V.Buldyrev,A.L.Goldberger,S.Havlin,F.Sciortino,M.SimonsandH.E.Stanley,\Long-RangeCorrelationsinNucleotideSequences,"Nature,vol.356,pp.168-171,1992. [54] D.L.Gilden,T.Thornton,andM.W.Mallon,\1/fnoiseinhumancognition",Science,vol.267,pp.1837-1839,1995. [55] Y.H.Zhou,J.B.Gao,K.D.White,I.Merk,andK.Yao,\PerceptualDominanceTimeDistributionsinMultistableVisualPerception,"BiologicalCybernetics,vol.90,pp.256-263,2004. [56] J.B.Gao,V.A.Billock,I.Merk,W.W.Tung,K.D.White,J.G.Harris,andV.P.Roychowdhury,\Inertiaandmemoryinambiguousvisualperception,"CognitiveProcessing,vol.7,pp.105-112,2006. [57] Y.Chen,M.Ding,andJ.A.S.Kelso,\LongMemoryProcesses(1/falphaType)inHumanCoordination",Phys.Rev.Lett.,vol.79,pp.4501-4504,1997. 108

PAGE 109

J.J.CollinsandC.J.DeLuca,\RandomWalkingduringQuietStanding,"Phys.Rev.Lett.,vol.73,pp.764-767,1994. [59] V.A.Billock,\Neuralacclimationto1/fspatialfrequencyspectrainnaturalimagestransducedbythehumanvisualsystem,"PhysicaD,vol.137,pp.379-391,2000. [60] V.A.Billock,G.C.deGuzman,andJ.A.S.Kelso,\Fractaltimeand1/fspectraindynamicimagesandhumanvision",PhysicaD,vol.148,pp.136-146,2001. [61] M.Wolf,\1/fnoiseinthedistributionofprimes,"PhysicaA,vol.241,pp.493-499,1997. [62] J.B.Gao,YinheCao,andJae-MinLee,\PrincipalComponentAnalysisof1=fNoise,"Phys.Lett.A,vol.314,pp.392-400,2003. [63] B.B.MandelbrotandV.Ness,\FractionalBrownianmotions,fractionalnoisesandapplications,"SIAMRev.,vol.10,pp.422-437,1968. [64] M.S.Taqqu,W.Willinger,andR.Sherman,Proofofafundamentalresultinself-similartracmodeling,ACMSIGCOMMComputerCommunicationReview,vol.27,pp.5-23,1997. [65] B.B.Mandelbrot,FractalsandScalinginFinance,NewYork:Springer,1997. [66] T.Solomon,E.Weeks,andH.Swinney,\ObservationofAnomalousDiusionandLevyFlightsinaTwoDimensionalRotatingFlow,"Phys.Rev.Lett.,vol.71,pp.3975-3979,1993. [67] T.Geisel,J.Nierwetberg,andA.Zacherl,\AcceleratedDiusioninJosephsonJunctionsandRelatedChaoticSystems,"Phys.Rev.Lett.,vol.54,pp.616-620,1985. [68] S.Stapf,R.Kimmich,andR.Seitter,\ProtonandDeuteronField-cyclingNMRrelaxometryofLiquidsinPorousGlasses:EvidenceofLevy-walkStatistics,"Phys.Rev.Lett.,vol.75,pp.2855-2859,1993. [69] G.M.Viswanathan,V.Afanasyev,S.V.Buldyrev,E.J.Murphy,P.A.Prince,andH.E.Stanley,\Levyightsearchpatternsofwanderingalbatrosses,"Nature,vol.381,pp.413-415,1996. [70] G.M.Viswanathan,S.V.Buldyrev,S.Havlin,M.G.E.daLuz,E.P.Raposo,andH.E.Stanley,\Optimizingthesuccessofrandomsearches,"Nature,vol.401,pp.911-914,1999. [71] J.B.Gao,J.Hu,W.W.Tung,Y.H.Cao,N.Sarshar,andV.P.Roychowdhury,\Assessmentoflongrangecorrelationintimeseries:Howtoavoidpitfalls,"Phys.Rev.E,vol.73,pp.016117,2006. 109

PAGE 110

C.K.Peng,S.V.Buldyrev,S.Havlin,M.Simons,H.E.Stanley,andA.L.Goldberger,\Mosaicorganizationofdnanucleotides,"Phys.Rev.E,vol.49,pp.1685-1689,1994. [73] A.JanickiandA.Weron,\Canonesee-stablevariablesandprocesses,"StatisticalScience,vol.9,pp.109-126,1994. [74] M.Kanter,\Stabledensitiesunderchangeofscaleandtotalvariationinequalities,"AnnalsofProbability,vol.3,pp.697-707,1975. [75] J.M.Chambers,C.L.Mallows,B.W.Stuck,\Methodforsimulatingstablerandom-variables,"JournaloftheAmericanStatisticaLAssociation,vol.71,pp.340-344,1976. [76] S.ResnickandG.Samorodnitsky:Fluidqueues,on/oprocesses,andteletracmodelingwithhighlyvariableandcorrelatedinputs,inSelf-SimilarTracandPerformanceEvaluation,eds,K.ParkandW.Willinger,John-Wiley&Sons(2000),pp.171-192. [77] J.B.GaoandI.Rubin,\MultifractalmodelingofcountingprocessesofLong-Range-DependentnetworkTrac,"ComputerCommunications,vol.24,pp.1400-1410,2001;\MultiplicativeMultifractalModelingofLong-Range-Dependent(LRD)TracinComputerCommunicationsNetworks,"J.NonlinearAnalysis,vol.47,pp.5765-5774,2001;\MultiplicativemultifractalmodelingofLong-Range-Dependentnetworktrac,"Int.J.Comm.Systems,vol.14,pp.783-801,2001. [78] R.AlbertandA.L.Barabasi,\Statisticalmechanicsofcomplexnetworks,"Rev.ModernPhys.,vol.74,pp.47-97,2002. [79] C.Labovitz,G.R.Malan,andF.Jahanian,\Internetroutinginstability,"IEEE/ACMTransactionsonNetworking,vol.6,pp.515-528,1998. [80] J.C.R.Bennett,C.Partridge,andN.Shectman,\Packetreorderingisnotpathologicalnetworkbehavior,"IEEE/ACMTransactionsonNetworking,vol.7,pp.789-798,1999. [81] A.ErramilliandL.J.Forys,\TracsynchronizationEectsinTeletracSystems,"Proc.ITC-13,Copenhagen,1991. [82] R.Srikant,TheMathematicsofInternetCongestionControl,Birkhauser,2004. [83] C.Li,G.Chen,X.Liao,andJ.Yu,\Experimentalqueueinganalysiswithlong-rangedependentpackettrac,"Chaos,Solitons,&Fractals,vol.19,pp.853-868,2004. [84] A.VeresandM.Boda,\ThechaoticnatureofTCPcongestioncontrol,"Proc.ofInfocom,Piscataway,NJ,pp.1715-1723,2000. 110

PAGE 111

N.S.V.RaoandL.O.Chua,\Ondynamicsofnetworktransportprotocols,"Proc.WorkshoponSignalProcessing,Communications,ChaosandSystems,2002. [86] P.Ranjan,E.H.Abed,andR.J.La,\NonlinearinstabilitiesinTCP-RED,"IEEE/ACMTransactionsonNetworking,vol.12,pp.1079-1092,2004. [87] N.S.V.Rao,J.GaoandL.O.Chua,\Ondynamicsofnetworktransportprotocols,"inComplexDynamicsinCommunicationNetworks,editedbyL.KocarevandG.Vattay,2004. [88] J.GaoandN.S.V.Rao,\ComplicatedDynamicsofInternetTransportProtocols,"IEEECommun.Lett.,vol.9,pp.4-6,2005. [89] D.R.Figueiredo,B.Liu,V.Misra,andD.Towsley,\OntheAutocorrelationStructureofTCPTrac,"ComputerNetworks,vol.40,pp.339-361,2002;SpecialIssueon\AdvancesinModelingandEngineeringofLong-RangeDependentTrac",ComputerSci.Tech.Report00-55,Univ.ofMassachusetts,Nov.2002. [90] J.B.GaoandN.S.V.Rao,\SynchronizedOscillations,Quasi-Periodicity,andChaosinTCP,"Dept.ofElectricalandComputerEngineering,UniversityofFlorida,Technicalreport. [91] S.FloydandE.Kohler,inFirstWorkshoponHotTopicsinNetworks,28-29October2002,Princeton,NewJersey,USA. [92] K.ParkandW.Willinger,Self-SimilarNetworkTracandPerformance Evaluation,Wiley,NewYork,2000. [93] V.FiroiuandM.Borden,\Astudyofactivequeuemanagementforcongestioncontrol,"Proc.ofInfocom,vol.3,pp.1435-1444,2000. [94] V.Misra,W.Gong,andD.Towsley,\Astudyofactivequeueforcongestioncontrol,"Proc.ofSigcomm,2000. [95] C.Hollot,V.Misra,D.Towsley,andW.Gong,\AcontroltheoreticanalysisofRED,"Proc.ofInfocom,vol.3,pp.1510-1519,2001. [96] P.Kuusela,P.Lassilaand,andJ.Virtamo,\StabilityofTCPREDcongestioncontrol,"inProc.ofITC-17,pp.655-666,Elsevier. [97] N.S.V.Rao,Q.WuandS.S.Iyengar,\Onthroughputstabilizatinofnetworktransport,"IEEECommun.Lett.,vol.8,pp.66-68,2004. [98] T.Sauer,\ReconstructionofDynamical-SystemsfromInterspikeIntervals,"Phys.Rev.Lett.,vol.72,pp.3811-3814,1994. 111

PAGE 112

J.B.Gao,\Recognizingrandomnessinatimeseries,"PhysicaD,vol.106,pp.49-56,1997. [100] S.HaykinandS.Puthusserypady,\Chaoticdynamicsofseaclutter,"Chaos,vol.7,pp.777-802,1997. [101] S.Haykin,Chaoticdynamicsofseaclutter,JohnWiley,1999. [102] J.L.Noga,\BayesianState-SpaceModellingofSpatio-TemporalNon-GaussianRadarReturns,"Ph.Dthesis,CambridgeUniversity,1998. [103] M.Davies,\LookingforNon-LinearitiesinSeaClutter,"IEERadarandSonarSignalProcessing,Peebles,Scotland,UK,July1998. [104] M.R.CowperandB.Mulgrew,\NonlinearProcessingofHighResolutionRadarSeaClutter,"Proc.IJCNN,vol.4,pp.2633-2638,1999. [105] C.P.Unsworth,M.R.Cowper,S.McLaughlin,andB.Mulgrew,\Falsedetectionofchaoticbehaviorinthestochasticcompoundk-distributionmodelofradarseaclutter,"Proc.10thIEEEWorkshoponSSAP,August2000,pp.296-300. [106] C.P.Unsworth,M.R.Cowper,S.McLaughlin,andB.Mulgrew,\Re-examiningthenatureofseaclutter",IEEProc.RadarSonarNavig.,vol.149,pp.105-114,2002. [107] J.B.GaoandK.Yao,\Multifractalfeaturesofseaclutter,"IEEERadarConference,LongBeach,CA,April,2002. [108] J.B.Gao,S.K.Hwang,H.F.Chen,Z.Kang,K.Yao,andJ.M.Liu,\Canseaclutterandindoorradiopropagationbemodeledasstrangeattractors?"The7thExperimentalChaosConference,SanDiego,USA,Augustpp.25-29,2002. [109] P.GrassbergerandI.Procaccia,\Measuringthestrangenessofthestrangeattractor,"PhysicaD,vol.9D,pp.189-208,1983. [110] S.Haykin,R.Bakker,andB.W.Currie,\Uncoveringnonlineardynamics-thecasestudyofseaclutter,"Proc.IEEE,vol.90,pp.860-881,2002. [111] M.McDonaldandA.Damini,\Limitationsofnonlinearchaoticdynamicsinpredictingseaclutterreturns,"IEEProc.RadarSonarNavig.,vol.151,pp.105-113,2004. [112] F.A.Fay,J.Clarke,andR.S.Peters,\Weibulldistributionappliedtosea-clutter,"Proc.IEEConf.Radar,London,U.K.,pp.101-103,1977. [113] F.E.Nathanson,Radardesignprinciples,McGrawHill,pp.254-256,1969. 112

PAGE 113

E.JakemanandP.N.Pusey,\AmodelfornonRayleighseaecho,"IEEETransAntennas&Propagation,vol.24,pp.806-814,1976. [115] S.SayamaandM.Sekine,\Log-normal,log-WeibullandK-distributedseaclutter,"IEICETrans.Commun.,vol.E85-B,pp.1375-1381,2002. [116] F.Gini,A.Farina,andM.Montanari,\Vectorsubspacedetectionincompound-Gaussianclutter,PartII:performanceanalysis,"IEEETransactionsonAerospaceandElectronicSystems,vol.38,pp.1312-1323,2002. [117] G.DavidsonandH.D.Griths,\Waveletdetectionschemeforsmalltargetsinseaclutter,"ElectronicsLett.,vol.38,pp.1128-1130,2002. [118] T.BhattacharyaandS.Haykin,\NeuralNetwork-basedRadarDetectionforanOceanEnvironment,"IEEETrans.AerospaceandElectronicSystems,vol.33,pp.408-420,1997. [119] N.Xie,H.Leung,andH.Chan,\Amultiple-modelpredictionapproachforseacluttermodeling,"IEEETran.GeosciRemote,vol.41,pp.1491-1502,2003. [120] C.P.Lin,M.Sano,S.Sayama,andM.Sekine,\Detectionofradartargetsembeddedinseaiceandseaclutterusingfractals,wavelets,andneuralnetworks,"IEICETrans.Commun.,vol.E83B,pp.1916-1929,2000. [121] C.P.Lin,M.Sano,S.Obi,S.Sayama,andM.Sekine,\DetectionofoilleakageinSARimagesusingwaveletfeatureextractorsandunsupervisedneuralclassiers,"IEICETrans.Commun.,vol.E83B,pp.1955-1962,2000. [122] T.Lo,H.Leung,J.LitvaandS.Haykin,\Fractalcharacterisationofsea-scatteredsignalsanddetectionofsea-surfacetargets,"IEEProc.,vol.F140,pp.243-250,1993. [123] C.-P.Lin,M.Sano,andM.Sekine,\Detectionofradartargetsbymeansoffractalerror,"IEICETrans.Commun.,vol.E80-B,pp.1741-1748,1997. [124] B.E.Cooper,D.L.Chenoweth,andJ.E.Selvage,\Fractalerrorfordetectingman-madefeaturesinaerialimages,"ElectronicsLetters,vol.30,pp.554-555,1994. [125] J.Hu,W.W.Tung,andJ.B.Gao,\Detectionoflowobservabletargetswithinseaclutterbystructurefunctionbasedmultifractalanalysis,"IEEETransactionsonAntennas&Propagation,vol.54,pp.135-143,2006. [126] E.Aurell,G.Boetta,A.Crisanti,G.Paladin,andA.Vulpiani,\Predictabilityinthelarge:AnextensionoftheconceptofLyapunovexponent,"J.PhysicsA,vol.30,pp.1-26,1997. [127] A.Wolf,J.B.Swift,H.L.Swinney,andJ.A.Vastano,\DeterminingLyapunovexponentsfromatimeseries,"PhysicaD,vol.16,pp.285-317,1985. 113

PAGE 114

M.Cencini,M.Falcioni,E.Olbrich,H.Kantz,andA.Vulpiani,\Chaosornoise:Dicultiesofadistinction,"Phys.Rev.E,vol.62,pp.427-437,2000. [129] G.J.Ortega,\Anewmethodtodetecthiddenfrequenciesinchaotictimeseries,"Phys.Lett.A,vol.209,pp.351-355,1995. [130] G.J.Ortega,\InvariantmeasuresasLagrangianvariables:Theirapplicationtotimeseriesanalysis,"Phys.Rev.Lett.,vol.77,pp.259-262,1996. [131] J.L.Chern,J.Y.Ko,J.S.Lih,H.T.Su,andR.R.Hsu,\Recognizinghiddenfrequenciesinachaotictimeseries,"Phys.Lett.A,vol.238,pp.134-140,1998. 114

PAGE 115

JingHureceivedtheB.S.andM.E.degreesfromtheDepartmentofElectronicsandInformationEngineeringfromHuazhongUniversityofScienceandTechnology(HUST),Wuhan,China,in2000and2002,respectively.InMay2007,ShewasawardedthePh.D.degreefromtheDepartmentofElectricalandComputerEngineeringattheUniversityofFlorida,Gainesville,Florida.Herresearchinterestsincludesignalandimageprocessing,informationscience,nonlineartimeseriesanalysis,biologicaldataanalysis,bioinformatics,andbiomedicalengineering. 115


Permanent Link: http://ufdc.ufl.edu/UFE0017765/00001

Material Information

Title: New Approaches to Multiscale Signal Processing
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0017765:00001

Permanent Link: http://ufdc.ufl.edu/UFE0017765/00001

Material Information

Title: New Approaches to Multiscale Signal Processing
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0017765:00001


This item has the following downloads:


Full Text





NEW APPROACHES TO MULTISCALE SIGNAL PROCESSING


By
JING HU



















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

2007
































S2007 Jing Hu




































To my family, teachers and friends










ACKENOWLED GMENTS

First of all, I would like to thank my advisor, Professor .Jianho Gao, for his invaluable

guidance and support. I was lucky to work with someone with so many original ideas

and such a sharp mind. Without his help, my Ph.D. degree and this dissertation would

have been impossible. During my years at the University of Florida, I was moved by his

magnificent personality. A famous Chinese adage ; .va, "One d .< as my advisor, entire life

as my father." Professor Gao highly qualifies.

I also want to thank my coninittee nienters (Professors Su-Shing C'I. in~ Yuguang

Fang, .Jose C. Principe, and K~eith D. White), for their interest in my work and valuable

-a-----Hul-My thanks of course also go to all the faculty, staff and my fellow students in

the Department of Electrical and Computer Engineering. I extend particular thanks to my

office mates .Jaentin Lee, Yi Z1n in_ and IUngsik K~in, for their helpful discussions.

I express my appreciation to all of my friends who offered encouragement. I specially

thank Qian Zhan, Xingyan Fan, Lingyu Gu and his wife (.Jing Zhang), Yannmin Xu, Yan

Liu, and Yana Liang. They gave me a lot of help during my early years in the University

of Florida. They taught me to cook and made my life more colorful.

Finally, I owe a great debt of thanks to my parents, grandparents, and my boyfriend,

.Jing Ai, for their pillar-like support and their unwavering relief and confidence in me

without this I do not think I would have made it this far!










TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS ......... . . 4

LIST OF FIGURES ......... .. . 7

ABSTRACT ......... ..... . 9

CHAPTER

1 INTRODUCTION ......... . 10

1.1 Overview of Dynamical Systems . ..... .. 11
1.2 Examples of Multiscale Phenomena . .... .. 15
1.3 Brief Introduction to C'!s I... ......... .. 20
1.4 Brief Introduction to Fractal Geometry ... .. . . 21
1.5 Importance of Connecting ('!, I. .s and Random Fr-actal Theories .. .. 2:3
1.6 Importance of the Concept of Scale . ..... .. 24
1.7 Structure of this Dissertation ........ .. .. 24

2 GENERALIZATION OF CHAOS: POWER-LAW SENSITIVITY TO INITIAL
CONDITIONS (PSIC) ......... ... 27

2.1 Dynamical Test for C'!s I... .... .. . .. .. 27
2.2 General Computational Framework for Power-law Sensitivity to Initial
Conditions (PSIC) .. . .. .. .. .. 3:3
2.3 C'll I l .terizing Edge of C'I!s I. by Power-law Sensitivity of Initial Conditions
(PSIC) .... ........ ............ :36

:3 CHARACTERIZING RANDOM FRACTALS BY POWER-LAW SENSITIVITY
TO INITIAL CONDITIONS (PSIC) . ..... .. 41

:3.1 C'll I l .terizing 1/ fP Processes by Power-law Sensitivity to Initial Conditions
(PSIC) .... ........ .. .. ...... ..... 41
:3.1.1 C'll I l .terizing Fractional Brownian Motion (fBm) Processes by
Power-law Sensitivity to Initial Conditions (PSIC) .. .. .. .. 4:3
:3.1.2 C'll I l .terizing ON/OFF Processes with Pareto distributed ON and
OFF Periods by Power-law Sensitivity to Initial Conditions (PSIC) 50
:3.2 C'll I l .terizing Levy Processes by Power-law Sensitivity to Initial Conditions
(PSIC) .... ........ ............ 54

4 STITDY OF INTERNET DYNAMICS BY POWER-LAW SENSITIVITY TO
INITIAL CONDITIONS (PSIC) ........ ... .. 60

4.1 Study of Transport Dynamics by Network Simulation .. .. .. .. 61
4.2 Complicated Dynamics of Internet Transport Protocols .. .. .. .. 69










5 STUDY OF SEA CLUTTER RADAR RETURNS BY POWER-LAH
SENSITIVITY TO INITIAL CONDITIONS (PSIC) .. .. .. .. 76

5.1 Sea Clutter Data ........ ... .. 78
5.2 Non C'!s I..tic Behavior of Sea Clutter .... .. .. 79
5.3 Target Detection within Sea Clutter by Separating Scales .. .. .. .. 81
5.4 Target Detection within Sea Clutter by Power-law Sensitivity to Initial
Conditions (PSIC) ........ . .. 86

6 MITLTISCALE ANALYSIS BY SCALE-DEPENDENT LYAPITNOV EXPONENT
(SDLE) ........... ......... .. 89


6.1 Basic Theory
6.2 Classification of Complex Motions.
6.2.1 C'! I ..-~ Noisy (I I ..-~ and Noise-induced ChI ... .-
6.2.2 Processes Defined by Power-law Sensitivity to Initial
(PSIC).
6.2.3 Complex Motions with Multiple Scaling Behaviors.
6.3 C'!I. .) ..terizing Hidden Frequencies


Conditions


7 CONCLUSIONS ......... ... .. 104

REFERENCES .. .......... ........... 105

BIOGRAPHICAL SK(ETCH ...... .. 115










LIST OF FIGURES


Figure page

1-1 Example of a trajectory in the phase space ..... .. .. 14

1-2 Giant ocean wave (tsunami). ......... .. 17

1-3 Example of sea clutter data. ......... .. 18

1-4 Example of Heart rate variability data for a normal subject. .. .. .. 19

2-1 Time-dependent exponent A(k) vs. evolution time k curves for Lorenz system. .32

2-2 Time-dependent exponent A(k) vs. evolution time k curves for the Mackey-Glass
system. ......... ..... 33

2-3 Time-dependent exponent A(t) vs. t curves for time series generated from logistic
map. ......... ............ .... 37

2-4 Time-dependent exponent A(t) vs. t curves for time series generated from Henon
map. ............ ............... 40

3-1 White noise and Brownian motion . ...... .. 46

3-2 Several fBm processes with different H. . ... .. 48

3-3 Time-dependent exponent A(k) vs. In k curves for fractional Brownian motion. 51

3-4 Example of ON/OFF processes. .... ... .. 51

3-5 Time-dependent exponent A(k) vs. In k curves for ON/OFF processes with Pareto
distributed ON and OFF periods. ... ... .. 53

3-6 Examples for Brownian motions and Levy motions. .. .. .. 57

3-7 Time-dependent exponent A(k) vs. In k curves for Levy processes. .. .. .. 58

4-1 Example of quasi-periodic congestion window size W(i) time series. .. .. .. 63

4-2 Examples of chaotic T(i) time series. . ..... .. .. 65

4-3 Complementary cumulative distribution functions (CCDFs) for the two chaotic
T(i) time series of Figs. 4-2(a,c). . ... ... .. 66

4-4 The time series T(i) extracted from a W(i) time series collected using net100
instruments over ORNL-LSU connection. ..... .. .. 68

4-5 Time series for the congestion window size cwnd for ORNL-GaTech connection. 72

4-6 Time-dependent exponent A(k) vs. k curves for cwnd data corresponding to
Figf.4-5. ......... ..... 73










4-7 Time-dependent exponent A(k) vs. In k curves for cwnd data corresponding to
Figf.4-5. ......... ..... 75

5-1 Collection of sea clutter data ......... ... 79

5-2 Typical sea clutter amplitude data. ........ .. .. 79

5-3 Two short segments of the amplitude sea clutter data severely affected by clipping. 80

5-4 Examples of the time-dependent exponent A(k) vs. k curves for the sea clutter
data ........ ... . .... 81

5-5 Variations of the Lyapunov exponent A estimated by conventional methods vs.
the 14 range bins for the 10 HH measurements. .... .. .. 84

5-6 Variations of the scale-dependent exponent corresponding to large scale vs. the
range bins for the 10 HH measurements. ..... .. .. 85

5-7 Examples of the A(k) vs. In k curves for the sea clutter data. .. .. .. .. 87

5-8 Variation of the P parameter with the 14 range bins. .. .. ... .. 87

5-9 Frequencies of the bins without targets and the bins with primary targets for
the HH datasets. .. ... .. .. 88

6-1 Scale-dependent Lyapunov exponent A(e) curves for the Lorenz system and logistic
map ........ ... . .... 93

6-2 Scale-dependent Lyapunov exponent A(e) curve for the Mackey-Glass system. .. 94

6-3 The functions F(x) and G(x). ......... ... .. 97

6-4 Scale-dependent Lyapunov exponent A(e) for the model described by Eq. 6-9. .97

6-5 Power-spectral density (PSD) for time series generated from Lorenz system. 100

6-6 Lorenz attractor. .. ... .. .. 101

6-7 Hidden frequency phenomenon of laser intensity data. ... .. .. .. 102









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

NEW APPROACHES TO MITLTISCALE SIGNAL PROCESSING

By

Jing Hu

May 2007

C'I I!r: Jianho Gao
Major: Electrical and Computer Engineering

Complex systems usually comprise multiple subsystems with highly nonlinear

deterministic and stochastic characteristics, and are regulated hierarchically. These

systems generate signals with complex characteristics such as sensitive dependence on

small disturbances, long nienory, extreme variations, and nonstationarity. Chaos theory

and random fractal theory are two of the most important theories developed for analyzing

complex signals. However, they have entirely different foundations, one being deterministic

and the other being random. To synergistically use these two theories, we developed a new

theoretical framework, power-law sensitivity to initial conditions (PSIC), to encompass

both chaos and random fractal theories as special cases. To show the power of this

framework, we applied it to study two challenging and important problems: Internet

dynamics and sea clutter radar returns. We showed that PSIC can readily characterize

the complicated Internet dynamics due to interplay of nonlinear AINID (additive increase

and niultiplicative decrease) operation of TCP and stochastic user behavior, and robustly

detect low observable targets from sea clutter radar returns with high accuracy.

We also developed a new niultiscale complexity measure, scale-dependent Lyapunov

exponent (SDLE), that can he computed from short noisy data. The measure readily

classified various types of complex motions, and simultaneously characterized behaviors of

complex signals on a wide range of scales, including complex irregular behaviors on small

scales, and orderly behaviors, such as oscillatory motions, on large scales.









CHAPTER 1
INTRODUCTION

A dynamical system is one that evolves in time. Dynamical systems can be stochastic

(in which case they evolve according to some random process such as the toss of a coin)

or deterministic (in which case the future is uniquely determined by the past according

to some rule or mathematical formula). Whether the system in question settles down to

equilibrium, keeps repeating in cycles, or does something more complicated, dynamics are

what we use to analyze the behavior of systems.

Complex dynamical systems usually comprise multiple subsystems with highly

nonlinear deterministic and stochastic characteristics, and are regulated hierarchically.

These systems generate signals with complex characteristics such as sensitive dependence

on small disturbances, long memory, extreme variations, and nonstationarity [1]. A

stock market, for example, is strongly influenced by multi- l-1~ we decisions made by

market makers, as well as influenced by interactions of heterogeneous traders, including

intradwl~i traders, short-period traders, and long-period traders, and thus gives rise to

highly irregular stock prices. The Internet, as another example, has been designed in a

fundamentally decentralized fashion and consists of a complex web of servers and routers

that cannot be effectively controlled or analyzed by traditional tools of queuing theory

or control theory, and give rise to highly bursty and multiscale traffic with extremely

high variance, as well as complex dynamics with both deterministic and stochastic

components. Similarly, biological systems, being heterogeneous, massively distributed, and

highly complicated, often generate nonstationary and multiscale signals. With the rapid

accumulation of complex data in life sciences, systems biology, nano-sciences, information

systems, and physical sciences, it has become increasingly important to be able to analyze

multiscale and nonstationary data.

Multiscale signals behave differently depending upon which scale the data are looked

at. In order to fully characterize such complex signals, the concept of scale has to be










explicitly considered, and existing theories need to be used synergistically, instead of

individually, since on different scale ranges different theories may be most pertinent.

This dissertation aints to build an effective arsenal for niultiscale signal processing by

synergistically integrating approaches based on chaos theory and random fractal theory,

and going beyond, to complement conventional approaches such as spectral analysis and

machine learning techniques. To make such an integration possible, two important efforts

have been made:

* Power-law sensitivity to initial conditions (PSIC): We developed a new
theoretical framework for signal processing, to create a coninon foundation for chaos
theory and random fractal theory, so that they can he better integrated.

* Scale-dependent Lyapunov exponent (SDLE): We developed an excellent
niultiscale measure, which is a variant of the finite size Lyapunov exponent (FSLE).
We proposed a highly efficient algorithm for calculating it, and showed that it can readily
classify different types of motions, including truly low-dintensional chaos, noisy chaos,
noise-induced chaos, random 1/ fP and cc-stable Levy processes, and complex motions with
chaotic behavior on small scales but diffusive behavior on large scales. The measure can
aptly characterize complex behaviors of real world niultiscale signals on a wide range of
scales.

In the rest of this chapter, we shall first present an overview of dynamics systems,

especially we shall point out why nonlinear systems are much harder to analyze than

linear ones. We then give a few examples of nmultiscale phenomena to show the difficulties

and excitement of niultiscale signal processing. After that, we shall briefly introduce some

background knowledge about chaos and fractal geometry. Then we discuss the importance

of connecting chaos theory and random fractal theory for characterizing the behaviors

of niultiscale signals on a wide range of scales. We also emphasize the importance of

explicitly incorporating the concept of scale in devising measures for characterizing

niultiscale signals. Finally, we outline the scope of the present study.

1.1 Overview of Dynamical Systems

There are two main types of dynamical systems: differential equations and iterated

maps (difference equations). Differential equations describe the evolution of systems









in continuous time, whereas iterated maps arise in problems where time is discrete. To

show how systems evolve, we take differential equations as examples. When confining our

attention to differential equations, the main distinction is between ordinary and partial

differential equations. For instance, the equation for a damped harmonic oscillator


m +1 b + kx = 0 (1-1)

is an ordinary differential equation, because it involves only ordinary derivatives dx/dt and

d2 ld2. That is, there is only one independent variable, the time t. In contrast, the heat

equation
Bu 82U
dt 8x2

is a partial differential equation: it has both time t and sapce x as independent variables.

Our concern in this proposal is with purely temporal behavior, and so we deal with

ordinary differential equations almost exclusively.

A very general framework for ordinary differential equations is provided by the system




(1-2)



Here the overdots denote differentiation with respect to t. Thus xi = dxi/dt. The

variables xl, x, might represent concentrations of chemicals in a reactor, populations

of different species in an ecosystem, or the positions and velocities of the planets in the

solar system. The functions fl, f, are determined by the problem at hand.

For example, the damped oscillator (Eq. 1-1) can be rewritten in the form of

(Eq. 1-2), thanks to the following trick: we introduce new variables xl = x and x2 x










Then xl = Z2. Hence the equivalent system (Eq. 1-2) is


xi = x2
b k
xa2=--x 2- 1--i
m m

This system is said to be linear, because all the xi on the right-hand side appear to

the first power only. Otherwise the system would be nonlinear. Typical nonlinear terms

are products, powers, and functions of the Xi, such aS xlx2, 1 3), Or COSZ2.

For example, the swinging of a pendulum is governed by the equation


x + -sinx = 0,


where x is the angle of the pendulum from vertical, g is the acceleration due to gravity,

and L is the length of the pendulum. The equivalent system is nonlinear:


x z = x2

Z2 = L~smnxl.

Nonlinearity makes the pendulum equation very difficult to solve analytically. The

usual way around this is to fudge, by invoking the small angle approximation sinx e x for

x << 1. This converts the problem to a linear one, which can then be solved easily. But

by restricting to small x, we are throwing out some of the physics, like motions where the

pendulum whirls over the top. Is it really necessary to make such drastic approximation?

It turns out that the pendulum equation can be solved analytically, in terms of elliptic

functions. But there ought to be an easier way. After all, the motion of the pendulum is

simple: at low energy, it swings back and forth, and at high energy it whirls over the top.

There should be some way of extracting this information from the system directly -using

geometric methods.

Here is the rough idea. Suppose we happen to know a solution to the pendulum

system, for a particular initial condition. This solution would be a pair of functions xl(t)

and x2(t), representing the position and velocity of the pendulum. If we construct an















I (t), X 2(t))


(x (0), X2(0))


Figure 1-1. Example of a trajectory in the phase space


abstract space with coordinates (xl,,Z2), then the solution (xl(t),2 xa1)) COTTOSponds to

a point moving along a curve in this space, as shown Fig. 1-1. This curve is called a

trajectory, and the space is called the phase space for the system. The phase space is

completely filled with trajectories, since each point can serve as an initial condition.

Our goal is to run this construction in reverse: given the system, we want to draw

the trajectories, and thereby extract information about the solutions. In many cases,

geometric reasoning will allow us to draw the trajectories without actually solving the

system!

The phase space for the general system (Eq. 1-2) is the space with coordinates

xl, x,. Because this space is n-dimensional, we will refer to Eq. 1-2 as an n-dimensional

system or an nth-order system. Thus a represents the dimension of the phase space.

As we have mentioned earlier, most nonlinear systems are impossible to solve

analytically. Why are nonlinear systems so much harder to analyze than linear ones?

The essential difference is that linear systems can be broken down into parts. Then each

part can be solved separately and finally recombined to get the answer. This idea allows a










fantastic simplification of complex problems, and underlies such methods as normal modes,

Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear

system is precisely equal to the sum of its parts.

But many things in nature do not act this way. Whenever parts of a system interfere,

or cooperate, or compete, there are nonlinear interactions going on. Most of every w- life

is nonlinear, and the principle of superposition fails spectacularly. If you listen to your two

favorite songs at the same time, you will not get double the pleasure! Within the realm of

physics, nonlinearity is vital to the operation of a laser, the formation of turbulence in a

fluid, and the superconductivity of Josephson junctions.

1.2 Examples of Multiscale Phenomena

Multiscale phenomena are ubiquitous in nature and engfineeringf. Some of them are,

unfortunately, catastrophic. One example is the Tsunami that occurred in south Asia

around C!~!s-I it -~ 2004. Another example is the gigantic power outage occurred in North

America on August 14, 2003, which affected more than 4000 megawatts, was more than

300 times greater than mathematical models would have predicted, and cost between $4

billion and 71 billion, according to the U.S. Department of energy. Both events involved

scales that, on one hand, were so huge that our human being could not easily fathom,

and on the other hand, involved the very scales that were most relevant to individual life.

Below, we consider three more examples.

Multiscale phenomena in computer networks: Large-scale communications

networks, especially the Internet, are among the most complicated systems that man

has ever made, with multiscales in terms of a number of aspects. Intuitively speaking,

these multiscales come from the hierarchical design of protocol stack, the hierarchical

topological architecture, and the multipurpose and heterogeneous nature of the Internet.

More precisely, there are multiscales in

*Time, manifested by the prevailing fract al, multifract al, and long-range- dependent
properties in traffic,










* Space, essentially due to topology and geography and again manifested by scale-free
properties,

* State (e.g., queues, windows),

* Size (e.g., number of nodes, number of users).

Also, it has been observed that the failure of a single router may trigger routing

instability, which may be severe enough to instigate a route flap storm. Furthermore,

packets may be delivered out of order or even get dropped, and packet reordering is not

a pathological network behavior. As the next generation Internet applications such as

remote instrument control and computational steering are being developed, another facet

of complex niultiscale behavior is beginning to surface in terms of transport dynanxics.

The networking requirements for these next generation applications belong to (at least)

two broad classes involving vastly disparate time scales:

* High handwidths, typically multiples of 10Ghps, to support bulk data transfers,

* Stable handwidths, typically at much lower handwidths such as 10 to 100 1\bps, to
support interactive, steering and control operations.

Multiscale phenomena in sea clutter: Sea clutter is the radar backscatter

front a patch of ocean surface. The complexity of the signals contes front two sources,

the rough sea surface, sometimes oscillatory, sometimes turbulent, and the nmultipath

propagation of the radar backscatter. This can he appreciated by imagining how radar

pulses massively reflecting front the wavetip of Fig. 1-2. To be quantitative, in Fig. 1-3,

two 0.1 s duration sea clutter signals, sampled with a frequency of 1 K(Hz, are plotted

in Fig. 1-:3(a,h), a 2 s duration signal is plotted in Fig. 1-:3(c), and an even longer signal

(about 1:30 s) is plotted in Fig. 1-:3(d). It is clear that the signal is not purely random,

since the waveform can he fairly smooth on short time scales (Fig. 1-:3(a)). However, the

signal is highly nonstationary, since the frequency of the signal (Fig. 1-:3(a,b)) as well as

the randoniness of the signal (Fig. 1-:3(c,d)) change over time drastically. One thus can

perceive that naive Fourier analysis or deterministic chaotic analysis of sea clutter may



































Figure 1-2. Giant ocean wave (tsunami). Suppose our field of observation includes the
wavetip of length scale of a few meters, it is then clear that the complexity of
sea clutter is mainly due to massive reflection of radar pulses from a wavy and
even turbulent ocean surface.


not e very,,,,,, useul From Fig 1-3(e,\ where Xt(m) is the non-overlapping running mean of

X over block size m, and X is the sea clutter amplitude data, it can be further concluded

that neither autoregressive (AR) models (As an example, we give the definition for the

first-order AR model, which is given by x, I = ax, + rl,, where the constant coefficient a

satisfies 0 / |a| < 1 and rlo is a white noise with zero mean.) nor textbook fractal models

can describe the data. This is because AR modeling requires exponentially decaying
autcorrelation (which,1 amounts, to Var(Xe(m)) ~ m-l, or a Hurst parameter of 1/2. See

later chapters for the definition of the Hurst parameter), while fractal modeling requires

the variation between Vanr(Xe"m)) and m to follow a power-law~. However, neither behavior

is observed in Fig. 1-3(e). Indeed, albeit extensive work has been done on sea clutter,

the nature of sea clutter is still poorly understood. As a result, the important problem of

















80 80.02 80.04 80.06
Time (sec)


80.08 8C


Time (sec)


8
log2 m


U 2U 4U bU0 tU 1UU 12U
Time (sec)


Figure 1-3.


Example of sea clutter data. (a,b) two 0.1 s duration signal; (c) a 2 s duration
sea clutter signal; (d) the entire sea clutter signal (of about 130 s); and (e)
log2 m2Var(X(m))] vs. log2 m, Where
Xim) {Xe(m) : X(m) (Xtm-m+l + + Xtm)/m, t = 1, 2, } is the
non-overlapping running mean of X = {Xt : t = 1, 2, } over block size m,
and X is the sea clutter amplitude data. To better see the variation of
Vawr(X~) with mVa(e) is multiplied by m2. When the autocorrelation
of the data decays exponentially fast (such as modeled by an autoregressive
(AR) process), V'ar(Xe(m")) ~ m-1 Here Var(Xe(m)) decaty-s much faster. A
fractal process would have m2"Var(Xe"m)) ~ m ". However, th~is is not thle case.
Therefore, neither AR modeling nor ideal textbook fractal theory can be
readily applied here.


target detection within sea clutter remains a tremendous challenge. We shall return to sea

clutter later.

Multiscale and nonstationary phenomena in heart rate variability (HRV):

HRV is an important dynamical variable of the cardiovascular function. Its most salient

feature is the spontaneous fluctuation, even when the environmental parameters are


(b)


(d)
a,6
4
E


















0 1 2 3 4 5 6 7
Index n4
x 104
100 110
(b) (c)
100
95

90
I80

2.66 2.67 2.68 2.69 2.7 2.71 3.4 .9 396 .7 398 .9
Index n x 104 Index n x 104
(d) (e)
104 105


S102 0

10-2 10- 10-2 10-
f f

Figure 1-4. Example of Heart rate variability data for a normal subject. (a) the entire
signal, (hsc) the segments of signals indicated as A and B in (a); (d,e) power
spectral density for the signals shown in (b,c).


maintained constant and no perturbing influences can he identified. It has been observed

that HRV is related to various cardiovascular disorders. Therefore, analysis of HRV is

very important in medicine. However, this task is very difficult, since HRV data are highly

complicated. An example is shown in Fig. 1-4, for a normal young subject. Evidently,

the signal is highly nonstationary and niultiscaled, appearing oscillatory for some period

of time (Figs. 1-4(h,d)), and then varying as a power-law for another period of time

(Figs. 1-4(c,e)). The latter is an example of the so-called 1/ f processes, which will be

discussed in depth in later chapters. While the niultiscale nature of such signals cannot

he fully characterized by existing methods, the nonstationarity of the data is even more

troublesome, since it requires the data to be properly segmented before further analysis










hy methods from spectral analysis, chaos theory, or random fractal theory. However,

automated segmentation of complex biological signals to remove undesired components is

itself a significant open problem, since it is closely related to, for example, the challenging

task of accurately detecting transitions from normal to abnormal states in physiological

data.

1.3 Brief Introduction to Chaos

Imagine we are observing an aperiodic, highly irregular time series. Can such a

signal arise from a deterministic system that can he characterized by only very few state

variables instead of a random system with infinite numbers of degrees of freedom? A

chaotic system is capable of just that! This discovery has such far reaching implications

in science and engineering that sometimes chaos theory is considered as one of the three

most revolutionary scientific theories of the twentieth-century, along with relativity and

quantum mechanics.

At the center of chaos theory is the concept of sensitive dependence on initial

conditions: a very minor disturbance in initial conditions leads to entirely different

outcomes. An often used metaphor illustrating this point is that a sunny weather in New

York could be replaced by a rainy one some time in the future after a butterfly flaps

its wings in Boston. Such a feature contrasts sharply with the traditional view, largely

based on our experience with linear systems, that small disturbances (or causes) can only

generate proportional effects, and that in order for the degree of randomness to increase,

the number of degrees of freedom has to be infinite.

No definition of the term chaos is universally accepted yet, but almost everyone would

agree on the three ingredients used in the following working definition:

C'!s I. .s is periodic long-term behavior in a deterministic system that exhibits sensitive

dependence on initial conditions.

*"Aperiodic long-term b. !! lit.-! I means that there are trajectories which do not settle
down to fixed points, periodic orbits, or quasi-periodic orbits as t oc. For practical
reasons, we should require that such trajectories are not too rare. For instance, we could










insist that there he an open set of initial conditions leading to periodic trajectories, or
perhaps that such trajectories should occur with nonzero probability, given a random
initial condition.

* "Deterministic" means that the system has no random or noisy inputs or parameters.
The irregular behavior arises from the system's nonlinearity, rather than from noisy
driving forces.

* "Sensitive dependence on initial conditions" means that nearby trajectories separate
exponentially fast (i.e., the system has a positive Lyapunov exponent).

C'!s I. .s is also commonly called a strange attractor. The term attractor is also difficult

to define in a rigorous way. We want a definition that is broad enough to include all

the natural candidates, but restrictive enough to exclude the imposters. There is still

disagreement about what the exact definition should he.

Loosely -lo' I1:;0s an attractor is a set to which all neighboring trajectories converge.

Stable fixed points and stable limit cycles are examples. 1\ore precisely, we define an

attractor to be a closed set ,4 with the following properties:

1. A is an invariant set: any trajectory x(t) that starts in ,4 stays in ,4 for all time.

2. A attracts an open set of initial conditions: there is an open set U containing 24 such
that if x(0) E U, then the distance from x(t) to A tends to zero as t 00. This
means that ,4 attracts all trajectories that start sufficiently close to it. The largest
such U is called the basin of attraction of ,4.

3. A is minimal: there is no proper subset of ,4 that satisfies conditions 1 and 2.

Now we can define a strange attractor to be an attractor that exhibits sensitive dependence

on initial conditions. Strange attractors were originally called strange because they are

often fractal sets. ?- ue- ul .vis this geometric property is regarded as less important than

the dynamical property of sensitivity dependence on initial conditions. The terms chaotic

attractor and fractal attractor are used when one wishes to emphasize one or the other of

those aspects.

1.4 Brief Introduction to Fractal Geometry

Euclidean geometry is about lines, planes, triangles, squares, cones, spheres, etc. The

common feature of these different objects is regularity: none of them is irregular. Now let










us ask a question: Are clouds spheres, mountains cones, and islands circles? The answer is

obviously no. Pursuing answers to such questions, Alandelbrot has created a new branch of

science -fractal geometry.

For now, we shall be satisfied with an intuitive definition of a fractal: a set that shows

irregular but self-similar features on many or all scales. Self-similarity means a part of

an object is similar to other parts or to the whole. That is, if we view an irregular object

with a microscope, whether we enlarge the object by 10 times, or by 100 times, or even

by 1000 times, we ah-- i-n find similar objects. To understand this better, let us imagine

we were observing a patch of white cloud drifting .l.- li- in the sky. Our eyes were rather

passive: we were staring more or less at the same direction. After a while, the part of the

cloud we saw drifted away, and we were viewing a different part of the cloud. Nevertheless,

our feeling remains more or less the same.

Mathematically, fractal is characterized by a power-law relation, which translates to

a linear relation in log-log scale. For now, let us again resort to imagination -we were

walking down a wild and very j I__- d mountain trail or coastline. We would like to know

the distance covered by our route. Suppose our ruler has a length of e which could be

our step-size, and different hikers may have different step-sizes -a person riding a horse

has a huge step-size, while a group of people with a little child must have a tiny step-size.

The length of our route is

L = N(e) (1-3)

where NV(e) is the number of intervals needed to cover our route. It is most remarkable

that typically NV(e) scales with e in a power-law manner,


N~e) E-D e 0(1-4)


with D being a non-integer, 1 < D < 2. Such a non-integral D is often called the fractal

dimension -to emphasize the fragmented and irregular characteristics of the object under

study.










Let us now try to understand the meaning of the non-integral D. For this purpose, let

us consider how length, area, and volume are measured. A common method of measuring

a length, a surface area, or a volume is by covering it with intervals, squares, or cubes

whose length, area, or volume is taken as the unit of measurement. These unit intervals,

squares, and volumes are called unit boxes. Suppose, for instance, that we have a line

whose length is 1. We want to cover it by intervals (hoxes) whose length is e. It is clear

that we need NV(e) ~ e-1 hoxes to completely cover the line. Similarly, if we want to cover

an area or volume by boxes with linear length e, we would need NV(e) ~ E-2 to cover the

area, or NV(e) ~ e-3 hoxes for the volume. Such D is called the topological dimension, and

takes on a value of one for a line, two for an area, and three for a volume. For isolated

points, D is zero. That is why a point, a line, an area, and a volume are called an 0 D,

1 D, 2 D, and 3 D objects, respectively.

Now let us examine the consequence of 1 < D < 2 for a j I__- d mountain trail. It is

clear that the length of our route increases as e becomes smaller, i.e., when e 0 L o.

T> he more concrete, let us visualize a race between the Hare and the Tortoise on a fractal

trail with D = 1.25. Assume that the length of the average step taken by the Hare is 16

times that taken by the Tortoise. Then we have


LHare 2 ,Lartoise


That is, the Tortoise has to run twice the distance of the Hare! Put differently, if you

were walking along a wild mountain trail or coastline and tired, slowing down your pace,

shrinking your steps, then you were in trouble! It certainly would be worse if you also got

lost!

1.5 Importance of Connecting Chaos and Random Fractal Theories

Multiscale signals behave differently depending upon which scale the data are

looked at. How can the behaviors of such complex signals on a wide range of scales

he simultaneously characterized? One strategy we envision is to use existing theories










synergistically, instead of individually. To make this possible, appropriate scale ranges

where each theory is most pertinent need to be identified. This is a difficult task, however,

since different theories may have entirely different foundations. For example, chaos theory

is mainly concerned about apparently irregular behaviors in a complex system that are

generated by nonlinear deterministic interactions of only a few numbers of degrees of

freedom, where noise or intrinsic randomness does not pIIl w any important role. Random

fractal theory, on the other hand, assumes that the dynamics of the system are inherently

random. Therefore, to fully characterize the behaviors of multiscale signals on a wide

range of scales, different theories, such as chaos and random fractal theories, need to be

integrated and even generalized.

1.6 Importance of the Concept of Scale

Complex systems often generate highly nonstationary and multiscale signals because

of nonlinear and stochastic interactions among their component systems and hierarchical

regulations imposed by the operating environments. Rapid accumulation of such complex

data in life sciences, systems biology, nano-sciences, information systems, and physical

sciences, has made it increasingly important to develop complexity measures that

incorporate the concept of scale explicitly, so that different behaviors of signals on varying

scales can be simultaneously characterized by the same scale-dependent measure. The

most ideal scenario is that a scale-dependent measure can readily classify different types

of motions based on analysis of short noisy data. In this case, one can readily see that the

measure will not only be able to identify appropriate scale ranges where different theories,

including information theory, chaos theory, and random fractal theory, apply, but also be

able to automatically characterize the behaviors of the data on those scale ranges. The

importance of the concept of scale will be further illustrated in later chapters.

1.7 Structure of this Dissertation

We emphasized that to develop new approaches for multiscale signal processing, it is

most desirable to integrate and even generalize different theories. Recently, the defining










property for deterministic chaos, exponential sensitivity to initial conditions (ESIC) has

been generalized to power-law sensitivity to initial conditions (PSIC). In C'!s Ilter 2,

we first introduce the new concept of PSIC. Then we extend the concept of PSIC to

high-dimensional case. We shall present a general computational framework for assessing

PSIC from real world data. We also apply the proposed computational procedure to

study noise-free and noisy logistic and Henon maps at the edge of chaos. we show that

when noise is absent, PSIC is hard to detect from a scalar time series. However, when

there is dynamic noise, motions around the edge of chaos, he it simply regular or truly

chaotic when there is no noise, all collapse onto the PSIC attractor. Hence, dynamic

noise makes PSIC observable. In (I Ilpter 3, we demonstrate that the new framework of

PSIC can readily characterize random fractal processes, including 1/ fP processes with

long-range-correlations and Levy processes. From our study in C'!s Ilters 2 and 3, it is

clear that the concept of PSIC not just bridges standard chaos theory and random fractal

theory, but in fact provides a more general framework to encompass both theories as

special cases. To illustrate the power of the framework of PSIC, in OsI Ilpters 4 and 5, we

apply it to study two important but challenging problems: Internet dynamics and sea

clutter radar returns, respectively. Both are outstanding examples of complex dynamical

systems with both nonlinearity and randomness. The nonlinearity of Internet dynamics

comes from AllMD (additive increase and multiplicative decrease) operation of TCP

(Transmission Control protocol), while the stochastic component comes from the random

user behavior. Sea clutter is a nonlinear dynamical process with stochastic factors due

to interference of various wind and swell waves and to local atmospheric turbulence.

We shall show that the new theoretical framework of PSIC can effectively characterize

the complicated Internet dynamics as well as to robustly detect low observable targets

from sea clutter radar returns with high accuracy. In Chapter 6, we shall introduce

the basic theory of an excellent multiscale measure -the scale-dependent Lyapunov

exponent (SDLE), and develop an effective algorithm to compute the measure. To










understand the SDLE as well as appreciate its power, we apply it to classify various types

of complex motions, including truly low-dintensional chaos, noisy chaos, noise-induced

chaos, processes defined by PSIC, and complex motions with chaotic behavior on small

scales but diffusive behavior on large scales. We also discuss how the SDLE can help

detect hidden frequencies in large scale orderly motions. Finally we suninarize our work in









CHAPTER 2
GENERALIZATION OF CHAOS: POWER-LAW SENSITIVITY TO INITIAL
CONDITIONS (PSIC)

The fornialism of nonextensive statistical niechanics (NESM) [2] has found numerous

applications to the study of systems with long-range-interactions [3-6] and niultifractal

behavior [7, 8], and fully developed turbulence [8-10], among many others. In order to

characterize a type of motion whose complexity is neither regular nor fully chaotic/randons

recently the concept of exponential sensitivity to initial conditions (ESIC) of deterministic

chaos has been generalized to power-law sensitivity to initial conditions (PSIC) [7, 11].

Mathematically, the formulation of PSIC closely parallels that of NESM. PSIC has been

applied to the study of deterministic 1-D logisticlike maps and 2-D Henon niap at the

edge of chaos [7, 11-20], yielding considerable new insights. In this chapter, we first

briefly describe some background knowledge about chaos. In particular, we will discuss a

direct dynamical test for deterministic chaos developed by Gao and Zheng [21, 22]. This

method offers a more stringent criterion for detecting low-dintensional chaos, and can

simultaneously monitor motions in phase space at different scales. Then we introduce the

more generalized concept of PSIC, and extend the new concept to high-dintensional case.

Specifically, we present a general computational framework for assessing PSIC in a time

series. We also apply the proposed computational procedure to study two model systems:

noise-free and noisy logistic and Henon maps at the edge of chaos.

2.1 Dynamical Test for Chaos

C'!s I. .s is also coninonly called a strange attractor. Being an I11 .. 1 ..1-~", the

trajectories in the phase space are bounded. Being -lI I.ng.-", the nearby trajectories,

on the average, diverge exponentially fast. The latter property is characterized hv the

exponential sensitivity to initial conditions (ESIC), and can he niathentatically expressed

as follows. Let d(0) be the small separation between two arbitrary trajectories at time

0, and d(t) be the separation between them at time t. Then, for true low-dintensional










deterministic chaos, we have


d(t) ~ d(0)ex' (2-1)

where X1 is positive and called the largest Lyapunov exponent. Due to the boundedness

of the attractor and the exponential divergence between nearby trajectories, a strange

attractor typically is a fractal, characterized by a simple and elegant scaling law as defined

by Eq. 1-4. A non-integral fractal dimension of an attractor contrasts sharply with the

integer-valued topological dimension (which is 0 for finite number of isolated points, 1 for

a smooth curve, 2 for a smooth surface, and so on).

Conventionally, it has been assumed that a time series with an estimated positive

largest Lyapunov exponent and a non-integral fractal dimension is chaotic. However, it has

been found that this assumption may not he a sufficient indication of deterministic chaos.

One counter-example is the so-called 1/ fP processes. Such processes have spectral density


S(f ) f -(2H+1) (2-2)


where 0 < H < 1 is sometimes called the Hurst parameter. This type of processes

will be further discussed in much depth in ChI Ilpter 3. A trajectory formed by such a

process has a fractal dimension of 1/H. As we shall explain later, with most algorithms

of estimating the largest Lyapunov exponent, one obtains a positive number for the

"Lyapunov expei ill ~! Therefore, such processes could be interpreted as deterministic

chaos .

Due to the ubiquity of noise, experimental data are ahr-l- :- corrupted by noise to some

degree. mathematically speaking, a noisy system, no matter how weak the noise is, has

infinite dimensions. Experimentally speaking, one would be more interested in a certain

range of finite scales. If the noise is very weak, then its influence on the dynamics may be

limited to very small scales, leaving the dynamics on finite scales deterministic-like.

The above discussions motivate us to employ a more sophisticated method, the

direct dynamical test for deterministic chaos [21, 22], to determine whether the data









under investigation is truly chaotic or not. The method offers a more stringent criterion

for low-dimensional chaos, and can simultaneously monitor motions in phase space at

different scales. The method has found numerous applications in the study of the effects

of noise on dynamical systems [23-27], estimation of the strength of measurement noise

in experimental data [28, 29], pathological tremors [30], shear-thickening surfactant

solutions [31], dilute sheared aqueous solutions [32], and serrated plastic flows [33].

Now let us briefly describe the direct dynamical test for deterministic chaos [21, 22].

Given a scalar time series, x(1), x(2),..., x(NV), (.I--II11.11.- for convenience, that they have

been normalized to the unit interval [0,1]), one first constructs vectors of the following

form using the time delay embedding technique [34-36]:


1K = [x(i), x(i + L), ..., x(i + (m 1)L)] (2-3)

with m being the embedding dimension and L the delay time. For example, when m = 3

and L = 4, we have VI = [x(1), x(5), x(9)], VW = [x(2), x(6), x(10)], and so on. For the

analysis of purely chaotic signals, m and L has to be chosen properly. This is the issue of

optimal embedding (see [21, 22, 37] and references therein). After the scalar time series is

properly embedded, one then computes the time dependent exponent A(k) curves


A(k) = n +k-V (2-4)


with r < ||%~ Vy || < r + ar, where r and Ar are prescribed small distances. The angle

brackets denote ensemble averages of all possible pairs of (1M, Vj). The integer k, called

the evolution time, corresponds to time kbt, where 6t is the sampling time. Note that

geometrically (r, r + Ar) defines a shell, and a shell captures the notion of scale. The

computation is typically carried out for a sequence of shells. Comparing Eq. 2-4 with

Eq. 2-1, one notices that ||%~+k Vy, || phI i-4 the role of d(t), while ||%~ Vy1 || phi-4 the role

of d(0). Figure 2-1(a) shows the A(k) vs. k curves for the chaotic Lorenz system driven by









stochastic forcing:


dx
=-16(x y) + Drll(t)

= -xz + 45.92x y + Drl2(t) (2-5)

= xy 4z + Dr/3(

with < rli(t) > 0,- < i6i,(')> b(t-t'), i, j = 1, 2, 3. Note that D2 is the variance of

the Gaussian noise terms, and D = 0 describes the clean Lorenz system. We observe from

Fig. 2-1(a) that the A(k) curves are composed of three parts. These curves are linearly

increasing for 0 < k < k,. They are still linearly increasing for k, < k < ky, but with a

slightly different slope. They are flat for k > k. Note that the slope of the second linearly

increasing part gives an estimate of the largest positive Lyapunov exponent [21, 22]. k,

is related to the time scale for a pair of nearby points (1M, Vyj) to evolve to the unstable

manifold of 1M or Vyj. It is on the order of the embedding window length, (m 1)L. kg

is the prediction time scale. It is longer for the A(k) curves that correspond to smaller

shells. The difference between the slopes of the first and second linearly increasing parts

is caused by the discrepancy between the direction defined by the pair of points (1K, Vyj)

and the unstable manifold of 1M or Vyj. This feature was first observed by Sato et al. [38]

and was used by them to improve the estimation of the Lyapunov exponent. The first

linearly increasing part can be made smaller or can even be eliminated by adjusting

the embedding parameters such as by using a larger value for m. Note that the second

linearly increasing parts of the A(k) curves collapse together to form an envelope. It is

this very feature that forms the direct dynamical test for deterministic chaos [21, 22].

This is because the A(k) curves for noisy data, such as white noise or 1/ f processes,

are only composed of two parts, an increasing (but not linear) part for k < (m 1)L

and a flat part [21, 22]. Furthermore, different A(k) curves for noisy data separate from

each other, hence an envelope is not defined. We also note, most other algorithms for

estimating the largest Lyapunov exponent is equivalent to compute A(k) for r < ro, where










ro is selected more or less arbitrarily, then obtain A(k)/kbt, for not too large k, as an

estimation of the largest Lyapunov exponent. With such algorithms, one can obtain a

'pm enLyapunov exponent for white noise and for 1/ f processes. However, the value of

the Lyapunov exponent estimated this way sensitively depends on the parameter ro chosen

in the computation, hence, typically is different for different researchers. We thus observe

a random element here! With these discussions, it should be clear why 1/ f processes may

be interpreted as chaotic.

One can expect that the behavior of the A(k) curves for a noisy chaotic system lies

in between that of the A(k) curves for a clean chaotic system and that of the A(k) curves

for white noise or for 1/ f processes. This is indeed so. An example is shown in Fig. 2-1(b)

for the noisy Lorenz system with D = 4. We observe from Fig. 2-1(b) that the A(k)

curves corresponding to different shells now separate. Therefore, an envelope is no longer

defined. This separation is larger between the A(k) curves corresponding to smaller shells,

indicating that the effect of noise on the sniall-scale dynamics is stronger than that on the

large-scale dynanxics. Also note that k, + k, is now on the order of the embedding window

length, and is almost the same for all the A(k) curves. With stronger noise (D > 4), the

A(k) curves will be more like those for white noise [21, 22].

Finally, we consider the Mackey-Glass delay differential system [39]. When a=

0.2, b = 0.1, c = 10, F = 30, it has two positive Lyapunov exponents, with the largest

Lyapunov exponent close to 0.007 [21].

axr(t +F
dxr/dt = bxr(t) (2-6)
1 + r(t +F)"

Having two positive Lyapunov exponents while the value of the largest Lyapunov exponent

of the system is not much greater than 0, one might he concerned that it may be difficult

to deal with this system. This is not the case. In fact, this system can he analyzed as

straightforwardly as other dynamical systems including the Logistic nmap, Henon niap and

the Rossler system. An example of the A(k) vs. k curves for the chaotic Alackey-Glass

















"tI
3I





O 50


200


200


Time-dependent exponent A(k) vs. evolution time k curves for (a) clean and
(b) noisy Lorenz system. Six curves, from bottom to top, correspond to shells
(2-(i+1)/2, 2-i/2) With i 8, 9, 10, 11, 12, and 13. The sampling time for the
system is 0.03 sec, and embedding parameters are m = 4, L = 3. 5000 points
are used in the computation.


Figure 2-1.


100 150


0 50 100 150
Evolution timne k












3.5 -



2.5



1.5



0.5


0 100 200 300 400 500 600
Evolution time k

Figure 2-2. Time-dependent exponent A(k) vs. evolution time k curves for the
Mackey-Glass system. Nine curves, from bottom to top, correspond to shells
(2-(i+1)/2, 2-i/2) With i 5, 6, and 13. The computation was done with
m = 5, L = 1, and 5000 points sampled with a time interval of 6.


delay differential system is shown in Fig. 2-2, where we have followed [21] and used

m = 5, L = 1, and 5000 points sampled with a time interval of 6. Clearly, we observe that

there exists a well defined common envelope to the A(k) curves. Actually, the slope of the

envelope estimates the largest Lyapunov exponent. This example illustrates that when

one works in the framework of the A(k) curves developed by Gao and Zheng [21, 22], one

does not need to be very concerned about non-uniform growth rate in high-dimensional

systems .

2.2 General Computational Framework for Power-law Sensitivity to Initial
Conditions (PSIC)

To understand the essence of power-law sensitivity to initial conditions (PSIC),

let us first consider the 1-D case and consider Eq. 2-7, where Ax(0) is the infinitesimal

discrepancy in the initial condition, Ax(t) is the discrepancy at time t > 0.


((t) =lim (2-7)









When the motion is chaotic, then


((t) = exlt (2-8)

((t) satisfies



Tsallis and co-workers [7, 11] have generalized Eq. 2-9 to


di = A,((t)V (2-10)

where q is called the entropic index, and X, is interpreted to be equal to K,, the

generalization of the K~olmogorov-Sinai entropy. Eq. 2-10 defines the PSIC in the 1-D
case. Obviously, PSIC reduces to ESIC when q 1 The solution to Eq. 2-10 is


s'(t) =[1 +(1 q)Agt] 'il) (2-11)

When t is large and q / 1, ((t) increases with t as a power-law,


((t) Ct 7 -')(2-12)

where C = [(1 q) A,] 1 (-V). For Eq. 2-12 to define an unstable motion with A, > 0, we

must have q < 1. Later we shall map different types of motions to different ranges of q.

To apply PSIC to the analysis of time series data, one can first construct a phase

space by constructing embedding vectors 1M as defined by Eq. 2-3. Eq. 2-11 can then be

generalized to high-dimensional case [40],

((t) = Ilimo~l =lvo~l lv t 1 + (1 q)X()t (2 13

where | |a V(0) | is the infinitesimal discrepancy between two orbits at time 0, | |a V(t) | is

the distance between the two orbits at time t > 0, q is Ithe entropic index, and A i) is the

first q-Lyapunov exponent, corresponding to the power-law increase of the first principle

axis of an infinlitesimral ball in thle phase space. AI1 mayi not be equal to K,. TIhisi is

understood by recalling that for chaotic systems, the K~olmogorov-Sinai entropy is the sum









of all the positive Lyapunov exponents. We conjecture a similar relation may hold between

the q-Lyapunov exponents and K. When there are multiple unstable directions, then in

general A I) maty- not be equal to Ki,. Whe~n t is large andc q / 1, Eq. 2-13 again gives a

power-law increase of ((t) with t. Note that under the above framework, the motion may

not be like fully developed chaos, thus not ergodic [41].

We now consider the general computational framework for PSIC. Given a finite

time series, the condition of ||av(0)|| 0 cannot be satisfied. In order to find the law

governing the divergence of nearby orbits, one thus has to examine how the neighboring

points, (1K, Vj), in the phase space, evolve with time, by forming suitable ensemble

averages. Notice that if (1MI, by) and (1%2, V/j2) arT tWO pairs of nearby points, when

| |%1 Vjll | |
|I |%2+t Vj2+tl CRllllOt be simply averaged to provide estimates for q anld Af ). Inl fact, it

would be most convenient to consider ensemble averages of pairs of points (1M, V4j) that

all fall within a very thin shell, rl < ||%~ g||1 < r2, Where rl and T2 arT ClOSe. These

arguments so-----~ -1 that the time-dependent exponent curves defined by Eq. 2-4 provide a

natural framework to assess PSIC from a time series. This is indeed so. In fact, we have


In I(t) a A(t) = In (||+ +t) (2-14)


Now, by the discussions in Sec. 2.1, it is clear that PSIC is a generalization of ESIC:

as long as the A(k) curves from different shells form a linear envelope, then q = 1 and the

motion is chaotic. The next question is: Does PSIC also include random fractals as special

cases? The answer is yes. It will be given in the next chapter. There, we will also gain

a better undlerstanding of the meaning of A l. The re~st of this chapter shows examples

of characterizing the edge of chaos by the new framework of PSIC. Specifically, we study

time series generated from noise-free and noisy logistic and Henon maps.









2.3 Characterizing Edge of Chaos by Power-law Sensitivity of Initial
Conditions (PSIC)

Let us first examine the logistic map around the edge of chaos. Considering that

dynamic noise is often an important component of a system (e.g., spontaneous emission

noise in semiconductor lasers [26], and physiological noise in biological systems [30]), we

study both the deterministic and the noisy logistic map:


X,+1 = ax,(1 x,) + arl, (2-15)


where a is the bifurcation parameter and rlo is a white Gaussian noise with mean zero

and variance 1. The parameter a characterizes the strength of noise. For the clean system

(a = 0), the edge of chaos occurs at the accumulation point, a, = 3.569945672 We

shall study three parameter values, al = a, 0.001, a,, and a2 = a, + 0.001. When

noise is absent, al corresponds to a periodic motion with period 25, while a2 COTTOSponds

to a truly chaotic motion. We shall only study transient-free time series. In Figs. 2-3(a-c),

we have plotted the A(t) vs. t curves for parameter values al, a,, and a2, TOSpectively.

We observe from Fig. 2-3(a) that the variation of A(t) with t is periodic (with period 16,

which is half of the period of the motion) when the motion is periodic. This is a generic

feature of the A(t) curves for discrete periodic attractors, when the radius of the shell is

larger than the smallest distance between two points on the attractor (when a periodic

attractor is continuous, A(t) can be arbitrarily close to 0). It has been found by Tsallis

and co-workers [11] that at the edge of chaos for the logistic map, ((t) is given by Eq. 2-11

with q m 0.2445. Surprisingly, we do not observe such a divergence in Fig. 2-3(b). In

fact, if one plots A(t) vs. In t, one only observes a curve that increases very slowly (similar

to that shown in Fig. 2-4(a)). The more interesting pattern is the one that is shown in

Fig. 2-3(c), where we observe a linearly increasingly A(t) vs. t curve. In fact, shown in

Fig. 2-3(c) are two such curves, corresponding to two different shells. Very interestingly,

the two curves collapse to form a common envelope in the linearly increasing part of the











6
(d)






1
05

6
(e)




O


4



1-
O
0 1 23 456
In


Time-dependent exponent A(t) vs. t curves for time series generated from the
noise-free logistic map with (a) al a, 0.001, where the motion is periodic
with period 2s, (b) a, 3.569945672 and (c) a2 a, + 0.001, where the
motion is chaotic. Plotted in (d-f) are A(t) vs. In t curves for the noisy logistic
map with o- = 0.001. Very similar results were obtained when o- = 0.0001.
Shown in (c-f) are actually two curves, corresponding to two different shells.
104 pOintS Were used in the computation, with embedding parameters
m = 4, L = 1. However, so long as m > 1, the results are largely independent
of embedding. When m = 1, the A(t) curves are not smooth, and the
estimated 1/(1 q) value is much smaller than the theoretical value.


Figure 2-3.


curve. The slope of the envelope gives a good estimate of the largest positive Lyapunov

exponent. This is a generic feature of chaos [21, 22], as explained in Sec. 2.1. Since the

chaos studied here is close to the edge of chaos, the curves shown in Fig. 2-3(c) are less

smooth than those reported earlier [21-23, 26].

Why cannot the theoretical prediction of PSIC at the edge of chaos for the logistic

map be observed from a clean time series? In a recently published very interesting and


2.0
1.5 -





-0.5

0 50 100 150 200 250
2.0 '

1.5






0.0 .
0 50 100 150 200 250











0 50 100 150 200 250










insightful paper, Beck [42] so__~- -;- that dynamic noise may be of fundamental importance

to the Tsallis NESM. May dynamic noise pl ai similarly significant role for PSIC? As is

shown below, the answer is yes. In Figs. 2-3(d-f), we have shown the A(t) vs. In t curves

for the three parameters considered, with noise strength al = 0.001. In fact, shown

in each figure are two curves, corresponding to two different shells. They parallel with

each other. The slopes of those curves are about 1.20, close to the theoretical value of

1/(1 0.2445) a 1.32. While it is very satisfactory to observe PSIC at the edge of chaos,

it is more thrilling to observe the collapse of regular as well as chaotic motions onto the

PSIC attractor around the edge of chaos. This signifies the stableness of PSIC when there

is dynamic noise. It is important to emphasize that the results shown in Figs. 2-3(d-f)

are largely independent of the noise strength, so long as noise is neither too weak nor too

strong. For example, very similar results have been observed with a2 = 1/10 = 0.0001.

Before we move on to discuss the Henon map near the edge of chaos, let us comment

on the difference between the ESIC and the PSIC. When the ESIC is the case, the A(t)

vs. t curves are straight for T, < t < T,, where T, is a prediction time scale, and T, is a

time scale for the initial separation to evolve to the most unstable direction. When A(t)

vs. t curves corresponding to different shells are plotted together, they collapse together

for a considerable range of t. When the PSIC is the case, the A(t) vs. In t curves are

fairly straight, and the A(t) vs. In t curves corresponding to different shells separate and

parallel with each other. This feature is the fundamental reason that ensemble averages

are most conveniently formed by requiring neighboring pairs of points to all fall within

a thin shell. For example, if two distinct thin shells, described by rl < ||Xi Xj| < T2 r

and T2 ||II Vy|| < T 3, arT joined together to form a single thicker shell, described by

rl < ||V~ Vy||1 < T3, and one averages the separation between Xi and Xj before taking

the logarithm, then the resulting slope between A(t) and In t, assumed to be still linear,

will have to be smaller than that estimated from the two thin shells separately. At this

point it is also worth noting that the linearly increasing part of the A(t) vs. In t curves










may only contain a small interval of t. Actually, for the simple logistic nmap, A(t) saturates

around i s 20, when the radius of the shell is about 10-4 and the length of the time series

is 104. For experimnertal tilie series, the t for A(t) to saturate may be even smaller (we

suspect it would be about 10 samples -irrespective of the sampling time, because of

the logarithmic timre scale). Tob get more accurate estimnates of the parametersi q and A ',

it may be better to fit ((t) using Eq. 2-13 when dealing with experimental time series.

In fact, we have found that if we do so, the estimated 1/(1 q) value from Figs. 1(d-f)

increases to about 1.27, much closer to the theoretical value of 1.32.

Next let us consider the Henon nmap, where ty., and ty, are white Gaussian noise, and

are uncorrelated with each other. The parameter a measures the strength of the noise.


r,z+l = 1 axr + Utz + wif, (77)

Un+l = br,z + aty,(iv)

Tirnakli [20] studied the deterministic version of this nmap at the edge of chaos, for

parameter values a,. = 1.40115518 b = O; a,. = 1.39966671 b = 0.001; and

a,. = 1.:386:37288 b = 0.01. We have studied both the noise-free and noisy nmap

for parameter values listed above, and found very similar results to those presented in

Fig. 2-:3. In Figs. 2-4(a,h), we have plotted A(t) vs. In t curves for the noise-free and noisy

nmap for a,. = 1.:386:37288 b = 0.01. As can he observed clearly, Fig. 2-4(b) is very

similar to Figs. 2-:3(d-f), while the slope for the curves in Fig. 2-4(a) is much smaller than

that in Fig. 2-4(b). Again, we have observed (but not shown by figures here, since they are

very similar to Figs. 2-:3(d-f) and Fig. 2-4(b)) that dynamic noise makes the regular and

chaotic motions to collapse onto the PSIC attractor for parameters around those definingf

the edge of chaos, and that such transitions are largely independent of the noise strength.

To suninarize, we have described an easily intplenientable procedure for computationally

examining PSIC front a time series. By studying two model systems: noise-free and noisy

logistic and Henon maps near the edge of chaos, we have found that when there is no










2.0 (a)''""'""''"" 6 (b
1.5



S0.5 -

0.0 -1 2

-0.5- -1 1
1.0O ~......... ......... ......... ......... .......... ............ O .
0123456 0123456
Int Int

Figure 2-4. Time-dependent exponent A(t) vs. t curves for time series generated from (a)
the noise-free Henon map with ac = 1.38637288 b = 0.01. Plotted in (b)
are two A(t) vs. In t curves (corresponding to two different shells) for the noisy
map with a 0.001. Similar results were obtained with a 0.0001. 104 pOintS
were used in the computation, with embedding parameters m = 4, L = 1.


noise, the PSIC attractor cannot be observed from a scalar time series. However, when

dynamic noise is present, motions around the edge of chaos, be it simply regular or

truly chaotic when there is no noise, all collapse onto the PSIC attractor. While these

examinations signify the ubiquity of PSIC, they also highlight the importance of dynamic

noise. The existence of the latter is perhaps the very reason that truly chaotic time series

can seldom be observed.









CHAPTER 3
CHARACTERIZING RANDOM FRACTALS BY POWER-LAW SENSITIVITY TO
INITIAL CONDITIONS (PSIC)

In C'!s Ilter 2, we have discussed that PSIC is a generalization of ESIC: as long as

the A(t) curves from different shells form a linear envelope, then q = 1 and the motion

is chaotic. We have also shown that edge of chaos can be well characterized by PSIC.

In this chapter, we shall study two 1!! I r~~ types of random fractals: 1 fP processes with

long-range-correlations and Levy processes. We shall show both analytically and through

numerical simulations that both types of processes can be readily characterized by PSIC.

3.1 Characterizing 1/ f Processes by Power-law Sensitivity to Initial
Conditions (PSIC)

A continuous-time stochastic process, X = {X(t), t > 0}, is said to be self-similar if


X(At) d AHX(t), t > 0 (3-1)

for A > 0, O < H < 1, where u~~denotes equality in distribution. H is called the

self-similarity parameter, or the Hurst parameter.

Before proceeding on, we note that, more rigorously speaking, processes defined by

Eq. 3-1 should be called self-affine processes [43] instead of self-similar processes, since X

and t have to be scaled differently to make the function look similar. A simple physical

explanation for this is that the units for X and t are different. In this dissertation,

however, we will continue to call such processes self-similar, to follow the convention in

some of the engineering disciplines.

The following three properties can be easily derived from Eq. 3-1:

E[X(At)]
E [X(t)] Mean (3-2)

Var [X(At)]
Var[X(t)]= 2HVariance (3-3)

R, (t, s) = 2HAutocorrelation (3-4)









Note that Var[X(t)] = R,(t, t); hence, Eq. 3-3 can be simply derived from Eq. 3-4; we

have listed it as a separate equation for convenience of future reference. If we consider

only second-order statistics, or if a process is Gaussian, Eq. 3-2 to Eq. 3-4 can be used

instead of Eq. 3-1 to define a self-similar process.

A very useful way of describing a self-similar process is by its spectral representation.

Strictly -p.' I1:;14 the Fourier transform of X(t) is undefined, due to the nonstationary

nature of X(t). One can, however, consider X(t) in a finite interval, ;?i, O < t < T:


X (t, T) = t)0< t< T (3-5)
otherwise

and take the Fourier transform of X(t, T):


F(f, T) = lX(t)e-2xjftd (36)

|F(f, T)|12df is the contribution to the total energy of X(t, T) from those components with

frequencies between f and f + df. The (average) power spectral density (PSD) of X(t, T)
is then

S(f T) = FfT

and the spectral density of X is obtained in the limit as T oo


S( f) = lim S( f ,T)
T->oo

Noting that the PSD for A-HX(At) is

A-2H limX(t-2ft _-2-S/)
T->oo AT$ I 'o X.X)ZiLt 2 X"-sfX

and that the PSD for A-HX(At) and X(t) are the same [44], we have


S(f) = S(f/A)A-2H-1









The solution to the above equation is


S( f) ~ f -P (3-7)

with

/7 = 2H + 1 (3-8)

Processes with power spectral densities as described by Eq. 3-7 are called 1/ f

processes. Typically, the power-law relationships that define these processes extend over

several decades of frequency. Such processes have been found in numerous areas of science

and engineering. Some of the older literatures on this subject can be found, for example,

in Press [45], Bak [46], and Wornell [47]. Some of the more recently discovered 1/ f

processes are in traffic engineering [48-50], DNA sequence [51-53], human cognition [54],

ambiguous visual perception [55, 56], coordination [57], posture [58], dynamic images [59,

60], and the distribution of prime numbers [61], among many others. It is further observed

that principle component analysis of such processes leads to power-law decaying eigenvalue

spectrum [62].

Two important prototypical models for 1/ f processes are the fractional Brownian

motion (fBm) processes and the ON/OFF intermittency with power-law distributed ON

and OFF periods. Below, we apply the concept of PSIC to both types of processes.

3.1.1 Characterizing Fractional Brownian Motion (fBm) Processes by Power-
law Sensitivity to Initial Conditions (PSIC)

We first briefly introduce Brownian motion (Bm). Brownian motion is defined as a

(nonstationary) stochastic process B(t) that satisfies the following criteria:

1. All Brownian paths start at the origin: B(0) = 0.

2. For 0 < 1 independent.

3. For all (s, t) > 0, the variable B(t + s) B(t) is a Gaussian variable with mean 0 and
variance s.









4. B(t) is a continuous function of t.
We may infer from the above definition that the probability distribution function of B

is given by:

Pt { B(t + ) B(t) ]
This function also satisfies the scaling property:


P{[B(t + s) B(t)] < x} = P{B(At + As) B(At) < X1/2X (310)

In other words, B(t) and A-1/2B(At) have the same distribution. Thus, we see from

Eq. 3-1 that Bm is a self-similar process with Hurst parameter 1/2.

Suppose we have measured B(tl), B(t2) 2~ 1 > 0. What can we ;?i about

B(s), 1 < s < 2? The answer is given by the Levy interpolation formula:

(S tl) (t2 s) 8 1l]~ 1/21
B(s) = B(tl) +[B(t2) 1 ~t
(t2 1l 2a 1

where W is a zero-mean and unit-variance Gaussian random variable. The first two terms

are simply a linear interpolation. The third term gives the correct variance for B(s)-B(t )

and B(t2) B 8), Which are s 1 and t2 8, TOSpectively.

A popular way of generating Bm is by the following random midpoint displacement

method, which is essentially an application of the Levy interpolation formula: suppose we

are given B(0) = 0 and B(1) as a sample of a Gaussian random variable with zero mean
and variance O.2. We can obtain B(1/2) from the following formula:


B(1/2) B(0) = n[B(1) B(0)] + D1

where D1 is a random variable with mean 0 and variance 2-2 2. D1 is simply the third

term of Eq. 3-11, and the coefficient 2-2 equals (t2 s) 8 1 2~t 1l), With tl = 0, t2 =

1, a = 1/2. Similarly, we have


B(1/4) B(0) = ,[B(1/2) B(0)] + D2









where D2 1S a random variable with zero mean and variance 2-3 2. We can apply the

same idea to obtain B(3/4). In general, the variance for D, is 2-("+l)a2.

We note that a Brownian path is not a differentiable function of time. Heuristically,

this can be understood in this way: consider the variable, B(t + s) B(t), with variance

s. Its standard deviation, which is a measure of its order of magnitude, is ~ 2. Thus, the

derivative of B at t behaves like the limit Z/S = S-1/2 aS s 0.

Although B(t) is almost surely not differentiable in t, symbolically one still often

writes

B(t) = w~(T)dv (3-12)

where w(t) is stationary white Gaussian noise, and extends the above equation to t < 0

through the convention,

JO~ Jt
It should be understood that integrals with respect to the differential element w(t)dt

should be interpreted more precisely as integrals with respect to the differential element

dB(t), in the Riemann-Stieltjes integral sense. This has profound consequence when

numerically integrating Eq. 3-12.

To see how, let us partition [0, t] into a equally spaced intervals, at = t/n. Since w(t)

are Gaussian random variables with zero mean and unit variance, in order for B(t) to have

variance t, we should have


i= 1
That is, the coefficient is (At)1/2 inStead of at, as one might have guessed. Typically,

at at. Thus, if one incorrectly uses at instead of (At)1/2, One is

severely underestimating the strength of the noise.









(a) White noise







(b) Brownian motion









Figure 3-1. White noise (a) and Brownian motion (b). Axes are arbitrary.

When the time is genuinely discrete, or the units of time can be arbitrary, one can
take at to be 1 unit, and Eq. 3-13 becomes


B(t) = w~7(i) (3 -14)
i= 1

Eq. 3-14 provides perhaps the simplest method of generating a sample of Bm. An example
is shown in Fig. 3-1.

Eq. 3-14 is also known as a random walk process. A more sophisticated random
walk (or jump) process is given by summing up an infinite number of jump functions:

Je(t) = AiP(t ti), where P(t) is a unit-step function

1t>0
P(t) = O < (3-15)


and Ai, ti are random variables with Gaussian and Poisson distributions, respectively.









Next we discuss fractional Brownian motion (fBm) processes. A normalized fBm

process, Z(t), is defined as follows:

Z(t) WiH (t > 0) (3-16)

where W is a Gaussian random variable of zero mean and unit variance, and H is the

Hurst parameter. When H = 1/2, the process reduces to the standard Brownian motion

(Bm). It is trivial to verify that this process satisfies the defining Eq. 3-1 for a self-similar
stochastic process.

Fr-actional Brownian motion BH 1) is a Gaussian process with mean 0, stationary
increments, variance

E [(BH )2] t2H n7

and covariance:

E[B 8)H 2H 2PH _2HJ I

where H is the Hurst parameter. Due to its Gaussian nature, according to Eq. 3-2

to Eq. 3-4, the above three properties completely determine its self-similar character.

Fig. 3-2 shows several fBm processes with different H.

Roughly, the distribution for an fBm process is [63]:


BH~l TTii~ H-/2 (319)

where F(t) is the gamma function:


0/OO"t-e~d 7

It is easy to verify that fBm satisfies the following scaling property:


P(BHt + ) BH~t I) = P 8H(X + ) BXH XHX} (3-20)

In other words, this process is invariant for transformations conserving the similarity

variable X/tH




















(c) H=0.75





(d) H=0.90





Figure 3-2. Several fBm processes with different H.

The integral defined by Eq. 3-19 diverges. The more precise definition is


BH t) BH(0) =r( K1 -7dB0 (3-21)

where the kernel K(t -r) is given by [63]


K(t r) =l l (3-22)


The increment process of fBm, Xi = BH(( + )t BH i 1), i > i, Where at can

be considered a sampling time, is called fractional Gaussian noise (fGn) process. It is a

zero mean stationary Gaussian time series. Noting that


E(XXik) = E {[BH(Z + )t BH 8t~[ H ((i + 1 + k) at) BH ((i + k) at)]}









by Eq. 3-18, one obtains the autocovariance function y(k) for the fGn process:

q~~~~k)2` =. E(4ik/(X)= ( )2H- 2k2H + |k- 1|2H, k >0 (3 23)


Notice that the expression is independent of at. Therefore, without loss of generality, one

could take at = 1. In particular, we have


y(1) = 22-2

Let us first note a few interesting properties of y(k):

(i) When H = 1/2, y(k) = 0 for k / 0. This is the well-known property of white
Gaussian noise.

(ii) When 0 < H < 1/2, y(1) < 0.

(iii) When 1/2 < H < 1, y(1) > 0.

Properties (ii) and (iii) are often termed anti-persistent and persistent correlations [43],

respectively.

Next, we consider the behavior of y(k) when k is large. Noting that when |x|

(1 + x)" a 1 + a~x + x

we have, when k > 1,


(k + 1)2H + |k~ 1|2H = k~2H[(1 + 1/k)2H + (1 1/k~)2H] = k~2H[2 + 2H(2H 1)k-2]

hence,

y(k) ~ H(2H 1)k2H-2, Sk 00(3-24)

when H / 1/2. When H = 1/2, y(k) = 0 for k > 1, the Xi's are simply white noise.

Now we are ready to characterize 1/ f processes in the framework of PSIC. Recalling

that the defining property for a 1 fP process is that its variance increases with t as t2H









Irrespective of which embedding dimension is chosen, this property can be translated to


I(t) = tH (3-25)

Therefore ,
dt)= HtH-1 (3-26)

Expressing t in terms of (, we have


= H( -$ H3-27)


Comparing with the defining equation of PSIC (Eq. 2-10), we find that

1 2
q =1- 1- (3-28)
H P-1

X( ) = H= (3-29)

Noticing 0 < H < 1, from Eq. 3-28, we have -oo < q < 0.

Computationally, the key is to demonstrate behaviors described by Eq. 3-25. This

can be readily done by calculating the time dependent exponent curves defined by

Eq. 2-14 (see also Eq. 2-4). In Fig. 3-3 we have shown a few examples for the fBm

processes with H = 0.25, 0.5, and 0.75. Clearly, the slopes of the straight lines correctly

estimate the H parameter used in the simulations.

3.1.2 Characterizing ON/OFF Processes with Pareto distributed ON and
OFF Periods by Power-law Sensitivity to Initial Conditions (PSIC)

ON/OFF intermittency is a ubiquitous and important phenomenon. For example,

a queueing system or a network can alternate between idle and busy periods, a fluid

flow can switch from a turbulent motion to a regular (called laminar) one. Fig. 3-4

shows an example for an ON/OFF traffic model with three independent traffic sources

Xi(t), i = 1, 3, where each Xi(t) is a 0/1 renewal process.

Let us denote an ON period by 1 and an OFF period by 0. We study ON/OFF trains

where ON and OFF periods are independent and both have heavy-tailed distribution.












































































Time

Figure 3-4. ON/OFF sources with NV = 3, X (t), X2(t), X3(t), and their summation S3 -)


SON OFF ON OFF ON OFF


H = 0.25
H = 0.50
H = 0.75


___
--


-6.5


2 2.5 3 3.5 4


Ink


Figure 3-3. Time-dependent exponent A(k) vs. In k curves for fractional Brownian motion.


I I


X ,(t)


I I


I I


X 2t)



X3,t




S3(t









A heavy-tailed distribution is commonly expressed as


f (x) ~ x-"-1,- x 00


Equivalently, one may write:


P X > x] ~ x-". x 0 o 3-30)


This expression is often called the complementary cumulative distribution function

(CCDF) of a ]!. l.-i--r I.l:d distribution. In fact, it is more popular for expressing a

heavy-tailed distribution, since it emphasizes the tail of the distribution. It is easy to

prove that when p < 2, the variance and all higher than 2nd-order moments do not exist.

Furthermore, when p < 1, the mean also diverges.

When the power-law relation extends to the entire range of the allowable x, we have

the Pareto distribution:


P[X J > ~ I x]= >biU > 2> 2> (3-31)

where p and b are called the shape and the location parameters, respectively. It can be

proven [64] that when 1 < p < 2, the Hurst parameter of the ON/OFF processes with

Pareto distributed ON and OFF periods is given as


H = (3 p-) /2 (3-32)


Our purpose here is to check whether the ON/OFF processes with Pareto distributed

ON and OFF periods can be characterized by PSIC. In Fig. 3-5 we have shown a few

examples for p = 1.2, 1.6, and 2.0. Noticing Eq. 3-32, we see that the slopes of the

straight lines again correctly estimate the H parameter. Also note that when 0 < p < 1,

1 < H < 1.5. Therefore, Eqs. (3-25) and (3-28) are still correct when 1 < H < 1.5. The

entire range of H = (3 p)/2, with 0 < p I 2, determines that -1 I q < 1/3 for Pareto

distributed ON/OFF processes.













4L = 2.0
4 = 1.6
4 = 1.2


-2.8






-3.4







.53 3.5 4 4.5
Ink

Figure 3-5. Time-dependent exponent A(k) vs. In k curves for ON/OFF processes with
Pareto distributed ON and OFF periods.


Before ending this section, we would like to emphasize that the possibility of

misinterpreting 1/ f processes being deterministic chaos never occurs if one works

under the framework of PSIC. Under this framework, one monitors the evolution of two

nearby trajectories. If two nearby trajectories diverge exponentially fast, then the time

series is chaotic, if the divergence increases in a power-law manner, then the trajectories

belong to 1/ f processes. These ideas can be easily expressed precisely. Let | |% Vy | |

be the initial separation between two trajectories. This separation is assumed to be not

larger than some prescribed small distance r. After time t, the distance between the two

trajectories will be ||%~+t Vj+t||. For true chaotic systems,


|| K~+t Vj+t|| oc ||K 1,- Vj ||ext









where A > 0 is called the largest Lyapunov exponent. On the other hand, for 1/ f

processes,



The latter equation thus provides another simple means of estimating the Hurst

parameter.

3.2 Characterizing Levy Processes by Power-law Sensitivity to Initial
Conditions (PSIC)

We now consider Levy processes, another important type of random fractal

models that have found numerous applications [65-70]. There are two types of Levy

processes [71]. One is Levy flights, which are random processes consisting of many

independent steps, each step being characterized by a stable law, and consuming a

unit time regardless of its length. The other is Levy walks, where each step takes time

proportional to its length. A Levy walk can be viewed as sampled from a Levy flight

with a uniform speed. The increment process of a Levy walk, obtained by differencing

the Levy walk, is very similar to an ON/OFF train with power-law distributed ON

and OFF periods. Therefore, in the following, we shall not be further concerned about

it. We shall focus on Levy flights. Note that Levy flights, having independent steps,

are memoryless processes characterized by H = 1/2, irrespective of the value of the

exponent a~ characterizing the stable laws [71]. In other words, methods such as detrended

fluctuation analysis (DFA) [72] cannot be used to estimate the a~ parameter.

Now we define Levy flights more precisely. A (standard) symmetric ac-stable Levy

process {L,(t), t > 0} is a stochastic process with the following properties [73]:

1. L,(t) is almost surely 0 at the origin t = 0;

2. L,(t) has independent increments; and

3. L,(t) L,(s) follows an ac-stable distribution with characteristic function e-(t-s) lo~

where 0 < s < t < 00.










A random variable Y is called (strictly) stable if the distribution for CE is the same as

that for niggY


i= 1
where Yi, Y2, are independent random variables, each having the same distribution as

Y. From Eq. 3-33, one then finds that nVarY = n2/"VarY. For the distribution to be

valid, O < a~ < 2. When a~ = 2, the distribution is Gaussian, and hence, the corresponding

Levy flight is just the standard Brownian motion. When 0 < a~ < 2, the distribution is

heavy-tailed, P[X > x] ~ x-", x 00o, and has infinite variance. Furthermore, when

0 < a~ < 1, the mean is also infinite.

At this point, it is worth spending a few paragraphs to explain how to simulate an

ac-stable Levy process. The key is to simulate its increment process, which follows an

ac-stable distribution. The first breakthrough for the simulation of stable distributions

was made by K~anter [74], who gave a direct method for simulating a stable distribution

with a~ < 1 and p = 1. The approach was later generalized to the general case by

C'I I1!.1wers et al. [75]. We first describe the algorithm for constructing a standard stable

random variable X ~ S(a~, p; 1).

Theorem Simulation of stable random variables: Let 8 and W be independent

with 8 uniformly distributed on (-xr/2, xr/2), W exponentially distributed with mean 1.

For 0 < a~ < 2, -1 < p < 1, and 8o = ard i I[Sltan(;ac/2)]/a,


Z (cosina(00+8) cos[aeo+(a-1)8] 1a/ 34
(cos000cos8)1/a



has a S(a~, p; 1) distribution.

For symmetric stable distribution with P = 0, the above expression can be greatly

simplified. To simulate an arbitrary stable distribution S(a~, P, y, 5), we can simply take









the following linear transform,


Y yZ + 6 a rJ 1 (3-35)
yZ+( S"qln n+6 a~= 1

where Z is given by Eq. 3-34.

We have simulated a number of symmetric ac-stable Levy motions with different

a~. Figure 3-6 shows two examples of Levy motions with a~ = 1.5 and 1. The difference

between the standard Brownian motions and Levy motions lies in that Brownian motions

appear everywhere similarly, while Levy motions are comprised of dense clusters connected

by occasional long jumps: the smaller the a~ is, the more frequently the long jumps appear.

Let us now pause for a while and think where Levy flights-like esoteric processes

could occur. One situation could be this: a mosquito head to a giant spider net and got

stuck; it struggled for a while, and luckily, with the help of a gust of wind, escaped. When

the mosquito was struggling, the steps it could take would be tiny; but the step leading

to its fortunate escape must be huge. As another example, let us consider (American)

football games. For most of the time during a game, the offense and defense would be

fairly balanced, and the offense team may only be able to proceed for a few yards. But

during an attack that leads to a touchdown, the hero getting the touchdown often "flies"

tens of yards -somewhat he has escaped the defense. While these two simple situations

are only meant as analogies, remembering them could be helpful when reading research

papers searching for Levy statistics in various types of problems such as animal foraging.

At this point, we also wish to mention that Levy-flights-like patterns have been used as a

type of screen saver for Linux operating systems.

The symmetric ac-stable Levy flight is 1/a~ self-similar. That is, for c > 0, the

processes {L,(ct), t > 0} and {cl/oL,(t), t > 0} have the same finite-dimensional

distributions. By this argument as well as Eq. 3-33, it is clear that the length of the





















(a) (b)













(c) (d)











(e) (f)

Figure 3-6. Examples for Brownian motions and Levy motions. 1-D and 2-D Brownian
motions (a,b) vs. Levy motions with a~ = 1.5 and 1 for (c,d) and (e,f),
respectively.













o a =1.0
-4.5 0 a =1.5



-5-



-5.5



-6




06~.5 1 1.5 2 2.5 3
Ink

Figure 3-7. Time-dependent exponent A(k) vs. In k curves for Levy processes.


motion in a time span of at, AL(At), is given by the following scaling:


AL(At) oc Atl/o (3-36)


This contrasts with the scaling for fractional Brownian motion:


ABHat oc OC (33H


We thus identify that 1/a ]I phi- the role of H. Therefore,


q = 1 a~ (3-38)


A 0 = 1/al (3-39)

Noticing 0 < a~ < 2, from Eq. 3-38, we have -1 < q < 1.










We have applied the concept of PSIC to Levy processes with different a~. Examples

for Levy processes with a~ = 1 and 1.5 are shown in Fig. 3-7. The slopes of the straight

lines correctly estimate the values of a~ used in the simulations.









CHAPTER 4
STUDY OF INTERNET DYNA1\ICS BY POWER-LAW SENSITIVITY TO INITIAL
CONDITIONS (PSIC)

Time series analysis is an important exercise in science and engineering. One of the

most important issues in time series analysis is to determine whether the data under

investigation is regular, deterministically chaotic, or simply random. Along this line, a

lot of work has been done in the past two decades. A hit surprisingly, quite often the

exponential sensitivity to initial conditions (ESIC) in the rigorous mathematical sense

cannot he observed in experimental data. While current consensus is to attribute this

fact to the noise in the data, we shall show that the more general concept of power-law

sensitivity to initial conditions (PSIC) provides an interesting framework for time series

analysis. We illustrate this by studying the complicated dynamics of Internet transport

protocols.

The Internet is one of the most complicated systems that man has ever made.

In recent years, two types of fascinating multi-scale behaviors have been found in the

Internet. One is the temporal-domain fractal and multifractal properties of network traffic

(see [76, 77] and references therein), which typically represents .I_ _regation of individual

host streams. The other is the spatial-domain scale-free topology of the Internet (see [78]

and references therein). Also, it has been observed that the failure of a single router may

trigger routing instability, which may be severe enough to instigate a route flap storm [79].

Furthermore, packets may be delivered out of order or even get dropped, and packet

reordering is not a pathological network behavior [80]. As the next generation Internet

applications such as remote instrument control and computational steering are being

developed, it is becoming increasingly important to understand the transport dynamics

in order to sustain the needed control channels over wide-area connections. Typically, the

transport dynamics over the Internet connections are a result of the nonlinear dynamics

of the widely-deploi- II Transmission Control Protocol (TCP) interacting with the Internet

traffic, which is stochastic and often self-similar. This leads one to naturally expect the










transport dynamics to be highly complicated. To tackle the hard problem of studying

the transport dynamics, we first systematically carry out network simulation using a

coninonly used network simulator, the ns-2 simulator (we have chosen the Tahoe version

of TCP, see http://www.isi. edu/nsnam/ns/), and then carefully study the dynamics of

the TCP congestion window-size traces collected over the real Internet connections.

4.1 Study of Transport Dynamics by Network Simulation

By carrying out network simulation, we critically examine whether TCP dynamics

can he chaotic, and if yes, to identify a route to chaos. We shall also present Internet

measurements to complement the simulation results. Since the advent of the Internet,

it has been perceived that the dynamics of complicated coninunication networks may

be chaotic [81]. If transport dynamics are indeed chaotic, then steady-state analysis and

study of convergence to a steady-state will not he the sole focus in practice, particularly

for remote control and computational steering tasks. Rather, one should also focus on the

Il II-! 1!1" non-convergent transport dynanxics. Although recently a lot of research has

been carried out [82-88] to better understand this issue, a lot of fundamental problems

remain to be answered. For example, the observation of chaos in [84] cannot he repeated,

which leads to the suspicion that the chaotic motions reported in [84] may correspond to

periodic motions with very long periods [89]. Fundamentally, we shall show, by developing

a 1-D discrete nmap, that the network scenarios with only a few competing TCP flows

studied in [84] cannot generate chaotic dynanxics. However, with other parameter settings,

chaos is possible.

In order to examine the temporal evolution of a dynamical systent, one often

measures a scalar time series at a fixed point in the state space. Takens embedding

theorem [34-36 ensures that the collective behavior of the dynamics described by the

dynamical variables coupled to the measured one can he conveniently studied by the

measured scalar time series. To illustrate the qualitative aspects of TCP dynamics by only

measuring a scalar time series, it is most effective to consider a single bottleneck link, as









such a link must be tightly coupled with the rest of the network. For simplicity, we shall

follow the setup of Veres and Boda [84] consisting of long lived TCP flows on a single link.

This setup has four parameters, the link speed C in Mbps, the link propagation delay d in

ms, the buffer size B in unit of packets of 1000 bytes, and the number of competing TCP

flows NV. We have found that NV acts as a critical bifurcation parameter. For example,

when C = 0.1, d = 10 ms, B = 10 and NV is small, we have observed only periodic

and quasi-periodic motions (Figs. 4-1(a-d)). When NV is large, such as NV = 17, chaos is

observed (see Fig. 4-2(a,b)). We thus conclude that one route to chaos in TCP dynamics

is the well-known quasi-periodic route. In order to show more diversified dynamical

behavior such as quasi-periodic motions with more than two incommensurate (i.e.,

independent) frequencies, below, however, we shall vary all the parameters. In particular,

we shall show that when NV is small, chaos cannot happen.

For each of the NV competing TCP flows, one can record its congestion window

size data. Let us only record one of them, and denote it by W(i). An example of

quasi-periodic W(i) is shown in Fig. 4-1(a), where C = 0.1 Mbps, delay d = 10

ms, B = 10, NV = 3. Its power spectral density (PSD) is shown in Fig. 4-1(b). We

observe many discrete sharp peaks. Note that when only around 105 points are used

to compute the PSD, one does not observe ]rn lw: peaks in the spectrum. Rather one

observes a broad spectrum, and hence, is tempted to interpret the W(i) time series

to be chaotic. Therefore, the W(i) time series is not ideally suited for the study of

(quasi-)periodic motions with long periods. This motivates us to define a new time

series, T(i), which is the time interval between the onset of two successive exponential

backoff (i.e., multiplicative decrease) of TCP, as indicated in Fig. 4-1(a). The start of an

exponential backoff indicates triggering of a loss episode or heavy congestion. Hence, the

T(i) is related to the well-known round trip time (RTT). In fact, it can be considered

to be more or less equivalent to the time interval between two successive loss bursts.

The T(i) time series for the W(i) of Fig. 4-1(a) is shown in Fig. 4-1(c), together with its









1015

1010

0 105

100

10
0 0.05 0.1
Frequency

10'" (d)


0 1000 2000 3000 4000 5000
index i


50
index i


Frequency


105

100


105
0 0.1 0.2 0.3 0.4 0.5
Frequency


Figure 4-1.


Example of quasi-periodic congestion window size IT(i) time series sampled by
an equal time interval of 10 ms (a) and its power spectral density (PSD) (b).
The parameters used are C = 0.1 Mbps, delay d = 10 ms, B = 10 packets,
where a packet is of size 1000 bytes, and the number of TCP streams NV = :3.
(c) The T(i) time series corresponding to (a) and (d) its PSD. (e) Another
T(i) time series corresponding to C = 0.5 Mbps, d = 10 ms, B = 10, NV = :3,
and (f) its PSD.


PSD in Fig. 4-1(d). We observe that this T(i) time series is a simple periodic sequence,

and hence, the IT(i) time series is quasi-periodic. However, T(i) time series can still be


quasi-periodic. An example is shown in Fig. 4-1(e), with C


0.5 Mbps, delay d


ms, B = 10 packets, NV = :3, together with its PSD is Fig. 4-1(f). Note that there are still

many discrete sharp peaks in Fig. 4-1(f). To appreciate how many independent frequencies


100

90

80
0 50 100 150
index i


I--rui-lu~









may be contained in the T(i) time series of Fig. 4-1(e), we have further extracted two new

time series, Til) (i), which is the interval time series between the successive local maxima

of the T(i) time series, and T(2) i), Which is the interval time series between the successive

local maxima of the Til) (i) time series. We have found that the T(2) i) iS Simply periodic.

Hence, we conclude that the W(i) time series is quasi-periodic with four independent

frequencies.

Before we proceed to discuss the chaotic TCP dynamics, we note that simple periodic

or quasi-periodic W(i) time series with less or more than four independent frequencies

can all be observed. In fact, we have found the "chaotic" congestion window size data

studied in [84] to be quasi-periodic with two independent frequencies. Furthermore, the

probability distributions for the constant, periodic, and quasi-periodic T(i) time series are

only composed of a few discrete peaks. This -II__- -; that the observed quasi-periodic T(i)
time series constitutes a discrete torus.

Let us now turn to the discussion of chaotic TCP dynamics. Shown in Figs. 4-2(a,c)

are two irregular T(i) time series. The parameters C, d, and B used for Fig. 4-2(a) are the

same as those for Figs. 4-1(a-d). Noting from Fig. 4-2 that T(i) often can be on the order

of 104 to 10s, hence, a not too long W(i) time series is even more ill-suited for the study of

the underlying irregular dynamics.

To show that the T(i) time series in Figs. 4-2(a,c) is indeed chaotic, we employ

the direct dynamical test for deterministic chaos developed by Gao and Zheng [21, 22].

The A(k) curves for the T(i) time series of Figs. 4-2(a,c) are shown in Figs. 4-2(b,d),

where the four curves, from bottom to top, correspond to shells of sizes (2-(i+1)/2, 2-i/2)

i = 14, 15, 16, 17. we have simply chosen m = 6 and L = 1. Very similar curves have

been obtained for other choices of m and L. We observe a very well defined linear common

envelope at the lower left corner of Figs. 4-2(b,d). The existence of a common envelope

guarantees that a robust positive Lyapunov exponent will be obtained no matter which

shell is used in the computation. Hence, the two T(i) time series are indeed chaotic.


























(c)

5


1


5


x14
8 5











2 15


xr 101


(d)
5

4

3

2


0 0
0 2000 4000 6000 0 10 20 30 40 50


Figure 4-2. Two examples of chaotic T(i) time series obtained with parameters (a)
C = 0.1, delay d = 10 ms, B = 10, and the number of TCP streams NV = 17
and (c) C = 0.1, d = 10 ms, B = 12, and NV = 19. The A(k) curves for (a) and
(c) are shown in (b) and (d), respectively.

We have remarked that the probability distributions for the T(i) time series of regular

motions are comprised of a few discrete peaks. What are the distributions for the chaotic

T(i) time series such as shown in Figs. 4-2(a,c)? They are power-law-like, P(T < t) ~ t Y,
for almost two orders of magnitude in t, as shown in Figs. 4-3(a,b). The exponent y is not

a universal quantity, however.

We now develop a simple 1-D map to describe the operation of TCP. TCP relies

on two mechanisms to set its transmission rate: flow control and congestion control.













10' 10-'


-10 a 10-


10-3~ 10-3


1-41 1-4
102 103 104 105 102 103 104 105
t t

Figure 4-3. Complementary cumulative distribution functions (CCDFs) for the two chaotic
T(i) time series of Figs. 4-2(a,c).

Flow control ensures that the sender sends no more than the size of the receiver's last

advertised flow-control window based on its available buffer size while congestion control

ensures that the sender does not unfairly overrun the network's available bandwidth.

TCP implements these mechanisms via a flow-control window ( fwnd) advertised by the

receiver to the sender and a congestion-control window (cwnd) adapted based on inferring

the state of the network. Specifically, TCP calculates an effective window (ewnd), where

ewnd = min( fwnd, cwnd), and then sends data at a rate ewnd/RTT, where RTT is the

round trip time of the connection. Recently Rao and Chua [85] developed an analytic

model describing TCP dynamics. By assuming fwnd being fixed and the TCP's slow start

only contributing to transient behavior, Rao and Chua have developed the w-up~date map

M~: [1, Wmax]i [1, Wmax],

I, + 1/l, if It, E1 RI, r tE R1

M~re)= < It < n /2"" if It, E R1, I tE R2
I /2"" if n-, 6R2, I, 1 1 6

I, + 1/l, if no Ea R I, 1I R1 1










In region R1, there is no packet loss, as = 0, and the TCP is in additive increase mode. In

R2, there are ni > 0 packet losses, where ni may be a complicated function of time, and

the TCP is in multiplicative decrease model.

The above map (Eq. 4-1) can be simplified [90],




I / 2"' if n -, > Wmax

Since fwnd is not assumed to be a constant, the modified map is more general than

the original Rao and Chua's map. To apply the above map to the study of a TCP

flow competing with NV 1 other TCP flows, one may lump the effect of the NV 1

other TCP flows as a background traffic. Hence, Wmax is determined by cwnd and the

background traffic, and typically is a complicated function of time, noticing that the

available bandwidth of a bottleneck link can vary with time considerably due to the

dynamic nature of background traffic. We have carried out simulation studies of Eq. 4-2

by assuming Wmax to be periodic and quasi-periodic, and have indeed observed chaos.

We are now ready to understand why chaos cannot occur when the number of

competing TCP flows is small, such as 2. To see this, let us assume that I, and no +

al, as well as I,. It and I, t + Al, t are all smaller than Wmax. It is then easy to

see that 1An' ,I t| < An- |. That is, small disturbance decays. This means during the

additive increase phase of TCP, nearby trajectories contract instead of diverge. Hence,

the dynamics are stable. In order for the dynamics to be unstable so that the Lyapunov

exponent is positive, the transition from the additive increase phase to the multiplicative

decrease phase has to be fast. This can be true only when the number of competing TCP

streams is large. We thus conclude that the network scenarios considered in [84] cannot

generate chaotic TCP dynamics.

How relevant is our simulation result to the dynamics of the real Internet? Are the

irregular T(i) time series shown in Figs. 4-2(a,c) an artifact of Tahoe version of TCP
















S10-
400

200

0 CIVYUU yIYYUU"UU VRl V'IIIIYUII IIIVV 111 10-
0 50 100 150 200 250 300 350 100 101 102 103


Figure 4-4. The time series T(i) extracted from a W(i) time series collected using net100
instruments over ORNL-LSU connection. (a) the time series T(i), (b) the
complementary cumulative distribution function (CCDF) for the T(i) time
series.


we used or more generally of the ns-2 simulator? To find an answer, we collected W(i)

measurements between ORNL and Louisiana State University at millisecond resolution

using the net100 instruments and computed T(i) as shown in Fig. 4-4(a), whose profile is

qualitatively quite similar to Fig. 4-2(a). In fact, it also follows a (truncated) power-law,

as shown in Fig. 4-4(b). The exponent for the power-law part is even smaller than the

time series of Fig. 4-2. The truncation in the power-law could be due to the shortness

of the T(i) time series. As we have pointed out, the T(i) time series is related to RTT.

Cottrell and Bullot have been experimenting with many advanced versions of TCP,

and also observed RTT time series very similar to these. We also have observed from

RTT and loss data measured on geographically dispersed paths on the Internet (with

a resolution of 1 sec) by researchers at the San Diego Supercomputer Center that the

probability distributions for the time interval between successive loss bursts typically

follow a power-law-like distribution, with the exponent y also smaller than 1. Thus,

we have good reason to believe that the observed irregular T(i) time series is not an










artifact of the ns-2 simulator, but reflects reality to some degree. In fact, the complicated

dynamics described here is much simpler than the actual dynamics of the Internet,

considering that there are all kinds of randomness in the Internet, especially that typical

TCP flows are not long lived [91], but vary considerably in durations determined by

specific applications. In the next section, we will analyze the actual dynamics of the

Internet measurements.

4.2 Complicated Dynamics of Internet Transport Protocols

As we have discussed earlier, it is very difficult to understand the dynamics of a

transport protocol over Internet connections. The 1! in r~ difficulties lie in two aspects.

First, it has been technologically hard to collect high quality measurements of network

transport variables on hli,-' Internet connections. Secondly, it is very difficult to

analytically handle the interaction between the deterministic and stochastic parts of

transport dynamics. The deterministic dynamics are due to the highly non-linear behavior

of TCP's Additive Increase and Multiplicative Decrease (AIMD) method emploi-x & for

congestion control. The stochastic component is due to the often self-similar traffic [92]

with which TCP competes for bandwidth and router buffers. Roughly speaking, the

interaction between the two is in terms of the additions to TCP congestion window-size,

denoted by cwnd, in response to acknowledgments, and multiplicative decreases in

response to inferred losses (ignoring the initial slow-start phase). Any protocol, however

simple its dynamics are, is generally expected to exhibit apparently complicated

dynamics due to its interaction with the stochastic network traffic. TCP in particular

is shown (albeit in simulation) to exhibit chaotic or chaos-like behavior even when the

competing traffic is much simpler such as a single competing TCP [84] or User Datagram

Protocol (UDP) stream [85]. The difficulty is to understand the dynamics when both the

phenomena are in effect, and equally importantly to understand their impact on actual

Internet streams. Historically, the methods from standard chaos and stochastic theories

have been unable to offer much new perspective on the transport dynamics.










In this section, we utilize the recently developed net100 instruments (obtained front

http://www.net100. org) to collect high quality TCP cwnd traces for actual Internet

connections. We analyze the measurements using the concepts of time dependent exponent

curves [21, 22, 37]. Our major purpose is to elucidate how the deterministic and stochastic

components of transport dynamics interact with each other on the Internet. It is of

considerable interest to note that recently there have been several important works on

TCP dynamics with the purpose of improving congestion control [93-96]. In fact, there

has been considerable effort in developing new versions and/or alternatives to TCP so

that the network dynamics can he more stable. Conventional v- 0-<~ Of analyzing the

network dynamics are unable to readily determine whether newer methods result in stable

transport dynamics or help in designing such methods. Our analysis shows two important

features in this direction:

(a) Randoniness is an integral part of the transport dynamics and must he explicitly
handled. In particular, the step sizes utilized for adjusting the congestion parameters
must he suitably varied (e.g. using Robbins-1\onro conditions) to ensure the eventual
convergence [97]. This is not the case for TCP which utilizes fixed step sizes.

(b) The chaotic dynamics of the transport protocols do have a significant impact on
practical transfers, and the protocol design must he cognizant of its effects. In
particular, it might he worthwhile to investigate protocols that do not contain
dominant chaotic regimes, particularly for remote control and steering applications.


The traditional transport protocols are not designed to explicitly address the above two

issues, but justifiably so since their original purpose is data transport rather than finer

control of dynanxics.

In the rest of this section, we shall first briefly describe the ownd data studied here.

Then we shall briefly explain the analysis procedure and describe typical results of the

analysis.

We have collected a number of cwnd traces using single and two competing TCP

streams. This data was collected on two different connections front Oak Ridge National

Laboratory (ORNL) to Georgia Institute of Technology (GaTech) and to Louisiana State









University (LSU). The first connection has high-bandwidth (OC192 at 10Gbps) with

relatively low backbone traffic and a round-trip time of about 6 milliseconds. Four traces

were collected for this connection, two with a single TCP stream and the others with two

competing TCP streams. The second connection has much lower bandwidth (10 Mbps)

with higher levels of traffic and a round trip-time of about 26 milliseconds. The sampling

time is approximately 1 millisecond, with an error of 10s of microseconds (due to a "busy

waitingt measurement loop). Eight traces were collected for the second connection. The

results based on these eight traces are qualitatively very similar to the ORNL-GaTech

traces, and we only discuss the results for the latter. To appreciate better what these data

look like, we have plotted in Figs. 4-5(a-d) segments of four datasets for ORNL-GaTech

connection. Power spectral analysis of these data does not show any dominant peaks,

and hence, the dynamics are not simply oscillatory. Since our data was measured on the

Internet with Int -' background traffic, it is apparently more complicated and realistic

than the traces obtained by network simulation.

Next let us analyze cwnd data, x(i), i = 1, n, using the concepts of time

dependent exponent A(k) curves [21, 22, 37]. As has been discussed earlier, we first

need to employ the embedding theorem [34-36, 98] to construct vectors of the form:

K = [x(i), x(i + L), ..., x(i + (m 1)L)], where m is the embedding dimension and L the

delay time. The embedding theorem [34-36] states that when the embedding dimension

m is larger than twice the box counting dimension of the attractor, then the dynamics

of the original system can be studied from a single scalar time series. We note that the

embedding dimension used in [84] is only two, which has to be considered not large enough

(this may call for a closer examination of their conclusions).

Before we move on, we point out a few interesting features of the A(k) curves:

(i) For noise, only for k up to the embedding window size (m 1)L will A(k) increase.
Thus, whenever A(k) increases for a much larger range of k, it is an indication of
non-trivial deterministic structure in the data.

(ii) For periodic signals, A(k) is essentially zero for any k.










x 104
4


-rl


qX104
(b) 1 TCP source


-


3





1


0
0


2000 4000 6000


2000 4000 6000


x 104



3~ i


X 104
41(d) 2 TCP sources
[Hdi n8 n, Ellrfli rfl


TCP sources


1


2000 4000
sample n


6000


2000 4000
sample n


6000


Figure 4-5. Time series for the congestion window size cwnd for ORNL-GaTech
connection.


(iii) For quasi-periodic signals, A(k) is periodic with an amplitude typically smaller than
0.1, hence, for practical purposes, A(k) can be considered very close to 0.

(iv) When a chaotic signal is corrupted by noise, then the A(k) curves break themselves
away from the common envelope. The stronger the noise is, the more the A(k)
curves break away till the envelope is not defined at all. This feature has actually
been used to estimate the amount of both measurement and dynamic noise [99].


Now we are ready to compute and understand the A(k) curves for cwnd traces. The

set of A(k) curves, corresponding to Fig. 4-5 are plotted in Figs. 4-6. In the computations,


3 x 104 pOintS are used, and m


10, L = 1. The seven curves, from the bottom to


(a) 1 TCP source













3





1


0
0 50 100 150 200 250



4
(c 12 TC urces

3


2


1


0 50 100 150 200 250


50 100 150 200
Evolution time k


50 100 150 200
Evolution time k


250


Figure 4-6.


Time-dependent exponent A(k) vs. k curves for cwnd data corresponding to
Fig. 4-5. In the computations, 3 x 104 pOintS are used, and m 10, L 1 .
Curves from the bottom to top correspond to shells with sizes (2-(i+1)/2, 2-i/2)
i = 2, 3, ---,8.


top, correspond to shells of sizes (2-(i+1)/2, 2-i/2), i


2, 3, 8. We make the following


observations:

(i) The dynamics are complicated and cannot be described as either periodic or
quasi-periodic motions, since A(k) is much larger than 0.

(ii) The dynamics cannot be characterized as pure deterministic chaos, since in no case
can we observe a well-defined linear envelope. Thus the random component of the
dynamics due to competing network traffic is evident and can not simply be ignored.










(iii) The data is not simply noisy, since otherwise we should have observed that A(k) is
almost flat when k > (m 1)L. Thus, the deterministic component of dynamics
which is due to the transport protocol phI i- an integral role and must be carefully
studied. The features (ii) and (iii) indicate that the Internet transport dynamics
contains both chaotic and stochastic components.

(iv) There are considerable differences between the data with only 1 TCP source and
with 2 competing TCP sources. In the latter case, the A(k) curves sharply rise when
k just exceeds the embedding window size, (m 1)L. On the other hand, A(k)
for Fig. 4-6(a) with only one TCP source increases much slower when k just exceeds
(m-1)L. Also important is that the A(k) curves in Figs. 4-6(c,d) are much smoother
than those in Figs. 4-6(a,b). Hence, we can ;?i that the deterministic component of
the dynamics is more visible when there are more than 1 competing TCP sources
(along the lines of [84]).

Since the increasing part of the A(k) curves are not very linear, let us next examine if

the cwnd traces can be better characterized by the more generalized concept of power-law

sensitivity to initial conditions (PSIC). Figure 4-7 shows the A(k) vs. In k curves for

ORNL-GaTech connection. Very interestingly, we have now indeed observed a bunch of

better defined linear lines, especially for small scales. In particular, the transport dynamics

with more than 1 competing TCP sources are better described by the concept of PSIC.

To summarize, by analyzing a number of high quality congestion window-size data

measured on the Internet, we have found that the transport dynamics are best described

by the concept of PSIC, especially for small scales. It is interesting to further examine

how one might suppress the stochasticity of the network by executing more controls on

the network when making measurements, such as using a large number of competing TCP

sources together with a constant UDP flow. Our analysis motivates that both chaotic and

stochastic aspects be paid a close attention to in designing Internet protocols that are

required to provide the desired and tractable dynamics.

















































02 4


Time-dependent exponent A(k) vs. In k curves for cwnd data corresponding to
Fig. 4-5. In the computations, 3 x 104 pOintS are used, and m 10, L 1 .
Curves from the bottom to top correspond to shells with sizes (2-(i+1)/2, 2-i/2)
i = 2, 3, ---,8.


Figure 4-7.










CHAPTER 5
STITDY OF SEA CLITTTER RADAR RETITRNS BY POWER-LAW SENSITIVITY TO
INITIAL CONDITIONS (PSIC)

Understanding the nature of sea clutter is crucial to the successful modeling of

sea clutter as well as to facilitate target detection within sea clutter. To this end, an

important question to ask is whether sea clutter is stochastic or deterministic. Since

the complicated sea clutter signals are functions of complex (sonietintes turbulent) wave

motions on the sea surface, while wave motions on the sea surface clearly have their

own dynamical features that are not readily described by simple statistical features,

it is very appealing to understand sea clutter hv considering some of their dynamical

features. In the past decade, Haykin et al. have carried out analysis of some sea clutter

data using chaos theory [100, 101], and concluded that sea clutter was generated hv an

underlying chaotic process. Recently, their conclusion has been questioned by a number of

researchers [102-108]. In particular, Unsworth et al. [105, 106] have demonstrated that the

two main invariants used by Haykin et al. [100, 101], namely the in! I::!!Itsinl likelihood of

the correlation dimension alI lin II. and the "false nearest neighbors" are problematic in

the analysis of measured sea clutter data, since both invariants may interpret stochastic

processes as chaos. They have also tried an improved method, which is based on the

correlation integral of Grassherger and Procaccia [109] and has been found effective in

distinguishing stochastic processes front chaos. Still, no evidence of deternxinisni or chaos

has been found in sea clutter data.

To reconcile ever growing evidence of stochasticity in sea clutter with their chaos

hypothesis, recently, Haykin et al. [110] have -II_t-r-- -1.. that the non-chaotic feature of sea

clutter could be due to many types of noise sources in the data. To test this possibility,

1\kDonald and Dantini [111] have tried a series of low-pass filters to remove noise; but

again they have failed to find any chaotic features. Furthermore, they have found that

the coninonly used chaotic invariant measures of correlation dimension and Lyapunov

exponent, computed by conventional r- .--4, produce similar results for measured sea










clutter returns and simulated stochastic processes, while a nonlinear predictor shows little

intprovenient over linear prediction.

While these recent studies highly ell_-- -r that sea clutter is unlikely to be truly

chaotic, a number of fundamental questions are still unknown. For example, most of the

studies in [102-106] are conducted by comparing measured sea clutter data with simulated

stochastic processes. We can ask: can the non-chaotic nature of sea clutter he directly

demonstrated without resorting to simulated stochastic processes'? Recognizing that

simple low-pass filtering does not correspond to any definite scales in phase space, can we

design a more effective method to separate scales in phase space and to test whether sea

clutter can he decomposed as signals plus noise'? Finally, will studies along this line he of

any help for target detection within sea clutter'?

In this chapter, we employ the direct dynamical test for deterministic chaos discussed

in Sec. 2.1 to analyze 280 sea clutter data measured under various sea and weather

conditions. The method offers a more stringent criterion for detecting low-dintensional

chaos, and can simultaneously monitor motions in phase space at different scales.

However, no chaotic feature is observed front any of these data at all the different scales

examined. But interestingly, we show that scale-dependent exponent corresponding to

large scale appears to be useful for distinguishing sea clutter data with and without

targets. This -II_--- -is that sea clutter may contain interesting dynamic features and that

the scale-dependent exponent may be an important parameter for target detection within

sea clutter. Furthermore, we find that sea clutter can he conveniently characterized by the

new concept of power-law sensitivity to initial conditions (PSIC). We show that for the

purpose of detecting targets within sea clutter, PSIC offers a more effective framework.

Below, we shall first briefly describe the sea clutter data. Then we study the

sea clutter data by employing the direct dynamical test for low-dintensional chaos

developed by Gao and Zheng [21, 22]. And we show that the scale-dependent exponent

corresponding to large scale can he used to effectively detect low observable targets within










sea clutter, while the conventional Lyapunov exponent fails in such a task. Finally, we

apply the new concept of PSIC to detect targets within sea clutter.

5.1 Sea Clutter Data

14 sea clutter datasets were obtained from a website maintained by Professor Simon

Haykin: http: //soma.ece.mcmaster. ca/ipix/dartmouth/datasets.html. The measurement

was made using the McMaster IPIX radar at Dartmouth, Nova Scotia, Canada. The

radar was mounted in a fixed position on land 25-30 m above sea level, with the operating

(carrier) frequency 9.39 GHz (and hence a wavelength of about 3 cm). It was operated at

low grazing angles, with the antenna dwelling in a fixed direction, illuminating a patch

of ocean surface. The measurements were performed with the wave height in the ocean

varying from 0.8 m to 3.8 m (with peak height up to 5.5 m), and the wind conditions

varying from still to 60 km/hr (with gusts up to 90 km/hr). For each measurement, 14

areas, called antenna footprints or range bins, were scanned. Their centers were depicted

as B1, B2, B14 in Fig. 5-1. The distance between two .Il11 Il:ent range bins was 15 m.

One or a few range bins (;?-,, Bi_l, Bi and Bi 1) hit a target, which was a spherical block

of styrofoam of diameter 1 m, wrapped with wire mesh. The locations of the three targets

were specified by their azimuthal angle and distance to the radar. They were (128 degree,

2660 m), (130 degree, 5525 m), and (170 degree, 2655 m), respectively. The range bin

where the target is strongest is labeled as the primary target bin. Due to drift of the

target, bins .Il11 Il:ent to the primary target bin may also hit the target. They are called

secondary target bins. For each range bin, there were 217 complex numbers, sampled with

a frequency of 1000 Hz. Amplitude data of two polarizations, HH (horizontal transmission,

horizontal reception) and VV (vertical transmission, vertical reception) were an~ llh-. 1

Fig. 5-2 shows two examples of the typical sea clutter amplitude data without and

with target. However, careful examination of the amplitude data indicates that 4 datasets

are severely affected by clipping. This can be readily observed from Figs. 5-3(a, b). We










































40 60 80 100 120 140
Time (sec)


11 11_1 11* I)


h : Antenna height
S: Grazing angle
R. : Range (distance from the radar)
B1 ~ B14 : Range bins

Target

BB..B. Bi B.+ ... B,
1 2 1-1 1 111
(Secondary)(Primary)(Secondary)

I-~R----------Ri

Figure 5-1. Collection of sea clutter data


25


20


o 15


E 10

5


(b)


E 1


0 20


rrl


Figure 5-2. Typical sea clutter amplitude data (a) without and (b) with target.


discard those 4 datasets, and >.1, lli. .. the remaining 10 measurements, which contain 280

sea clutter time series.

5.2 Non Chaotic Behavior of Sea Clutter

In this section, we employ the direct dynamical test for deterministic chaos developed

by Gao and Zheng [21, 22], to 2.1, lli. .. sea clutter data. The method offers a more

stringent criterion for low-dimensional chaos, and can simultaneously monitor motions

in phase space at different scales. The method has found numerous applications in the


0 20 40 60 80 100 120 140
Time (sec)












2.5 2.5


02 02

E 1.5 E 1.5


1 1


0.5 0.5
4.00 4.01 4.02 4.03 4.04 5.22 5.23 5.24 5.25 5.26
Time (sec) Time (sec)

Figure 5-:3. Two short segments of the amplitude sea clutter data severely affected hv
clipping.


study of the effects of noise on dynamical systems [2:3, 24, 26, 27], estimation of the

strength of measurement noise in experimental data [28, 29], pathological tremors [:30],

shear-thickening surfactant solutions [:31], dilute sheared aqueous solutions [:32], and

serrated plastic flows [:33]. In particular, this method was used by Gao et al. [107, 108] to

analyze one single set of sea clutter data. While chaos was not observed front that dataset,

no definite general conclusion could be reached, due to lack of large amount of data at

that time. Here we systematically study 280 sea clutter data measured under various sea

and weather conditions, and examine whether any chaotic features can he found front

these sea clutter data.

The explicit incorporation of scales in the Gao and Zheng's test [21, 22] enables us to

simultaneously monitor motions in phase space at different scales. We have systematically

analyzed 280 amplitude sea clutter time series measured under various sea and weather

conditions. However, no chaotic feature has been observed front any of these data at

all the scales examined. Typical examples of the A(k) vs. k curves for the sea clutter

amplitude data without targets are shown in Figs. 5-4(a, c, e) and the curves for the data

with targets shown in Figs. 5-4(h, d, f), respectively. We have simply chosen m = 6 and










4

3








4r

3

r2

1

0


4

3



1


(a)
5 10 15 20 25 30


(b)
5 10 15 20 25 30


(c)
5 10 15 20 25 30


(d)
5 10 15 20 25 30


Oi(e '" (f)c
0 5 10 15 20 25 30 0 5 10 15 20 25 30
k k

Figure 5-4. Examples of the time-dependent exponent A(k) vs. k curves for the sea clutter
data (a, c, e) without and (b, d, f) with the target. Six curves, from bottom to
top, correspond to shells (2-(i+1)/2, 2-i/2) With i 13, 14, 15, 16, 17, and 18.
The sampling time for the sea clutter data is 1 msee, and embedding
parameters are m 6, L 1 217 data points are used in the computation.


L = 1. Very similar curves have been obtained for other choices of m and L. We have

not observed a common envelope at any scales. In fact, the results of Fig. 5-4 are generic

among all the 280 sea clutter data on~ li. .1 here. Hence, we have to conclude that none of

the sea clutter data is chaotic.

5.3 Target Detection within Sea Clutter by Separating Scales

Robust detection of low observable targets within sea clutter is a very important issue

in remote sensing and radar signal processing applications, for a number of reasons: (i)









identifying objects within sea clutter such as submarine periscopes, low-flying aircraft,

and missiles can greatly improve our coastal and national security; (ii) identifying small

marine vessels, navigation buoys, small pieces of ice, patches of spilled oil, etc. can

significantly reduce the threat to the safety of ship navigation; (iii) monitoring and

policing of illegal fishing is an important activity in the environmental monitoring. Due

to the rough sea surface, which is a consequence of energy transfer from small scale to

large scale of sea surface wave, and the multipath propagation of the radar backscatter,

sea clutter is often highly non-Gaussian, even spiky, especially in heavy sea conditions.

Hence, sea clutter modelling is a very difficult problem, and a lot of effort has been made

to study sea clutter, both through the analysis of the distributions of sea clutter, including

Weibull [112], log-normal [113], K( [114, 115], and compound-Gaussian distributions [116],

as well as using chaos theory [100-104, 107, 108, 110, 111], wavelets [117], neural

networks [118, 119], wavelet-neural net combined approaches [120, 121], and the concept of

fractal dimension [122] and fractal error [123, 124]. However, no simple method has been

found to robustly detect low observable objects within sea clutter [125].

In this section, we examine whether the Lyapunov exponent estimated by conventional

methods and the scale-dependent exponent may be helpful for target detection within sea

clutter.

We first check whether the Lyapunov exponent A estimated by conventional methods

can be used for distinguishing sea clutter data with and without targets. As pointed

out earlier, the conventional way of estimating the Lyapunov exponent is to compute

A(k)/kbt, where A(k) is defined as in Eq. 2-4, subject to the conditions that ||Xi-Xj|| < r

and ||Xi+k Xj,, || < R, where r and R are prescribed small distance scales. The condition

||Xi Xj || < r amounts to our smallest shell in computing the A(k) curves. The condition

||Xi+k Xj+k I < R is presumably to set the time scale k smaller than the prediction

time scale k. Our analysis finds that the Lyapunov exponent estimated this way fails to










detect targets from the sea clutter data on~ lli. here. This motivates us to try a modified

approach, as described below.

The modified approach for estimating the Lyapunov exponent A uses the least-squares

fit to the first few samples of the A(k) curve where the A(k) curve increases linearly or

quasi-linearly. This is done for all the 280 sea clutter time series. To better appreciate the

variation of the Lyapunov exponent A among the 14 range hins of the sea clutter data,

we have first subtracted the A parameter of each hin by the minimum of A values for that

measurement, and then normalized the obtained A values by its maximum. The variations

of the A parameters with the 14 range hins for the 10 HH measurements are shown in

Figs. 5-5(a)-(j), respectively, where open circles denote the range hins with the target,

and asteroids denote the hins without the target. The primary target hin is explicitly

indicated hv an arrow. Similar results have been obtained for the 10 VV measurements.

While Figs. 5-5(a) and (h) indicate that the primary target hin can he separated from bins

without the target, in general, we have to conclude that the Lyapunov exponent estimated

by methods equivalent or similar to conventional means cannot he used for distinguishing

sea clutter data with and without targets.

Next let us examine whether the scale-dependent exponent corresponding to large

scale may be useful for detecting targets within sea clutter data. By scale-dependent

exponent, we mean that if we use the least-squares fit to the linearly or quasi-linearly

increasing part of the A(k) curves of different shells, the slopes of those lines depend

on which shells are used for computing the A(k) curves. In other words, if we plot the

exponent estimated this way against the size of the shell, then the exponent varies

with the shell size or the scale. The scale-dependent exponent has been used in the

study of noise-induced chaos in an optically injected semiconductor laser model to

examine how noise affects different scales of dynamic systems [26]. We now focus on the

exponent corresponding to large shell or scale. Figures 5-6(a)-(j) show the variations of

the scale-dependent exponent corresponding to large scale with the 14 range hins for the











Primary
JTarget Bin


&z 05


0 5 10 15

(d)

05


0 5 10 15

1x
(f)~*


0 5 10 15

(c) J




0 5 10 15

(e) +




0 5 10 15


5 10 15


00 5 10 15 00 5v 10 `15
Bln Number Bin Number

Figure 5-5. Variations of the Lyapunov exponent A estimated by conventional methods vs.
the 14 range bins for the 10 HHI- measurements. Open circles denote the range
bins with target, while denote the bins without target. The primary target
bin is explicitly indicated by an arrow.


same 10 HH measurements as studied in Fig. 5-5. We observe that the primary target bin

can be easily separated from the range bins without the target, since the scale-dependent

exponent for the primary target bin is much larger than those for the bins without the

target.

It is thus clear that the scale-dependent exponent corresponding to large scale is

very useful for distinguishing sea clutter data with and without targets. This -II__- -;

that sea clutter may contain interesting dynamic features and that the scale-dependent

exponent may be an important parameter for target detection within sea clutter. This












(a) Primar (b)
Target Bin

0505 /q

0g 5 10 15 0 5 10 15

(c) (d)

4z 05 0


0 5 10 15 0 5 10 15

(e) ? (f) I

A 05 /05


0 5 10 15 0O 5 10 15

(g) / (h)

Az 05 05


0 0
0 5 10 15 0 5 10 15



4z 05 / 05

O' 0
0 5 10 15 0 5 10 15
Bln Number Bin Number

Figure 5-6. Variations of the scale-dependent exponent corresponding to large scale vs. the
range bins for the 10 HHI- measurements. Open circles denote the range bins
with target, while denote the bins without target. The primary target bin is
explicitly indicated by an arrow.


example clearly signifies the importance of incorporating the concept of scale in a measure.

In fact, the concept of scale is only incorporated in the time dependent exponent A(k)

curves [21, 22] in a static manner. In ('! .pter 6, we shall see that when a measure

dynamically incorporates the concept of scale, it will become much more powerful.

Note that one difficulty of using the scale-dependent exponent for target detection

within sea clutter is that we may need to choose a suitable scale for estimating the

exponent. This makes the method not easy to use, and it is hard to make the method

automatic.










5.4 Target Detection within Sea Clutter by Power-law Sensitivity to Initial
Conditions (PSIC)

In this section, we show how the concept of PSIC can he applied to detect target

within sea clutter data. Examples of the A(k) vs. In k curves for the range hins without

target and those with target of one measurement are shown in Figs. 5-7(a) and (b),

respectively, where the curves denoted by asteroids are for data without the target,

while the curves denoted by open circles are for data with the target. We observe that

the curves are fairly linear for the first a few samples. Also notice that the slopes of

the curves for the data with the target are much larger than those for the data without

the target. For convenience, we denote the slope of the curve by the parameter /9. To

better appreciate the variation of the /9 parameter with the range hims, we normalize /9

of each hin by the nmaxinial /9 value of the 14 range hins within the single measurement.

Fig. 5-8 shows the variation of the /9 parameter with the 14 range hims. It is clear that

the /9 parameter can he used to distinguish sea clutter data with and without target.

Interestingly, the feature shown in Fig. 5-8 is generically true for all the measurements. It

is worth pointing out that usually the A(k) vs. In k curves for different scales are almost

parallel, thus the estimated /9 parameter is relatively less dependent on the scale. This is a

very nice feature of this method.

Let us examine if a robust detector for detecting targets within sea clutter can he

developed based on the /9 parameter. We have systematically studied 280 time series

of the sea clutter data measured under various sea and weather conditions. To better

appreciate the detection performance, we have first only focused on hins with primary

targets, but omitted those with secondary targets, since sometimes it is hard to determine

whether a hin with secondary target really hits a target or not. After omitting the range

hin data with secondary targets, the frequencies for the /9 parameter under the two

hypotheses (the hins without targets and those with primary targets) for HH datasets





























0.5'
0 0.5 1 1.5 2 2.5 3 3.5

In k


In k


Figure 5-7.


Examples of the A(k) vs. In k curves for range bins (a) without and (b) with
target. Open circles denote the range bins with target, while denote the bins
without target.


Primary
Target Bin


0.9


co. 0.85


0.75


x


0.7'
0 2 4


8
Bin Number


10 12 14


Figure 5-8.


Variation of the parameter with the 14 range bins. Open circles denote the
range bins with target, while denote the bins without target.


are shown in Fig. 5-9. We observe that the frequencies completely separate for the HH

datasets. This means the detection accuracy can be 1011' .

































no targets
12 primary targets

10


a,

LL


n mm


t0.4 0.5 0.6 0.7 0.8 0.9 1


Figure 5-9. Frequencies of the bins without
the HH datasets.


targets and the bins with primary targets for









CHAPTER 6
1\ULTISCALE ANALYSIS BY SCALE-DEPENDENT LYAPUNOV EXPONENT
(SDLE)

In C'!. Ilter 1, we have emphasized the importance of developing scale-dependent

measures to simultaneously characterize behaviors of complex multiscaled signals on a

wide range of scales. In this chapter, we shall develop an effective algorithm to compute

an excellent multiscale measure, the scale-dependent Lyapunov exponent (SDLE), and

show that the SDLE can readily classify various types of complex motions, including truly

low-dimensional chaos, noisy chaos, noise-induced chaos, processes defined by power-law

sensitivity to initial conditions (PSIC), and complex motions with chaotic behavior on

small scales but diffusive behavior on large scales. Finally, we shall discuss how the SDLE

can help detect hidden frequencies in large scale orderly motions.

6.1 Basic Theory

The scale-dependent Lyapunov exponent (SDLE) is a variant of FSLE [126]. The

algorithm for calculating the FSLE is very similar to the Wolf et al's algorithm [127].

It computes the average r-fold time by monitoring the divergence between a reference

trajectory and a perturbed trajectory. To do so, it needs to define 1.. I.rest neighbors",

as well as needs to perform, from time to time, a renormalization when the distance

between the reference and the perturbed trajectory becomes too large. Such a procedure

requires very long time series, and therefore, is not practical. To facilitate derivation of

a fast algorithm that works on short data, as well as to ease discussion of continuous but

non-differentiable stochastic processes, we cast the definition of the SDLE as follows.

Consider an ensemble of trajectories. Denote the initial separation between two

nearby trajectories by co, and their averagee separation at time t and t + at by et and et as,

respectively. Being defined in an average sense, et and et as can he readily computed from

any processes, even if they are non-differentiable. Next we examine the relation between et









and et+nt, where at is small. When at 0 we have,


et+nt = etex(tt)nt (6-1)


where A(et) is the SDLE. It is given by

In et+nt In et
A(et) = (6-2)


Given a time series data, the smallest at possible is the sampling time 7r.

The definition of the SDLE so_~- -;- Ma simple ensemble average based scheme to

compute it. A straightforward way would be to find all the pairs of vectors in the phase

space with their distance approximately e, and then calculate their average distance after a

time at. The first half of this description amounts to introducing a shell (indexed as k),


Ek | Vs Vy | < k Ek(6-3)


where 1%, Vj are reconstructed vectors, Ek, (the radius of the shell) and A~rk (the width

of the shell) are arbitrarily chosen small distances. Such a shell may be considered as

a differential element that would facilitate computation of conditional probability. To

expedite computation, it is advantageous to introduce a sequence of shells, k = 1, 2, 3, -

Note that this computational procedure is similar to that for computing the so-called

time-dependent exponent (TDE) curves [21, 22, 37].

With all these shells, we can then monitor the evolution of all of the pairs of vectors

(1M, Vj) within a shell and take average. When each shell is very thin, by assuming that

the order of averaging and taking logarithm in Eq. 6-2 can be interchanged, we have





where t and at are integers in unit of the sampling time, and the angle brackets denote

average within a shell. Note that contributions to the SDLE at a specific scale from

different shells can be combined, with the weight for each shell being determined by the









number of the pairs of vectors (1M, Vyj) in that shell. In the following, to see better how

each shell characterizes the dynamics of the data on different scales, we shall plot the A(e)

curves for different shells separately.

In the above formulation, it is implicitly assumed that the initial separation, ||% 1- Vy ||l,

aligns with the most unstable direction instantly. For high-dimensional systems, this is

not true, especially when the growth rate is non-uniform and/or the eigenvectors of the

Jacobian are non-normal. Fortunately, the problem is not as serious as one might be

concerned, since our shells are not infinitesimal. When computing the TDE curves [21,

22, 37], we have found that when difficulties arise, it is often sufficient to introduce an

additional condition,

I i> m 1 L6-5

when finding pairs of vectors within each shell. Such a scheme also works well when

computing the SDLE. This means that, after taking a time comparable to the embedding

window (m 1)L, it would be safe to assume that the initial separation has evolved to the

most unstable direction of the motion.

Before proceeding on, we wish to emphasize the 1!! ri ~ difference between our

algorithm and the standard method for calculating the FSLE. As we have pointed out,

to compute the FSLE, two trajectories, one as reference, another as perturbed, have to

be defined. This requires huge amounts of data. Our algorithm avoids this by employing

two critical operations to fully utilize information about the time evolution of the data: (i)

The reference and the perturbed trajectories are replaced by time evolution of all pairs of

vectors satisfying the inequality (6-5) and falling within a shell, and (ii) introduction of

a sequence of shells ensures that the number of pairs of vectors within the shells is large

while the ensemble average within each shell is well defined. Let the number of points

needed to compute the FSLE by standard methods be NV. These two operations imply

that the method described here only needs about z/Vpoints to compute the SDLE. In the










following, we shall illustrate the effectiveness of our algorithm by examining various types

of complex motions.

6.2 Classification of Complex Motions

T> understand the SDLE as well as appreciate its power, we apply it to classify

various types of complex motions.

6.2.1 Chaos, Noisy Chaos, and Noise-induced Chaos

Obviously, for truly low-dimensional chaos, A(e) equals the largest positive Lyapunov

exponent, and hence, must he independent of e over a wide range of scales. For noisy

chaos, we expect A(e) to depend on small e. T> illustrate both features, we consider the

chaotic Lorenz system with stochastic forcing described by Eq. 2-5. Figure 6-1(a) shows

five curves, for the cases of D = 0, 1, 2, 3, 4. The computations are done with 10000 points

and m = 4, L = 2. We observe a few interesting features:

For the clean chaotic signal, A(e) slightly fluctuates around a constant (which
numerically equals the largest positive Lyapunov exponent) when e is smaller than a
threshold value which is determined by the size of the chaotic attractor. The reason
for the small fluctuations in A(e) is that the divergence rate varies from one region of
the attractor to another.

When there is stochastic forcing, A(e) is no longer a constant when e is small, but
increases as -y In e when the scale e is decreased. The coefficient ]* does not seem to
depend on the strength of the noise. This feature -II__- -is that entropy generation is
infinite when the scale e approaches to zero. Note that the relation of A(e) ~ -y In e
has also been observed for the FSLE and the e-entropy. In fact, such a relation can
he readily proven for the e-entropy.

When the noise is increased, the part of the curve with A(e) ~ -y 1n e shifts to the
right. In fact, little chaotic signature can he identified when D is increased beyond
:3. When noise is not too -1i~~e.: this feature can he readily used to quantify the
strength of noise.

Next we consider noise-induced chaos. To illustrate the idea, we follow [2:3] and study

the noisy logistic map

r,w+1 = p~r, (1 r,w) + P,, O < r,z < 1 (6-6)










2.5 0.7
(a) LorenZ0. (b) Logistic


2 -05- (1): slope =0.28

S0.4 h (2): slope = 0


D. = o0.
D = 3 0.1


~D=4
0.5 -v 0
-1.5 -1 -0.5 -2 -1.5 -1 -0.5 0
log ,E log ,E

Figure 6-1. Scale-dependent Lyapunov exponent A(e) curves for (a) the clean and the
noisy Lorenz system, and (b) the noise-induced chaos in the logistic map.
Curves from different shells are designated by different symbols.


where p is the bifurcation parameter, and P, is a Gaussian random variable with

zero mean and standard deviation o-. In [23], we reported that at p = 3.74 and

o- = 0.002, noise-induced chaos occurs, and thought that it may be difficult to distinguish

noise-induced chaos from clean chaos. In Fig. 6-1(b), we have plotted the A(et) for this

particular noise-induced chaos. The computation was done with m = 4, L = 1 and 10000

points. We observe that Fig. 6-1(b) is very similar to the curves of noisy chaos plotted in

Fig. 6-1(a). Hence, noise-induced chaos is similar to noisy chaos, but different from clean

chaos .

At this point, it is worth making two comments: (i) On very small scales, the effect

of measurement noise is similar to that of dynamic noise. (ii) The A(e) curves shown in

Fig. 6-1 are based on a fairly small shell. The curves computed based on larger shells

collapse on the right part of the curves shown in Fig. 6-1. Because of this, for chaotic

systems, one or a few small shells would be sufficient. If one wishes to know the behavior

of A on ever smaller scales, one has to use longer and longer time series.










1x 10-


35-








-5
10- 100


Figure 6-2. Scale-dependent Lyapunov exponent A(e) curve for the Mackey-Glass system.
The computation was done with ni = 5, L = 1, and 5000 points sampled with a
time interval of 6.


Finally, we consider the Mackey-Glass delay differential system [39], defined by

Eq. 2-6. The system has two positive Lyapunov exponents, with the largest Lyapunov

exponent close to 0.007 [21]. Having two positive Lyapunov exponents while the value of

the largest Lyapunov exponent of the system is not much greater than 0, one might he

concerned that it may be difficult to compute the SDLE of the system. This is not the

case. In fact, this system can he analyzed as straightforwardly as other dynamical systems

including the Henon map and the Rossler system. An example of the A(e) curve is shown

in Fig. 6-2, where we have followed [21] and used ni = 5, L = 1, and 5000 points sampled

with a time interval of 6. Clearly, we observe a well defined plateau, with its value close to

0.007. This example illustrates that when computing the SDLE, one does not need to be

very concerned about non-uniform growth rate in high-dimensional systems.

Up till now, we have focused on the positive portion of A(es). It turns out that when t

is large, A(e) becomes oscillatory, with mean about 0. Denote the corresponding scales by










ea, and call them the limiting or characteristic scales. They are the stationary portion of

et, and hence, they may still be a function of time. We have found that the limiting scales

capture the structured component of the data. This feature will be further illustrated

later, when we discuss identification of hidden frequencies. Therefore,the positive portion

of A(et) and the concept of limiting scale provide a comprehensive characterization of the

signals .

6.2.2 Processes Defined by Power-law Sensitivity to Initial Conditions (PSIC)

In C'!s Ilter 3, we have seen that power-law sensitivity to initial conditions (PSIC)

provides a common foundation for chaos theory and random fractal theory. Here, we

examine whether the SDLE is able to characterize processes defined by PSIC. Pleasingly,

this is indeed so. Below, we derive a simple equation relating the A, and q of PSIC to the

SDLE.

First, we recall that PSIC is defined by


( = lim = 1 ( ) )


Since et = ||av(0)||s,, it is now more convenient to express the SDLE as a function of (c.

Using Eq. 6-2, we find that
(()=In $t+nt In (a(67


When at 0, 1+ (1 q)Agt > (1 q)XAft. Simplifying Eq. 6-7, we obtain


A((s) = Aq tq-1 (6-8)

We now consider three cases:

*For chaotic motions, q = 1, therefore,


A((s) = Xq = COnStant









F'or 1/ fB noise, we, hae = 1_ (Eq. :328) and A," = H (Eq. :329). TIherefor~el


A((s) = Hg, H

For Le~vy fightsl weit have q = 1- a? (Eq. :3-38) and A,"l = 1/0~ (Eq. :3-39). Therefore,





6.2.3 Complex Motions with Multiple Scaling Behaviors

Some dynamical systems may exhibit multiple scaling behaviors, such as chaotic

behavior on small scales but diffusive behavior on large scales. To see how the SDLE can

characterize such systems, we follow Cencini et al. [128] and study the following map,


Xrn+l = [Xrn] + F(xrn [Xrnl) + aife (6-9)


where [r,z] denotes the integer part of r,t, r7< is a noise uniformly distributed in the interval

[-1, 1], o- is a parameter quantifying the strength of noise, and Fly) is given by


Fly) = (2 + A) y if y E [0, 1/2) (6-10)
(2 + A) y (1 + a) if y E (1/2, 1]

The map Fly) is shown in Fig. 6-3 as the dashed lines. It gives a chaotic dynamics with

a positive Lyapunov exponent In(2 + a). On the other hand, the term [:re] introduces a

random walk on integer grids.

It turns out this system is very easy to analyze. When a = 0.4, with only 5000 points

and m = 2, L = 1, we can resolve both the chaotic behavior on very small scales, and the

normal diffusive behavior (with slope -2) on large scales. See Fig. 6-4(a).

We now ask a question: Given a small dataset, which type of behavior, the chaotic

or the diffusive, is resolved first? To answer this, we have tried to compute the SDLE

with only 500 points. The result is shown in Fig. 6-4(b). It is interesting to observe that

the chaotic behavior can he well resolved by only a few hundred points. However, the




















(30.6

S0.4
LL.


1
T


-0.2'
0


Figure 6-3.


The function F(x) (Eq. 6-10) for a = 0.4 is shown as the dashed lines. The
function G(x) (Eq. 6-11) is an approximation of F(x), obtained using 40
intervals of slope 0. In the case of noise-induced chaos discussed in the paper,
G(x) is obtained from F(x) using 104 HinerValS Of Slope 0.9.


100











10-2


100











10-2


clean data
noisy data


10-4 10-3 10-2 10 1


10-4 10-3 10-2


Scale-dependent Lyapunov exponent A(e) for
(a) 5000 points were used, for the noisy case,
used.


Figure 6-4.


the model described by Eq. 6-9.
o- = 0.001. (b) 500 points were









diffusive behavior needs much more data to resolve. Intuitively, this makes sense, since

the diffusive behavior amounts to a Brownian motion on the integer grids and is of much

higher dimension than the small scale chaotic behavior. Therefore, more data are needed

to resolve it.

We have also studied the noisy map. The resulting SDLE for a = 0.001 is shown in

Fig. 6-4(a), as squares. We have again used 5000 points. While the behavior of the SDLE

-II- -_ -r ;noisy dynamics, with 5000 points, we are not able to well resolve the relation

A(e) ~ In e. This indicates that for the noisy map, on very small scales, the dimension is

very high.

Map (6-9) can be modified to give rise to an interesting system with noise-induced

chaos. This can be done by replacing the function Fly) in map (6-9) by G(y) to obtain

the following map [128],

xt+l = [Xt] + G(xt [xt]) + arlt (6-11)

where Orl is a noise uniformly distributed in the interval [-1, 1], a is a parameter

quantifying the strength of noise, and G(y) is a piecewise linear function which approximates

Fly) of Eq. 6-10. An example of G(y) is shown in Fig. 6-3. In our numerical simulations,

we have followed Cencini et al. [128] and used 104 interValS Of Slope 0.9 to obtain G(y).

With such a choice of G(y), in the absence of noise, the time evolution described by

the map (6-11) is nonchaotic, since the largest Lyapunov exponent In(0.9) is negative.

With appropriate noise level (e.g., a = 10-4 or 10-3), the SDLE for the system becomes

indistinguishable to the noisy SDLE shown in Fig. 6-4 for the map (6-9). Having a

diffusive regime on large scales, this is a more complicated noise-induced chaos than the

one we have found from the logistic map.

Before proceeding on, we make a comment on the computation of the SDLE from

deterministic systems with negative largest Lyapunov exponents, such as the map (6-11)

without noise. A transient-free time series from such systems is a constant time series.

Therefore, there is no need to compute the SDLE or other metrics. When the time series









contains transients, if the time series is sampled with high enough sampling frequency,

then the SDLE is negative. In the case of simple exponential decay to a fixed point, such

as expressible as e-xt, where A > 0, one can readily prove that the SDLE is -A. Since such

systems are not complex, we shall not be further concerned about them.

6.3 Characterizing Hidden Frequencies

Defining and characterizing large scale orderly motions is a significant issue in many

disciplines of science. One of the most important types of large scale orderly motions is

the oscillatory motions. An interesting type of oscillatory motions is associated with the

so-called hidden frequency phenomenon. That is, when the dynamics of a complicated

system is monitored through the temporal evolution of a variable x, Fourier analysis of

x(t) may not -II---- -1 any oscillatory motions. However, if the dynamics of the system is

monitored through the evolution of another variable, 11-, z, then the Fourier transform of

z(t) may contain a well-defined spectral peak, indicating oscillatory motions. An example

is the chaotic Lorenz system described by Eq. 2-5. In Figs. 6-5 (a-c), we have shown the

power-spectral density (PSD) of the x, y, z components of the system. We observe that the

PSD of x(t) and y(t) are simply broad. However, the PSD of z(t) shows up a very sharp

spectral peak. Recall that geometrically, the Lorenz attractor consists of two scrolls (see

Fig. 6-6). The sharp spectral peak in the PSD of z(t) of the Lorenz system is due to the

circular motions along either of the scrolls.

The above example illustrates that if the dynamics of a system contains a hidden

frequency that cannot be revealed by the Fourier transform of a measured variable

( 11-, x(t)), then in order to reveal the hidden frequency, one has to embed x(t) to a

suitable phase space. This idea has led to the development of two interesting methods

for identifying hidden frequencies. One method is proposed by Ortega [129, 130], by

computing the temporal evolution of density measures in the reconstructed phase space.

Another is proposed by C'I. I i. et al. [131], by taking singular value decomposition of local

neighbors. The Ortega's method has been applied to an experimental time series recorded












10 "

104

102 10-4~
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
10 Frequency f 14Frequency f
1(b) (e)

106
,8100
4
104

1021 10-4
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
109Frequency f 14Frequency f

106 100





1031 1 10-4
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Frequency f Frequency f

F'igur~e 6-5. P~ower-spectral density (PSD>) for (a) x(t), (b) y(t), (c) z(t), (d) ei)(t), (e)



from a far-infrared laser in a chaotic state. The laser dataset can be downloaded from

the link http://www-psy ch. stanf ord .edu/~ andreas/Time- Ser ies /Sant aFe .htm1. It

contains 10000 points, sampled with a time interval of 80 ns. Fig. 6-7(a) shows the laser

dataset. The PSD of the data is shown in Fig. 6-7(b), where one observes a sharp peak

around 1.7 MHz. Fig. 6-7(c) shows the PSD of the density time series, where one notes

an additional spectral peak around 37 kHz. This peak is due to the envelope of chaotic

pulsations, which is discernable from Fig. 6-7(a).