
Full Citation 
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Title: 
Ensemble characteristics of visual evoked cortical potentials in noise 

Alternate Title: 
Visual evoked cortical potentials in noise 

Physical Description: 
viii, 97 leaves. : illus. ; 28 cm. 

Language: 
English 

Creator: 
Doyle, Timothy C., 1943 

Publication Date: 
1969 

Copyright Date: 
1969 
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Subject: 
Brain ( lcsh ) Electrophysiology ( lcsh ) Noise ( lcsh ) Electrical Engineering thesis Ph. D Dissertations, Academic  Electrical Engineering  UF 

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bibliography ( marcgt ) nonfiction ( marcgt ) 
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Thesis: 
ThesisUniversity of Florida, 1969. 

Bibliography: 
Bibliography: leaves 9296. 

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Also available on World Wide Web 

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Manuscript copy. 

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Vita. 
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UF00097755 

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VID00001 

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University of Florida 

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University of Florida 

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All rights reserved by the source institution and holding location. 

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alephbibnum  000955719 oclc  16969575 notis  AER8348 

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ENSEMBLE CHARACTERISTICS OF VISUAL EVOKED
CORTICAL POTENTIALS IN NOISE
By
TIMOTHY C. DOYLE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT O'F THE REQUIREMENTS FOR THE
DEGRIF OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1969
ACKNO1LE I [CIEiTS
The author wishes to express his sincere appreciation to
Dr. D.G. Childers, chairman of his supervisory committee, for his
counsel and encouragement. He also wishes to thank Dr. N.W. Pernr
and Dr. J.R. Smith, the other members of his supervisory ccmvittee,
and Dr. A.H. Nevis for their guidance. He wouldd like to acknowledge
the Visual Sciences Laboratory and staff for their facilities and
help.
TABLE OF CONTENTS
Page
ACKNOWLEDG NTS . . . . . . . . .. .. ii
LIST OF TABLES . . . . . . . . ... .. v.
LIST OF FIGURES . . . . . . . . . . . vi
ABSTRACT . . . . . . . . . . . . vi ii
CHAPTER
1 INTRODUCTION .............. ......... 1
2 SEQUENTIAL AND ENSEMBLE AVERAGING . . . . . 4
2.1 Sequential Averaging . . . . ... . 4
2.2 Variability of the VER . . . . . . . 7
2.3 Theoretical Models . . . . . . .. 10
2.4 MultiSubject Ensemble Model . . . . . . 12
2.5 Ensemble Averaging . . . . . . .... .13
2.6 Methods . . . . . . . .... ..... .15
2.7 Objectives . . . . . . . . ... . 16
3 RESULTS OF ENSEMBLE AVERAGING . . . . .... .19
3.1 Introduction . . . . . . . . 19
3.2 Selection of Bandwidth . . . . . . .. 21
3.3 Ensemble Evoked Response . . . . . . .. 22
3.4 Noise in the Ensemble Evoked Response . . .. 28
3.5 Quality of the Ensemble Evoked Response Estimate 32
3.6 Variations in the Ensemble Evoked Response with
Stimulus Number . . . . . . . . .. 36
3.7 Average of Ensemble Evoked Responses . . . .. 39
3.8 Summary . . . . . . . .... ..... .41
4 THEORETICAL ASSUMPTIONS UNDERLYING RESPONSE AVERAGING .. 43
4.1 Stationarity of the Background Noise . . ... .44
4.2 Dependence of Signal and Noise . . . . .. 56
4.3 Additivity of Signal and Noise . . . . .. 59
4.4 Summary . . . . . . . .... ..... .67
5 SULMMARY . . . . . . . . . . . . 68
iv
Page
APPENDIX
A EXPERIMENTAL PROCEDURE ................. 75
B SLIPPLEMENTARY FIGURES AND TABLES . . . . ... 78
REFERENCES. .. . . . . . . . . . . .. 92
BIOGRAPHICAL SKETCH ......... ............. 97
LIST OF TABLES
Page
3.1 Average Correlations for Figure 3.2 . . . ... 25
3.2 Average Correlations for Figure 3.4 . . . . .. 28
4.1 Maximum Number of Standard Deviations of Ensemble Control
Average about a Zero Mean . . . . . . .... 47
4.2 Maximum Number of Standard Deviations of + Estimate
about a Zero Mean . . . . . . . .... 48
4.3 Maximum Number of Standard Deviations (times 0.1) of
EER Estimate about Zero Mean . . . . . .... .49
4.4 RMS Values in uv of + Averages Under Stimulus and
NonStimulus Conditions . . . . . . .... .57
B.1
through
B.7 Response Correlations Coefficients . . . . .. 88
B.8
through
B.13 EER Correlation Coefficients for the 0.3 to 50 Hz
Bandwidth . . . . . . . . ... ...... 89
B.14
through
B.20 Control Correlation Coefficients . . . . .... 90
B.21 Values for Cf(t, tT) in Figure 4.1 . . . . .. 91
LIST OF FIGURES
Page
2.1 Ensemble Versus Sequential Averaging . . . . .. 14
3.1 Bandwjidth Coinparison . . . . . . . .... 21
3.2 Ensemble Evoked Response . . . . . . .... 23
3.3 Superimposed Ensemble Evoked Response . . .... 27
3.4 Ensemble Control Averages . . . . . .... .29
3.5 Noise Amplitude in EER . . . . . . . . .. 31
3.6 Superimposed Ensemble Standard Deviations ........ 32
3.7 EER Confidence Intervals . . . . . . .... .. 34
3.8 \ER Confidence Intervals .. . . . . . ..... 35
3.9 Amplitude Fluctuations of the Primary Response ...... 38
3.10 Latency Fluctuations of the Three Primnary Components .. 38
3.11 Avcrage of Response and Control Ensemble Averages . . 10
4.1 Ensemble Autocovariance Function for First Control 51
4.2 Superimposed Ensemble Standard Deviations ....... 53
4.3 Histogram Estimates of Ensemble pdf for First Control .55
4.4 S/N Ratio Versus Number of SLurations . . . . .. 65
B.1 EER for Two Bandwidths . . . . . ..... .. 79
B.2 Ensemble Control Average for Two Bandwidths ...... SO
B.3 + Ensemble Responses for Tw'.o Bandwidths ........ 81
B.4 + Ensemble Control Averages for Two Bandwidths ..... 82
B.5 Ensemble Standard Deviations Under Stimulus and
NonStimulus Conditions. . . . . . ..... 83
B.6 Ensemble Autocovariance Function for First Recpoinse . 84
Vi1
Page
B.7 Ensenmble Autocovariance Function for Fourth Pesponse .
B.S Histogram Estimates of Ensemble pdf for First Response 86
B.9 S/N Ratio Versus Number of Summations . . . .... 87
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
E'SEMBLE CHARRACTEPJSTICS OF VISUAL
EVOKED CORTICAL POTENTIALS IN NOISE
By
Timothy C. Dole
August, 1969
Chairman: Dr. D.G. Childers
Major Department: Electrical Engineering
To date, most investigations of evoked potentials monitored at
the human scalp have utilized the average of a series of responses
recorded from a single subject to define components of the evoked
response, ER. This technique has provided a fairly consistent estimate
of an individual's average ER. However, with this method it is not
uncommon to obtain quite dissimilar averages from different subjects
under identical experimental conditions. This contrast in the ER
estimates across a population has hindered the interpretation of the
effects of stimulus parameters on the ER and frustrated the attempt
to use this average for clinical diagnosis.
The present study investigated one method of obtaining informa
tion about the character of the response to visual stimuli common to
a considered normal subject population. The estimation technique
utilized was to average across an ensemble of fifty subject EEG's
containing evol.ed activity. This method permitted estimation of
common intersubject response activity evoked by a single stimulus.
viii
By averaging the resultant waveforms of this ensemble average across
stimuli, a reliable, relatively noisefree intersubject response
template was obtained.
The resemblance of the ensemble of cortical potentials to a
classical stochastic process enabled the author to investigate
three prevalent assumptions about the statistical character of the
evoked response and background noise activity. These three assumptions
are: (1) the response and background noise are independent; (2) the
background noise is stationary; and (3) the response and noise are
additive. The results presented in this dissertation lend some
measure of support to each of the three assumptions.
CHAPTER 1
I NTRODUCT ION
The timevarying electroencephalographic potential specifically
elicited by a single controlled sensory stimulus is referred to as
an evoked potential or evoked response (ER). An invariant evoked
response to controllable stimulus parameters, monitored from the
human scalp, would provide an excellent empirical diagnostic tool
for brain research. How..ever, this has eluded all investigators
because of tw:o inherent limitations. First, the parameters affecting
the ER, such as physiological variability of the subject during
stimulation, are usually unkrcno.n and uncontrollable. Second, the
observability of a single response is severely handicapped by the
interaction of the desired signal (ER) w..ith the background bioelectric
ongoing activity.
The physiological variability, or state of the subject, intro
duces a random variable into i..lat micht othenrise have been a
deterministic response. The ongoing background activity masks and
distorts the ER such that it usually is undetectable at the scalp.
Thus, these tw.o sources of variability deny the investigator anything
but tenuous conclusions about the relationship of a single bioelectric
response to specific stimulus parameters. In order to determine the
general characteristics of an ER, several data processing techniques
have been utilized. The most co!r.on technique is the sequential
average (see (Capter 2) of a series of responses recorded from an
individual subject. This average is fairly consistent for a parti
cular subject under identical stimulus conditions. However, the
averages obtained from several subjects often appear to be quite
dissimilar. Thus, it is difficult to establish a certain average
response w.aveform, elicited by specific stimulus parameters across a
subject population.
In the present study the average of a single response across a
subject population is investigated. This average provides an estimate
of the characteristics of the response to a particular stimulus which
are common to an ensemble of subjects. From these common character
istics one can develop a model or template of the average response
activity across the ensemble. The changes in this response average
with repetitions of a specific stimulus can also be investigated. In
addition, the collection of subject EEG w.aveforms can be viewed as an
ensemble of independent sample functions from a stochastic process.
Thus, assumptions underlying the averaging process can be examined
by classical techniques. Tlese assumptions generally involve the
statistical nature of the evoked signal and background noise.
Therefore, the two major objectives of this investigation are:
(1) to develop a better understanding of the common characteristics
or norm of the evolved responses within a subject population; and
(2) to provide an empirical technique to evaluate the prevalent
statistical assumptions about the character of the evoked response
uand background potentials. Since at this time the response signal
cannot be separated from the background ncise, the results are
limited to some combination of the signal and noise rather than either
by itself. The characteristics of the noise under nonstimulus condi
tions are observed, but these may provide little insight into the
background noise activity during stimulation. Therefore, the second
objective is theoretically unattainable at the outset. However, the
results obtained in this study should contribute to the mkowledge of
the statistical character of evoked ongoing activity.
The results of this investigation are divided into tw.o chapters.
Chapter 3 explores the intersubject norm or ensemble average and the
transient changes in this average. Also examined are the nonstimulus
control estimates of the average background activity. Chapter 4 con
tains an examination of three assumptions about the signal and noise
which are necessary to theoretically justify results obtained by these
averaging techniques. In this examination the statistical character
of the ensemble of ER's with background noise and the ensemble of
resting bioelectrical potentials is investigated. To facilitate
understanding of the results presented in these chapters, Chapter 2
provides an overview of the background material and methodology for
this investigation.
CHAPTER 2
SEQUENTIAL AD!L) ENSEMBLE AVERAGING
The pea,topeak amplitude of the potentials recorded from the
scalp usually have a range of 9 to 50 v\' (Cobb, 1963). In most cases
the amplitude of the evoked activity in these potentials is below the
threshold of observability. It therefore becomes necessary for the
researcher to preprocess the scalp potentials to extract some charac
teristics of the ER. Generally, these processing techniques require
reiterative measures and estimate the response characteristics in
terms of probability statements or statistical averages.
2.1 Sequential Averaging
To date the most powerful statistical descriptor of the ER is the
sequential average which was implemented photographically by Galanbos
and Davis (1943) and later electromechanically by Dau.son (1954) and
electronically by Barlow (1957). Today this average is usually cal
culated by a general or special purpose digital computer. The sequen
tial average is essentially the cumulative\ sum or average of successive
trains of the scalp potentials which attend a series of repeated
identical stimuli. The mathematical expression for this average is
Nl
(t) f.(t+jT) for O
j=0
where fi(t) is the time varying EEG (electroencephalogram) recorded
from subject i and the stimulus cccurs periodically N times at t=0, T,
2T . .. The ~ indicates the above is an estimate of the true mean
5
m(t) for the range of t between 0 and T. If the stimuli are visual,
the result of this average is called a visual evoked response (VER).
Since the present study employs visual stimuli, most of the discussion
will focus on responses evoked in this modality.
The basic advantage of sequential averaging over other data pro
cessing methods is that it extracts an estimate (e.g., the VER) of the
response waveshape from the random potential fluctuations monitored
at the scalp. Other data processing techniques such as autocorrelation
(Barlow, 1959), spectral analysis (Childers, et al., 1968), matched
filtering (Negin, 1968), and synchronous correlation (Regan, 1966a)
may detect the periodicity, frequency composition, presence, or phase
locked components of the ER, respectively, but they do not produce the
w:aveshape information provided by averaging. In addition, the apparent
underlying integration mechanisms noted in most complex neurological
systems tend to justify or lend a degree of rationale to the averaging
technique, as Rosenblith (1959, p. 540) reasons:
In the handling of sensory information, organisms
behave as if they were acting on the basis of
activity averaged over substantial regions of the
nervous system, such a weighted averaging process
could be carried out by comparing the outputs of
a large number of neural elements. Since there are
at present no devices of performing this task
(even if one knew exactly where and how to perform
it), one chooses to average over an ensemble of
responses to repeated identical stimuli in order
to bring out certain typical aspects of behavior
of the nervous system.
The major disadvantages of sequential averaging are: (1) only
those components of the ER which are time locked to the stimulus will
be extracted; (2) the responses to succeeding stimuli may differ and
distort the average response estimate because of the effects of those
preceding them; and (3) the resultant average of many responses is
not identical under invariant stimulus conditions. By extracting only
the time locked components, the averaging process obscures any phase
and frequency deviations of the individual responses. In addition,
the degree of amplitude variation of these time locked components is
hidden within the resultant waveform. However, it is impossible to
improve the averaging process by weighting and phase locking each res
ponse without the a priori knowledge of the specific angle or amplitude
modulation which may affect a series of responses. Recent studies
by Harris and W'oody (1969) and Woody (196') have attempted to estimate
the degree of angle modulation of the ER by a correlation technique.
From this estimate, they' have suggested an adaptive filter to improve
signal extraction by averaging. However, to most investigators such
modulation is a perturbing factor distorting the estimate obtained by
averaging.
The second disadvantage is imposed by the practical limitation
that the large number of stimuli necessary to extract an average res
ponse is generally presented over a relatively short period of time.
This limitation introduces twno disturbing factors. The first factor
can be termed overlap interference, i.e., the transient tail of one
response overlaps and distorts the initial components of the subse
quent response. This usually occurs only with relatively high stimu
lus rates. The critical stimulus frequency at which this interaction
occurs is not clearly established, but the consensus of evidence
indicates that it lies in the region of 0.7 Hz (Barlow, 1960) to
2.7 Hz (Kitzsato, 1966). There is some indication that higher rates
are permissible if tie stimulus is presented at random intervals
(Ruchkin, 1965).
The second disturbing factor is habituation, which can generally
be described as the decrease in the subject's attention to monotonous
repetitive stimulation, h.'hich changes the successive responses. Since
the extraction by averaging of a reliable ER requires many repetitions
of the stimulus (see Perry and Childers, 1969) most investigators
discuss habituation in terms of the changes in consecutive averages of
large groups of successive responses. Therefore short term habituation,
particularly transient changes of die ER's to the initial stimuli, is
difficult to record. However, "erlin and Davis (1967) utiliing a
highly responsive subject, have noted a systematic doi.. i.'ard drift in
the amplitude of individual responses to successive auditory stimuli.
This result agrees with the less specific gradual .waxing and waning
decrease in aiplitude of consecutive averages (VER's) found by many
authors (e.g., Perry and Copenhaver, 1965; Haider, et al., 1964; and
GarciaAustt, et al., 1963). Bogac:, et al. (1960) have also noted
that habituation to visual stimuli is also reflected in increased
phase lag (latency) of certain averaged response components.
2.2 Variability of the VER
The third shortcoming of sequential averaging, the variability of
the resultant, poses the greatest dilemma to the researcher using this
tool. The VER's taken on different occasions from a single subject
are relatively stable over a period of time (Dustman and Beck, 1965);
how.ever, when r:e compared s the V'ER's obtained under identical stimulus
conditions from a population of subjects, variability of the waveform
makes it difficult to interpret the effect of certain stimulus para
meters on the human visual system. Some investigators have attempted
to reduce this intersubject variability by expanding the resultant
waveform into orthogonal sets for comparison, with little success
(see Donchin, 1966 and John, et al., 1964). Others have organized a
system of classification of the certain maximna and minima of the \ER
waveform in terlms of amplitudes and time delays latenciess) from stim
ulus onset (see Bergamini and Bergamasco, 1967 and Kooi and Bagchi,
1964). With this latter technique, investigators have been able to
occasionally detect trends of change common to a subject population
with gross variations in the stimulus parameters. Eut such findings
are again hampered severely by the variability of the VER. This
variability appears to be the product of changes in the psychological
state of the subject (e.g., attention) and the inherent statistical
character of the ER and background activity @.'erre and Smith, 1964).
In an attempt to understand the statistical properties of the ER
and background noise, a vast amount of effort has been concentrated
on the classification of the physiological origin of these scalp
potentials. However, to date there is only a rudimentary knowledge
of the mechanisms which underlie the generation of these EEG wave
forms (Elul, 1967). The present evidence indicates that EEG poten
tials arise from some complex integration of excitatory and inhibitory
post synaptic potentials, psp (Humphrey, 1968; Creutzfeldt and Kuhnt,
1967; Eccles, 1966 and Amassian, et al., 1964). The relationship
between these subcranial potentials and the bioelectric waveform
monitored at the scalp has been investigated in animals by De Lucchi,
et al. (1962). They have hypothesized that the scalp acts as a
further average of psp's. In similar investigations involving human
subjects, Cooper, et al. (1965) demonstrate that only an average of
those potentials which are widely synchronized can be observed in the
scalp. It is therefore interesting, as Rosenblith (1959) notes, that
even with this gross averaging of psp's by the scalp electrodes, we
still find variability of the ER to identical stimuli.
The amount of this variability produced by the interaction of the
response signal with the background activity (noise) is generally
unknown. To date, most investigations of this interaction have con
centrated on only one component of the noise (resting EEG), the
alpha rhythm. This component is defined by Brazier, et al. (1961,
p. 647) as the "rhythm, usually with frequency of 833 c/sec., of
almost sinusoidal form, in the posterior areas, present during relax
ation when the eyes are closed, attenuated during attention, particu
larly visual." The physiological origin of this regular rhythmic
activity is yet undetermined. However, Andersen and Andersson (1968)
point to the thalamTus as the source of the rhythmic impulses which
trigger the psp's monitored at the scalp. This agrees with the
observations of Creutzfeldt, et al. (1966). The alpha rhythm is the
most striking compncent of the resting EEG of the majority of normal
adults. This compcnont is so prevalent that a resonant peak: around
10 H: can readily be seen in the EEG frequency spectrum 0 alter,
et al., 1967). In addition, the late rhythmic component (after
discharge) of the ER and \TR has the general form and dimensions of
Lhe 3pcrOtartnec.s alpha wave (Bishop and Clare, 1953 and Barlow, 1960).
It is tiherefcre not unexpected tha: studies involving the relationship
of this predominant EEG component to evoked responses, particularly
those elicited by visual stimuli, received precedence.
A review of the studies focUtsing on this interaction with the VER
is provided in the recent work by Childers and Perry (1969). They
note that many investigators suspect two fcrms of S to 13 Hz.
activity: the first, the predominant nonprovoked EEG activity, and
the second, the components of the evoked response which are alpha
like in character. The evidence presented indicates that if two dis
tinct forms of activity exist, they probably share common neural
ensembles. However, this relationship is unclear and Childers and
Perry stress the need for further experimentation to secure the
additional information necessary to determine the interaction of
these components and other background activity with the ER.
2.3 Theoretical Models
Several investigators (e.g., Ruchkin, 1969; Kitasato and Hatsuda,
1965; Bendat, 1964; and Goldstein and Weiss, 1962) attempt to under
stand the mechanisms of the VER and hence its variability by developing
theoretical models of the statistical character of scalp potentials.
These studies generally rely on the hypothesis that the signal (ER)
and noise (background activity) are additive. This assumption permits
a certain amount of mathematical tractability to these models but
contains little physiological basis. However, it is justified in
part because of the ability to extract a somewhat reproducible esti
mate of the signal from the scalp potentials by sequential averaging.
For example, if the signal and noise were multiplicative, this
extraction technique would provide completely inconsistent results.
In order to facilitate statistical analysis of the scalp poten
tials, these model developers generate an artificial stochastic
process. This process is constructed by segmenting (usually at the
points of stimulation) the provoked EEG record into an ensemble of
shorter records. The statistical moments of this ensemble can then
be calculated from the empirical data and compared to the theoretical
model (e.g., the first moment is the sequential average). Unfortu
nately, few authors have progressed past the modeling stage to
compare the theory with practice. This gap exists, because, as
Barlow (1967) notes, the theoretical ability to separate the statis
tical components (i.e., signal and noise) does not have an experimen
tal counterpart. Such a separation would require a much greater
knowledge of the character of either the signal and/or noise and
their interdependence than is available at the present.
In addition, the assumptions made about the stochastic model are
generally violated in the empirical ensemble. For example, the
empirical statistics are derived from temporally related members,
which requires a stationary ergodic random process. This is highly
improbable if one notes the existence of a time varying mean (the
'ER signal). Also the members would probably have some statistical
dependence, reducing the quality of the empirical statistics.
Finally, each subject would have to be examined separately, .which
would eliminate the estimation of the intersubject statistics.
lHo.wever, is there a better stochastic model and is such a model
feasible? This question initiated the present study and led to the
development of the following model of a multisubject random process.
2.4 MillltiSubiect Ensemble Model
It was decided that a more appropriate empirical stochastic
process might be generated from an ensemble of individual EEG's
(i.e., each EEG which contains evoked activity is recorded from a
different subject). Such an ensemble would eliminate the temporal
and probably the statistical dependence of its members. Thus, this
ensemble would more accurately comply with the classical definition
of a stochastic process as a process composed of an ensemble of inde
pendent simultaneously monitored sample functions. The empirical
ensemble statistics would then provide a more precise evaluation of
the statistical assumptions necessary to theoretically justify
averaging to extract a consistent response estimate. In addition,
these ensemble statistics, particularly the first moment (average)
of the ensemble, would provide information about common intersubject
characteristics of the ER.
The inherent limitations of this artificial process are twofold.
First, the individual records are not simultaneously monitored, but
serially recorded over a several month period. Second, the evoked
activity in the member functions may result from several stochastic
processes instead of simply one. The first restriction becomes
significant if the nature of a normal adult EEG is dependent upon
the day of the year or the hour of the day. Fortunately, however,
the present evidence points to a relatively stable individual EEG,
particularly in frequency composition (Berkhout and Walter, 1968 and
Johnson and Ulett, 1959). The second limitation becomes important if
the population of ER's can be divided into discrete classes. To
date, there has been some indication (Perry, et al., 1968) that such
subdivisions might exist: however, the method of classification and
the number of classes is yet undetermined. Therefore, the present
research effort will assume a continuum of EEG and response types
which can be modeled tender the same stochastic process.
2.5 Ensemble Averaging
The ensemble mean or average provides an estimate of the common
intersubject response to a single stimulus. The mathematical
expression for this average is
1 N
m (t) Z f (t) (2.2)
Si= 1
where f.(t) is the EEC containing evoked activity recorded from
subject i out of a population of N subjects. This ensemble average,
VER, and unprocessed scalp potentials are schematically represented
in Figure 2.1. This figure demonstrates the basic difference in the
ensemble and sequential average (i.e., \TR). The ensemble average
provides an estimate of the intersubject average response to a single
stimulus and the VER provides the interstimulus average response for
a single subject. The ensemble average eliminates the intersubject
response variability while it reflects the changes in the average
response to a series of individual stimuli. The \VER average elimi
nates the interstimulus response variability while it reflects the
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individuality of a subject's average response.
The average of a set of \TER's from a subject ensemble equals the
average of the set of ensemble averages for a series of stimuli, pro
vided both averages are obtained from the same inclusive collection
of data. This double average provides an overall estimate of the
response activity by sacrificing all variability information. This
overall average has been suggested by Dawson, et al. (1968) to reduce
all excess information about the response. Therefore, this single
estimate might provide a general intersubject, interstimulus
response norm to specific stimulus parameters.
2.6 Methods
In the initial stages of this investigation it w.as decided that
an ensemble size (subject population) of fifty would provide
reasonable statistical accuracy and remain iithin the realm of experi
mental practicability. Both stimulus (signal) and nonstimiulus
(noise) trials were recorded from each of the subjects. The stimulus
was a monocular periodic flash of light 'which had a repetition rate
of about one flash per second. The stimulus was visually occluded
during noise (control) trials. The potentials recorded in these
trials were monitored from a pair of scalp electrodes vertically
oriented on the occiput midline. These potentials were amplified
by a factor of 100 K; separated into t.'o data channels (i.e., a
wide band, 0.3 to 50 H:, channel and a narrow band, 0.3 to 15 Hz,
channel); and recorded on magnetic tape.
These recorded potentials were then digitized and stored in an
IBM 1S00 computer in order to facilitate necessary mathematical
computation. Computer memory limitations restricted the amount of
data analyzed. Therefore, most of the results presented in this
study are derived from activity following ten stimulus triggers (both
signal and noise trials) selected from the periodic (approximately
1 Hz) stimulus train. The triggers chosen are the first through
fifth, tenth, twentieth, thirtieth, fortieth and fiftieth in the
repetitive stimulus train. All of the computation was performed on
the first 950 milliseconds of data following each trigger. A more
detailed discussion of the data collection and analysis procedure
is provided in Appendix A.
2.7 Objectives
The two objectives of this study, as stated in the introductory
chapter, are: (1) to develop a better understanding of the common
characteristics or norm of the evoked responses within a subject
population and (2) to provide mi empirical technique to evaluate
the prevalent statistical assumptions about the character of the ER
and background potentials. The intersubject ensembles recorded under
stimulus and nonstimulus conditions provide data from which both
these objectives can be pursued. The average across the ensemble
under stimulus conditions furnishes an estimate of the common inter
subject evoked activity. The data provided by these ensembles is
appropriate for the investigation of theoretical assumptions about
the EEG stochastic process since these potentials are a reasonable
facsimile of such a process.
In pursuit of the first objective the ensemble average across
fifty subjects under stimulus conditions is examined. It is antici
pated that the results from this average will reveal the components of
the evoked activity which are the most consistent across the subject
population. These components should reappear after each stimulus with
some degree of variability. The variability would probably be the
result of transient changes (e.g., habituation) in the intersubject
response with the number of stimuli, variations introduced by the
inherent error of the estimation process, or background noise poten
tials which may remain in the average. A transient variation should
result in some overall trend of change in the response average while
noise interference and estimation error should affect the average in
a more random manner. The transient trends of response estimate and
confidence in this estimate across a series of stimuli will be inves
tigated. The ensemble average under ronstimulus conditions is
expected to provide an approximation of the amount of noise contained
in the response estimate. The average of a collection of ensemble
averages should eliminate transient variability, reduce the components
of the background noise and provide a fairly stable intersubject
response template.
In order to theoretically justify the above response estimation
by averaging,it is necessary that (1) the background noise come from
a relatively wide sense stationary random process; (2) the signal and
noise processes be statistically independent; and (3) the signal and
noise be additive. Since the signal and noise potentials cannot be
separated at the present time, the above properties are generally
assumed to be true. The second objective of this study, evaluation
of these assumptions, is hindered by the inability to segregate the
signal and noise. However, by hypothesizing one or tw.o of these
assumptions an attempt can be made to empirically assess the third.
For example, by hypothesizing the signal and noise to be statistically
independent, the intersubject ensemble tnder nonstimulus1 conditions
should approximate the noise activity under stimulus conditions. The
stationarity (i.e., time dependence of the ensemble statistics) of
the background noise can then be examined using the resting EEG as
noise approximation. Evaluation of the second assumption can be
obtained in a similar manner by hypothesizing the third assumption
and examining the + ensemble average and ensemble variance under
stimulus and nonstimulus conditions. The assessment of the third
assumption is more difficult. However, by hypothesizing the first
t.o assumptions w.e can examine the growth of the signal to noise
(S/N) ratio, speculating the signal to be the ensemble average and
the noise the ensemble average under stimulus conditions. If the
signal and noise are additive, the S/N;. ratio should grow linearly
with the number of averages.
The results presented in the t1w.o succeeding chapters attempt to
realize the above objectives. The response activity common to the
subject ensemble is first examined, followed by the evaluation of the
statistical assumptions about the character of the ER and background
noise.
aIAPTER 3
RESULTS OF ENSEMBLE AVERAGING
3.1 Introduction
In this chapter the intersubject average response to a monocular
visual stimulus is examined. The average across the fifty subject
ensemble record (see Figure 2.1) is utilized to extract an estimate
of the response activity from the random scalp potentials. This
ensemble average should provide an intersubject response norm if:
(1) the assumptions mentioned in the previous chapter hold; (2) there
exist stimulus locked response components which are common to the
subject population; and (3) the amnplitude of the average background
activity in this estimate is much less than the amplitude of these
response components. The first condition, tie validity of assiriptions
about the response signal and background noise necessary to justify
averaging, is the subject of Chapter .1. The present chapter will
h.pothesi:e these assumptions to be true and examine the ensemble
average for any intersubject stimulus locked response components. In
addition, an estimate of the amount of background noise contained in
this average is investigated.
In, order to siLplify discussion of the ensemble average evoked
response estimate the term ensemble evoked response (EER) will be
utilized. This te:i is analogous to the term visual evoked response
(VER) in that it is not intended to iiply that the average response
estimate contains only evoked activin'. Theoretically the \ER and EER
contain noise components those variance decreases directly with the
number of sums in the average (see Perrn and Childers, 1969). An
estimate of the amount of noise in the EER is obtained from the average
across the ensemble muder nonstimulus conditions. This average pro
vides a series of control waveforms, each attending a particular
occluded stimuli, whose amplitudes should provide an estimate of the
background noise in the EER. This background noise activity which is
not related to the stimulus will introduce variations in the EER from
one stimulus to the next.
As noted in the previous chapter, other parameters ..hich could
affect the EER across stimuli are deviations of the estimate about
the true mean and variations in this response mean with stimulus
niuTiber. The deviation of the EER from the true mean is the result of
the variance of the potentials across the ensemble (see Bendat and
Piersol, 1966). Both the response signal and background noise contri
bute to this .variance since it is unlikely that the response activity
from every subject is identical and the noise is random. However,
since the signal activity generally has less amplitude (Perrn and
Childers, 1969) and greater sirmilarity across the ensemble than the
noise potential, the latter should account for most of the ensemble
variance. Therefore the estimation deviations of the EER should be
primarily the result of the background noise. The range of these
deviations can be predicted utilizing the Tchebycheff inequality and
ensemble variance. This prediction takes the form of confidence
intervals about the EER.
In order to examine the second parameter, the variations in the
EER caused by changes in the actual intersubject response, it is
necessary to separate response signal from the noise activity in the
ensemble average. Since this is not possible at this time, the
present investigation will generally be restricted to variations in
the more prominent components of the EER's which are fairly consistent
across stimuli. Since the stimuli are identical, these variations
should result in some trend of change rather dtan random fluctuations.
3.2 Selection of Bandwidth
The EER and other results presented in this chapter and the
succeeding chapter are obtained from the stimulus provoked and resting
EEG ensembles mentioned in Chapter 2. In addition, most of the dis
cussion of these results is limited to narrow band filtered (0.3 to
15 H:) potentials. Although the fidelity of this more restrictive
2 1
S4 Standard Deviation
o 3
0I 
Eandiuidth Comparison
Thi ensemble mean and standard deviation
for nro bandJwidths are SLpecr imposed.
bandwidth is not as good as the wider band potentials (0.3 to 50 Hz),
it still retains tie significant EEG potential fluctuations, as
Figure 3.1 demonstrates. In this figure the first and second ensemble
moments for both bandwidths are superimposed and reflect only minor
variations, including a slight phase shift introduced by the filter.
In addition, the upper limit of 15 Hz exceeds the minimum range of
ten times the frequency of stimulation suggested by Ciilders, et al.
(1968) to recover a VER average with good fidelity. Therefore, only
the narrow band averages which provide an accurate description of the
EEG activity will usually be discussed. However, many of the results
presented here are duplicated for the wider band potentials in
Appendix B. This appendix also contains other results which are not
pertinent to the discussion. Any reference to these results will
have a prefix B.
3.3 Ensemble Evoked Response
The ensemble evoked response, EER, to the selected photic stimuli
is provided in Figure 3.2. Each average is designated by the number
signifying which stimulus within the train evoked it. In each
waveform the flash stimulus occurs at the extreme left of each 950
millisecond response average. The first four ensemble response
averages can also be found in Figure 2.1, which depicts the method by
which the averages were obtained. The waveforms are segmented into
seven 135 millisecond intervals to facilitate comparison and dis
cussion. Again, it should be pointed out that each EER is only an
estimate of the actual evoked potential and contains both evoked and
background activity.
Interval
4
C
z 5
c,
' 1'
FIGURE 3.2
Ensemble Evoked Response
Ensemble averages for ten
selected stii:muli are
divided into 7 intervals
of 135 ;rlliseconds.
A cursory glance at the EER's in
Figure 3.2 provides the reader with little
indication that the fluctuating waveforms
contain any response activity. However,
if one utilizes the inverted sinusoid in
the second interval of the fourth res
ponse as an ER template, the most common
attribute of the w1aveform set becomes
more apparent. This inverted sinusoid,
although sometimes partially distorted,
appears in the second interval of each
response in the figure. The potential
fluctuations across the ten EER's also
share a degree of likeness in the latter
portion of the first and initial part of
the third intervals. However, neither
approaches the conformity found in the
second interval.
In order to better establish the
bounds of the response activity it is
necessary to find some quantitative
measure of this similarity across the
EER's. The degree of similarity between
two responses for a specific interval
can be quantified by a correlation
coefficient which would d provide a total of 45 correlations per
interval for the ten EER's of Figure 3.2. This coefficient, p, can
be expressed as:
1 x'
1= r (3.1)
where P is the time covariance between EER's x and v in interval
xy
i and Y~ and Y are the time standard deviations of these w.aveforms
x y
in this interval. How to reduce these paired correlations to a single
measure of the similarity across the entire set of EER's becomes
the major concern. The method chosen in this study a.'s to average
the correlation coefficient for one interval. This average provides
a conservative measure of the conformity across the ten responses
since most of the correlation coefficients between w\aveform pairs
must be significant to obtain a significant average correlation. The
average correlation reduces the effects of sporadic deviations in
these correlation coefficients, which randomly appear between
paired waveform segments. The significance of these correlations is
questionable, since they do not have any consistent relation to the
stimulus.
The correlation coefficients for each paired EER (Figure 3.2) at
a specific interval are provided in Tables B.1 through B.7 in
Appendix B. The averages and maxima and minima of these correlations
are provided in Table 3.1. The average correlations reinforce the
previous observation that the activity in the second interval across
the ten EER's is the most consistent. In fact, even the minimum
TABLE 3.1
Average Correlations for Figure 3.2
INTERVAL 1 2 3 4 5 6 7
Pave .30 .88 .12 .05 .06 .04 .07
Pmax .89 .99 .88 .88 .94 .87 .98
min .45 .58 .71 .92 .87 .92 .87
correlation of .58 is above the 1 per cent significance level, i.e.,
assuming samples are normally distributed and independent (Undenrood,
et al., 1954). The average of the correlation coefficients across
the segmented EER's obtained from the wider band potentials were
generally less than those in Table 3.1 (see Tables B.8 through B.13).
Thus filtering the data appears to eliminate some of the variability
across the response estimates.
The first interval of the EER's is the only other 135 millisecond
time period where the waveforms demonstrate any degree of consistent
correlation. This consistency appears to be the result of the more
common components in the latter half of this interval rather than
the initial deflections after stimulus onset. In intervals 3 through
7 (i.e., 270 to 950 milliseconds after the stimulus) the EER's
demonstrate negligible correlation. This variability of these late
components of the EER seems to indicate that: (1) the average back
ground activity is much greater than the average response activity
during this period; (2) the individuality of the response for different
subjects is greater, which increases the deviations of the estimate
in this region; or (3) the variations with stimulus number are greater
in the late response. In similar correlation tests, Perry and
Childers (1969) have demonstrated a significant decrease in the
average correlation coefficients of intersubject VER's following
the first 300 milliseconds of the response, and a decrease in intra
subject correlations after 500 Inilliseconds. They also note that
the highest correlations are generally found in the second and third
100 millisecond intervals, which is in close agreement with the
previous results. Therefore, the EER and \ER results seem to reject
the reasoning that it is the individuality of the evoked response
which introduces the variability in the late components of the
ensemble average.
It is interesting to note the occurrence of alphalike activity
in the latter portion of response 40. This singular occurrence may
be the result of a particular combination of the random EEG potentials.
However, these potential fluctuations have a strong resemblance to
the afterdischarge activity noted in the studies mentioned in
Chapter 2.
In Figure 3.3 the ten EER waveforms are superimposed with the
time mean removed from each average. In this figure the consistency
of the responses in the region from about 60 to 300 milliseconds
after the stimulus can easily be seen. Because of the similarity of
the EER in this region (R2) it will be designated the primary
response region. In this region there are three prominent peaks.
These peaks are identified in Figure 3.3 as components one, two
uad three, and occur approximately 90 Tsec, 150 msec, and 250 msec
after stimulus onset, respectively.
The average correlation coefficient in the primary response
region is 0.75, which is slightly less than the average correlation
in the second interval of Figure 3.2. This decrease is probably the
result of the increase in interval width from 135 to 240 rrec. Thus,
the second region of Figure 3.3 has more components and greater
variability than the second interval of Figure 3.2. The average
correlation of the third region in Figure 3.3 is 0.23, which indi
cates a minor degree of similarity not previously found in the corres
ponding correlations in Table 3.1. This similarity is possibly the
result of some low frequency activity comrimon to the tails of the EER's
which would not be detected by the correlations of the 135 millisecond
3
/"
2
i
R., R R,
FIGURE 3.3
Superimposed Ensemble Evoked Response
Overlapping EER's are divided into three regions for comparison.
R, = 0 to 60 rrsec; P2 = 60 to 300 rrsec; R = 300 to 950 msec.
28
intervals. However such a low frequency component is not visually
detectable, except for a very slight negative drift in the late
response components.
3.4 Noise in the Enserble Evoked Response
The EER's in the previous figures contain an unl no.,n amount of
background noise. One estimate of the amplitude of the background
activity which remains in the EER is the ensemble average under
nonstimulus (control) conditions. The accuracy of this estimate is
dependent on the relationship between the signal and the noise. For
example, if the signal and noise are independent and the noise is
stationary, this average provides a good measure of the amount of
background activity in the EER. The waveshape of this average is
important only if a particular component appears time locked to the
occluded stimulus, indicating the existence of some experimental
artifact.
In Figure 3.4, this average is subdivided into ten control ..,ave
forms attending the ten occluded stimuli. The format of this figure
is identical to that of Figure 3.2. The average correlations of the
TABLE 3.2
Average Correlations for Figure 3.4
INTERVAL 1 2 3 4 5 6 7
ave .07 .02 .03 .01 .07 .00 .07
Pmax .81 .92 .95 .99 .87 .92 .81
Pmin .79 .99 .93 .92 .98 .93 .93
Interval
FIGURE 3.4
Enserble Control Averages
Ensei a'le averages for ten
occluded stimuli are
dividcd into 7 intervals
of 155 milliseconds.
intervals of Figure 3.4 given in Table 3.2
demonstrate the lack of correlation between
the control w.aveforms. These correlations
are similar to the average EER correlations
in the last four intervals of Table 3.1.
This supports the previous observation
that there may be little response activity
in this region.
In every interval the root mean square
of the control w.aveforms, Cr is less
than the rrrs value of the responses, R rMs
in the second interval of Figures 3.2
(where these rms values have the dc removed).
The iiniinum RP is only 1.03 times the
maximum Cs and the magnitude of the
largest R nis 9.7 times the smallest
Crm. The average of all ratios of R
to C is 2.6. Therefore, assuming that
the ensemble control average provides a
good estimate of the amount of background
activity remaining in the averaged response
and that the signal and noise are additive,
what is the signal to noise ratio (S/N) in
the EEF? To date, the answer to this
question cannot be resolved, since it is
necessary to also know the phase of the average background activity.
How,,ever, with only a magnitude estLmate, the upper and lower bounds of
this signal to noise ratio can be found by adding and subtracting the
magnitude of the control from the EER and dividing by the control
amplitude, i.e.,
R C R + C
TES TmtS Tllms ITS
Ms < S/N < lrs ms (3.2)
C C
riS lIms
It should be noted that in the case when the lower bound is negative
(which is a theoretically impossible event) the lower bound is set
to zero.
Rather than establish these bounds, a more common practice is
to estimate the signal to noise ratio in the average response by
dividing the magnitude of the average response, Rs, by the average
noise, Cms (Perry, 1966), i.e.,
S R
S = rms (3.3)
N C
rms
Using this estimate, the relative signal to noise measures on the
previous page provide the maximum, minimum and average S/N. Thus, one
would predict that the EER's in the second interval of Figure 3.2
contain approximately 10 to 50 per cent noise with an expected value
of 28 per cent. In Chapter 4 it is shown that under certain assump
tions these predicted S/N values are reinforced by the degree of
similarity (i.e., correlation) previously found between the EER's.
Therefore, the average amplitude under control conditions probably
provides a reasonable estimate of the amount of noise distorting the
31
EEP. waveform.
If the overall rms value of the control w aveforms approximatelyy
0.5 pv) represents the expected noise distortion in the response
average, then Figure 3.5 provides an upper and lower bound of the
noise distortion exTected in the EER. The actual magnitude of the
background noise potentials at any instant could be several times
their rms value. Hown.ever, the bounds in Figure 3.5 provide an
average approximation of the amplitude of the background noise which
could be contained in the EEP estimate. That is, the average
response without noise should be expected to be within these bounds
if the control w.aveforms provide a good estimate of background
noise and if tde EER is a good estimate of the average signal plus
noise activity. The latter stipulation, the quality of the EER
estimate, provides further confidence limits in the average evoked
activity. Thus a more accurate bound on the evoked activity is a
combination of the noise bounds and the confidence limits on the
EER.
1 Ayv
1 2 3 4 5
Response Number
10 20 30 40 50
FIGURE 3.5
Noise Amplitude in EER
Overall pr s control value is added and subtracted from the center
EER waveform.
24~
:2
'U
32
3.5 Quality of the Enserable Evoked Pesponse Estimate
The EER is a good estimator of the signal plus noise mean if the
deviations of this estimate about the mean are relatively minor. A
measure of this estimation deviation is provided by the standard
deviation, of(t), of the underlying process. The actual value of
Of(t) is unknown. However, an unbiased estimate of of(t) is the
ensemble standard deviation, rf(t), which is calculated from the
equation
1 N 1/2
Jf(t) = { N i1 [fi(t) ff(t)]} (3.4)
.where N is the nmabuer of fi(t)'s, subject EEG's, and mf(t) is the
ensemble average given by equation 2.2.
A plot of 3f(t) for the potentials attending e3ch of the ten
is provided in Figure B.5. These ten waveforms are superimposed in
Figure 3.6 in order to observe any stimulus effects on 8f(t). As
S
0 4
> I HI2
S *i ' /iV /
' FIGURE 3.6
1 Superimposed Ensemble Standard Deviations
Each standard deviation attends one of the ten
selected stimuli which occur at the extreme
left of the figure.
can be seen in this figure, the fluctuations of the af(t) waveforms
do not appear to have any consistent relationship to the stimulus,
other than a slight average increase around the primary response
region.
If the ensemble standard deviations in Figure 3.6 provide a good
estimate of the standard deviations of the process, then the standard
deviation of the mean estimate, mf(t), is
o (t)
sd[lf(t)] ~ (3.5)
This standard deviation can be incorporated into the Tchebycheff
inequality to provide reliability or confidence bounds on the EER
estimate. This inequality provides a lower bound on the probability
that the true mean and estimated mean will differ by less than n
standard deviations of the estimate. For example, we can be at least
89 per cent confident that the true fiean and estimated mean (EER)
will differ by no more than three standard deviations (calculated
by Equation 3.5) of the estimate, regardless of the underlying
probability distribution. If the underlying probability distribution
is Gaussian, the confidence increases to 99 per cent.
The S9 per cent confidence intervals for ten EER's are plotted
in Figure 3.7. The bounds in this figure are constructed from ten
EER waveforms plus or minus threesevenths of the ensemble standard
deviations superimposed in Figure 3.6. As can be seen in Figure
3.7, or calculated from Figure 3.6, the overall width of the intervals
varies from about 2 to 4 Dv. Although this appears to be a relatively
large amount of variability, the shape of the bounding waveforms
generally conforms to the EER waveshape, with a few exceptions. The
most notable exception occurs in the first response, around 300
milliseconds after stimulus onset. This variation could have been
anticipated from Figure 3.6, since the standard deviation of the
potentials in this region of the first response partially stands
above the remaining response deviations.
If the confidence intervals in Figure 3.7 are compared with the
VER confidence intervals in Figure 3.8, the reliability of the VER
is greater than the EER in two out of the three cases shown (J.M.
and B.A.). In the third case (J.F.) the reliabilities are approxi
mately equal, if the greater amplitude of the VER is taken into
consideration. In all three cases the VER's were obtained by
averaging the subject's response to fifty stimuli. Therefore, for
V I 1 grv
1 2 3 4 5
Response Number
10 20 30 40 50
FIGURE 3.7
EER Confidence Intervals
The above intervals are plus and minus three standard deviations of
the EER estimate. This is about 3/7 of the deviations in Figure 3.6.
the same number of averages, the EER appears to provide as good an
estimate of the response activity as some VER's.
The VER amplitude is about one and onehalf times that of the
EER's in all of the cases show.n. This larger amplitude and greater
reliability of the VER estimate implies that the intersubject
response variability is greater than the variability of responses
obtained from a single subject. This is i.hat one would predict from
the comparison of \VR's obtained from a single subject and those
obtained from several subjects. In addition, the background noise
variability would influence tie reliability of the estimate. Thus
intersubject EEG variability (e.g., amplitude and frequency deviation
in the background activity) probably reduces the confidence in the
EER estimate. One variable parameter which might reduce confidence
in the VER and not in the EER would be a general change in the evoked
activity with the rnmber of stimuli. In order for this variation not
to affect the FER it is necessary for this change to be fairly con
sistent across the stimuli for the subject population.
J.F. J.M. B.A.
I\ ;' l: llJ Y
FIGURE 3.8
VER Confidence intervals
Scale and procedure is the same as in Figure 3.7. Each of the above
suhiects are members of the ensemble.
3.6 Variations in the Enseirble Evoked Response .withi Stimulus Number
As noted previously, the EEP provides an estimate of the changes
in tl. evoked rcsonse (cofuion to the subject population) with repe
titions of the stimulus. That is, each EER in Figure 3.2 is an
estimate of the activity which is time locked to a specific stimulus
within the periodic train. In order to obtain an accurate estimate
of the response changes related to the stimulus number, it is necessary
to remove the EER variations resulting from the backCground noise.
Since this cannot be accomplished at this time, the EER's across
stimuli will be examined for trends of change, noting that variations
caused by the background noise should be random in nature.
In a preliminary investigation of the EER's in Figure 3.2 as
a function of stimulus number, little transient change is discernible.
The only noticeable change is a general decrease in amplitude between
the first and remaining responses. The first EER appears to contain
a low frequency component, particularly evident in the latter portion
of this response, which is not as predominant in the EER's that
follow. In the superimposed EER's in Figure 3.3, this low frequency
activity causes the initial components of the first response to rise
above the other responses and the later components to descend below
the remaining responses. In addition, the EER to the first stimulus
contains a sustained.positive component about 300 msec, which does
not have a counterpart in any of the following EER's. However, the
reliability of this component is questionable because of the increase
in the ensemble standard deviation in this region. It therefore
appears that not only does the average response to the first stimuli
differ from those that follow, but that the predominant component
(around the third peak: of Figure 3.3) of this response is more
variant than any other.
If the activity outside of the primary response region (60 to
300 msec) in Figure 3.3 is ignored and the means within the correlated
section are removed from each response, the amplitude of the first
response in this region does not stand out. For example, if the
magnitude of the waveforms in the primary response region are plotted
as a function of stimulus number, the amplitude of the first response
is no longer so predominant. A plot of both the nns and peak to
peal:, ptp, amplitude variations for the individual response in the
interval is provided in Figure 3.9. For this figure the nms value is
about a :ero mean and the ptp value was measured from the second and
third primary deflections indicated in Figure 3.3.
Since the responses are very similar in this region, the rms and
ptp measures provide essentially the same information. Both demon
strate a slight decreasing trend in amplitude with the number of
stimulus presentations. However, the magnitude of the amplitude
fluctuations is not mUuch smaller than the overall decrease; therefore
the significance of the latter appears questionable. In a plot
(Figure 3.10) of the changes in the time delay or latency of the
three peaks in the 60 to 600 millisecond interval no significant
trend is discernible. Therefore, the above results imply that little,
if any, transient change exists in the first fifty responses for
0
rO
> in
4 r4'
ii .,
G
1.1
1.0
0.9
0.S
0.7
0.6
10 20 30 40 50
FIGURE 3.9
Amplitude Fluctuations of the Primary Response
260 T
240 4
Component 3
Component 2
Component 1
Response
!_L __J__I.I I .C
1 2 3 4 5
1 2 3 4 5
10 20 30 40 50
FIGURE 3.10
Latency Fluctuations of the Three Primary Components
 ITS
Sptp
A
Response
2 3
220
200
180
160
140
120
*U
r4
orH
,
0
100 
responses averaged across a subject population. An exception to this
observation is the difference in the overall character between the
first and remaining responses. This may be evidence of some initiali
zation of the visual system either at the receptor or higher neural
processing level. At the receptor level, Brazier (1967) has noted
that there is a large difference in the amplitudes of the electro
retinogram attending the first and second stimuli in a repetitive
train. However, she also notes that the amplitude of the second ERG
is smaller than any that follow, indicating that the transient change
at the receptor input exists beyond the first flash. The persistent
transient variations (11 to 20 seconds) noted by Regan (1966b) in
responses recorded at the scalp also imply a more sustained initiali
zation period than is found in the above EER results. However, Regan
utilized a continuous sinusoidal stimulus which restricts extra
polation of his results to the present study. If initialization is
not a factor then a high degree of variability in the first response
might account for the relative uniqueness of the EER to the first
flash. The ensemble standard deviation following the first stimulus
partially supports this reasoning but it does not explain all of the
component differences between the first and succeeding EER's. There
fore, system initialization and intersubject response deviation may
both contribute to this difference.
3.7 Average of Ensemble Evoked Responses
The results of the previous sections indicate that the EER provides
a relatively stable response estimate in die region from 60 to 300
40
a,
a
o O
:), r
o 0 c
CD 0 v
.. , O O
oL .
S 0 r
. [" O i ,
C) LUC
S 0
0 *r
C ) L C }
I 0I
/ ro
o
1 ^ Ci 'f
msec. If this region is stipulated to contain the primary activity
of the evoked response, then very little response infonnation should
be lost by averaging EER estimates across stimuli. This average
should further reduce the average background noise activity and
increase the quality of the response estimate. That is, the variance
of the background noise and the variance of the EER estimate should
both decrease with the increase in the total nLumber of averages
from fifty to five hundred.
Figure 3.11 provides the average of the ten EER's and the ten
control waveforms shown previously. As can be seen in this figure,
this average seems to extract the near sinusoidal complex in the
primary region and reduce the remaining components. The averaged
control is a general overall reduction of the individual control wave
forms presented earlier. The S/N ratio in the primary response
region of the averaged EER is estimated from the averaged control to
be about 3.6. This is a twofold improvement over the average S/N
ratio estimated for the EER to a single response from the individual
control waveforms. If confidence intervals were provided for the
averaged FER,each interval would be less than 0.5 iv from this
average. Therefore the average of the ten EER's provides a reliable,
relatively noisefree intersubject response norm.
3.8 Summnar'
The average over an ensemble of subject EEG's (EER) extracts
a relatively consistent estimate of the response to visual stimuli.
1Te ccnsistenc cf ten EER's is found to be greatest for components
in the 60 to 300 msec region following stimulus onset. The variations
of these components with stimulus nurmber are generally insignificant,
indicating little change, either habitual or otherwise, in the activity
evoked by the first fifty stimuli. However, a general modification
in the character of the entire average waveform is noted following the
EER to the first stimulus. It is suggested that this change is the
result of initialization of the visual system and a greater variability
of the first response across the subject population. This variability
of the first response is demonstrated in confidence intervals about
the ten EER estimates.
The amount of noise activity in the more consistent region of the
EER's is estimated by ensemble control averages. From these estimates
the average S/i' ratio in the EER is found to be 2.6. In order to
further extract the response signal from the noise, the average of
the ten EER's is calculated. This averaged EER provides a more
reliable response estimate than single EER's and improves the S/N
ratio. The response estimate extracted by tiis average is almost an
inverted sinusoid, beginning approximately 100 msec after stimulus
onset.
CHAPTER 4
THEORETICAL ASSUMFTIONiS UNDERLYING RESPONSE AVERAGING
In order to theoretically justify the results obtained from many
EEG data processing techniques, it is necessary to assume certain
hypotheses about the statistical character of the fluctuating poten
tials. Investigators who employ the averaging techniques to extract
time locked signals from the incoherent background noise generally
assume that: (1) the noise is a wide sense stationary random process;
(2) the signal and noise are independent; and (3) the signal and
noise are additive. The latter assumption is probably the most
important since nonadditive components cannot be separated by the
linear process of averaging. However, a signal dependent on non
stationary noise can be extracted if the degree of dependency and
nonstationarity is minor.
In this chapter an attempt will be made to gain positive or
negative evidence about these assumptions by examining the data
obtained from the subject ensemble. The results of these examinations
will not, however, enable the experimentor to negate or to confirm
these assumptions because of the lack of knowledge about the poten
tials being tested. Thus, to examine one of the above assumptions,
the test utilized must hypothesize the nature of the signal and
noise parameters. In many cases the tst hypotheses depend on one
or n.to of the above assumpTticons, conditioning the conclusLiveness of
the result. The present study is therefore intended to provide
insight into the possible workability, rather than ultimate veracity
of these assumptions. Sudc an examination may then provide a foun
dation for further studies into the statistical properties of evoked
signals in background noise.
4.1 Stationaritv of the Background Noise
A random process is said to be stationary if the statistics of
the process are time invariant (see Davenport and Root, 1958). For
example, a large nwrmber of statistically independent sample functions
which are simultaneously recorded constitute a wide sense stationary
process if the mean and autocorrelation function at one instant of
time equals those at all other times. In the present study the
sample functions are the EEG records of the fifty subjects. The
time varying function of particular interest is the ongoing activity
under stimulus conditions. However, these potentials cannot be
separated from the evoked response. Therefore, this investigation
must rely on the resting potentials under nonstimulus conditions to
estimate the background activity. It then becomes necessary to
work under the constraint of the second assumption (i.e., that the
signal and noise are independent) to examine the first.
Since an average is an estimate of the first moment of the
ensemble, it is the stationarity of background noise mean which is
the most critical to the averaging technique. If the noise mean is
highly nonstationary (e.g., the averaged potential fluctuations
exceed those of the response average) then the averaging process
cannot extract a detectable signal from the background noise.
Utilizing the resting EEG potentials to approximate the background
activity, the noise mean estimate (ensemble control average) seen in
Figure 3.4, is not time invariant. IHow.ever, are these fluctuations
the result of an underlying nonstationary process or caused by the
variations inherent in the estimation process? In order to answer
this question it is necessary to determine the inherent variance of
the mean estimate of N sample functions. If the N sample functions
are independent, which is plausible since individual subject EEG's
should not depend on one another, then this variance of the estimate
is (see Bendat and Piersol, 1966)
o}(t)
VAR\[if(t)]= N (4.1)
where o'(t) is the variance of the process which is wuknown and
must be estimated. This latter estimate also has variance associated
with it :.hich again must be estimated, and so fcrth. In order to
prevent endless estimation of these statistics, the estimate of the
variance, .2(t), will be assumed equal to the actual variance,
oa(t), i.e.,
62(t) = 2(t) (4.2)
N;ow. utilizing the Tchebycheff inequality, the range of the mean
estimate can be predicted without knowing its exact probability
distribution. This irne,:uality states that the probability that the
difference between the true m'an and estimate mean will be greater
than C standard deviations of the estimate will be less than 1/C2,
i.e.,
PROB [Imf(t)mf(t)l C{sd[r f(t)]}] > 1 (4.3)
C2
In Chapter 3 this inequality was used to provide S9 per cent confi
dence limits (C=3) about the EEP and \EP estimate, noting that
OfCt)
sd[mf(t)] = 3 (4.4)
Similar confidence limits can be obtained by hypothesizing a
mean and using 3f(t) to obtain confidence limits. For example, if
the noise mean, mf(t), is hypothesized to be stationary and equal to
zero, then Equation 4.3 becomes
PPOB[Jlf(t)> C{ f(t)/N] < 1 (4.5)
C 2
Again setting C=3, the 89 per cent confidence intervals about the
hypothesized stationary mean can be established. In addition, from
Equation 4.5 another value of C can be calculated from the ensemble
value of mf(t the ensemble control average, and 3f(t), the control
standard deviation, i.e.,
mf(t)
C < (4.6)
,(t)
The maximum value of C found in Equation 4.6 can then be compared
to the threshold value of 3  rejecting the stationary hypothesis
for C>3 and accepting the hypothesis for C<3. This is equivalent
TABLE 4.1
Maximum Number of Standard Deviations of
Ensemble Control Average About a Zero Mean
Control 1 2 3 4 5 10 20 30 40 50
Cmax 2.56 1.62 2.53 2.43 2.24 3.00 2.37 3.07 3.69 2.37
max
to testing the h),yothesis at a significance level of greater than
5.5 per cent since the probability of acceptance of this twotailed
test is greater than 11/C2, if the hypothesis is true. If the under
lying distribution is Gaussian, this significance becomes greater
than 1 per cent. The maximum value of C for each of the control
trials, whose waveform is provided in Figure 3.4 is given in
Table 4.1. From this table the stationary mean hypothesis is
accepted seven times, rejected twice, with one indecision. These
results would therefore iLply that the ensemble mean of the resting
EEG, and perhaps the background activity, is nearly time invariant
with an occasional nonstationary trend.
Another estimate of the degree of fluctuation of the first
moment of the background activity is the plusminus average suggested
by Schimmel (19671. This estimate is similar to the EER except that
alternate sample function are subtrtacted in the averaging process
rather than added. This estimate cam be expressed mathemratically by
odd even
ft) = p f (t 7 f ( t)] (4.7)
If the evoked response is assL'ed to be nearly the same for each
subject, i.e., near deterministic, then the resultant average should
contain mainly background activity. The reliability of this estimate
is, ho\.ever, contingent upon the third assumption (i.e., that the
signal and noise are additive) in addition to the second.
With this restriction in mind, the preceding test for station
arity can also be applied to the plusminus average. The standard
deviation of this estimate about the true + average is the same as
the standard deviation of mf(t) provided in Equation 4.4 since +o (t)
is the same as the variance, a o(t), of the process (see SchiTmmel)
under stirmTlus conditions. By assuming the response signal to be
near deterministic, this variance is mainly the result of the back
ground activity. If i.we again assume that the estimate of the variance
is very near the actual variance, the values for C can be calculated.
The values for Cma provided in Table 4.2 again indicate that the
max r
first moment of the background activity is stationary, i.e., the
stationary hypothesis is accepted for every + average.
If the signal is assumed to be nondeterministic, it contributes
to the estimated variance of the process and the below Cmax values
TABLE 4.2
Maximum Number of Standard Deviations
of + Estimate about Zero Mean
Response 1 2 3 4 5 10 20 30 40 50
1.93 2.34 2.69 2.00 2.21 2.14 2.30 2.49 2.27 2.82
C
max
increase. That is, the noise variance is now the estimated ensemble
variance, 8.(t), minus the signal variance. In addition, the average
a ef ra.e
will contain some signal activity. In order to accept the stationary
mean hypothesis for the ten estimates the standard deviation of the
signal cannot be greater than 10 per cent of the total, Of(t). Such a
limitation is reasonable since present magnitude estimates of the
evoked response indicate that the background EEG activity has an
amplitude about five to ten times as great as the response (Kitasato
and Hatsuda, 1965; Perry and Childers, 1969). Therefore, with the
incoherent nature of the noise and time locked character of the
signal, the ensemble variability of the former should be at least
ten times greater than the variations of the latter.
In order to check the validity of the previous test, it is
interesting to examine the stationarity of the response average, i.e.,
EER. Again, assuming that the signal and noise are additive and
independent, and that the signal standard deviation is at most 10 per
cent of the total ensemble variance, then the values of Cm for the
max\
ten EER's in Figure 3.2 are given in Table 4.3.
TABLE 4.3
lMaimumT Number of Standard Deviations
(times 0.1) of EER Estimate About Zero Mean
Response 1 2 3 4 5 10 20 30 40 50
C 42.5 36.1 28.1 30.1 :2.6 38.3 33.0 35.8 28.6 19.0
m, x
These results clearly demonstrate that the mean of the signal with
noise appears to be nonstationary, which is imperative for the signal
to be extracted by averaging. Even if the signal variance accounts for
the entire ensemble variance, the Ca values calculated (i.e., 1/10
max
of the above results) would d indicate the EER to be nonstationary
sLx out of ten times. Therefore, the preceding stationary mean test
can detect nonstationary trends under the worst case conditions,
which indicates that it is a valid test.
In order for the background activity to be wide sense stationary
the autocorrelation function must also be time invariant. The auto
correlation function, R(t, tT), is a function of both time, t,
and the time shift of the correlation, T. For R(t, tr) to be time
invariant the autocorrelation function at time t1 must be the same at
all other time t, for a given T, i.e.,
Rf(tl, tlr) = Rf(t2, t2r) (4.8)
The estimate of this function is obtained utilizing Equation 4.9.
Rf(t, tT) = 1 f.(t)f.(tT) (4.9)
1 1
i=l
Using Equation 4.9, a threedimensional plot of the ensemble auto
correlation function estimate versus t and tT for the resting EEG
ensemble (Control 1) is provided in Figure 4.1. Since the previous
test indicated that the mean is stationary, this average is sub
tracted from the autocorrelation to eliminate any sampling
0
(U
4J r
4
0.
cm
O E
u
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4 J
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fluctuations of tiis first moment. Therefore the autocorrelation
function in Figure 4.1 is more correctly an autocovariance function
which has the form
N
Cf(t, tT) = N [fi(t) mi(t)][f(tT) mi(tT)] (4.10)
The magnitude of this function, like the autocorrelation function, is
a maximum for T=0, which occurs along the main diagonal of Figure 4.1.
It also should be noted that the covariance function is symmetrical
about T=0; therefore the super and subdiagonal values are equivalent.
In order to examine the stationarity of this function one muLst
examine the fluctuations of Cf(t, tT) for a fixed T and varying t,
which is analogous to viewing the variation C along the main and off
diagonal terms in Figure 4.1. Fluctuations of Cf(t, tT) w itJ time
are noticeable in this figure; however, their significance is
questionable. From the actual values of Cf(t, tT) provided in
Table B.21 of Appendix B, one does notice that the magnitude of these
fluctuations is generally independent of the diagonal used, i.e.,
the time shift, T. Therefore fluctuations of the main diagonal
for T=0, which is equal to the ensemble variance, should provide as
good a measure as any other Cf(t, tT1) of the time fluctuation of
the correlation function.
The ensemble standard deviations of the ten control trials are
superimposed in Figure 4.2. These deviations are overlapped to pro
vide a simultaneous estimate of the degree of fluctuation and to
provide a comparison with the standard deviations presented in the
sane maimer in Figure 3.6. The reader interested in the individual
ensemble standard deviations of each control is referred to Figure B.5.
The maximum, minimum, and average values of the waveforms in
Figure 4.2 are 4.5 pv, 1.S pv, and 2.S uv, respectively. If the
standard deviation is hypothesized to be stationary and equal to the
average of these w\aveforms, then the fluctuations of the estimated
values exceed a range of 61 to 160 per cent of this average.
This variation is probably greater than one would expect from the
estimation error; however, without knowledge of the higher moments
one cannot further quantify this statement.
These higher moments are essentially descriptors of the finer
detail of the uLnderlying probability structure of the process.
Therefore the profile variations of the ensemble probability density
function, pdf, should provide a rough evaluation of the degree of
5
FIGURE 4.2
1 I */ Iz
Superimposed Ensemble Standard Eeviations
Each standard deviation attends one of the ten
occluded stimuli %which occur at the extreme
left of the figure.
time dependence of these morrents. Previous investigations of the
amplitude probability density function of individual EEG records
ha'e provided conflicting results. Elul (196S) and Saunders (1963)
note that EEC amplitude histograms appear to approximate the normal
(GaLussian) distribution. However, the results of a study by Campbell,
et al. (1967) generally reject the hypothesis that the EEC amplitude
is normlly distributed. A comparison of the ensemble histograms to
the normal distribution may provide additional information about the
underlying EEC pdf.
Histogram estimates of the ensemble pdf at every 10 millisecond
interval after the first occluded stimulus are shown in Figure 4.3.
Each of the seven row.'s in this figure approximately correspond to
the seven 135 millisecond intervals used previously to segment
the ensemble control averages (Figure 3.4). As can be seen in
Figure 4.3, there appears to be little consistency between the
ensemble histograms at different time instances. The significance
of these pdf deviations with time is conditioned by the variance of
the estimation process (i.e., the histogram approximation). However,
the w.ide contrast in contour between individual pdf estimates appears
to exceed the limitation of sample variability. For example, the
histograms in Figure 4.3 vary from unimodal (t=60 msec) to bimodal
(t=90 msec), asymmetric (t=710 msec) to near s\rmmtric (t=100 msec),
and leptokurtic (t=490 msec) to platykurtic (t=330 msec).
In addition to this variability, few of the histograms reserrble
the normal distribution. An attempt to quantify the degree of
rcL
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normality utilizing the KolmogorovSmirnov test proved unsuccessful.
Although the }olmogorovSmirnov test is generally more efficient than
the chisquare goodness of fit test (f1iller and Freund, 1965) for the
number of available samples (50), the necessity of estimating the
mean and variance reduced the sensitivity of the test (see Lilliefors,
1967). That is, there was a very high probability of accepting
the hypothesis of normality when it w.as false.
4.2 Dependence of Signal and Noise
In the previous section .'e noted that if the signal and noise
are additive and the signal near deterministic, then the plusminus
average should provide an average estimate of background activity.
If the same averaging process is performed on the resting EEG,
then a similar estimate of the noise under nonstimulus conditions
is obtained. A comparison of the general character of these
processed potentials should provide an estimate of variations in
the average noise under stimulus and nonstimulus conditions. How
ever, hot. does one make such a comparison? The w.aveshape of these
potentials for ten signal and ten control averages (see Figures B.3
and B.4) has little meaning, since they were recorded at different
times. But the degree of fluctuation in the ms measure of the
average should provide an estimate of average energy under both
conditions. Table 4. provides the rms values of each plus minus
control and response w.aveform. The range of rms values is nearly
equal under both conditions. Thus, the time fluctuating averages
TABLE 4.4
rZ'S Values in j' of Averages
Under Stimulus and NonStimulus Conditions
Waveform 1 2 3 4 5 10 20 30 40 50
Response 0.46 0.37 0.42 0.28 0.39 0.3S 0.32 0.36 0.27 0.46
Control 0.26 0.36 0.40 0.38 0.32 0.28 0.49 0.58 0.45 0.67
of the background activity and resting activity are of the same
magnitude. Since the signal potentials are present in one case and
not the other, the general implication is that the magnitude of the
average noise is independent of t]e signal.
Another property of the background and resting EEG which can
be investigated under the assumptions of a near deterministic signal
added to noise (s+n), is the ensemble standard deviation. If the
signal and noise are independent, the standard deviation of the
resting EEG ensemble (Figure 4.2) should have a character similar
to the standard deviation of the stimulus provoked activity
(Figure 3.6). That is, the ensemble standard deviation of signal
plus noise, as+n(t), is given by
c. (t) = [ (t) + o2(t) + 20 (t)]l/2 (4.11)
+n n sn
If the signal is near detenninistic and independent from the noise,
then the signal variance, a (t), should be much less than the noise
variance, oa(t), and the covariance between the signal and noise,
o2 (t), equals zero, which reduces Equation 4.11 to the following:
sn1
S ( (t) = n(t) (4.12)
n
Thus, the standard deviation of the signal plus noise ensemble
is primarily the result of the background noise and should be similar
to the standard deviation of the resting EEG, if the stimulus does not
affect the noise (i.e., signal and noise are independent). As noted
previously, the tine fluctuations of the standard deviations should
not be the same for both cases, since these deviations were recorded
at different times. If w.e compare the standard deviations, 3 s+(t)
and n(t), provided in Figures 3.6 and 4.2 respectively, the overall
average and peak to peak values are very similar, i.e.,
ave[ s+n(t)] = 2.82 ou ave[6n(t)] = 2.86 p\
ptp[8 (t)] = 3.1 uv ptp[n (t)] = 2.7 u\
From these measurements one could predict that the variance of the
noise is independent of the signal. However, in Figure 3.6 a slight
increase in the average standard deviation, 8 (t) w.as noted around
s+n I
the primary response region. Since similar trends do not appear in
the standard deviations, 8 (t), in Figure 4.2, this increase could
imply some signal and noise dependence.
If the near deterministic signal and noise are dependent and the
average value of the noise is small, then the ensemble standard
deviation of signal plus noise becomes
O (t) = {o2(t) + E[s(t)n(t)]} (4.13)
s+n n
where E[s(t)n(t)] is the mathematical expectation or stochastic
average of the product s(t) n(t).
The ensemble standard deviation for noise dependent on a deter
ministic signal should then increase when the amplitude of the signal
increases. This is the case in Figure 3.6, particularly for the
standard deviation of the first response (see Figure 3.1). However,
if the signal is relatively nondeterministic, then the standard
deviation of independent signal and noise would also tend to increase
in the region of greatest signal variability. Since it is possible
that the standard deviation of a random signal could be proportional
to the magnitude of the signal, the existence of a random signal would
also explain the slight increase in Figure 3.6. In either case thiis
correspondence between signal magnitude and the ensemble standard
deviation is minor, except for the first response; this would imply
that the standard deviation of the noise is relatively independent
of the signal.
A comparison of autocovariance functions under stimulus and
nonstimulus conditions did not reveal any additional information
about the dependence of the signal and noise. The variations of this
function appeared to depend primarily on the variance, o2(t), as noted
previously (see Figures 4.1, B.6 and B.Y). The histogram estimates
of amplitude probability density functions under stimulus and non
stimulus conditions were too variable to attempt a comparison (see
Figures 4.3 and B.8).
4.3 Additivity of Sinal and Noise
In the previous chapter the maximLun (9.68), minimum (1.03) and
average (2.6) nms signal to noise, S/N, ratios were computed. These
ratios were calculated from the ratio of the rms value of the EER's
in the second interval to the rIms value of the ensemble control
average in any interval. In addition, the correlation coefficients
1.ere found between EER w.aveforms. Since the S/N ratio estimates
the ajrount of noise remaining in the EER w.aveforms, a decrease in
the S/N' ratio should be reflected by a decrease in the correlation
coefficients between waveforms. 1T1at is, if tie signal and noise
are additive, then the correlation between EER's should depend on
the relative amount of uncorrelated noise and correlated signal
contained in these averages.
In Chapter 3 the correlation coefficient between EER wiaveforms
x and y was calculated from the time standard deviations, T' and 4 ,
and the time covariance 'f' i.e.,
P~' = f (4.14)
x V
The S/l! ratios were found using the equation
R
S/N = (.15)
uns 4
n
where Ys is an estimate of the average signal time standard deviation
and 7' is an estimate of the average noise deviation under stimulus
conditions (assuming the signal and noise are independent). Both of
these functions are dependent on the time variance,, a, nd a mathe
matical relationship between the two can be established with certain
constraints.
If the EER is composed of signal plus noise, i.e.,
x = s5 + n1 (4.16)
y = 52 + n2 (.17)
then the time covariance is
= T 'T + T (4.18)
xv sis: Sn s2nI nln2
If the signal and noise are statistically independent, which insures
linear independence, then the t.o covariances between the signal and
noise in Equation 4.18 are zero and Equation 4.14 becomes
'" + 'T
SiS2 nin2
p = 1/ (4.19)
[() + (E+ )]1
5s n1 2 n2
If the noise is stationary then the time standard deviation, ',' of
its estimated average over equal periods of time should be the same,
providing the period is sufficiently long. If we issuTme that the
period between stimuli is sufficiently long and that the ensemble
average signal is near deterministic for each stimulus, then
'll n = m'
sI 2 s (4.20)
and
T T = T (4. 21)
ni n2 n
The correlation coefficient between EFR w.aveforms can then be written
as in Equation 4.22.
'l' + 4'
SIS2 n1n2
S= (4. 2)
y'2 + 112
s n
which reduces to
PS(S/N)2 + p
p = (4.23)
1 + (S/N)2
where ps and pn are the correlation coefficients between the signal
and noise and S/N is defined by Equation 4.15 with the ~ deleted.
Since we assumed previously that the average signal was near deter
ministic, ps is almost unity. The noise should be incoherent with
the stimulus; thus p is near zero and Equation 4.23 reduces to
(S/N)2
P = (4.24)
1 + (S/N)2
Therefore, if a pair of identical signals are added to stationary
noise, which is independent from the signal, then the correlation
coefficient between the resultant waveforms can be calculated from
the S/N ratio.
Using the three S/N ratios given on page 59, the following
values of p are found from Equation 4.24:
Pmax = 0.989
Pmn= 0.515
Pmin
Pave = 0.871
Therefore, subject to the above stipulations, one would predict the
above values for correlation coefficients between the EER responses
in the second interval. A comparison of these predicted correlations
with those calculated in the previous chapter, i.e.,
ma = 0.99
Pmin = 0.53
P min O.S
Pave = 0.88
discloses that the former and latter are nearly equivalent.
The implications of this strong similarity between the predicted
and measured coefficients are twofold. First, the R to C
11RS ImS
signal to noise ratio provides a good estimate of the background
activity remaining in the EER. Second, the assumption of an additive
signal to noise is further supported. One argument which might be
presented against the validity of the predicted coefficients is that
the ;assumptions of pn=0 and p.=1.0 are not realistic. How;.ever, the
average correlation coefficients calculated across segments of the
control w aveforms were very near :ero. This would partially reject
the argument, particularly in the case of the average predicted
correlation coefficient.
Another method to investigate the additivity of the signal and
noise is to observe the growth of the signal to noise ratio with the
number of averages. For example, if a near deterministic signal and
random background noise are additive and independent, then the
waveform of the average signal will remain essentially constant and
the variance of the average noise will decrease directly with the
nulnber of stumiations in the average. If we assume that the noise is
stationary, then the true noise mean is a constant, which in the
present study is preset to zero. The nIs value of the average noise
\.ill approach this zero value as the number of sLmmmiations, N, becomes
very large (noting that the variance of this average about the zero
mean decreases by a factor of N). The rms value of the average signal
should be relatively invariant with the number N. Therefore the
signal to noise ratio, S/N, should be expected to increase as the
number of summations in the average increases.
The minimum rate of the S/N ratio increase can be specified in
terms of the true ensemble variance, oa(t), and a probability deter
mined from the Tchebycheff inequality (see Equation 4.3). Thus, we
should expect the estimated S/N ratio to reflect a random increasing
trend with the number of summations in the average. However, how
can this increase be examined? The technique chosen to observe this
growth was to calculate the rms value (in the 60 to 300 millisecond
region) of the EER at every even number of summations and to compare
this value to the rms value of the average of the same data. After
a number of summations, the average of the data should provide an
amplitude estimate for the background noise and the EER a signal
amplitude estimate. The S/N ratio can then be formed by dividing the
EER by the average at every even number of summations, N, within
the average. This procedure was repeated for each set of data
following the ten selected stimuli.
A plot of the S/N ratio versus the number of summations N for
the EER's to the first five stimuli is provided in Figure 4.4. Although
the S/N ratio demonstrates an overall slight increase with N in each
6 t S/N
N
24 50
Response 1
Response 2
3 t S/N
N
  j 1 i
24 50
Response 3
S/N
Response 5
"x/
N
24 50
S/N
N
esponse 4
24
Response 4
FIGURE 4.4
S/IN Ratios Versus ILmuber of
Sumabnat ions
S/iN Ratios for the EER are
calculated at every even
nLmTber of summations, N.
+r
of the plots in this figure, the random fluctuations of S/N over
shadow this gro.th. In addition, it should be noted that the S/N
ratio fluctuations of the first response are nearly twice those of the
remaining responses. In a similar plot of the S/N ratio for EER's to
later stimuli (see Figure B.9), an overall decrease in this ratio with
N can be noted. These plots appear to discredit the signal plus
noise assumption. However, this conclusion contradicts the previous
results in this chapter and questions the results achieved by the
established technique of response averaging. Therefore, the adequacy
of the above test becomes highly suspect.
The inadequacy of this test appears to be the result of: (1) the
large variations in amplitude of individual EEG's which appear to
introduce abrupt changes in the noise estimates; and (2) the S/N
ratio estimate being probably inaccurate except for a large number
of sums (i.e., EER is more noise than signal). The first short
coming might be eliminated by performing the same test on a single
subject (i.e., using sequential as opposed to ensemble averaging);
the latter shortcoming could be reduced by beginning with forty to
fifty sums and increasing the number of summations. Still another
technique would be to utilize Equation 4.24 and discuss the improve
ment in S/N by the degree of correlation of the average after N
summations with a preset response template or norm. This correlation
technique would be more sensitive to the waveshape of the response,
which would help to separate signal and noise energy in the case
of fewer summations.
4.4 SLummair
The separation of the foregoing examinations into three groups
is somewhat artificial because of the interdependence of the
results obtained. Therefore, many of the results obtained actually
provide tests for more than a single .assumption. The above examina
tions are grouped into three areas to help organize the test results
and produce a more readable text.
The results of these tests strongly imply that the mean of the
background noise is time invariant and that its amplitude is rela
tively independent of the signal activity. The higher order statistics
(i.e., standard deviation, autocovariance and pdf) of the noise appear
to be nonstationary. However, the time fluctuations of the estimates
of these higher order statistics are dependent on the variance of the
estimation process, which was not investigated. The average and
peaktopeak fluctuations of the ensemble standard deviations under
stimulus and nonstimulus conditions were very similar, which further
indicates that the signal is independent of the noise. The average
correlation coefficients of a theoretical signal plus noise were
nearly identical to those calculated from the actual averaged res
ponses. Thus, the assumption of signal plus noise was partially
supported. However, a subsequent test of the theoretical increase
in the S/N ratio by averaging signal plus noise data provided incon
clusive results.
CHAPTER 5
SU.NA4RY
To date, most investigations of evoked potentials monitored at
the htuian scalp have utilized the average of a series of responses
recorded from a single subject to define components of the evoked
response. This technique has provided a fairly consistent estimate
of an individual's average ER. However, with this method it is not
uncommon tc obtain quite dissimilar averages from different subjects
under identical experimental conditions. This contrast in the ER
estimates across a population has hindered the interpretation of the
effects of stimulus parameters on the ER and frustrated the attempt
to use this average for clinical diagnosis. It may be possible that
the character of an ER is, like a signature, unique for each subject.
If this analogy is valid, utilization of the ER as a research or
diagnostic tool might be limited. However, if one could establish a
particular feature of ER estimates common to a considered normal
subject population, then this template or norm might identify that
quality associated with a population of responses evoked by a specific
stimulus. This would be analogous to identifying the characteristics
attributed to a specific letter of the alphabet in a set of signatures.
The influence of variations in stimulus parameters and physiological
abnormalities might then be recognizable in the degree of deviation
of a subject's ER estimate about this norm.
The present, study investigated one method of estimating such
a response norm. The estimation technique utilized was an average
across an ensemble of subject EEG's containing evoked activity. The
advantages of this method were twofold. First, it permitted estima
tion of common intersubject response activity evoked by a single
stimulus. Second, the collection of subject EEG's strongly resembled
an ensemble of independent sample functions from a stochastic process.
This second advantage permitted an investigation of several prevalent
assumptions about the statistical nature of the evoked signal and
background noise.
The ensemble average revealed a fairly consistent, recurring
component complex following each of a series of photic stimuli, i.e.,
ten stimi.uli chosen between the first and fiftieth stimuli in a
periodic stimulus train. This averaged complex or primary response
mainly consisted of a positive to negative to positive potential,
near sinusoidal in form, which began approximately 60 milliseconds
after the onset of each stimulus. Significant correlation coefficients
were found between all pairings of the ten ensemble averages in this
primary region. Outside of this region most of the paired correlations
between wavefonrms were insignificant. The average of these paired
correlations provided a reliable measure of the conformity across
semnents of the ten responses. Control estimates of the amount of
background noise contained in the primary response region predicted
that this complex could contain from 10 to 50 per cent noise, with
an expected amount of 2S per cent.
Under the assumption that die evoked signal and background noise
are additive, this prediction was further substantiated by the range
of the correlation coefficients between paired response estimates.
Therefore, these results implied that the control estimate provided
a good evaluation of the amount of distorting noise remaining in
the ensemble evoked response, EER. Since the primary response was
detectable in each EER, these results also imply that even with
a signal to noise ratio of unity the average response is observable.
The ensemble average technique also provided an estimate of
the changes in the evoked response 'ith the nLunber of stimulus
repetitions. That is, each EER w'as the average of the activity
attending a single stimulus. An examination of the changes in the
response estimates following each stimuli disclosed moderate com
ponent fluctuations in amplitude and latency of the primary response.
Since these fluctuations did net exhibit any specific trends or
transient variations wiith the number of stimulus repetitions, they
appeared to be the result of background noise distortion rather than
response changes. The one noted exception, the uniqueness of the
EER tc the first flash, is hypodthesized to be the result of inter
subject response variability or some initialization of the visual
system. This first response estimate appeared to contain a low
frequency ccimpnent which was not evident in the remaining estimates.
In addition, the time fluctuations of the ensemble standard deviation
aboit the ensc:i.ee average resembled the waveshape of this average
for the first 15 :.,illisecor.s foll'...'ling ite first stimulus. This
was a characteristic relatively unique to the standard deviation
attending the first stimulus and resulted in a decrease in the con
fidence of the first EER. Another singular property of the potentials
attending the first stimulus was demonstrated in plots of the EER
signal to noise ratio as a function of the number of subjects in the
average. The magnitude of the variations of the S/N ratio in the
plot for the first response was nearly twice that found in the
remaining graphs. Thus, these graphs reflected the greater varia
bility of the initial potentials. Because of these unique properties
of the first response estimate it may be advisable to delete this res
ponse when comparing \TER's across subjects.
The average of the ten EER's waveform appeared to provide a
reliable, relatively noisefree response template. In this template
the primary sinusoidal complex was enhanced and the amplitudes of
the remaining components were reduced. The character of these reduced
components strongly resembled that of a similar single control
template. Thus, this response template further supported the hypo
thesis that the common intersubject response activity generally
is restricted to the primary response region. This average response
template also implied that investigations which compare the sequential
average obtained from different subjects should possibly restrict
this comparison to the 60 to 300 milliseo.nd region after stimulus
onset. However, since the above template would not extract inter
subject signal components whose latency varies from one subject to
the next, this restriction might be inappropriate for some studies.
Three common assumptions about the statistical character of the
signal and noise necessary to theoretically justify response averaging
were examined using the ensemble of subject EEG's. The results of
these examinations generally supported the assumptions that: (1) the
noise mean is stationary: (_) the signal and noise are independent;
and (3) scalp potentials consist of signal plus noise. The first
assumption was investigated by testing tw:o amplitude estimates of
the average background activity against a hypothesized stationary
mean. Of the twenty times tested, the stationary hypothesis was
rejected only twice. The conclusiveness of these tests was dependent
on the second assumption, i.e., the independence of signal and noise.
The combination of signal and noise activity is demonstrated to be
generally nonstationary. If the signal and noise are dependent, the
hypothesis tests on the + average should demonstrate this dependence
by rejecting the stationary hypothesis. Since this was not the case,
the results of the hypothesis tests appear to definitely support the
assumption of a stationary noise mean. An investigation of the
stationarity of the higher noise moments was generally unsuccessful.
Since the variance of the estimate of these statistics was unknown,
the time dependence of these functions could not be evaluated. However,
the relative degree of the time fluctuations of the variance, auto
covariance, and pdf functions implied that the higher moments of the
noise might be nonstationary.
An investigation of the second assumption, the independence of
signal and noise, was accomplished by comparing the ensemble + average
and standard deviation under stimulus and nonstimulus conditions.
The average provided an estimate of the noise amplitude with and
without the evoked activity. The range of amplitude fluctuations
of this average was nearly equal under both conditions. The average
and peaktopeak fluctuations of the ensemble standard deviation
under stimulus and nonstimulus conditions were very similar.
Therefore, both the noise average and the deviation about this
average appear to be independent of the signal activity.
An examination of the third assumption, additivity of signal
and noise, was generally hindered by the inability to separate these
tv.'o components. This shortcorrng thwarted an attempt to empirically
demonstrate the theoretical improvement of the signal to noise ratio
by averaging the assuTmed signal plus noise potentials. However,
the relative stability of the average estimates and the previous
prediction of the signalnoise ratio by the correlation coefficient
between w.aveforms supported this signal plus noise assui ption.
The results of this investigation of an ensemble of subject
EEC's which contained evoked responses have provided an estimate
of the common intersubject response. The estimated response template
found in this study was limited to responses evoked by visual stimuli,
specifically a monocular, periodic flash of light. The applicability
of this teniplate to other studies of responses in this modality
is at present uncertain. However, the method utilized in this
investigation has provided at least one technique for obtaining an
intersubject response norm to a specific stimulus. It is impractical
to utilize this ensemble technique in most experimental or clinical
investigations because of the large subject population required.
Therefore, the applicability of the technique is restricted to
isolated studies. The results achieved in such studies, however,
should be useful in many other investigations which compare the intra
subject response average across a subject population.
The ensemble of subject EEG's appears to provide a worthwhile
tool by which to evaluate theoretical assumptions about the statistics
of the EEG. However, the degree of sample variation noted in this
study seems to indicate that a greater number of sample functions
would provide a more reliable evaluation. This is particularly true
in the case of the estimation of the higher ensemble moments
and the ensemble probability density function. In order to obtain a
significant improvement, a fair amount of increase in the number of
sample functions would be necessary. The advantages of such an
increase would probably be overshadowed by the difficulties presented
in acquisition of this larger ensemble.
APPENDIX A
EXPERIMJf.TAL PROCEDURE
A.1 Data Acquisition
The data used in this experiment was collected from fifty subjects
(32 males, IS females). The right eye acuities of all subjects were
normal or correctable to normal as measured on a Snellen chart. Sub
jects were asked their hand preference and given a far vision sighting
test for eye dominance. One pair of bipolar electrodes (Lexington
C100) were placed S cm apart on the scalp, located vertically on the
occiput midline with the lower electrode over the inion. This con
figuration was chosen becuLse it has been noted to produce more
reliable \ER's across subject populations. Each subject \was seated
upright in an electrically shielded Faraday cage and asked to concen
trate on the light seen through the right eyepiece of a modified
troposcope. During a stimulus (signal) trial the light consisted
of a xenon flash of intensity 3.6 log ftlamberts subtending 215' in
an invariant incandescent surround of intensity 1.3 log ftlamberts
subtending 18024'. The flash was presented periodically (approximately
0.9 H:) in t.o trials and periodically (average rate of approximately
0.4 H:) in a third. Each signal trial was followed by a control
(noise) trial identical to the preceding stimulus trial with the
xenon flash occluded. After each trial the subject was permitted
a rest period of two to three minutes: total time required of each
subject was usually less than fortyfive minutes.
The potential difference monitored bet\.een the scalp electrodes
w.as fed simultaneously through two Grass P5111 differential amplifiers
(channels 1 and 2) with a gain of 100K and bandpass of 0.3 to 50 Hz.
The bandpass of the signal in channel 2 was further restricted by a
KhronHite 330B electronic filter from 0.3 to 15 H=. These amplified
signals were then fed into a Sanborn 2000 F1 magnetic tape recorder
and a Mnemotron 400C computer of average transients (CAT). Also
recorded on magnetic tape was the output of a Crass SS stimulator
which provided an external trigger for the CAT and a Grass PS2
photostimulator which powered the xenon flash. In order to detect
changes in equipment gain from one subject to the next a calibrate
signal from a Medistor ClA Calibrator was processed in a manner
similar to EEG data prior to the experirrnntal run for each subject.
A.2 Data Processing
In each trial the CAT sequentially summed each one second epoch
of the bioelectric potentials following each stimulus trigger. In
this manner VER and control wvaveforms were provided for each subject.
These waveforms were permanently registered on a Sanborn 701A Strip
Chart Recorder. From these records the "best" signal and noise trials
(under periodic stimulus conditions) were chosen. Judgment of "best"
was obtained strictly on an amplitude basis, disregarding the v.aveshape
of the summed response to avoid biased results. That is, the larger
sequential averaged signal and smaller sequential averaged noise
amplitudes were judged to be the best.
The unprocessed analog signals from these better trials were
then digitized and stored on a memory disc of an IBM 1SOO computer.
Since the disc could only accommodate a total of approximately 4.4
(105) data words, it became necessary to limit the amount of analog
data stored. Therefore, the storage of analog potentials (digitized
at a rate of 200 samples per second) was restricted to ten near one
second epochs of data for each subject and each channel. The ten
epochs chosen followed the first through fifth, tenth, twentieth,
thirtieth, fortieth and fiftieth stimuli in the periodic stimulus
train. These ten epochs were chosen to provide a measure of both
the initial (first few seconds) and prolonged (fort)' to fifty seconds)
changes in the evoked activity with repetitions of the stimulus. Each
epoch consisted of the first 190 digitized points following each of
the ten stimulus triggers; thus a single epoch is 950 milliseconds
in duration.
In order to eliminate any near DC potentials introduced by equip
ment base line shifts, the mean was removed from each 190 point epoch.
Equipment gain fluctuations across the subject population were also
eliminated by multiplying the digitized potentials by a constant
inversely proportional to the magnitude of the recorded calibrate
signal. Computaticns on this adjusted array of data provided most
of the results contained in this study. The programs utilized to
obtain the data were fairly simple in nature and are therefore omitted
from the text. Tne numerical results of these computations were
converted to graphical displays by a Computer Industries 135 OnLine
Incremental Plotter.
APPENDIX B
SUPPLEMENTARY FI CQURES XrD TABLES
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FIGURE B.9
S/N Ratios Versus irumber of
Summations
S/N Ratios for the EER are
calculated at every even
nuTmber of summations, N.
N
21 Sd
3 I S/N
1 c ? 10 i
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TABLES B.1 through B.7
Response Correlation Coefficients
The above tables provide the cor
relation coefficients beti..een the
ten response i.avefoims in Figure
3.2 for each 135 msec interval.
I
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TABLES B.S through B.13
EER Correlation Coefficients for the 0.3 to 50 Hz Bancdwidth
The above tables provide the correlation coefficients between the
ten wide band response waveforms in Figure B.1. These response wave
forms were shifted to compensate for phase differences introduced by
the filtering process. The above intervals therefore correspond to
those segmrnted to narrowband EEP's.
C l I 5 1
1 I.O 2
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I 0 .00
> O. i. 1.00
C.,6 0. :: 0. :. 0.65
0 '1J 0 0O 0 "I 0 i.
0 10 ', '( 0 0 'I.
*O.]? 0 L 0 r "(' 1)
J. 100
0 '] 0 IO
hNT I
: 0 1
1 "0. :
S 0
C '10 0.
CC. .
60 0 0
0 *0 !1 '
*C' 0. 1.
1.00
I 10 :0 7o0 L0 0
1 00
0. 1.00
0. 0. 1' 1.00
0 :5 0 1: 0. 1 1 00
0. !8 0 C0.8. 0 (: 1.00
,* 0" o.7 o 0 7 o 1 0 7.
0 0 0 ! 0 l* 0 18 0 1:7
0 :: 0 7' 0.70 0 1' 8 0.7 1
I C :10 1'0 O, 0
2 2.001
: 0. 1.00
5 0.0. 0.'" 1 00
0 87 0 : 0 1'
0 1. 0.78 0 :'
:0 0.69 0 5, O.li.
0 "0 :' 0.'.O 0 '7
0 0 *., 0 1: 0 I
0 0.i C. 0 C L
c.e .BL *
r. 0'
J.CO
*O. : 1.00
0 5 0 1: 1.00
o0.: 0o i. o0 6
0. 0 71 0.'
C0. : 0. 0 C.
I '00'
0. :1 1.00
0 0' '. ].0o'
C TL ]
I I 00
: 0.88 I 00
7 0 '' 0 '1'
i. 0 28 0 '.7
0.0'' 0.1:
10 0 7: 0 7:
0 ''. I t
70 0 '1 0 '1
0 0 02 0 .'
:O C. C.
10 7 0 0 '.0 0
d e
1.00
0.0' 1 00
0. 0 0 :*
*O. i' 0 1
( '" 0 1 .
1 00
0 :0 2.00
, 8 I 0 80 1.00
O. B 0 0C 0 1i 1 ..O
0.,: 0 5. 1.1' 0 ,8 I 00
CnlL : I
J 1 OC,
0 : ..0 I 0 I
.0 .5 0. 0 0 81
S0 10 0 ., CL
CO C.'. : ( C. :1
O 0.' 0 1 C0 C.
'0 0. 0 0.0 '.
1 10 :0 1' S .0 G 0
Il i.4L '
1. 00
0 !o 0 0 100
0 : : 0 '2 0 10 1.00
0 1" 0 1a (.'0 0.7O 12 00O
TABLES B.14 through B.20
Control Correlation Coefficients
The above tables provide the cor
relation coefficients between the
ten control i.avefonns in Figure
3.4 for each 135 msec interval.
I,7i2.al a
c e 0'
.:a 00
0.81 0I ,
0 80 0 4I. J 00
$ I.
T
.00 .05 .10 .15 .20
.25 .30 .35 .40 .45
.00
.05
.10
.15
.20
.25
.30
t .353
.40
.45
.50
.55
.60
.65
.70
.75
.SO
.S5
.90
TABLE B.21
Values for Cf(t, tT) in Figure 4.1
The following values are measured relative to Cf(0,0) and the
values of t and T are given in seconds.
1.00
0.90
0.SS
0.79
0.71
0.86
1.07
0.58
0.63
0.57
0.51
0.62
0.S9
0.53
0.73
0.77
0.59
0.57
0.79
0.41
0.53
0.30
0.10
0.03
0.04
0.10
0.07
0.20
0.0S
0.13
0.20
0.27
0.25
0.30
0.26
0.20
0.36
0.52
0.2S 0.22
0.35 0.27
0.17 0.03
0.38 0.07
0.34 0.06
0.21 0.17
0.06 0.00
0.01 0.01
0.22 0.02
0.32 0.09
0.12 0.04
0.18 0.05
0.13 0.01
0.2S 0.12
0.1S 0.18
0.14 0.01
0.28
0.09 0.11
0.06 0.09 0.00
0.07 0.06 0.09 0.00
0.10 0.12 0.04 0.0S 0.16
0.06 0.01 0.OS 0.01 0.13 0.15
0.21 0.12 0.10 0.00 0.19 0.05
0.06 0.16 0.00 0.24 0.27 0.23
0.05 0.02 0.23 0.2S 0.2S 0.35
0.12 0.OS 0.05 0.14 0.17 0.24
0.02 0.16 0.09 0.09 0.29 0.32
0.02 0.13 0.16 0.14 0.15 0.30
0.03 0.02 0.05 0.21 0.14 0.16
0.02 0.07 0.10 0.19 0.12 0.12
0.OS 0.01 0.01 0.09 0.03 0.13

