ON THE DETECTION OF PERIODICITIES
IN THE RESPONSE TIMES OF SIMPLE VISUAL TASKS

Thesis for the Degree of M. S.
MICHIGAN STATE UNIVERSITY
DUANE G. LEET
1968

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LIBRARY ”:5-

Michigan State j
I

University
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T116318

 

ON THE DETECTION OF PERIODICITIES IN THE

RESPONSE TIMES OF SIMPLE VISUAL TASKS

By

I

.I I.“
Duane GI Leet

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Biophysics

1968

Chapter

I.

II.

III.

IV.

TABLE OF CONTENTS

Page
INTRODUCTION . . . . . . . . . . . l
1.10 The Information Sciences and the
Brain I . . . . . . . . . 1
1.20 The Thesis. . . . . . . . . . 2
BACKGROUND EXPERIMENTATION . . . . . . A
11.10 A Statistical Model for the Function
Which Generates Response Times. . . A
11.20 Criteria for a Good Experimental
Task Design . . . . . 6
11.30 Some Designs for Experimental
Visual Tasks. . 10

11.A0 The Model and Power Spectral Analysis 13
11.50 The Analytical Methods, Results, and

Conclusions . . . . . . . . . 18
DISCUSSION: PART 1. . . . . . . . . 26
111.10 Criticisms of the SEN Experimental

Procedure I . 26
111.20 Criticisms of the Eoregoing Model: . 27
EXPERIMENTATION .' . . . . . . . . . 29
IV.10 Introduction. . . . . . . . . 29
1V.20 Experiment B—l . . . o . . . . 29
1V.30 Experiment B-2 . . . . . . . . A8
EXPERIMENT B—3 . . . . . . . I . . 83
V.10 Introduction . . . 83
V.20 The Experimental Procedure and

Results . . . . . 8A
V.30 The PrOperties Established to

Interpret an Experimental PSE. . . 90
V.A0 A Demonstration of the Representative

Nature of the Generated PSE' s . . 91
v.50 An Interpretation of the Experimental

Power Spectral Estimates. . . . . 99

ii

Chapter Page

V1. DISCUSSION: PART II . . . . . . . 101
V1.10 Summary . . . . . 101

V1.20 Suggestions for Further
Experimentation. . . . . . . 106
APPENDICES. . . . . _ . . . . . . . . 109
BIBLIOGRAPHY . . . . . . . . . . . . . 151

iii

Table

11.1

IV.1

IV.lO

LIST OF TABLES

Summary of experimental types and designa—
tions and descriptions of the tasks
designed by Augestein (1)

Average time per symbol during the course of
experiment B—l: Subject DJ.

The average time per symbol, Experiment B-2

iv

Page

11

A8
71

Figure

II.

II.

II.

II

II.

II.

II

II

II

IV.

IV.

IV.

10

2O

30

.A0

50

60

.70

.71

.80

2O

30

A0

LIST OF FIGURES

The density P (-), where the component
density functions are defined in Eqns.
II.20.lO—II.20.l3

The density fT(.), where the component
density functions are defined by Eqns.
11.20.11—1120.1A

The density function P (.) developed by
Augenstein to fit the experimental data
From (1)

Power spectral estimate for subject SR in
experiment SO

Power spectral estimate for subject BR in
experiment SHN

The power spectral estimate which is the
normalized sum of the power spectral
estimates of Figure 11.71

A composite power spectral estimate con—
structed by normalizing the sum of 21
estimates obtained from six subjects
under six experimental conditions.

Power spectral estimates calculated from
histograms generated by the density
functions defined in Eqns. 11.A0.20-
11.A0.23 . . . . . . . . .

Histograms by position subject H. in
. 1
experiment SHN

Histogram with 10 msec. intervals
Examples of power spectral estimates and
their autocorrelations calculated from

histograms of Experiment B-l

Histograms of response times for selected
positions: Subject DJ

Page

20

21

21

22

22

23

25

33

3A

A3

Figure Page

1V.50 Histograms of response times for selected
positions: Subject PA . . . . . . . AA

1V.60 A history of response times for selected
positions. . . . . . . . . . . . A6

1V.70 Histogram with 10 msec. intervals . . . . 51

IV.80 Examples of power spectral estimates and
their autocorrelations calculated from
histograms of Experiment B—2F . . . . . 52

1V.90 Examples of power spectral estimates and
their autocorrelations calculated from

the histograms of Experiment B—2S. . . . 61
1V.100 Histograms of response times for selected
positions: Subject TJ . . . . . . . 72
1V.110 Histograms of response times for selected
positions: Subject DF . . . . . . . 76
1V.l20 Histories of response times for selected
positions: Experiment B—2 . . . . . . 79
V.10 Summary of PSE results, Experiment B-3. . . 93
V.20 Summary of PSE results, Experiment B-3. . 95
V.30 Summary of PSE results, Experiment B—3. . . 97
A. Power spectral estimates and their auto—
correlations calculated in Experiment B—3 . 111
B. Power spectral estimates and their auto-
correlations for a single histogram
generated during Experiment B—3 . . . . 128

vi

CHAPTER I

INTRODUCTION

1.10. The Information Sciences and
the Brain

 

The function the Egyptians attributed to the brain
is not known, but it could not have been very important.
This is obvious from the fact that when the Egyptians
embalmed their dead, the brain was retracted through the
left nostril and discarded. Today, even a youngster
would laugh if you told him that the function of the
brain was to cool the blood, as Aristotle had believed.
To him the brain is for "thinking." But what does he
mean? To answer the question "What is the function of
the brain?" satisfactorily requires a language that is
less than a quarter of a century old, the language of the
information sciences. Using this language, the function
of the brain is to collect "sensory information and
(correlate) this . . . with stored information to Compute
the daily course of bodily activity." (8)

The methods of information processing used by the
nervous systems of even primitive organisms are just
beginning to be understood. A basic philosophy used by
scientists to achieve this understanding is to divide

the nervous system of the organism being studied into

subsystems. The functions of these subsystems are
described and the connections between them are detailed

to give the organism as a complete system. The subsystems
are then explored to determine the techniques of infor-

mation processing used by the subsystem.

1.20. The Thesis

In this spirit, this report examines a thesis which
has been proposed to explain how humans process information
acquired by the visual system. The thesis was inspired by
a postulate of Stroud's: If continuous physical time is
represented in the experience of man as psychological time
T, then T is not a continuous variable. 0n the basis of
experimental evidence, Stroud has concluded that the reso-
lution or moment of T is between 0.05 and 0.2 seconds.
The phenomenon that leads most directly to his postulate
is flicker fusion (18). If a subject is asked to count the
number of flashes of light and the light is flashing at a
frequency above a critical frequency called the flicker
fusion frequency, then the number of flashes that the
subject sees is consistently lower than the actual number.
The flicker fusion frequency is usually somewhere between
20 and 70 cps., with 30 cps. being the mean. This type of
phenomenon has also been reported with sound and touch
inputs (20). If a noncontinuous information system is

hypothesized, then the phenomenon is easily explained.

Using Stroud's postulate as a starting point,
Augenstein wondered if the information processing technique
of humans might be analogous to the technique used by the
digital computer, for if Stroud had as evidence experiments
performed on a digital computer, he would have arrived
at an identical postulate. This is because the operations
of a computer are controlled by a discrete clock, called
the master clock. In the computer, if the master clock were
to die, or even become ill, computations and commands would
become hopelessly fouled up; in short, the computer could
not function.

After some preliminary experimentation, Augenstein
developed the following thesis (1): Information presented
to the human visual system is first coded into an internal
symbolism in an operation analogous to that performed by
the input black box of the basic information processing
system. It is then processed for meaning. This "processing
black box" operates in a manner similar to that of a
synchronous sequential system: it accepts an input or
makes a binary decision only at periodic* intervals.
Finally, the decision is relayed to the appropriate centers
for the execution of the decision (the output black box).

The following chapters discuss the eXperimentation
that led Augenstein to this thesis and also some experiments

which have been performed to test further its validity.

 

*The terms "period" and "periodic" are used in the
intuitive sense, not in the rigorous mathematical sense.

CHAPTER 11
BACKGROUND EXPERIMENTATION
11.10. A Statistical Model for the Function
Which Generates Response Times

Consider the simple visual task of deciding whether
some aspect of a visual pattern is or is not present. If
a naive attempt is made to predict the response time for
this task, it might be considered to be the sum of the
random variables corresponding to the acquisition and
accomodation time, Ta(-), the time to input the pattern,

Ti(-), the time necessary to reach a decision ,Tg(-) and

the time for the output of the corresponding response,

TO(-). The response time is, therefore, the random
function*

I T
Eqn 11.10.10. T(s) = Ta(s) + Ti(s) + Tp(s) + To(s).

 

*The notation used in this thesis follows the current
trend (6). A random variable in the implicit form will be
written T ( ). A random function in the implicit form will
be writteR in the same way. A random function written in
the explicit form (i. e., as an equation) will have an argu-
ment supplied with all the component random variables [e.g.,
T(s)]. Throughout, the argument commonly used is s, a
single trial of an experiment. The notation for (an implicit
density function will be of the form fT (. ) or p The f
signifies a continuous function and the p signigies a dis-
crete function. The subscript is the random variable
(function) associated with the density function. An argu—
ment is supplied when the function is written explicitly.

Now assume that the subsystems of the visual system per—
forming the inputting and decision—making Operations have
the property that they produce outputs sequentially and
that each Operates in a synchronous fashion (i.e., has

a cycle time). Furthermore, assume that the lengths of
the cycle times for the two subsystems are different.
Thus, an output from the input subsystems occurs bti
seconds after the initial input of information, where b
is an integer and ti is the cycle time for input; an out-
put from the decision-making subsystem occurs atp seconds
after the initial input, where a is an integer and tp is
the processing cycle time. By allowing both variability
in the number of cycles required by each subsystem and a
slight variability in their cycle time, the total times

for the inputting and decision—making Operations become

Eqn. 11.10.11. Ti(s) = B(s)Ti(s)
and
Eqn. 11.10.12. Tg(s) = A(s)Tp(s).

Finally, assume that the density function of the random
variables Ta(°) and TO(') are Gaussian. Since the sum of
random variables with Gaussian densities is a random
function with Gaussian density, Ta(-) and TO(-) can be

combined into the random function

Eqn 11.10.13. Ts(s) = Ta(s) + To(s).

Combining Eqns. 11.10.10.--11.10.l3.,

Eqn 11.10.1A. T(s) = Ts(s) + B(s)Ti(s) + A(s)Tp(s).

Note that the means of Ti(-) and Tp('), which will be

designated by ti

primary importance.

and tp respectively, are the values of

The following pages review experiments Augenstein
(l, 2, 3) developed to test the ability of T(-) to describe
the response times generated by humans performing simple
visual tasks.

11.20. Criteria for a Good Experimental
Task Design

 

The model has several properties that were used to
design tasks to emphasize, if possible, the values of the
parameters tp and ti’ The first of these can be seen in
an intuitive way by considering the following density

functions for each of the random variables of Eqn. 11.10.lA:

Eqn 11.20.10. pT (t1) = 1, t1 = A00
5
= 0, otherwise;
Eqn 11.20.11. pB(b) = l, b = 0

= 0, otherwise;

Eqn 11.20.12. pA(a) = .2, a = l,2,3,A,5
= 0, otherwise;
Eqn 11.20.13. pT (t2) = 1, t2 = 50
p

0, otherwise.

These density functions are simple enough that the

density function pT(-) can be constructed by inspection
(see Fig. 11.10). The important property to note is that
the value of tp can be determined directly from the

density function. It is the distance between the impulses.

Suppose that now, instead of Eqn 11.20.10.,*

Eqn 11.20.1u. fT (t

N(t , o ).
s s

l) = s

With a little thought Figure 11.20 can be shown to be the
general shape of fT('). Note that if the ratio of Os/tp

is "small enough," the value of tp can again be determined
by visual inspection. It would, therefore, be advantageous
to have this ratio and, in any more general analysis, the
ratios oi/tp and op/tp as small as possible. In order to
optimize this possibility experimentally, the following

criteria were established:

 

*This nOtation will be used for a Gaussian density
function of mean tS and standard deviation CS

1w)

0

—
L
I

 

' '— ‘ I l V ' mscc o
500 600 no t ’

 

:60 25° 306 450

Figure 11.10. The density pT(.), where the
component density functions are defined
in Eqns. 11.2O.lO—II.2O.12.

 

 

 

IE“)
fil-
.I. "I ” ~..__ *-~\
. . ~r'— M. - C .
O poo zoo 300 400 500 so: 706 t) ms c
Figure 11.20. The density fT(.), where the component
density functions are defined by Eqns.

II.20.11~11.20.1A.

Cl. Either a single response or very simple responses
should be required.
C2. The tasks should be self-paced.

Assuming the model is correct, if the cycle time is
to be detected by measuring the distance between peaks,
then the task should require more than one cycle. On-the
other hand, if the task requires complex memory references,
or the pattern to be acted upon contains unfamiliar
symbols or requires an unusual response, then the data may
contain elements not postulated by the model. Hence, the
following criteria:

C3. The patterns should consist of simple, familiar
symbols.

CA. Each response should be formulated by making a
sequence of simple unequivocal decisions.

C5. Each response should require many such decisions.

The next criterion was:

C6. Ths displays should be shown in a random order.
This was adOpted so that the subject could not anticipate
the exact composition of the display.

Stroud's prediction of the magnitude of the funda-
mental quantum (50-200msec) suggests a final criterion:
C7. The data and analysis should allow for the detec—

in the 25-300 msec range.

tion of values for tp and t1

10

11.30. Some Designs for EXperimental
Visual Tasks.

The tasks which Augenstein designed using the above
criteria are outlined in Table 11.1. One task, SHN, is of
particular interest and will be described in detail.

A pattern (hereafter called a stimulus) for SHN
consisted of a 26 element sequence of letters and numbers.

An example of such a sequence is

DSVCYKGMHPSZ2WX5RAK5M7Z38X

The stimulus variable of interest was the first position
from the left to contain a number (2 in this case). The
position (13 in this case) which contains this first
number will be called the target element.

All positions previous to the target element con-
tained a letter chosen at random from 23 equiprobable
letters (Q, 0 and 1 were not used because of their simi-
larity to numbers). The digit in the target element was
chosen at random from the equiprobable numbers 2 through
9. The positions following the target element contained
either a letter or number chosen at random from the 23
letters and 8 numbers, all equiprobable.

There were a total of 251 different sequences
typed on white cards. The target element occurred 25 times
in each of the positions 3, 8, 13, 18 and 23 of the

sequence. The target element occurred a total of six times

11

TABLE 11.l.--Summary of experimental types and designations
and descriptions of the tasks designed by Augenstein(l).

 

Type Symbols

 

1. Scanning SV

SHE

SHN

SO

SA

2. Minimal scanning WSA

W83
3. Muscular response MRS
MRT
A. All factors NN

minimized

Recognition of first letter
in a vertical list of mixed
numbers and letters-—all
positions equiprobable.

Recognition of left-most
letter in a horizontal list
of mixed numbers and letters
-—all positions equiprobable.

Same as SHE except all posi-
tions were not equiprobable.

Recognition of either 10 or
01 in a vertical column of
00 and 11 symbols-—all
positions equiporbable.

Addition of vertical columns
of numbers. -

Synthesis of a A-letter
English word from a scrambled
tetragram.

Synthesis of the A-letter
English word from one of its
scrambled permutations.
Skeet shooting.

Typing random text.

Null response

 

12

in each of the remaining 21 positions. The cards were
shuffled at the beginning of each eXperimental run.

The lighting in the room where the task was per-
formed was described as being comfortable for reading.
To begin a trial, the subject was required to focus his
eyes on a fixation point on a Gerbrand TachistOSOOpe
screen. When the subject felt that he was ready to begin,
he depressed an easy action microswitch. This turned off
the light illuminating the Spot and turned on the fluor-
escent light illuminating one of the stimulus cards from
the shuffled group of 251. All symbols fell to the right
of the fixation point as the subject viewed the stimulus.
A timer with an eXpressed accuracy of i3msec. was trig-
gered with the lights. The subject had been instructed
and trained beforehand on the method he was to use to
search for the target element. Specifically, when the
stimulus appeared the subject was to move his eyes from
the fixation point to the beginning of the sequence.
He was to scan the sequence from left to right until he
came to the target element. When the subject became
aware of the target element, he released the microswitch.
This stopped the timer and turned off the illumination
on the stimulus. The subject then identified the number
in the target element. The eXperimenter recorded the
response time to the nearest ten msec., replaced the
stimulus with a new one, reset the timer and signaled the

subject that he could begin a new trial.

13

The response times obtained for each subject were
assembled to give a histogram of the number of times a
given response time occurs. Note that a histogram can be
thought of as an unnormalized estimate of the density

function of the response times.

 

11.A0. The Model and Power Spectral
Analysis '

11.A0.10. An Aside on Digital Power Spectral

 

Analysis of Real Periodic Signals in Ergodic Noise.--A

 

standard technique for detecting signals in noise, and
particularly periodic signals, is to calculate the power
spectral estimate of the available signal. This para-
graph will heuristically develop from basic principles

a commonly used method for calculating the power spectral
estimate (PSE).

To begin with, define X(-,°) = X to be a real orgodic
stochastic process.* A member of the ensemble of time
functions will be signified by X(-,¢) and a random variable
can be defined for any particular t by using the notation
X(t,'). Weiner (10) has shown that the following are a

Fourier transform pair:
Eqn 11.A0.10. R(T)=E{X(t,-),X(t+T,°)}=£:P(w)eJWwa

ij

Eqn 11.40.11. P(w) = f% £:R(T)e- dT

 

*For definitions and discussion, see Reference 11,
pp. 323-332 and Reference A, p. 11.

1A

R(T) is called the autocorrelation of X(t,-) and P(w) is

 

called the power spectrum. Since ergodicity has been

 

assumed, an equivalent eXpression for the autocorrelation
is

Eqn 11.A0.15. R(T)=lim l fT/2X(t ¢)X(t+T,¢)dt

T+m T —T/2 .

Therefore, for the ergodic case, only one member and not
the entire ensemble is required in order to calculate R(T).
Several simplifications can be made in the calcula-
tion of the eXpressions for R(T) and P(w).r First, assuming
that X(-,0) is real, R(T) is real and P(w) can be written

in the form:

Eqn 11.A0.20. P(w) = §%_LTR(T)cos wTdT

R(T) and P(w) are even functions (because they are real),

Eqn II.A0.22. R1(T)=2%i2 {Ex(t,¢)x<t+1,¢)dt for 1:0
and
Eqn 11.A0.23. Pl(w)= % A) R1(T) cos wTdT for wZO.

 

Rl(T) is called the oneysided autocorrelation (AC) and

Pl(w) is called the one-sided power spectrum (PS).

 

15

From the statistical point of view, the AC and PS are
important invariant properties of the stochastic process X.
From a communications point of view, they allow for the
design of systems which can take advantage of the frequency
properties of the noise in that system. There would be
little more to be said at this point if it were not that in
practical situations it is impossible to record X(-,¢)
from t = —w to t = +w. Instead, the signal is recorded
over the interval [0,T]. An autocorrelation expression

for finite signals is*:

TJT
A - l < <
Eqn 11.40.A0. R1(T) - 1:1 A,X(t.¢)X<t+T,¢)dt O-T-m<Tmax
where Tmax is the greatest possible lag. There has been

much study concerning the comparison between finite and
true autocorrelation values for various types of signals.
For instance, the mean square error of Rl(T) increases
with increasing T and decreases with increasing T. The
eXpression for the finite power spectrum corresponding to
R1(T) is

T)coszdT ofiwsw.

Eqn 11.A0.A5. Pl(w) =

ISR

dIH

1(

Note that this is an estimate of the true power spectrum

and so it is called the power spectral estimate (PSE) of X.

 

*Note that a change of variable, t' = g + t, allows

this expression to conform to the previous arguments
concerning evenness.

16

The effect of R1(T) on the power spectrum has been studied
extensively. Several modifications of R1(T) have been
proposed as a result of these studies. A modification

which helps compensate both for the finite signal length

and the decreasing accuracy of R1(T) with increasing T is

to multiply R1(T) by a window function, D(T) before taking
its transform. A very common window function is the hamming

window, which is defined as

Eqn II.AO.50. D(T) = 0.5M + 0.u6 cos %§ 0

IA
'-3
IA
3

Another modification which is used specifically to correct
for error in R1(T) due to large lag is to use a correlation
coefficient expression instead of the standard autocorre-

lation (l3):
fi.<T>-Ev{x<-,¢>}E"{x<-,¢>1
/sz 0112
x x

where the prime superscript indicates that the value is

Eqn 11.A0.60. R1(T)

 

 

calculated over the interval teEo,T-T], and the double
prime superscript indicates that the value is calculated
over the interval te[T-T, T}. Note that -l s R1(T) 5 1.
1f the time function is recorded digitally, then
digital autocorrelation and PSE expressions can be
defined which include the modifications of Eqn 11.A0.50

and II.AO.60:

Eqn

11.A0.7O

l7

 

_X(1At)X[(i¥K7At]-X(1At) xTTITKTIEU'
R1(KAT)-

 

//%f(K) %:K (K)

where OfKEM, At is the time between samples, assuming

equispaced samples, N is the number of samples, AT is the

fundamental lag time, and m is the maximum lag count. Also,

 

 

 

 

 

N-K
Eqn. 11.A0.71. X(iAt)X[(i+K)At]=fi—%:T z X(iAt)X[i+K)At]
- i=0
1 N-K
Eqn 11.A0.72. X<iAt) = N-K+l 1:0 X(1At)
l N-K
Eqn II.AO.73. x[(1+KTAEI = z x[(1+K)At]
N-K+1 i=0
2 1 N-K N-K 2
Eqn 11.A0.7u. °1(K)=NIRIT z [X2(iAt)]- 1 [ 2 X(iAt)]
' i=0 (KIRIIT i=0
2 1 N—K‘ 1 N-K
Eqn II.AI0.75. 01+K(K)=N—-T(TT 1:0X[(1+K)ACJ-W 1:0[(1+K)At]
and
P(pfo) l m
Eqn 11.uo.80. Po(pfo)= P -P 2 D(KAT)R1(KAT)cosKATpfO
max K=0
max
where fO e 55%? , Pmax is the maximum P(pfo) values (this

means that Po(pfo) f l) and

Eqn 11.50.81.

D(KAT) = 0.5A + 0.A6 cos KATfl/m.

I"

18

Note that this PSE has an additional source of error.
Because of aliasing, the frequencies above 2%? are not
included in the PS. If there is significant power at a

frequency higher than this, it is not detected.

11.A0.20. The Reason for Using Power Spectral

 

Analysis on the Histograms.—-The density functions which

 

were plotted in Figures 11.10 and 11.20 are periodic

with period equal to the value of tp on the interval

where the density functions are non-zero. It was ex—
pected that the histograms would also be periodic, but

that the periodicityTwOuld be difficult to detect visu—
ally because of the noise caused by the small number of
data points. Since the power spectral technique has

been used successfully to detect periodic signals in

noise, Augenstein felt that it would be a useful analytical
technique for detecting the values of tp and t in his

1
histograms.

11.50. The Analytical Methods, Results,
andfiConclusions.

 

 

11.50.10. Power Spectral Analysis.--A computer

 

program was written to calculate Eqns. 11.50.70 and 11.50.80
from the histograms. Figures 11.A0 and 11.50 are PSE's

from two subjects under two different experimental condi-
tions. Twenty—one PSE‘s, including these two, from six
subjects under six experimental conditions were averaged

and normalized to give the PSE of Figure 11.70. The peaks

19

in the spectrum which are larger than background have been
labeled with the period corresponding to the frequency at
which the peak occurred.

Augenstein interpreted this PSE with the aid of a
computer program which generated data according to the
following density functions for the random variables that

made up T(-):

Eqn 11.50.10. pT = N(A00,0s)

s
Eqn 11.50.11. pB(b) = 0, for all b
Eqn 11.50.12. pr(t3) = N(tp, Op)

An attempt was made to match PSE's calculated from
data generated by varying cs, tp, Op and pACa). Augen—

stein found that by using the density functions

Eqn 11.50.20. pT (t1) = N(A00,l3)
S
Eqn 11.50.21 pT (t3) = N(33,0)
p
Eqn 11.50.22. pB(b) = 0, for all b
Eqn 11.50.23. pA(a) = see Figure 11.30.

he could generate data which resulted in the PSE's shown in
Figure 11.71. Figure 11.60 is the renormalized average of

these PSE's. Augenstein noted the resemblance of this plot

21

 

 

 

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Figure 11.70.

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to Figure 11.70. Based on this correspondence and on the
general shape of the individual eXperimental PSE's he
concluded that (l)
1. there was "strong evidence of‘a 100 msec.
periodicity in most of our subjects”;
2. "the fundamental 33 msec. periods probably occur
in groups of three";
3. "the 33 msec. value is probably for both ti and
t ."
p
It is important to recognize that the statistical

model is in the form

Eqn 11.50.30 T(s) = TS(s) + A(s)TP(s).

The interpretation of this eqn. resulted in the

thesis stated in Chapter 1.

11.50.20. Average time per symbol.--Recall that in
the SHN eXperiment the stimuli had many more target elements
in the 3rd, 8th, 13th, 18th and 23rd positions than in the
other positions. For each subject Augenstein collected the
appropriate response times and constructed a histogram for
each of these five positions. One such set of histograms
is Figure 11.80. The dotted line in the figure connects
the approximate average response time for each of the
positions. Using these averages, average times per symbol

of 61—72 msec. were obtained for the three subjects studied.

25

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CHAPTER III

DISCUSSION: PART I

III.lO. Criticisms of the SHN Experimental
Procedure

 

 

A critical review of the eXperimental method and
procedures used in SHN suggests two sources of data
artifacts.

To understand the first source of error, note that
when a continuous function is recorded at discrete inter-
vals (either by sampling or by quantising the data) it
is possible that the values of power density at frequen-
cies above fo = l/2At cps. will not be correct. The
analytical procedure that Augenstein used sidestepped this
issue by not calculating the power density in the fre-
quencies above fc. However, the true density function
could contain significant power at these frequencies. The
source of the second potential error was the fluorescent
lighting used in the Gerbrand Tachistosc0pe. Due to their
60 cps. flicker, fluorescent tubes have been observed to
cause a stroboscopic effect. Therefore, there was the
possibility that this flicker was introducing a periodic

artifact into the response times.

26

27

111.20. Criticisms of the Foregoing Model
Recall that Augenstein concluded that a sufficient

statistical model for his data was

Eqn III.20.lO. T(s) = TS(s) + A(s)Tp(s).

This equations says that a response time is the sum of an
acquisition and accomodation time plus an integral number
of fundamental quanta. "Within each fundamental quantum
either data are input or one binary decision can be made"
(1). Note, however, that the equation requires that the
quantum remain constant for the duration of the perform-
ance of the task. Such a requirement is not necessary

and indeed is not consistent if it is assumed that (l) the
variability in the cycle time is due to random fluctuations
in the neural pathways and (2) the variability is rela-
tively independent of the nature of the information being
processed. Therefore, the following model is prOposed

to more completely describe the random function associated
with the generation of response times of simple visual
tasks:

A(s)

Eqn III.20.20. T(s) = T (s) + Z T (s)
S i=1 P1

This eXpression says that a response time is the sum of
the times necessary to input the stimulus and make the

response Ts(s) plus an integral number of processing cyles

28

T (s), with the possibility of each individual cycle having

pi

a slightly different length. If the same stimulus is
presented again under exactly the same conditions, the
time Ts(s), the number of cycles required for processing
A(s), and, of course, the length of the cycles will be

different. Hence the random nature of the expression.

CHAPTER IV

EXPERIMENTATION

IV.lO. Introduction

 

Augenstein's eXperiments and the criticisms made
of them inspired the series of three eXperiments reported
in this chapter and the next. The first series, named
B-l, was designed to eliminate from the eXperimental
procedure the lighting and data accuracy as sources of
error. The PSE's that were calculated from the data of
this eXperiment were not at all similar to the PSE's
calculated in SHN. A review of the differences in the
experimental procedures that were previously considered
unimportant lead to the design of the second series of
experiments, B-2. The third series, B—3, investigated

the problem of PSE interpretation.

IV.20. Experiment B-l

 

IV.20.lO. Experimental design and procedure.-—

 

This eXperiment was designed to duplicate eXperiment SHN
in principle, while at the same time the procedure was
modified to increase the resolution of the data and the
amount of data that could be collected. In addition,

incandescent rather than fluorescent lights were used.

29

30

The Stimuli. The stimulus format was the same as in

 

SHN. However, instead of typing the sequences on cards and
presenting them to the subject using a tachistos00pe, the
cards were photographed and made into 2"x2" glass slides.
The total number of stimuli was increased from 251 to 1235,
with 100 target elements in each of the 3rd, 8th, 13th,
18th and 23rd positions and 35 target elements in each of

the other positions.

The Experimental Equipment and the Task. The eXperi—

 

ment was automated through the use of a Kodak Carousel
Slide Projector (Model 500R) with a shutter arrangement,
solid state control circuitry, a digital clock and a paper
tape punch output. A description of a trial of the exper—
iment will serve to introduce the task as well as the
relationship of the equipment to the task.

The subject was seated in a secretarial chair behind
a low table (36"). His head was positioned on a headrest
and his right index finger was used to Operate a micro-
switch taped to the table tOp. A 60 watt incandescent
lamp was placed behind the subject to give indirect illum-
ination of the screen. When the subject felt that he was
ready to begin a trial he would focus on a 2"x2" "Everglo"
night lamp placed about seven degrees of visual angle to
the left of the point where the left most element of the
sequence was to be flashed. It should be noted that

special care was taken in the construction of the slides

31

and the selection of a projector to insure that the first
position would fall at approximately the same point on the
screen trial after trial (there was less than a 0.5 degree
of visual angle variation measured during a test run). The
logic system was brought into play when the subject
depressed the microswitch and held it down. Within the
next millisecond the timer was triggered and the Synchronome
shutter pulsed to reveal a sequence to the subject. The
subject was instructed to move his eyes to the beginning

of the sequence when he became aware of the display and to
scan it from left to right for the first number. Scanning
was necessary because only five to six elements were sub-
tended by the fovea at any one time. Upon recognition of
the target element, the subject was instructed to release
the switch and enunciate the digit. The release of the
switch turned off the timer and pulsed the shutter. At the
same time the switch was deactivated until the housekeeping
duties of the logic system were completed (about 3 sec.).
These housekeeping duties consisted of punching the time

on paper tape, clearing the timer and changing the slide.

The Experimental Procedure. Four subjects, all

 

college students*, were used in this eXperiment: The

subjects performed at a rate of one 80 slide tray every

 

*Subject DJ was the only female.

32

5 minutes. About 2 minutes wererequired to change trays
and tend to the paper tape. The subject rested during this
time. Each was given a practice session of between A00 and
500 trials to learn the correct procedure. Each subject
was required to complete 5600 trials during a series of
eXperimental sessions lasting one and one-half hours each.
At the conclusion of the eXperiment the data On the
paper tapes were transferred to magnetic tapes and edited

for errors, such as faulty punching and mistrials.

IV.20.20. Power spectral analysis.--The edited data
from each subject were arranged in histograms having 1
msec. intervals. Histograms with 3, 5, 7 and 10 msec.
intervals were compiled from the l msec. histograms by
using a lumping technique. For instance, the first entry
in the 10 msec. intervals histogram is obtained by adding
the values of the l msec. interval histogram between 0 and
9 msec. The 10 msec. interval histograms are shown in
Figure IV.20.

Each of the histograms was examined and the portion
TEa,b] with no long strips of zeroes was chosen for power
spectral analysis. PSE's were calculated as in SHN for
several values* of m and m'. Some of the most interesting

PSE's are shown in Figure IV.30. The major characteristic

 

*The parameters m=m' were set equal to .5., .3 and
.25 of the total number of points in the interval [a,b].

9'3.

57‘

Number of
Responses ,

'4 a
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DiSplay Duration, msec.

Figure IV.20.1. Histogram with 10 msec. intervals:
Subject DJ in Experiment B-l.

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Figure
Subject PA in Experiment B—l.

34

Figure IV.30. Examples of power spectral esti-
mates and their autocorrelations calculated from histo-
grams of Experiment B-l. The autocorrelation is the
upper or leftmost of the two graphs that make up
each subfigure. The abscissa for the a tocorrelation
is the lag number k. The ordinate is (kAT). See
Eqn -II.AO.70. Conversion to time lag AT can be made
using AT = KAt. The abscissa for the power spectrum
is p, which is related to frequency by f = p/2mAt.
The ordinate is P (p). See Eqn II.NO.80. T[a,b]
denotes the histogram interval analyzed, At is the
histogram interval size, m* = m-At/(b-a-l),

m'* = m'oAt/(b-a—l), and 8* is the average number of
responses/interval in the histogram over T[a,b].

35

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of all of these, and indeed all the PSE’s, is their lack
of feature. They are unlike most of the PSE's obtained
by Augenstein. Figure II.50, a PSE from SHN, is one of

those that was somewhat similar.

IV.20.30. The average time per symbol.--The response
times corresponding to the situation where the target
element was in the 3rd position of the sequence were
edited from each subject's data and 10 msec. interval hist—
ograms of these response times were constructed. The same
prodecure was used for the 8th, 13th, 18th and 23rd
positions,** two of these sets of histograms are shown in
Figures IV.HO and IV.50. The average response time per
symbol was obtained by plotting the average response time
for each of the five positions and calculating the slope
of the straight line of approximate mean square best fit.
Only one subject, DJ, with an average response time per
symbol of 68 msec., had a nonzero value. The other three
subjects were taking as much time to process a sequence
with a target element in the third position as it took
them to process a sequence with a target element in the

23rd poSition.

 

**Recall that these positions had three times the
number of target elements in them as in the other positions.
Therefore, there were a significant number of data points
corresponding to each of these positions.

Number of Responses

143

20

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45
IV.20.HO. A history of response times for selected

poSitions.--While pondering the reasons for the discrep-

 

ancies between the results of SHN and the results of this
experiment, it was noticed that subject DJ's data seemed
to fall into two classes: the data collected during the
first part of the experiment were a function of position,
the data collected during the last part were not. The
data from the other subjects did not have this prOperty.
To construct the graphs in Figure IV.6O the data were
arranged by position and in the correct chronological
order of their generation. The data were then grouped
into sets of 25 and the averages of these sets were cal—
culated. Figure IV.60.l indicates that DJ began the
eXperiment by using the anticipated scanning procedure:
she appears to have scanned the sequence from left to
right at a constant rate. However, as the experiment
progressed something began to happen to DJ's processing
technique. This change is illustrated by Table IV.l,
where the average time per symbol drops from 122 msec.
at the beginning of the experiment to 0 near the end.
Accompanying the decrease in average time per symbol,
there is an increase in the average time necessary to
process the sequences with target elements in the first
eight or so positions. The net result is that by the end
of the experiment, DJ appears to have used the same

processing technique that the other subjects used.

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48

TABLE IV.l.-—Average time per
symbol during the course of
experiment B-l: Subject DJ.

 

 

Set of Average time
data points per symbol
averaged* (msec.)

l-75 122
76-175 98
176-275 82
276-350 61
351-425 0
remaining 0

 

*For instance, 1—75 signifies that the first 75
data points (chronologically ordered) from each position
were used.

IV.30. Experiment B—2

 

IV.30.lO. Experimental design and procedure.—-With
very few exceptions, the power spectral estimates of
Exp. B—l did not have the expected "bumpy" form. Augen-
stein suggested that this discouraging lack of bumpiness
might have been due to an experimental variable that was
not identified at the conception of EXp. B-l, the amount
of time between trials.

It is known that a periodically varying electrical
potential, called the alpha rhythm, can be recorded from
the scalp (most strongly above the occipital lobes of the
brain) of most persons who are awake but whose mind is in

a resting state. Furthermore, if this resting state is

“9

disturbed, say by presenting something interesting into
the visual field, the alpha rhythm disappears. In the
eXperimental procedure of SHN there was an unintentional
30 second interval between trials. Conceivably, boredom
with the whole procedure could have imposed an alpha
rhythm during these intervals. On the other hand, in
B-2 there was only a very brief (the subjects averaged
four seconds per slide) interval between trials. Hence,
the probability of an alpha rhythm being present during
this interval is much smaller than during a 30 second
interval. Could it be that the same phenomenon which
generates the alpha rhythm is also responsible for the
periodic nature of the histograms? This was pure specu-
lation, but it did indicate a previously unforeseen dif-
ference between the two eXperimental procedures: the
time between trials.

The eXperiments developed to test this difference
were straightforward. In an experiment called B—2F, each
of five subjects* was given 1120 trials with a minimal
time between trials; this procedure duplicated that of
B-l. Then, in an eXperiment designated B—2S, each of the

subjects was given 1120 trials with a mandatory 30 second

 

*Subjects TJ, MV and DF were American. JT was from
the Far East and IK from the Middle East. All the subjects
were males. DF was very impatient and his data contained
many errors. He had a reputation for being very intel-
ligent. JT did not speak English as well as IK.

50

rest period between trials; this procedure was an attempt

to duplicate the procedure of SHN.

IV.30.20. Power spectral analysis.--The analytical

 

procedure for B-2F and B-2S was identical to that of B—l.
To enumerate the steps:

1. The raw data were edited.

2. Histograms of one millisecond intervals were
formed (10 histograms altogether, five from
B-2F and five from B-2S).

3. Histograms with three, five, seven and ten
millisecond interval histograms (there were a
total of five histograms per subject per eXper-
iment). Examples of the 10 msec. interval
histograms are shown in Figure IV.70.

u. Sections, T[a,b] of each of the histograms
except those with one millisecond intervals
were chosen.

5. Parameter values of m'* = m* = m/NAt = .50,

.33 and .25 were chosen and PSE's calculated
using Eqns. II.NO.70 and 11.40.80.
Figures IV.8O and IV.90 contain examples of the
PSE's of Exp. B—2F and B-2S respectively. After an inspec-
tion of these and the other PSE's it has been concluded
that the only visible qualitative difference between the

PSE's of the two experiments is that the PSE's calculated

51

Number of s
Responses

“Q_._.

9

 

I!

10‘ II II I [I
f "l‘I'I‘IIIKIIII‘ .I; ['II, .3. II‘“. II
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‘I, ‘ 0'" '
A‘ _. .. ..__ - -'.' ‘3‘-“ :1-V‘Au.,:_nd._.1_ -31 ...- ".14 ..... -
XXI) 2000 3cm

Display Duration, msec.

0‘ 1L

Figure IV.70.1. Histogram with 10 msec. intervals:
Subject TJ in Experiment B-2S.

3'.
Number of g
.-
Responses
‘3.
.I
I .
”I
”I
'1
+~—- - -- '
‘ .I ’ 3-- '.- ‘K
II}; fl$$fif~fquIg fix-1195'." . ,II. ..
aL‘..___.,‘_____--, _____, I. if" +3va AMI-ma L. -. _ .. _ _ _. _.~_.-I__.,__.____n-s __
3 an L'\.. ll." m m

DisPIay Duration, msec.

Figure IV.70.2. Histogram with 10 msec. intervals:
Subject TJ in Experiment B-QF.

52

Figure IV.80. Examples of power spectral estimates
and their autocorrelations calculated from histograms of
Experiment B-2F. The autocorrelation is the upper or
leftmost of the two graphs that make up each subfigure.
The abscissa forAthe autocorrelation is the lag number k.
The ordinate is R(kAT). See Eqn II.MO.70. Conversion to
time lag AT can be made using AT - KAt. The abscissa for
the power spectrum is p, which is related to frequency
by f = p/2mAt. The ordinate is Po(p). See Eqn II.NO.80.
TEa,b] denotes the histogram interval analyzed, At is the
histogram interval size, m* = m.At/(b-a—l), m'* = m'o
At/(b-a-l), and s* is the average number of responses/
interval in the histogram over T[a,b].

53

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61

Figure IV.90. Examples of power spectial estimates
and their autocorrelations calculated from the histograms
of Experiment B-2S. The autocorrelation is the upper or
leftmost of the two graphs that make up each subfigure.
The abscissa for the autocorrelation is the lag number k.
The ordinate is fi(kAT). See Eqn 11.40.70. Conversion
to time lag AT can be made using AT - KAt. The abscissa
for the power spectrum is p, which is related to fre—
quency by f = p/2mAt. The ordinate is P (p). See Eqn
II.uo.80. T[a,b] denotes the histogram Ointerval ana-
lyzed, At is the histogram interval size, m* - moAt/(b-a—l),
m'* = m‘oAt/(b-a-l), and s* is the average number of re—
sponses/interval in the histogram over T[a,b].

62

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~— 1 l I I 1‘
go I -—l0 l—i Lo I to I 100 150 1L0
Figure IV.§C.§. Subject if; At = x.5 msec.,
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3.9 response /inte:val.
at
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Figure 1V.SC.T. Subject TJ: ‘At = 7 msec.,
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8* = 5.1 rcspon ses/j nterval.
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TIa,b] = T1710,3509], m* = m'* = .5 and
5* = 4.3 responses/interval.

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Figure IV.§O.1?.
T[a,b] = T[¥
8* a 5.8 res

1!

h

70

from the histograms of B-2F with 10 msec. intervals tend
to be slightly less bumpy than their counterparts in B—2S.
Note that the PSE's of B—2F are, for the most part, not
similar to those of EXp. B—l. However, the PSE's of the
histograms with 10 msec. intervals from both B—2F and
B—2S are similar to the PSE's obtained by Augenstein.

The qualitative interpretations of the PSE's of
B—28 and B-2F (also B—1 and SHN) will be made in the
next chapter. Whatever interpretative method is used, it
will have to in some way contend with the following:

1. For a given subject, as the value of At increases,
the bumpiness of the PSE's decreases (e.g.,
Figures IV.90.l, IV.90.3-IV.90.5).

2. As the value of m' decreases there can be a
change in the relative magnitudes of the peaks
of PSE's, all other parameter values being equal
(for instance, see Figures IV.90.ll and IV.90.12).

3. For a given subject, the frequencies of the
largest peaks of the PSE's from the histograms
of one interval size usually do not correspond
to the frequencies of the largest peaks of the

PSE's from another interval size.

IV.30.30. The average time per symbol.——Using the

 

Same procedure that was used in EXp. B-l, 10 msec. interval
histograms were constructed of the response times corres—

pOnding to target elements falling in each of the positions

71

3, 8, 13, 18 and 23. Typical sets of histograms of a
subject for both B-2F and B-2S are shown in Figure IV.lOO.
Note in both sets the increase in both average response
time and dispersion of the data with increasing position.*
However, the dispersion of the B—2F set is greater position
by position than the dispersion of the B-2S set. This
indicates that if the model T(-) does hold true, the

values of the parameters as and Up of the density functions
are different (however, the difference is not great enough
to noticeably affect the shape of the PSE's).

The relationship between the position and the average
response time of that position was approximately linear for
most subjects and for both B-2S and B-2F. The lepe of the
line which is the approximate mean square best fit of the
average response time vs. position data is the average time
per symbol. These values are recorded in Table IV.lO for

each subject and for both eXperimental conditions.

TABLE IV.lO.--The average time per symbol, Experiment B-2.

 

Value (msec.)

 

 

Subject
Exp. B-2F Exp. B-ZS
TJ NH 53
MV 64 57
JT —large- 233
IK 67 67
DF 22 13

 

 

*This is also a property of the statistical model T(').

Figure IV.lOO. Histograms of response times for
selected positions: Subject TJ.

73

 

 

 

 

 

 

 

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75

There are no appreciable differences in the values
of the two experiments for each subject. Furthermore, the
values are in the general range of those reported by
Augenstein for SHN.

Two of the subjects have average times per symbol that
are not in the general range Augenstain reported. However,
in both cases there is a logical eXplanation for this
discrepancy. Subject JT was an Oriental and was not
accustomed to scanning from left to right. It would be
eXpected, therefore, that his average time per symbol is
greater. Subject DF had a large number of errors (saying
the wrong number). He seemed to be more interested in
maintaining a fast, constant time than in accuracy or

obeying the rules of the experiment. (See Figure IV.llO.)

IV.30.HO. A history of response times for selected

 

positions.--The graphs in Figure IV.l2O were obtained using

 

the same procedure which was used to obtain the graphs of
Figure IV.60. Note that there are no significant common
differences between histories of each subject obtained from
B-2F and the histories obtained from B-2S. Also, in no
case does the subject require, during the course of the
experiment, as much time to notice the target element when
it is in the third position as he does to notice the target

element when it is in the 23rd position.

Figure IV.llO. Histograms of response times for
selected positions: Subject DF.

Number of
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-- -_._..

...-1““ .. ...m ...-......“

Form -2

 

 

 

I
P 11-. Aunt; A ___ ,_____ .____.._.
f
l
=:s T :x 3
PI
,. I"
C ......— .__...._..L - ..LLLL 1 W. __,__ 0......
f- A
Dzsnlvvrluatacn, T, sac.

20

~20 I

0 O u——————L!T:t

20

I

 

Figure

fl

0 , L......_...L ..
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78

POSITION 23

\

 

 

 

. '1
2.71.} .1 “.4 L'Jl-l L A___ A ____
POSITION I8
M
- __.__‘3_d; .
POSITION :3
.. "A; 1
POSITION 8
h, ---. ..
..ulrg'ufi_._-_._.._._---:.. _:___.....:>- .._-_.fi.- “fl..- ..-- l r l
lv‘OSITION 3
‘1
«waflw - a- .. -_ - H _,
l , 2 .3 _ Lo -
D:;p y B; ;t3'n, T, :;C. -
IV.llC.2. Laps :nt f‘-2.“.

79

Figure IV.l20.—-Histories of response times
for selected positions: Experiment B—2. The data
are lumped into groups of twenty-five.

 

 

l» [I '
m
:5
0
H .
0 I
8 C L_—.+—+--— o- ——§—-.-—-+---4 - -—-—O—+~
a
[1‘ 2 L; b 8
63:9; From

ime

Fign'r 21.12;.1.

onse T

(Q
lop

Wm
-\

 

 

1:. I.

?
5:4.9.

4» D
r 2‘
I L/o—(p—Qfl
J
/ JD
.1
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a
1
5
0 e 4 4+ : e a :» e 5 e A
o 2 u b e 10
the reginning oi the Expe:in L:
I
' I? I‘ ..,.. ’. ,
SUbjCCt :u. 14‘.) 4.: 5/32.). B”c’::‘ and
B-gS.
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:3 ,
2 .
r

80

 

 

 

I... 5,. -_ - . ._ - ---. ---“---
O “ 2

o:~‘b I -n l: J.“
Beglx-1in( (L. the

(D

L

Avernoo Rorhonno Tina

J

81

 

 

 

 

 

I
- F
o 1’
b i- / B
,1; *13 .
1. y"
5 I A .3 $\\y//’
:5 x
13
k‘ --‘~ "I‘ ...—~—
I-4 L, J-
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3
2 2 l 3
.1 {p l I-
O .I.___+...+.__V....----,----...--4--._..+_ D s —% I : : : :
O 2 u b O 2 ‘+ to
Group From the Beginning of the Experiment

Figure IV.120.3. Subject JT. (A) is Exp. B-2F and
(B) is Exp. B~2S.

82

Group
w

,-

 

ponse Time Po
N

   

 

 

 

 

...4) 25
La . “ ,, m
p _0 l5
0’) 0
Q.‘
m1 1 ax.»* 9
Q‘ ”WM 3
IE? "' Jr \J"° ~13 3
E a- -’ I’
> ./
‘fi
0 LA -§--—o—~-+~— ¢ ¢ ; e t : 0 -. ; 4. A. : ¢ ¢ a
O 2 H b 8 10 D 2 L? E:- B
Group From the Beginning of the Experiment
Figure IlfllQOJh Subject :w. (A) is Exp. 13-2? and
(B) is Exp. B-2s.
04 I
:3

 

 

 

 

H
0
5'4
0 I .923 r
a
.fi -
a q ..a ,
c Q *I/ 2 . '
m 1
co: ”.0 1.3
04 v
"'\
91 “\\0
a: 1 h “~o 5 l
m
m
L;
g «L 0n
(1", R‘. .
0 L.-- +—----+~—-—+-- —+———+-‘--+- 0 -L-——+-—~—o — - fi'-*‘4 -—-+---+ -—+—-——
0 2 Lo La 0 2 L. 'b
Group From tho Beginnixg o: the E piri33nt
Figure IV.120.5. Subje-t'15. (A) 1” Exp. B-2F and
(B) is Exp. Bw2S.
. ,‘

CHAPTER V

EXPERIMENT B-3

V.lO Introduction

 

The analytical procedure used in the previous
eXperiments consisted of constructing histograms appoxi-
mating unknown density functions and calculating power
spectral estimates (PSE's) from these histograms. Recall
that the power spectrum transformation was chosen because
inductive reasoning suggested that the value of the
parameter tp of the density function fT(°) would corres—
pond to the frequency in the power spectrum with the
greatest power density. There is, however, no proof that
this is the case for PSE's. The purpose of this eXper-
iment, therefore, was to attempt to establish definite
properties to be used to interpret the PSE's obtained in
experiments SHN, B—1 and B—2. More specifically, the
purpose of Experiment B-3 was to determine properties
associated with the shape and frequency of maximum power
density by which the following questions concerning an

arbitrary eXperimental PSE can be answered:

Ql. Is the power spectral estimate a member of the
set Q' of all power spectral estimates of the fT(')

defined next?

83

8H

Q2. If it is, does the frequency of maximum power density

correspond to the value of tp?

Two computer programs were used in this eXperiment.

One program generated histograms from the density function

fT(°) with the component density functions

Eqn v.10.1. fT (t,) = N(ts,os),
S
Eqn v.1o.2. fT (t2) = N(tp,op),
p
and
Eqn V.lO.3. PA(a) = 1/30, a=l, 2, . . ., 30.

This program required the specification of the values of
the parameters At (the resolution of the histogram), ts,
t

o o and S (the total sample size) in order to

p’ S’ p

generate a histogram. The other program calculated the
PSE from a given histogram. To do this it was necessary
to specify the parameters T[a,b] (the interval of histo-
gram used for the estimate), m (the maximum lag of the
autocorrelation) and m' (the maximum resolution of the
PSE).

v.20 The Experimental Procedure
and Results

 

 

The experimental technique used to develOp the prop-

erties of Q' associated with question Q2 was to generate

PSE's for various combinations of the nine parameters

specified above and then, through trial and error
classification, to develOp a set of properties which
allowed the eXperimenter to place an arbitrary PSE
generated from the model into one of three groups.
Evaluation of the members of each of these groups revealed
that one group consisted entirely of PSE's whose maximum
power density occurred at the frequency corresponding to
tp,fp. It was concluded, assuming the generated set of
PSE's was representative of the entire set Q', that these
properties can be used to classify any eXperimentally
derived PSE such that if the PSE is placed in the first
group the value of the parameter tp can be stated with
some confidence.

The technique used to answer question Ql, was to
examine the PSE's generated in conjunction with question
Q2 for a property common to all of them. Assuming the
generated PSE's are representative of the set Q', this
property is a necessary condition for an eXperimental PSE
to be a member of Q'.

The critical aspect of the experimental technique
was the choice of the parameter values used to generate
the PSE's. They had to be so chosen that the generated
PSE's were representative of the set Q', while at the same
time the number of PSE's generated had to be kept to a
minimum. To see how this was accomplished, a brief descrip—
tion of the procedure used to choose the parameter values

is in order.

CD
0\

The first step in the experiment was the selection
of the values for the parameters ts, At, T[a,b] and tD.
The value for At was set equal to 10 msec. in order that
the results might be immediately applicable to an inter-
pretation of Augenstein's PSE's. The value for T[a,b] was
set equal to T[801,2900] because the histograms were
expected to have substantial values in this interval. The
value for tp was set equal to 50 msec. because this value
appeared to give the best balance between good resolution
and the ability to detect harmonics. A value of tS = MOO
msec. was chosen because this seemed to be close to the
true value. However, the value assigned to tS will not
have any effect on the PSE, since a power spectrum does
not retain the phase information of the signal.

The next set of parameter values specified was 8*,

m* and m'*. These parameters are defined by
Eqn. V.20.l. m* = m°At/(b-a-l)

Eqn. V.20.2. m'* = m'-At/(b-a-l)
and

Eqn. v.2o.3. 3* = S~At/(b—a—l)
Also define

Eqn. V.20.A. o * = oS/t

E n. V.20. . o * o t
q 5 p p/ p

for future convenience.

87

The computer programs were used to empirically
determine the best values for these. First, PSE's were
calculated for the following parameter settings; 8* = 3.6,
m* = m'* = 0.5, 08* = 0 and op* = 0.02, 0.06, 0.10. 0.16
_and 0.26. These PSE's are Figures A.l, A.2, A.lO, A.l2
and A.l8 of Appendix A. Note that the PSE corresponding
to op* = 0.02 has maximum power density at the frequency
corresponding to the value of the parameter tp(fp) and
very little power density at the other frequencies (the
background). As op* increases, the average magnitude of
the background increases and becomes more irregular. For
op* 3 0.10 the PSE is very irregular and the maximum power
dentiy no longer occurs at fp. Furthermore, the same
trend was found for the parameter combinations S* = 3.6,
m* = m'* = 0.33, op* = 0.1, 0.2, 0.3, 0.5, 0.7 and 0.9.

In this case, the PSE corresponding to 08* = 0.5 had many

large peaks, the largest being at fp. The PSE's corres—

ponding to 05* f 0.5 did nOt have maximum power density

at fp.
The effect of 8* on the shape of the PSE's was

tested by using the parameter combinations op* = 0,

08* = 0.3, m* = m'* = 0.33 and 8* = 3.6, 7.2, 10.8, 14.3

and 17.9. The results were that as the number of samples

per interval increased, the irregularity and average

power density of the background decreased.

88

Finally, the effect of m* on the shape of the PSE‘s
was noted. It was observed that as the autocorrelation
lag parameter increased, the AC became more irregular and
the average magnitude decreased. This was accompanied by
increasing irregularity and higher average power density
in the background of the PSE's.

S*, m* and m'* are parameters whose values are
specified by the eXperimenter. Nothing more can be asked
of their values than that they give the best possible
resolution; i.e., a sharp peak at fp and lower power
density elsewhere. Therefore, m'* and 8* should be as
large as possible and m* should be as small as practical.
The value chosen for S* was 17.9. From experience, this
is the largest practical number of trials to run on any
one subject. The conditions that m'* be large and m* be
small are antagonistic. The reason for this can be seen
from the eXpression for a given frequency in a power
spectrum, f = p/2mAt. The largest possible frequency
which can be detected in the histogram is f0 = l/2At.
This corresponds to p = m in the frequency eXpression.
Therefore, the largest p possible, and hence the
largest m' possible, is m. The best compromise that
could be found was m* = m'* = 0.5.

With these parameters so specified, the current
five dimensional system was reduced to an easily visualized

two dimensional system. Appendix A contains the majority

89

of the PSE's generated from the parameter combinations
S* = 17.9, m* = m'* = 0.5, tp = 50 msec., tS = A00 msec.,
TEa,b] = T[801, 2900] or TEAOl, 2900], At = 10 msec. and
08* and op* variable. (The reason two different histogram
intervals were used is that midway through the calcula~
tions it was found that sticking only to the front of the
histogram improved the resolution.)

At this point in the eXperiment some tentative
properties were established for use with question Q2. They
were tentative because of conditions under which these

properties were applicable: T[a,b] could only be the

entire histogram, fT (.) and fT (°) could only be Gaussian,
s

tp could only be around 50 msec?, tS = MOO msec., s* = 17.9,
pA(a) = 1/30, m* = m'* = 0.5 and At = 10 msec. It should
be noted that these restrictions are not as severe as they
might seem. As a matter of fact only the restraints on

tp, At, T[a,b] and pA(a) are serious. PSE's were generated
and analyzed for various combinations of values of the
first three of these parameters and the tentative proper-
ties revised until the restrictions on these parameters
were lifted (the restriction on pA(-) remains). Appendices
A and B contain the majority of the PSE's generated and
Appendix 0 outlines some of the miscellaneous prOperties
observed during this portion of the experiment.

The next step in the experiment was to use these

revised properties to demonstrate that the generated

\O
C)

subset of Q' was indeed representative of the entire set.
Section V.U0 is this demonstration. Finally, the property
common to all of the generated PSE's was defined.

V.30. The PrOperties Established to
Interpret an Experimental PSE

 

 

The necessary property for a PSE to be a member of
Q' is that it be irregular, or bumpy. However, just how
bumpy a PSE has to be is difficult to define. An equiva-
lent property for a sufficient condition for a PSE not to
be a member of Q' is that the PSE should be non—bumpy.
This property is more easily defined: By non-bumpy is
meant a shape where the maximum power density occurs at
zero frequency and low and reasonably uniform power
density is found at all other frequencies. The PSE's of
Experiment B—l are good examples.

The properties established to determine the value of
tp in an arbitary PSE were the sharpness of the peak of
greatest power density and the average value and bumpiness
of that portion of the background. These prOperties place
the PSE in one of three groups:

Group A——The maximum power density occurs at a non—
zero frequency. The associated peak is sharp and there is
low and fairly regular power density at other frequencies.
"Fairly regular" is defined to mean no peak in the back—
ground greater than about 0.2 above the average power
density. The value considered to be "low depends on At.

The larger the value of At, the higher this low.

91

Group B-—The PSE has two or more peaks with a power
density of at least 0.8. The topology of the PSE is
generally irregular and the average power density high.

Group C-—The maximum power density occurs at a non—
zero frequency. There may be other peaks of comparable
magnitude (on the order of 0.8) but there is no doubt as
to which is the principal peak.

Recall that the PSE is calculated from a histogram
generated by a density function of the random function

A(s)

T(s) = Ts(s) + .2 TP (s)

i=1 1
Also note that these properties assume that the density
functions fT (-) and fT (-) are Gaussian and that pA(-)
is fairly uniform. In :ddition, it is advisable to have
m'* = m* = 0.5, S* = 17.9 and T[a,b] equal to the first
third of the histogram (if necessary) in order to obtain
the best possible resolution.

V.u0. A Demonstration of the Representative
Nature of the Generated PSE's

 

In order to demonstrate that the PSE's generated
during the course of Experiment B-3 were representative
of the entire set 0', each of the generated PSE's was
classified into one of the three groups defined in the
previous section. Figures V.10-V.30 summarized the results

of this classification. If a PSE has been classified, then

92

the symbol corresponding to the group the PSE was placed
in is found at the coordinates of the values of 8*, 0p* and
05* associated with the given PSE. A square signifies that
the PSE was placed in a group A, an ellipse signifies that
the PSE was placed in group C, and no symbol signifies
that the PSE was placed in group B. The number "inside"
the symbol is the average power density of the background.
Note that all of the PSE's in group A had their maximum
power density at the frequency fp, all of the PSE's in
group B did not have their maximum power density at fp and
the PSE's in group C sometimes did and sometimes did not
have their maximum power density at fp. An examination of
Figures V.10—V.30 reveals that the properties under con-
sideration are indeed well behaved over the parameter
values covered by the figures. That is, close to the
origin of the figures and close to each axis the PSE's
classified in group A dominate. Near the fringes of the
explored space, on the other hand, the PSE's placed in
group B are dominant. There also appears to be a transi-
tory space between these two extremes where group C PSE's
are found. This well behavedness leads to the conclusion
that the available PSE's are representative of the entire

set Q'.

93

Figure V.10. Summary of PSE results, Experiment B—3.
The histogram interval analyzed in all cases except those
noted was T[a,b] = T[801,2900]. Other parameter values
were m* = m'* = .5 (exce t where noted), At = 10 msec.,

tp = 50 msec., and tS = 00 msec. If a PSE has been cal-

culated in Exp. B-3, a number, possibly inside a square or
ellipse, appears at the coordinates (03*, o *, 8*)
corresponding to the values used in generating the PSE. The
number is the average power density of the PSE. The square
signifies that the maximum power density occurred at a non—
zero frequency, the associated peak was sharp, and the
remainder of the estimate (the background) was fairly
regular. The ellipse signifies that the maximum power
density occurred at a non—zero frequency and that, although
there are peaks of comparable magnitude, there is no doubt
as to which is the principal peak. See the text for details.

 

 

b8. x® . 11me g
. E
«NW.
8. Q @ s.
6 a... ....
a,» ma +3 or.» u...» «h a? I»... ma. .3. Ilmnfi‘ Q\M vd on. n: N. m .T
.5. 26 ms. 8 ,ch Rs .30. . - _ - - ...... %
1
. at u ....................... n .....-” . .
"@ g a n to. %
w; ~ "mmm. a E H 31...
_ ”rm? ® 0m... m ...;
nOowu..3uku_o>81H ... .......................... . mm... 9 Ed
R? 1...?
1 s.
31 E
a an.
*5
Mm.h‘.§u&€§\ u*
on

 

......» .... 6

sm

95

Figure V.20. Summary of PSE results, Experiment B-3.
The histogram interval analyzed was T[a,b] = T[401,1U00].
Other parameter values were m* = m'* = .5 (except where
noted), At = 10 msec., tp = 50 msec., and tS = “00 msec.

If a PSE has been calculated in Exp. B-3, a number,
possibly inside a square or ellipse, appears at the coor—
dinates (08*, op*, 8*) corresponding to the values used

in generating the PSE. The square number is the average
power density of the PSE. The square signifies that the
maximum power density occurred at a non-zero frequency, the
associated peak was sharp, and the remainder of the esti-
mate (the background) was fairly regular. The ellipse
signifies that the maximum power density occurred at a non-
zero frequency and that, although there are peaks of
comparable magnitude, there is no doubt as to which is the
principal peak. See the text for details.

96

LVOF 2m .Hb

 

 

Du 9. 3 v3 «1 or an 3 vWIIWn on on on N «a 0K 3 t .2 Mn r.
is. 8
a e as
x«/. ,
a a a mm.
@ m8 .2.

 

97

Figure V.30. Summary of PSE results, Experiment B-3.
The histogram interval analyzed was TEa,b] = T[1401,2A00].
Other parameter values were m* = m'* = .5 (except where
notedh At = 10 msec., tp = 50 msec., and t8 = “00 msec.

If a PSE has been calculated in Exp. B—3, a number,
possibly inside a square or ellipse, appears at the coor-
dinates (08*, op*, 8*) corresponding to the values used

in generating the PSE. The square number is the average
power density of the PSE. The square signifies that the
maximum power density occurred at a non—zero frequency, the
associated peak was sharp, and the remainder of the esti-
mate (the background) was fairly regular. The ellipse
signifies that the maximum power density occurred at a non-
zero frequency and that, although there are peaks of
comparable magnitude, there is no doubt as to which is the
principal peak. See the text for details.

98

mfle\o ......Uo

E» ..v 2. on on 3. lawn chum on 3.. an 9.. 3.: t 3 9

as. an...

I //- z

E. as e

Er. u «M. m2.

3w. 3 mg

Nay

¢/”
’9

42>

 

99

V.50. An Interpretation of the EXperimental
Power Spectral Estimates

 

The properties and groups discussed in the previous
chapters can be used to place an arbitrary eXperimental
PSE into one of the following three categories:

Category A——The PSE was not calculated from a

 

histogram generated by a fT(°) with a reasonably uniform

pA(')o

Category B--The PSE was calculated from a histogram

 

which could have been generated by a fT(') with a reason-
ably uniform pA(').

Category C-—The PSE is a member of the set Q': the

 

density function fT(-) which generated the histogram has
a fairly uniform component density function pA(°). A
legitimate value for the parameter tp can be determined.

The PSE's published by Augenstein and the PSE's of
Exps. B-1 and B—2 have been placed in one of these cate—
gories. The results are:

The PSE's published by Augenstein: All three cate-
gories are present. The PSE's of SHN (e.g., Figure II.50)
were placed in category A. Two PSE's, one from Exp. WSA
and the other from Exp. SO (Figure 11.40) were classified
in category C. In both cases, the value for tp was 100
msec. All other PSE's fell into category B.

The PSE's of Exp. B-l: All PSE's of this experiment

were placed in category A.

100

The PSE's of Exp. B-2: There was no significant
difference in the classification of PSE's from B—2S and
B-2F. The PSE's calculated from the 3 msec. histograms
were classified into category B. As the interval size
increased, the PSE's began to be classified in category
A. The PSE's calculated from the histograms with ten msec.
intervals were placed, for the most part, in category A.

Two further comments can be made concerning the
PSE's of Exp. B—2. First, the PSE's calculated from the
histograms with ten msec. intervals were very similar to
the PSE's of SHN. Second, the general behavior of the
PSE's——i.e., the decrease in the bumpiness with increasing
At--was seen in the behavior of the model, but the change

was not drastic.

CHAPTER VI

DISCUSSION: PART II

VI.10. Summary

 

In an eXperiment SHN, designed by Augenstein,
subjects were required to scan random sequences of 26
letters and numbers for the first number from the left in
the sequence. The response time data collected from this
and other experiments were used to form histograms and
power spectral estimates (PSE's) were calculated from
these. The PSE's were averaged to form an "experimental
PSE." This PSE was compared to several PSE's calculated
by substituting parameter values into an equation
(II.10.1U) expressing a statistical model and generating
data in a Monte Carlo fashion. (This model was based on
the thesis that the visual system of man can be thought of
as an information processing system whose decision—making
is performed in an essentially periodic* manner.) The
value of the parameter tp used in the generated data
whose PSE most closely resembled the eXperimental PSE

was said to be a reasonable estimate of the cycle time

 

*Periodic is used here in the intuitive sense and
not in the rigorous mathematical sense.

lOl

102

associated with the periodic data processing in the visual
system.

Three eXperiments, all modifications of SHN are
described in this report. A fourth eXperiment examines
the interpretation to be given to the resulting PSE's.

The outcome of these experiments allow at least partial
answers to some important questions. These questions and
answers are now presented.

QUESTION: How should the power spectral estimates
of the relevant eXperiments be interpreted?

The objective of any analytical method used on the
response time data is to determine if the data are periodic
and if so the time duration of the fundamental period. As
has been pointed out in previous chapters, the power
spectrum is the obvious choice for this analysis. However,
reasons have also been given for the difficulties in con—
fidently interpreting the power spectral estimate available
from eXperimentation. In an eXperiment B—3 some heuristic
rules were deve10ped that partially eliminate the diffi-
culty. A description of these rules and the conditions
under which they apply are found in Section v.60.

QUESTION: What are the eXperimental variables and
what are their effects on the results of the scanning
eXperiments?

Four experimental variables have been examined or

found to be important during the present eXperimentation.

103

The illumination

 

In an experiment B-2F, SHN was duplicated as closely
as possible except that incandescent lighting was used in
place of fluorescent lighting. The PSE's calculated from
the data of B2F were similar to those of SHN. The con-
clusion is that the lighting is not a source of data

artifacts.

The data resolution

 

The data of SHN were recorded to the nearest 10
msec., making reliable detection of any cycle times below
20 msec. impossible. Augenstein questioned whether the 100
msec. cycle time he detected might not be the result of
ten 10 msec., or some other similar combination, cycle
times being lumped together. The data of all experiments
reported in this thesis were recorded accurate to 1 msec.
No evidence of cycle times in the range of 2 to 20 msec.

was found.

The trial interval

 

Because of the manual nature of the procedure of
SHN, there was an interval of approximately 30 seconds
between trials. In an eXperiment called B—2F this interval
was shortened to 3 to A seconds. The PSE's of B-2F were
not significantly different from the PSE's of either SHN
or B-2S. The conclusion is that the interval between

trials is not a significant source of data artifacts.

104

The type font and the method of scanning

 

The scanning method used by the subjects of SHN and
the subsequent experiments was assumed to be to scan the
stimulus sequence from left to right for the first
number. That this method was indeed used by the subjects
of B-2F and B-2S has been confirmed from two sources.

The first source was by visual observation of the subjects:
their eyes appeared to be moving from the left hand side
of the sequence to the right. The other source is the
histories of the response times for selected positions.
These histories show that the average response time
increased with increasing position of the target element
to the right in the sequence, and that, almost without
exception, this relationship was maintained throughout

the course of the eXperiments. However, in an eXperiment
that was a duplication of B-2F, the subjects' histories

do not show this relationship: the average response time
for stimuli with the target element in the first position
is the same as that for stimuli with the target element

in the last position. The subjects in B-l were apparently
not scanning from left to right.

A possible method that the subjects of B-l were
using depends on the fact that pica type was used in the
construction of the stimuli. A characteristic of this type
is that the tails of the numbers fall above or below the

average top or bottom of the letters. Thus, the first

105

number from the left in the sequence literally sticks out
like a "sore thumb." It is conceivable that the human
visual system could be dynamically programmed to detect
only the tail, disregarding most of the information that
would be processed according to the anticipated scanning
method. A random scan may well be the most efficient way
to detect a tail. Furthermore, it is entirely possible
that such a method would be performed using a separate,
nonperiodic "primitive” processor.

The above method is also important in its effect on
the data of the other eXperiments in the SHN series. If
the primitive processor were used for only a small portion
of the tasks by a given subject, the effects on the data
of that subject would still be disastrous in terms of
introducing noise into the data.

All things considered, the tails on the numbers in
the pica type used in the stimuli is an important new
candidate for a source of data artifacts. Unfortunately,
this possibility was recognized only during the final
analysis and thus was not tested eXplicitly.

QUESTION: Do humans show periodicity in the time
required to perform a simple visual task, and if so, what
is the period or cycle time?

Because of the stochastic nature of any available
data, it is impossible to answer the question as it is

written with any confidence. Therefore, the question has

106

been rephrased. A stochastic model has been constructed
and the question is whether the data could have been
generated by this model, and if so what is the period, or
cycle time. Under all eXperimental conditions, including
Augenstein's experiments, and using the methods of inter-
pretation deveIOped eXpressly for the analytical tech-
nique used, only a couple of instances were found (for
instance, see Figure 11.A0) where it could be concluded
that the data could have been generated by the model. The
cycle time in these cases was about 100 msec. For the
rest of the data it was concluded that although they might
have been generated by the statistical model, it was
impossible to determine the cycle time (see the rules in
Section v.60).

V1.20. Suggestions for Further
Experimentation

 

 

Future eXperimental efforts in this area of research
will depend on the motivation of the experimenter. If the
experimenter is motivated by the desire to determine, once
and for all, the model to eXplain existing data, or if new
experiments are planned that follow the basic procedures
of SHN, then it will be necessary for him to use different
density functions of pA(.) and attempt to generate more
general rules for the interpretation of the power spectral
estimates. There is a pessimistic note to be sounded if

this course is pursued: a pattern recognition computer

107

program (learning with a teacher) was written (19) that
could not separate histograms according to the different
functions used for pA(°) (no matter how different they
were).

If the eXperimenter is motivated by the desire to
determine both the sufficiency of the model and the value
of the clock cycle time, but wants to use existing inter-
pretational procedures then it will be necessary for him
to design new experimental tasks or severely overhaul the
old ones. For instance, he could use the basic procedure
of SHN and attempt to decrease what amounts to as by
designing a system that would open the shutter on the
projector with the movement of the subject's eyes and
close the shutter at the beginning of the utterance of the
name of the digit. Optimism for this approach is lessened
by the problem of not knowing what went wrong with Exp.
B-l.

Alternative experimentation might be to repeat some
of the eXperiments Augenstein conducted that did produce
good results (WSA or SO). If these experiments are redone,
it will be necessary to determine the reason these exper-
iments produce the acceptable power spectral estimates
while the others do not.

There was something about the presentation of the
stimulus in EXp. B-l that enabled the subjects (including

DJ during the last twenty per cent of the eXperiment) to

108

detect the target element in about the same average time,
regardless of the position of the target element. It is
also interesting that the subjects develOped the processing
procedure independently and without any encouragement

from the eXperimenter. The eXperimenter might attempt to
determine the variables in the eXperimental procedure of
the B series that enabled this phenomenon to express
itself. A suggested initial eXperiment is to use ERG
records to determine eye movements as a function of size

of the sequence image on the retina.

APPENDIX A

109

APPENDIX A

The figures in this appendix are the power spectral
estimates and their autocorrelations calculated in Exper—
iment B-3. In each figure the autocorrelation is on the
left and the power spectral estimate is on the right. The
abscissa for the autocorrelation is k, the lag number.
This axis can be converted to the lag time by multiplying
each value by At. The abscissa for the power spectral
estimate is p and can be converted to frequency using the
relationship f = p.2mAt.

Parameter values of At = 10 msec., t = 50 msec.,

p
t = A00 msec. and m* = m'* = .5 were used (unless other-

s
wise indicated) for all the figures. The value of the

parameter T[a,b] is T[801,2900] when 08* = 0 or op* = 0
and T[A01,2900] when 08* # 0 and op* # 0 (unless other-

wise indicated).

110

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APPENDIX B

126

APPENDIX B

Each figure in this appendix is a set of power
spectral estimates and their autocorrelations for a
single histogram generated during Experiment B-3. The
parameter values used but not indicated in the figures
are t = 10 msec., tp = msec., tS = 400 msec., m* = m'* =
.5 and 8* = 17.9.

Note that power spectral estimates are the upper
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most of the two graphs in the bottom row.

12?

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APPENDIX C

145

APPENDIX C

Miscellaneous Notes on PSE Behavior

 

During EXperiment B-3 a number of prOperties relating
parameter values to power spectral and histogram behavior
were noted. Some of these are discussed in this appendix.

C.lO. The Effect of Different Values of

tp on the PSE Shape

The effect of different values of tp on the PSE shape

 

 

was found to be intimately related to the value of the
parameter At. To see this, note that at least two points
of a sinusoid are required per period in order to determine
the frequency of the sinusoid. Applying this to a histo-
gram with intervals of width At, the highest frequency

which can be detected without error by any method is

Eqn C.l. fc = l/2At.

Therefore, the shortest period which can be detected is
2At. Experimentally, it was found that when the value

of tp was less than 2At, the PSE's had a shape which would
have placed them in group B (defined in Sec. V.30). This
is fortunate because it says that if the value of tp is
less than 2At, the PSE will not be mistaken for one that
indicates tp correctly (i.e., the maximum power density

is at f ).
p

1&6

lb?

It was also noted that when fp was set equal to odd
values (e.g., 21.u ops) either the maximum power density
occurred at that frequency (if fp was one of the discrete
values on the abscissa) or the maximum power density was
shared between the two frequencies on the abscissa
closest to f

p.

C.20. The Effect of Different Values
of At on the PSE Shape

 

The effect of the value of the parameter At on the
shape of the PSE's was tested by generating PSE's from
histograms where At = 3 msec. The results indicated that
the background of these PSE's tended to be more irregular
and were of lower average background than the PSE's gener-

ated for the histograms with At = 10 msec.

C.30. The Effect of Different Values
of Tfa,b] on PSE Shape

 

 

Several of the histograms which were generated for
the PSE's of Figure V.lO were used to test the effect of
different values of T[a,b] on PSE shape. Values of
T[uOl,lMOO], T[lu01,2UOO] and in some cases TE2401,3HOO]
were used, with all other values remaining the same as
those used for the PSE's calculated from the entire histo-
gram. The shape of the PSE's generated from the intervals
T[uOl,lUOO] and T[luOl,2UOO] have been summarized in Figures

V.2O and V.30. By comparing these Figures with each other

1&8

and Figure V.lO it is apparent that the first third of
the histogram has PSE's classified in group A over a much
wider range of 08* and op* values than does any other
portion of the histogram tested.

The source of this prOperty is demonstrated by
Figure 0.1. The Figure contains some of the histograms
generated in conjunction with EXperiment B-3. Note that
as op increases, the histograms become more uniform. The
uniformity is present only at the far right of the histo—
grams generated from small op and drifts to the left as
op increases. The first third of the histogram is the
last section to succumb to this tendency. Therefore, if
the eXperimental histograms exhibit this prOperty, the
first third of the histogram should be used for PSE
analysis.

C.HO. A Note on the Averaging of Power
Spectral Estimates ‘

 

 

Recall that a major step in Augenstein's analytical
procedure was to average several of the model PSE's. He
had, however, given no indication that this procedure was
valid. This procedure was tested using the PSE's of
Appendix A (those which were done in duplicate). The
results suggest that averaging does increase the resolution
of the PSE.

The reason for this is clear from the following argu—

ment. Consider all histograms generated from a single

ll

149

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150

density function. It would be eXpected that averaging

the histograms would give a better estimate of the density
function which generated them. If the parameter values

to be specified in order to calculate a PSE are kept
constant, then for each histogram there is one PSE.
Therefore, averaging the PSE's should give a better esti-
mate of the power spectrum of the density function. As

a matter of fact, the PSE's should converge faster than
the histograms because the PSE's do not contain phase

information.

BIBLIOGRAPHY

151

10.

BIBLIOGRAPHY

Augenstein, L. G. "Human Performance in Information
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Augenstein, L. G. "Sampling Distribution of the
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Augenstein, L. G. in Information Theory in Psychology,
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Bendat, J. S. Measurements and Analysis of Random
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Blackman, R. B. and Tukey, J. W. The Measurement of
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Dubes, R. Class notes. Summer, 1967.

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153

Papoulis, A. Probability, Random Variables and
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Seiwell, H. R. Reviews of Sci-Instruments, g1:
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Stroud, J. M. "The Fine Structure of Psychological
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Tukey, J. W. "The Sampling Theory of Power Spectrum
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Woods Hole, Mass., 13-1” June, 1959. ONR
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MSFC Vibration Committee. Vibration Manual,
George C. Marshall Space Flight Center, Hunts-
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Wallis and Moore, J. Am. Statist. Assoc., 36,
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Graham, C. H. Vision and Visual Perception
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Von Bekesy, G. Sensory Inhibition, Princeton Univ.
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