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COMPLEXITY INTERNATIONAL |
| ISSN 1320-0682 |
| Source: | http://www.complexity.org.au/ci/vol06/henmi/henmi.html | Received: | 01/07/1998 | ||
| Vol ?: | Copyright 1998 | Accepted for publication: | 15/10/1998 |
Takuo Henmi and Michael L. Kalish
Department of Psychology
University of Western Australia
Email: takuo@psy.uwa.edu.au kalish@boneyard.psy.uwa.edu.au
There is ample evidence to show that nonlinear
dynamical or chaotical
properties underlie aspects of physiology,
neurology, and even behavior. This
paper presents a psychophysical ``cascading''
experiment in which the response is passed on to the next trial as the
new stimulus. The time series of response is modeled by a nonlinear
psychophysical model based on an existing recursive cubic polynomial
function called the ``
recursion'' originated by Robert Gregson. The
responses in the cascading experiment are found to be classified into
three categories, and some show the trace of chaos. However, the
attempt to model the time series with the new model or the original
recursion resulted only in coarse approximations to the
data. In spite of its inadequacy at simulating the time series itself,
the new model managed to simulate the autocorrelation functions of the
original data. These results suggest that the model we propose is in
some sense within the same family of dynamical systems as the psychophysical dynamical system generating the observed
data although it is necessary to develop more subtle nonlinear
dynamical models.
Both the biological mechanisms of life, and their behavioral manifestations, have been found in many of their aspects to show the properties of nonlinear dynamics, or deterministic chaos [12, 13, 20, 32, 37, 39]. Examples are found in analysis of biological systems with electroencephalogram [10, 23] and electrocardiogram [27], or also at neurological level [1, 2, 3, 4, 11, 31]. Psychophysics also provides examples of nonlinearity. The fundamental principle of psychophysics is to investigate the subjective intensity of a given stimulus, and to provide a mathematical description of the relationship between subjective intensity and stimulus magnitude in terms of a psychophysical law. For example, Stevens [40] proposed a ``power law'' which is defined as
where k and
are constants,
denotes the stimulus magnitude, and
denotes the subjective
magnitude. That is, by assuming that changes in responses within the
organism are directly proportional to changes in the level of external
stimulation, the power law can plausibly produce the relationship of
any stimulus magnitude and subjective intensity response. The power
law is often referred to as a ``linear'' psychophysical law since the
stimulus-response relationship can be represented with a straight line
on log-log coordinates. The power law has been widely accepted as a
good approximation of such relationships; however, it also has met
with some skepticism. For example, Ross and Di Lollo [38]
reported the constant failure of the power law in an experiment in
which observers were asked to judge the magnitude of lifted
weights. The failure of the power function lead some researchers to
propose other expressions [29] while others (see chapter 6
of Uttal [43]) have proposed various ``nonlinear''
psychophysical functions. Indeed it is plausible to assume an
essential nonlinearity in the stimulus-response relation if one
assumes a nonlinearity in the biological substrate of perception.
Among the nonlinear psychophysical models that have been proposed,
perhaps the leading example is the ``
recursion'' function
[14]. While other nonlinear functions have been applied to
particular phenomena (e.g., Watson [46]), Gregson has shown
that his
recursion, which is a complex valued cubic iterative
function, can plausibly simulate a wide variety of psychophysical
effects. Having previously shown that the
recursion and its
extended models are valid nonlinear psychophysical models at a
practical simulation level [21], this research examines the
ability of the one-dimensional model to simulate behavioral
phenomena. In particular, we are interested in how a special property
of the , ``cascading'', might be used to model dynamic aspects
of line length perception, which, as reviewed below, is known to
produce illusory effects under a number of conditions. The main
characteristic of cascading is to transform the intensity of a
response from one step of the simulation before applying it as a
stimulus value at the next stage. This cascading process is vital for
the model's account of nonlinear psychophysical phenomena
[14]. We suggest that the cascading process has a natural
behavioral interpretation if we allow observers to produce their own
stimuli. In particular, we asked observers to estimate the length of a
briefly presented stimulus line over the course of a number of
trials. In each trial but the first, in which a standard line was
presented, the line presented to an observer was of the same length as
they had indicated their subjective estimate of the line length to be
on the preceding trial. If we assume nonlinear properties in human
perception, then we should observe dynamics in this experiment like
those predicted by a nonlinear cascaded model.
In this paper, we first describe the model to simulate the time series
obtained by the experiment. The model is based on the cascaded
recursion [14], but since it is different in some
respects, we refer to it as a "cascaded
recursion ("cascaded
") model". We then briefly review the literature on
perceived line length, and present the experiment. Apart from modeling
with a nonlinear cascaded model, we also present a descriptive
response time series analysis to see if the perceptual cascading
process possesses nonlinear (in particular, chaotic) dynamics.
The recursion is a well-established psychophysical function
that Gregson has shown not only in a one-dimensional stimulus-response
level [14], but also in a multi-dimensional vector form,
[15, 16, 17], and a lattice form,
[18, 19]. A theoretical
advantage of an iterative perceptual model such as Gregson's is that
it can reflect the iterative behavior of biological neural
networks. As we mentioned above, we adopt a cascaded cubic
polynomial recursion ("cascaded
" ) model which is based on Gregson's
one-dimensional
recursion. We therefore briefly review some
properties of the
.
The recursion is defined as
or equivalently
where a is real, ie is imaginary, and Y is complex. In order to
apply this function as a psychophysical function, the input (stimulus)
series U is scaled so that the parameter value a lies between 2
and 4. The imaginary component represents the internal activity level
within the system. The parameter value e represents, roughly
speaking, the sensitivity of the system to rates of change of inputs
in time (not the well-known constant
). It is
constrained to the range between 0 and 0.5 (0 < e < 0.5) or ae < 1.7 in
the region of 0.5 < e < 0.7, to avoid an explosive condition. The
initial value for this model,
, is fixed at the onset, namely,
(i.e. the
initial condition
). After an arbitrary
number of iterations (denoted as
), the real component of Y
(denoted as
which must be in the region of
in order to prevent the explosion) represents
the observable output or response magnitude. In short, the simulation
is composed of;
The difference between
recursion and the cubic polynomial of the "cascaded
" model is in their initial values. The initial value of the
recursion is
which gives the complex aspect to the model, but it is modified just to
, i.e.
for the "cascaded
" model. One can view the iterative cubic polynomial adopted by the "cascaded
" model as the special case of
recursion collapsed onto the real line.
In the ``cascading'' process, the output is transformed to the next stimulus, that is
Each point estimated with (1) corresponds to that of the response time series obtained by an actual experiment. More descriptively, a cascading model consists of two phases;
Mathematically, the "cascaded
" model can be described as follows. The estimation phase is the same as (1), that is,
The cascading phase is composed of two parts. First, the next input is computed as
where c and d are linear scaling
constants, and
is the number of iterations. The second stage of
the cascading phase concerns the initial value for the next estimation
phase. For the first estimation phase,
The
estimated value subsequently becomes the initial value of the
following simulation, i.e.,
although
in the original
cascaded
recursion. If we make the time series obtained from
an actual empirical (scaled) data set,
then the "cascaded
" model estimates each point and produces the corresponding time series,
The parameter values
, e, c, and d are free, and can be set at the beginning to minimize the sum of the squared deviation,
As was mentioned above, line length is often perceived distortedly, as in geometric illusions such as the Müller-Lyer illusion, the Ponzo illusion, the horizontal-vertical illusion, and the parallel-line illusion [5, 22, 24, 25, 33, 34, 35, 45], four of which are shown in Figure 1. Depending on context, then, the human visual process distorts the equal length of the horizontal parallel or the horizontal-vertical lines, so that they appear to be of unequal length. However the current research concerns distortion that happens in the course of successive comparisons of line length, rather than the perceptual distortion in any simultaneous comparison.
Woodworth [48] referred the error caused by successive comparison as ``time error'', and noted that such error was usually negative--a negative time error happens when an observer underestimates the stimulus magnitude of an object (such as weight, frequency, etc). Woodworth's review on successive comparison and time error referred to weight lifting, auditory, and some esthetic experiments, but not to line length estimation. We were unable to trace the first systematic experiment investigating this particular subject, but as early as 1957, Yokose, et al. [49] showed that the perceived vertical length of a line, in successive comparisons, depended on the exposure time of the test line. According to their results, the line was perceived to be shorter when the test line was presented for shorter durations (the minimum exposure duration for their experiment was 50 ms). Moreover the subjective length of the test line was shorter than its objective length. Erlebacher and Sekuler [9] conducted an analogous experiment, and their results replicated Yokose et al.'s finding; subjective length within the stimulus exposure duration was shorter than the objective length.
Figure 1: Examples of line length illusions. (a) Müller-Lyer illusion, (b) Ponzo illusion, (c) horizontal-vertical illusion, and (d) the parallel-lines illusion. The horizontal lines of (a) and (b) are the same length, but the top lines appear longer in both cases. In (c), the vertical line seems longer than the horizontal line, but they are also the same length. The lines in (d) are different length. However, the presence of the other line has a normalizing effect. The top line appears shorter and the bottom line appears longer than it is.
Tsal and Shalev [42] and Prinzmetal and Wilson [36] studied the effect of attention on subjective line length; they also mentioned the phenomenon just described above. That is, in a successive line length comparison experiment, the comparison lines were perceived to be shorter than the physical length of the standard line on average. Prinzmetal and Wilson [36] hypothesized that the cause of the underestimation could include a framing effect from the display monitor [28, 30]. Alternatively, the bias may have been due to one of their methods to control subjects' attention in their experiment. Although in a modified version of their experiment, Prinzmetal and Wilson were able to eliminate underestimation, they nonetheless noted that observers do show an overall tendency towards negative time error.
Previous research has also suggested the tendency to underestimate line lengths in successive comparisons (cited in Brigell and Uhlarik [6]), agreeing with Woodworth [48]. However, the fact of negative time error in estimating a line length raises some questions. For example, what will happen to the underestimation effect for stimuli near the lower limen of perception? If the stimulus is determined from the previous response, will stimuli converge to the lower limen, or will responses show some kind of more irregular behavior? We have hypothesized at first that the response time series in a cascading experiment would show a nonlinear behavior, however as we reviewed the previous literature on successive comparison, it should be adequate to modify the hypothesis. That is, if the human perceptual process possesses nonlinear dynamical properties, then:
Twenty participants (four males and 16 females) were recruited; nine psychology undergraduate students seeking to fulfil a course requirement, and eight psychology PhD students at the University of Western Australia together with three people of the experimenter's personal acquittance. The participants' ages varied from 17 years to 38 years old. All had normal or correct to normal vision. Age and sex were not expected significantly to affect line length estimations (see Verrillo [44]), so the data from all observers was treated equally.
The apparatus used in this experiment is a PC with a 14-inch color display, and a two-button mouse device. The program is written in C++ in MS-DOS.
Participants sat in front of a PC display monitor in a semi-darkened room. The chair was adjusted so that the observer's eyes were level with the center of the screen. The distance between the observer and the display was approximately 50 cm. The screen resolution was 640 pixels wide by 480 pixels high. The stimulus consisted of red vertical lines on a gray background. The lines were 3 pixels wide and could be adjusted from 0 to 440 pixels high. The stimulus line was presented 243 pixels from the left hand side of the screen for 750 ms. This was followed by a 350 ms inter-stimulus interval, during which time the screen was blank. Then the control line was presented 243 pixels from the right hand side of the screen (about 10.3 cm). The distance between the stimulus and control lines was 148 pixels (about 6.3 cm). After the observer made their judgement about the length of the line (described in detail below) there was a 700 ms delay before the next stimulus line was presented.
The origin of the both the standard and the control lines was fixed at the bottom of the display screen. Observers were asked to use a mouse to adjust the top of the control line to match the stimulus line in length, and then to click the left hand mouse button to record their response. The length of the standard line for the first trial was set so that the line extended from the bottom of the working screen to its center (220 pixels in length), while on subsequent trials, the standard line was set to the length of the control line from the previous trial, again anchored at the bottom of the screen.
The length of the control line, when it appeared on the screen, was set to be the length of the stimulus line (in pixels) plus or minus a random number of pixels between 0 and 40. Irrespective of the size of the control line determined from previous trials, line length was constrained between 0 and 440 pixels. Prior to the experiment, the experimenter advised the observers to be as precise as possible in their judgements. The experimenter also advised each observer to notify the experimenter if they accidentally clicked the mouse during the course of experiment in order to correct the mistakes immediately. Observers were naïve to the method used for determining the stimulus line length on each trial.
The outcome of the experiment indicated that observers' responses tended to be of three distinct forms, described below, although, of course, no individual gave exactly the same responses as any other. The three categories of response series, of which an illustration over the course of 1000 trials is shown in Figure 2, were:
Figure 2: Changes in adjusted line length (in pixels) for 1000 trials in the cascading experiment. Observers' data can be grouped into three distinct categories. In (a) and (b), the line converged to the lower limen. For (c) and (d), line length decreased almost uniformly, before oscillating within a small range around a given value that differed for each observer. Observers in the third category produced data that did not appear to converge to any particular value. Examples of the third category are shown in (e) and (f).
The numbers of observers whose data could be classified into each case are 6, 10, and 4 respectively. The overall average of the positive and negative error (overestimation and underestimation of the stimulus magnitude) can easily be deduced by looking at the asymptotic response; that is, if the asymptotic response is below the starting point, then the overall average is negative and mutatis mutandis for responses above the starting point. Of the 20 observers we tested, only three showed a tendency towards positive errors, with the dynamics of their results obviously being of the form of case3. This result agrees with previous reports that a line indeed is perceived to be shorter than its physical length on average.
As for the dynamics of the time series, it is obvious that those in
case1 would be unlikely to show the trace of chaotic
dynamics as they converged to the lower limen. Since the
dynamics of those in case2 consist of two phases we chose to pay
particular attention to the second, oscillatory phase which we thought
might be well modeled by a chaotic dynamical system. In regard to
applying a chaos detecting algorithm to the time series, we have to be
careful as Kantz and Schreiber [26] have drawn attention to
the need to distinguish between stochastic noise and chaos. In order
to determine whether the time series data that we collected was
chaotic or stochastic noise, we adopted a method described by Khadra
et al. [27]. Their algorithm is applicable to
relatively short time series such as one in this experiment, as
opposed to those methods that require a larger time series, such as
those that need to compute the Lyapunov exponent. The algorithm for relatively short time series was
introduced by Sugihara and May
[41], but Ellner (1991, cited in Casdagli
[7]) indicated their results were faulty. Khadra et
al.'s algorithm [27], however, overcomes the problems that
lie not only in the Sugihara and May's [41] algorithm,
but in other algorithms as well, making it more reliable. The basis of
their algorithm is to state a null hypothesis that the given time
series is not chaotic, and to derive the probability of obtaining some
test statistic, calculated from the test data, under the null
hypothesis. If this probability p is less than some predetermined
, then we reject the null hypothesis and assert that
deterministic chaos is detected. Here, we will only summarize their
algorithm, but the interested reader should review the technical and
theoretical background of the algorithm in Khadra et
al. [27]. The reader who is familiar with Efron's
bootstrap [8] will see that the algorithm developed by
Khadra et al. is a close relative. Khadra et al.'s
algorithm is as follows:
) of the original time series with the forecasting algorithm
with fixed embedding dimension d for fixed k nearest neighboring
states described in Khadra et al [27]. The delay
factor (usually denoted as
) is set to the minimal value
(i.e. 
) in order to use all the data points. Since the method
to choose optimal d and k was not specified in Khadra et
al. [27], we adapted the parameters, over a small range
( 
), that minimize the MAE.
) and standard deviation (
) of the 128 MAEs.
, where
.
).
We applied the algorithm to those data in case2 (last 800 points) and case3 (1000 points), and disregarded case1 data. Moreover, in order to show the validity of the algorithm, we applied it to three randomly generated time series that are given by,
Gaussian white noise where
(R-W1), and
random integer l in [-20, 20] where
(R-W2),
" model. The results of the analysis are summarised in Table 1. As can be seen from the table, evidence of chaotic dynamics was found in 8 instances out of the 14 data sets analyzed (two of such chaotic time series are shown in Figure 2(c) and (e)) whereas all the randomly generated data gave the converse results. The dummy series generated by the "cascaded
" model also showed a sign of chaos that suggests its potential to model the data.
In order to test the "cascaded
" model, we first fit it quantitatively to the data. We utilized the simulated annealing method to optimize the parameters
of the "cascaded
" model in terms of minimizing the sum of squared deviation between actual and predicted data values. Figure 3 shows the dynamics of a case2 observer (Figure 2(c)) with the entire 800 points (a), first 400 points (b), first 200 points (c), and first 100 points (d) (solid line) along with the dynamics estimated by the "cascaded
" model (dashed line). The response magnitude of each figures was scaled to [0, 1] as the actual minimum value (in pixels) of the time series to be 0 and the maximum to be 1. As one can easily see, the simulated annealing method tended to determine the parameters so that the "cascaded
" model would estimate close to the moving mean value throughout the series. The failure to fit the data with the "cascaded
" model lead us to try simulating the data with the cascaded
recursion, that is,
The results are shown in Figure 3 with dotted line. In this case too, the model could not simulate the data.
Figure 3: Example of an actual data set (solid line) with the corresponding optimised "cascaded
The next attempt is to minimize the sum of squared deviation of the autocorrelation function of the data and that of the time series generated by the "cascaded
" model. The sample data taken here are two from case2 with first 200 points dropped and two from case3--those four time series shown in Figure 2 (Figure 4-7(a)). The autocorrelation was computed at lags up to one quarter of the entire time series (i.e. 200 points for those in case2, and 250 points for case3) as suggested in Williams[47]. The autocorrelation function of the real data are shown in Figure 4-7(b). Those four figures roughly represent the range of autocorrelation functions of all the series. That is, for case2, the autocorrelation functions either decreased uniformly and rather unsteadily, or went down to negative then came back up to positive again, and for all case3, the functions decreased uniformly and rather smoothly. No distinction was detected in the autocorrelation function between chaotic and stochastic data.
In this case also, the parameters
of the "cascaded
"
model were estimated by the simulated annealing method. The time series generated by the "cascaded
" model are
shown in Figure 4-7(c), and its autocorrelation function along with
that of the real data are shown in Figure 4-7(d). Since the
parameters were estimated to minimize the error in their
autocorrelation function, the time series generated by the "cascaded
" model
do not necessarily bear resemblance to the original data. However,
the "cascaded
" model could produce close approximation of the observed
spectrum at least in its overall shape.
Figure 4: Comparison of autocorrelation functions for the data from the line length experiment and the "cascaded
Figure 5: Comparison of autocorrelation functions for the data from the line length experiment and the "cascaded
Figure 6: Comparison of autocorrelation functions for the data from the line length experiment and the "cascaded
Figure 7: Comparison of autocorrelation functions for the data from the line length experiment and the "cascaded
The primary interest of this study was to investigate how well we could model the dynamics of observers making iterated judgments of line length; first, with a descriptive time series analysis, and then with a cascaded nonlinear function based on Gregson's
recursion. The descriptive analysis is essentially an attempt to detect chaos, if it exists, within observers' responses. The use of "cascaded
" on the other hand was an attempt to develop a model of observers' responses that was capable of generating the same response train as the observers produced.
Our results from the descriptive analysis indicate that the observers, with the proportion of 8:6, show deterministic chaos in their successive judgements of line length. Furthermore, it is possible that the perceptions of some of those observers who made stabilized responses (case1) are actually chaotic, but that this is masked by the fact that a lower bound is imposed on the adjustment of the line by the physical limit of the screen. Nonetheless, the fact that it is some, and not all, of the observers who show evidence of chaotic behavior, suggests that there are considerable between-subject differences in the form of responses. That is, for this particular judgement task, individual behavior varies from chaotic to stochastic. Intuitively, the long-term behavior of other judgement tasks involving successive comparisons, such as weight lifting or judging the pitch of a tone, would show similar results.
Our attempts to detect chaos in observers' responses were relatively successful. However, the presence or absence of chaos is not the most crucial question here--what one would really like to know is the nature of the process underlying the phenomenon observed. It seems intuitively plausible that the process of iterated perception ensues from a simple underlying nonlinear dynamic, such as Gregson's
recursion, which has proved useful in other instances of psychophysical modeling. But, demonstrating this rigorously, by matching the empirical data with data obtained from nonlinear dynamical equations, is a nontrivial matter.
We have made some initial efforts in this direction, using simulated annealing as a method of estimating optimal parameter values for the real
recursion and then for the "cascaded
" . The real gamma failed utterly in this experiment, providing, as the optimal fit, a trajectory converging to a fixed point. The "cascaded
" fared somewhat better, suggesting that "cascaded
" is a more plausible model than the real gamma recursion; however the results here were not entirely satisfactory either. What the "cascaded
" produced, with optimal parameters, was a smooth curve approximation, not unlike a moving average of the original data.
One could reasonably conclude from these results that the "cascaded
" and cascaded
are unsuitable for modeling the kind of chaotic behavior that we found in some observers' responses, irrespective of what parameter values one chooses. On the other hand, it turns out that the "cascaded
" model did a fair job of simulating the autocorrelation function of the original data although the time series generated by the "cascaded
" to fit the autocorrelation function do not show any particular resemblance to the original data. This result is not surprising since our attempt here is to fit the autocorrelation function as a whole, and not to show the overall trend of original time series. This could suggest that the "cascaded
" is in some sense within the same family of dynamical systems as the psychophysical dynamical system generating the observed data. One could construct a somewhat plausible argument that the psychophysical phenomenon observed consists of the "cascaded
" plus noise. However, we believe that this is not the case, and that the data reveals a deterministic chaos which is not revealed by the
and the "cascaded
" models that have been the subjects of our computational experiments.
The model was introduced into psychophysics in order to provide a deterministic explanation for psychophysical phenomena which were previously written off as ``just noise''. In this case, however, we have an apparently chaotic psychophysical phenomenon which neither the
nor the "cascaded
" predicts in any simple way. The only clue we have is that the "cascaded
" does better than
--it at least gives the trend of the data. Further study will be required to determine the form of the nonlinear dynamic that predicts not only the trend but the chaotic nature of the iterated perception phenomenon. Just as the classical psychophysical experiments led to the invention of
, this experiment must lead to the development of more subtle nonlinear dynamical models.
In addition to the primary results described above, our experiments also provided some secondary results, of particular note being the same tendency towards negative error that were found in earlier judgement tasks reviewed in Woodworth [48], and in the line length judgement experiments reviewed in the third section of this paper. Moreover, our experiments revealed three types of long term trends in the judgements of line length. Specifically, while some observers continued underestimating the length to the lower limen, the majority behaved irregularly when the line was very short; some even showed such irregularity throughout the experiment. In this experiment, the line was anchored at the bottom of the display with the line length being adjusted only from the top. If, however, one end of the line had been anchored at the center of the display with the line capable of being adjusted either upwards or downwards, then it is plausible to assume that the majority of case1 observers would still have shown irregularity in their responses when the line was very short.
We would like to thank Dr. Ben Goertzel at IntelliGenesis Corporation, Associate Professor Les Jennings at the Math Department, Mr. Mark Diamond and Mr. Jason Forte at the Psychology Department in the University of Western Australia for useful discussions.
ookisa no shincyou
kakudaikatei'' [The relationship between visual phenomina and the stimulating time (I)--About the growing process of the perceived length and size], 1957 Japanese Journal of Psychology (Japanese), 28, 10-17.
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