[From Bill Powers (950223.0500 MST)]

RE: Stevens' Power Law, the Weber-Fechner Law, and PCT.

The following may be a reconstruction of something, or related to

something, I read a few years ago and can't find the reference for. I

would like the opinion of all you perception experts out there.

## ···

----------------------------------------

Given: stimulus magnitude s and perceptual signal p:

Given: Weber-Fechner law of the form dp =k*ds/s, or p = k*log(s).

The Stevens power law is found by adjusting two stimuli until they are

perceived as equal.

By Weber-Fechner Law,

p1 = a*log(s1)

p2 = b*log(s2)

if p1 = p2 then

a*log(s1) = b*log(s2), or

s1^a = s2^b

Which is Stevens' power law. Note that the power law expresses only the

relationship between perceptions of different stimuli; it does not

establish an overall scaling factor between perceptions and stimuli,

because

if s1^a = s2^b then

p*s1^(ma) = p*s2^(mb)

where p and m are any numbers.

Therefore there are unknown common factors m and p relating all

perceptions to all stimuli, and no absolute psychophysical scale can be

established.

When Stevens published an article on the power law in Science many years

ago, I wrote a letter asking whether this wasn't simply a version of the

logarithmic law. The letter was rejected because Stevens said that

logarithms have nothing to do with the power law.

---------------------------------

These ruminations arose in the context of setting up a control-system

model with a logarithmic input function. Bruce Abbott had used a model

of control of the ratio of two input quantities in which the error

signal was multiplied by the input quantity in the denominator of the

ratio. This was motivated by the desire to keep the integration factor

in the output quantity equal to 1.0, and to have the control model reach

its steady state in one iteration.

The aim of having the model correct its error in one iteration was

actually unnecessary, because the real person being modeled can't do

that; with a step-disturbance, the real person responds with a finite

time-constant equivalent to perhaps 10 to 30 iterations of the program.

When the controlled quantity is a ratio, however, we find that the best

integration factor (in the particular experiment being done) is over 200

times the best integration factor under the assumption that the

controlled quantity is the difference between the same two variables.

The reason has to do with the loop gain of the control system, which is

the product of the output gain, the external feedback gain, and the gain

in the input function.

In the model using the difference between the two variables, one pixel

of handle movement results in one pixel of change of the input variable

in the numerator of the ratio, which results in one pixel of error

(expressing all units in terms of effects on the display screen).

However, the two input variables had an average magnitude of about 200

to 300 pixels. When the definition of the input function was changed so

the ratio was perceived, one pixel of handle movement now created only

1/200 to 1/300 of a unit of perceptual change in the model.

This meant that the loop gain dropped by a factor of 200 to 300. To make

the model behave like the person, it was necessary to increase the gain

elsewhere in the loop. Bruce increased it in the comparator, but the

increase could just as easily have been put into the input function or

the output function. What mattered was not using the value of one input

variable in the denominator of the ratio as a multiplier, but using a

multiplier of about the right size (which could then remain constant as

the model ran).

The two input variables being modeled were the horizontal and vertical

elements of an inverted T. The horizontal element's length was varied by

the computer, and the participant was to use a mouse to keep the

vertical element the same length. In fact, the participant keeps the

vertical element significantly shorter than horizontal element, thinking

it is equal. This is the T-illusion.

Two hypotheses were tested: that the illusion consists of a constant

difference between the lengths of the elements, and that it consists of

a constant ratio of the two elements. This is how ratio control came

into the model, and how we discovered the need to compensate for the

drop in loop gain that occurs in going from absolute length measures to

ratios. There is a consistent improvement in predictions of the

participant's behavior using the ratio model of perception in comparison

with the length-difference model.

Direct perception of the ratio vertical/horizontal is one possibility.

However, an equivalent perception would be log(vertical) -

log(horizontal), with a different scaling factor. The log-perception

model matches the behavior of the direct-ratio-perception model,

predicting the same magnitude of the illusion within 0.3 per cent and

predicting the handle movements with the same RMS error, 10.2 pixels, or

3 to 5 per cent of the range of the handle movements.

Thus it seems reasonable to say that in all control-system models

involving lengths, we should use an input function that perceives a

constant times the log of the actual length. This leaves some questions.

In a pursuit-tracking experiment, for example, should the position of

both target and cursor be reduced to logs before subtracting target from

cursor to get the magnitude of the perceived relationship? Or should the

target position be subtracted from the cursor position first, and then

the log of the difference be taken to produce the perceptual signal?

Only simulations will show which version fits best, and whether either

one yields improvements in predictions over the linear-perception model.

There are, of course, problems in dealing with the logs of variables

that can go negative, which may have to be handled by using two

perceptual functions to handle deviations in two directions from zero.

---------------------------------------

Logarithmic perception predicts some effects not seen with linear

perception. The main prediction concerns the scale of the presented

figures on the computer screen. In a linear model, the viewing distance

affects the size of image movements on the retina. Also, if the screen

presentation is simply scaled up or down, the same effect occurs. In

either case, the linear model predicts that the loop gain will change

with a change in the retinal size of the presentation, so that the best-

fit value of integration factor determined in the usual way will depend

directly on the visual size of the presentation.

In a logarithmic-perception model, on the other hand, the loop gain

becomes independent of the size of the presentation. A change of scale

is equivalent to a constant added to or subtracted from the perceptual

signal, and thus behaves like a disturbance that the control system will

remove by its normal action. Where ratios are involved, there is no

effect at all because the constant is added to the log of the numerator

and denominator equally and is removed when the logs are subtracted in

the perceptual function. Most importantly, the loop gain will be

unaffected by changes in scale, and we will find the same best value of

the integration factor regardless of scale (until we reach the limits of

the range over which the logarithmic relationship holds true).

Further investigations of these effects seem called for.

------------------------------------------------------------------

Best to all,

Bill P.