[From Rick Marken (970521.1350 PDT)]

Some time ago, Bill Powers (970507.0500 MST) said:

I propose that Rick and Bruce...put their heads together and

develop a rigorous statistical analysis for both of these cases

[predicting responses from disturbance when the controlled

variable is not known oir predicting disturbance from responses

when controlled variable is known -- RM].

Since Bruce hasn't been pestering me to work this problem with him,

I thought I would pester Bruce to work on a different (but similar)

problem with me. The problem is one I ran into long ago while developing

the "Mind Reading" demo -- what is now called "The Test

for the Controlled Variable" in the Java demos. I just ran into it again

recently while trying to develop an improved version of the "Hierarchy

of Behavior and Control" demo.

The problem occurs when one tries to Test more than two hypotheses about

a controlled variable at the same time. In the "Mind Reading"

demo I was testing five hypotheses (just three in the Java version)

at the same time. The hypotheses are that the position of each of the

objects on the screen is under control. I test all hypotheses

simultaneously by appying a _different_ time varying disturbance to the

position of each object. The "Mind Reading" program concludes that the

object being controlled is the one with the _lowest_ correlation

between position (the hypothetical controlled variable) and

disturbance.

This approach to testing several simultaneous hypotheses about the

controlled variable works pretty well. The problem is that sometimes

the correlation between position and disturbance will be lowest for

one of the _uncontrolled_ objects; it looks like an object's position is

controlled when it is _not_. These spuriously low correlations result

from the fact that the disturbances to each of the hypothetical

controlled objects are _correlated with each other_ to some extent!

So occasionally the mouse movements that are negatively correlated with

the disturbance to the controlled object are even _more_ negatively

correlated with the disturbance to another object.

This problem would not arise if the disturbances were _not_ correlated

with each other. It's easy to select two disturbance waveforms that are

not correlated with each other -- a sine and cosine wave, for example.

But I think (and this is where a statistician would help)

that it is impossible to select more than 2 disturbances such

that the correlation between every disturbance and every other

disturbance is 0.0. That is, given three disturbance waveforms,

d1(t), d2(t) and d3(t), I think it is impossible to have it be true that

the correlations between d1(t) and d2(t), d1(t) and d3(t) and d2(t) and

d3(t) are _all_ 0.0.

If this is true, then one way to improve the accuracy of Tests of

more than two simultaneous hypotheses about controlled variables is

by "statistically" removing the covariation between disturbances that

might influence the results of these Tests. This is where Bruce

Abbott comes in. What I would like is a way to determine the correlation

between d1 and cv1 (controlled variable 1) while

factoring out any covariation between d1 and d2 that could

contribute to the existence of a spuriously low correlation between

d2 and cv2 (giving the impression that cv2 rather than cv1 is under

control). To do this, I think it would also be necessary to factor

out any correlation between d2 and m (mouse movements), d2 and cv2

and cv2 and cv1.

I think what I want to compute is called a "partial correlation"

between, say, d1 and cv1 with d2 (and all other disturbances and

other possible contributors to the corrlation between disturbance

and controlled variables) "partialled out". I think I can get

"running values" of all the relevant correlations; what I need is

the formula that will give me the partial correlations (between

d1 and cv1, d2 and cv2, etc), presumably as a function of the

relevant correlations. If I had such a method available I think I could

extend the capability of my "Mind Reading" program so that it

could test not only for object position but also for variables like

"distance between objects". I had problems testing for these more

complex variables _while_ testing for position because the disturbances

to these more complex variables tend to be _highly correlated_ with the

disturbances to position. If I could compute partial correlation I might

be able to include these more complex variables in the set of variables

that the subject could control.

Best

Rick

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Richard S. Marken Phone or Fax: 310 474-0313

Life Learning Associates e-mail: marken@leonardo.net

http://www.leonardo.net/Marken