[From Bill Powers (950624.0620 MDT)]

Bruce Abbott and Rick Marken --

It occurs to me that we've been missing our best bet for a task to

compare PCT and reinforcement theory: a simple compensatory tracking

task. Let's set up a somewhat novel version in which cursor position is

set by

C := C + k*(H + D)

We know that with a nice slow disturbance, the subject will be able to

keep the cursor very close to, although not exactly on, a stationary

target. The errors, however, will show a very low correlation with the

handle movements. We can use a running average of error-squared, shown

as 100 minus the average error, as the "reinforcer." The cursor position

relative to the target, presumably, would be the discriminative

stimulus.

We know what the general PCT prediction is: handle position equal and

opposite to disturbance, cursor remains near target. The reinforcement

explanation would have to be something like probability of rightward

handle movement given cursor left or probability of leftward handle

movement given cursor right increases when the score moves toward 100,

decreases when it moves away from 100. Bruce can work out the details.

What will make the difference is that with an easy disturbance the

person will be controlling with an error near the noise level. The PCT

model will predict control whether the reinforcer is present or not --

or even if it varies randomly (lying about the actual error). An actual

working model using reinforcement theory will control very poorly

because of the low correlations, even with a truthful reinforcer, while

the PCT model will control as well as the subject does. I think.

This is a velocity-control task, so the PCT model just needs a

proportional gain in the output function; a very simple model with one

parameter will do well enough.

If this task proves too easy, we can add another derivative:

dc := dc + k*(H + D)*dt

c := c + dc*dt

This will require a first derivative in the output function of the

control system.

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Best,

Bill P.