[From Bill Powers (960715.1700 MDT)]
One last thing which I am likely to forget if I don't write it now,
before I take my computer apart to go to the meeting.
The standard PCT model is in fact an adaptive model very much in the
same category as the model that Hans Blom is pushing, or the "backward
propagation" methods of neural networks. It uses a measure of error to
"reach back" and adjust the parameters of a model on the basis of the
error, so the model's behavior approaches a reference behavior. This
isn't obvious, because the models we present are the fully-adapted
result, not the method of getting there.
The reason it isn't obvious is that we call the "adaptive" part simply
the process of fitting the model to the data. Long ago I got tired of
doing this by manual adjustment of parameters, and devised several ways
to automate the process. Any of these automated processes starts with a
model that has a specific form, but with parameters that are left
undetermined (just like Hans' model). This model is run while its
behavior is compared with the behavior of a real person. The record of
the real person's behavior is the reference signal against which the
process compares the behavior of the model with its current set of
parameters. The parameters are systematically altered so as to make the
difference between the model's behavior and the real behavior as small
as possible.
I've used a number of methods to do this, none of which works as well as
I think must be possible. This has always bothered me -- I've run into
local minimum problems, discovered simply by fiddling with the
parameters and finding that I can get a better fit that way (in hours of
work) than the automated fitting process can do.
As the fitting process has worked until now, it is done by repeatedly
running the model and measuring the error for a whole run, one minute's
worth of data in most cases. Then the correction to the parameters is
made, and the model is run again and compared with the same data.
However, it would be just as easy to make the corrections on the fly, so
that the parameters were continuously adjusted on the basis of the
current error. Except for the problem of subject fatigue, it would also
be possible to run the model in real time against the ongoing
performance of a subject in a continuing experimental run. That would
make this method even more like the Kalman Filter approach, which is
designed to adjust parameters on the fly instead of waiting until the
whole data set is available. Back in the mid-80s I actually did this
with a predecessor of the Artificial Cerebellum. At one of our meetings,
I demonstrated a program that continuously modified the transfer
function of a model so its trace gradually grew closer and closer to the
trace being generated by a subject doing a tracking task. Of course
nobody understood what was interesting about this and I let it go.
At any rate, it's ironic that we should be arguing about adaptive model-
based control when the basic method of doing this is so obvious that
even I tumbled onto it independently. What would be much more productive
than arguing would be for Hans or whoever else understands the Kalman
Filter approach to work out a way of using it to fit PCT control models
to real subject data. It would probably work far better than my seat-of-
the-pants methods.
Right now Bruce Abbott and I are trying to fit models to rat behavioral
data, which is very noisy, and I am having troubles finding a good
automated method for determining the four parameters we are now using
for part of the model. There are two models involved. One is simply a
model for the influence of food and water intake on total measured
weight. This involves finding parameters for long-term weight gain and
loss with an intake that consists partly of usable food and partly of
food that is due to be excreted with some short-term half-life. The
other is a simple control-system model that adjusts food intake as a
function of weight error. The space in which we must fit the data is
very lumpy and with four parameters in one model and two in the other it
is almost impossible to find a best fit by cut-and-try methods. I should
think that the Kalman Filter approach might be just what we need -- if
anyone were willing to help us with a working program to apply it, and
didn't insist that we use some model other than the PCT model.
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This time I am really pulling the plug.
Until next week,
Best to all,
Bill P.