[From Bill Powers (940420.0615 MDT)]
Wayne Hershberger (940418) --
Couldn't one take this reasoning a step further (sideways,
backwards, whatever)? Consider the "simplest" ideal control
system which transmits information optimally in this >regenerative
fashion (beam me up Scotty).
I'm not sure what you mean by "a control system which transmits
information." What does a control system transmit information from,
and to? I can think of several possible meanings, but would rather
hear yours.
Presumably, the parameters of the
simplest ideal system would vary with the amount of information
available to be transmitted, and the rate of transmission.
Again, I can think of several meanings here. Are you speaking of
several different control systems, each idealized for a different
situation, or of a single control system which adapts its parameters
to whatever situation exists? An adaptive control system is not a
"simple" control system; it really consists of two control systems,
one that controls a perception of an environmental variable and the
other which senses and controls some measure of the quality of
control in the main system itself -- for example, the average square
of the error signal in the main system. The auxiliary, adapting,
control system varies the parameters of the main control system to
make the quality of control remain near some standard.
In a _simple_ control system with a given design, the parameters of
the system don't change with the nature of the disturbance -- that
is, with the information amount or rate that we might attribute to
the disturbance. I was pointing out (among other things) that since
we can use the model to predict quantitatively the actions of the
system given the states of the two independent variables
(disturbance and reference signal), we can compute the information
amount and rate in the output variable directly from the prediction
of its time-variations. It's not necessary to trace the flow or know
the state of information inside the system between the disturbance
and the output variable.
This is a somewhat confusing subject because in tracking experiments
we find that the best-fit parameters do change with the nature of
the disturbance, particularly its bandwidth but also its
predictability. However, there is a good chance that at least for
nominally unpredictable waveforms the parameters would not have to
be changed if we introduced the right sort of nonlinearity into the
system. Ideally, we should be able to make a model with one set of
fixed parameters fit behavior over a wide range of disturbance
bandwidths and amplitudes. Then, if we look at the difference
between predictable and unpredictable disturbance effects, we should
be able to handle that with a second level of control concerned only
with the predictability of the disturbances. For all unpredictable
disturbances, the lower level model would explain the behavior while
the upper level model maintained a constant reference signal for the
lower system. For predictable disturbances, the upper level would
control for the predictable aspect by varying the lower-level
reference signal. In the whole model, then, one fixed set of
parameters would serve to explain behavior under both predictable
and unpredictable conditions. I hope to work on developing the
nonlinear model this summer, with data obtained during Martin
Taylor's associates' sleep-deprivation experiment.
The difference between the ideal system and the real one
(however it is instantiated: wetware, hardware, or software)
would be the information that is not transmitted. The
complement would be the information transmitted, wouldn't it?
Again, the question is information about what transmitted to what?
In the context of my original remarks, the information would concern
the operation of the control system in relation to the disturbance,
and would be transmitted to the analyst trying to understand the
system. Alternatively, it would be information transmitted (somehow)
from disturbance to output action. The latter interpretation gives
problems, however. As Martin Taylor has pointed out, the information
content of any variable depends on the expectations of the receiver.
This makes little sense outside a psychological context, because
it's pretty hard to conceive what the environment's expectations
about the effects of the output would be.
If you mean information transmitted to the analyst, the analyst has
two sources of information available: the performance of the model
given the disturbance, and the performance of the real person given
the same disturbance. The difference between these two performances
is information about what is wrong with the model. This information
is then interpreted and used to modify the model. The end-point
occurs, ideally, when the systematic difference between the model's
performance and that of the real person becomes zero, so there is no
further transmission of information to the analyst about defects in
the model. When there is no systematic difference, all the remaining
differences are pure noise: they do not repeat on repeated trials of
an experiment.
ยทยทยท
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Martin Taylor (940418.1530) --
As you know by now, I had already figured out that we could take
advantage of the type of data being transmitted in figuring out a
compression method. The first obvious gain came from finding that
the difference between the model and the real handle position was
small enough to be represented as one byte instead of a 16-bit
integer -- most of the time. A margin of safety was obtained by
storing the first entry in the resulting data table as an integer,
and then converting the remainder to signed bytes that represented
the first differences in the data. For a first difference to exceed
the byte limit, the handle would have to move the equivalent of 128
pixels in 1/60 second, which I believe is physically impossible.
I then found that instead of actually running the model again, I
could compress just the tracking error, handle + disturbance. This
effectively removes MOST of the information about the model, under
the assumption that the person is a perfect control system. Applying
PKZIP to the outcome resulted in a high degree of compression: for
worst-case data taken with difficulty 7 (0.8 Hz, very difficult),
the final file was 29% as long as the original data file (2155 bytes
vs 7362 bytes), including the header. For an easy disturbance the
final size was 19.6% of the original size. The compression was
essentially the same whether I used a rerun of the model or just the
disturbance to obtain the tracking error. This greatly simplies the
compression program.
So, don't be discouraged by a reduction of only 13% in the
amount of data needed after compression.
Obviously, I'm not discouraged!
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Richard Thurman (940418.1600)--
Your program segment is right.
I do worry about how to explain it to others though. On
several ocassions when I have tried to explain the basic
concepts of PCT to cognitive psychologists, I have gotten a lot
of 'yes buts' and 'thats just the same as' comments.
Join the club. It seems to be inevitable that people will try to
cram PCT into the Procrustean bed of their previous knowledge. Why
don't you try to get the Air Force to adopt PCT? Then if cognitive
psychologists keep giving us a hard time, we could drop bombs on
them.
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Best to all,
Bill P.