[From Bill Powers (960222.1410 MST)]
I agree with the idea that (mostly) there is a gap between the
current perception and the reference standard and that this gap is
called the error or perceptual gap. I meant another gap, the gap
between an error value unequal zero and the selection of an action,
which might be called the selection gap.
This is part of what is learned when you learn to control something. You
learn how to convert errors to changes in lower-level reference signals.
Once you have learned this, you have a specific way of controlling a
specific perception, and as long as this way continues to work, you
continue to use the same mapping of error signals to lower reference
signals. When it stops working, you have to learn (reorganize to
acquire) some new connections. By "you" I just mean your brain; this
does not necessarily involve any conscious awareness of what is going
on.
That's the simple-minded answer.
In a more complete picture of what is proposed in HPCT, there are
typically multiple control systems at one level acting by mapping their
error signals onto reference signals of multiple systems at the next
lower level. So the net reference signal received by any one lower-level
system is really the sum of effects of error signals in many higher
systems. As disturbances come and go, the higher systems vary their
outputs, and thus their contributions to many lower-level reference
signals. There is no simple connection between a single higher-level
goal or error and the set of lower-level goals used to accomplish it or
correct the error.
The result is that we see different systems at the lower level coming
into action as the environment changes. At all times, the reference
signals at the lower level are just those that will satisfy the
requirements of all the higher-level systems at once. It looks as if the
higher systems are choosing different goals at the lower level to meet
changing circumstances. In fact, they are, but they're not doing it in
the either-or way we tend to speak of, nor is this an adaptive process.
Instead, all the lower-level goals are shifting at the same time,
changes in each lower goal simultaneously helping to keep several
higher-level perceptions at their own varying reference levels. Any one
lower-level goal (which specifies the desired amount of one specific
perception) is serving many higher-level purposes at the same time. To
explain how this can possibly happen requires some math or a simulation;
perhaps others can back me up on this to help convince you that this
really works.
In a system as complex as this, it is hard to distinguish learning from
the operation of a system with fixed properties. The Crowd program is an
excellent (non-hierarchical) example of how we see apparent problem-
solving or learning when there is nothing of that kind going on -- when
all the systems actually have constant properties and are not doing any
adapting at all. When actors in a crowded field of obstacles and other
actors make their way to a goal position or follow another actor, we see
them getting into difficulties, trying different routes, avoiding traps,
doubling back to look for alternate paths, and lots of other things that
aren't actually designed into the system. It's simply the result of all
the complex interactions among obstacles and five or ten (or more)
active agents that creates the appearance of intelligent problem-
solving.
What the HPCT model does is show us how a collection of simple
independent control systems can create behavior that looks very complex
and adaptive, when in fact no one active element of the model is either
complex or adaptive. The suggestion is that there is far less learning
going on than there may seem to be, and that what seems to be a very
complex pattern of behavior might be understandable in terms of
basically simple components, each behaving in a simple way.
This bears on your question because it can seem that a control system is
trying different connections between its error and lower-level reference
signals, when in fact it is working in an unchanging way through an
unchanging set of connections. The apparent complexity is generated by
the fact that many other control systems are acting through the same
lower-level systems at the same time. The apparent adaptation that we
see is simply a result of the shiftig balance among all the systems that
are acting at once.
Of course adaptation does occur, and it does change the way individual
control systems operate. But the HPCT model can, in theory, accomplish
through a fixed organization much of what would otherwise have to be
treated as learning or adaptation. This makes the problems that have to
be solved by adaptation simpler, and perhaps makes the system that
carries out the adaptations simpler also.
···
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Hans Blom, 960222 --
Most of HPCT is concerned with model building, in this case the
determination of which function with which parameters provides a
best fit with the actual data. Prediction, however, is something
else. You will have to freeze the model that you have obtained in
the model building stage and compare what it does with the results
of an _independent_ group of test subjects and show that the fit in
your "test population" is (almost) as good as the fit that you
obtained in your "learning population".
This is essentially what we do, although our predictions are all for
individuals, not populations per se. First we determine the parameters
of a model to make it fit a specific data set from one experiment. Then,
with the model parameters frozen, we change the experimental conditions
and use the model to predict the behavior that should be seen under the
new conditions. Then we do an experiment with the new conditions, using
the same participant, and compare the participant's new behavior with
the new behavior predicted by the model. So now we have a true
prediction.
Since control parameters differ among individuals, trying to do this
sort of experiment across subjects would be a mistake. That is, if we
used the average perceptual delay and the average integration factor
determined from one group of people to predict the average tracking
behavior of another group, not only would the prediction of the other
group's behavior be poor, but the model wouldn't even fit the behaviors
of the people in the original group very well.
Within an individual, the control parameters are very stable over time.
Tom Bourbon is doing an ongoing experiment which is now in (I think) its
12th year. Tom can correct my dates if they are wrong.
Back in about 1985, he did a control task and matched a model to the
data. He then stored the results and the parameters, and froze the
model. Then, still in 1985, he applied a number of new disturbance
patterns to the model and recorded the model's behavior for each new
disturbance pattern. These predictions were saved. Five years later, at
a CSG meeting, he played back one of the patterns of disturbance while
the same subject, Tom, did the experimental run again. The predicted
handle movements of the model correlated with Tom's new pattern of
handle movements better than 0.99. The correlation of the new
disturbance and handle movements with the old ones was down around 0.2
or less. Another five years later, the saved prediction using still
another disturbance pattern was compared weith another experimental run,
with the same kind of high correlation -- still in the 0.99's. I believe
Tom has two or three more predictions saved.
The point is that matching models to individual behavior and then
predicting new behavior for the same individual is a valid way to use
the models, because it is clear that individuals retain the same
characteristics over long periods of time. And the parameters that
characterize one individual are distinguishably different from those of
other individuals. If we want to compare populations, we can easily do
so by using the data and predictions from the individual determinations,
and we will avoid averaging behaviors together that are not really
averagable.
In your comment, you were referring to traditional psychological ways of
doing research, in which you have to use populations just to discern any
effect at all, and in which the nature of experiments often precludes
retesting the same group. Using the same methods with PCT experiments
would actually degrade the results terribly.
Not knowing the details of your study, especially the sizes of
learn- ing and test populations, I would tend to say at this moment
that its predictive power has not been established
Now that you know, would you change your opinion?
-----------------------------------
(1) effect = f(cause)
(2) effect = f(cause, effect)
Note that, if there is any noise or uncertainty in the system, an
expression like effect = f(cause, effect) forces the analysis to be
local to the point in time where the effect occurs. This is maybe
clearer when the expression is rewritten as effect (t) = f (cause
(t-1), effect (t-1)) where the time is made explicit.
If you use continuous differential equations, and if the noise level in
the system is very small, this problem doesn't exist. When you write a
differential equation (or use Laplace transforms) the expressions are
present-time functions which automatically take care of the cumulative
effects of delays and integrations. There is no such expression in a
differential equation as (t - 1). You are thinking strictly in terms of
digital representations of the variables, where "1" represents one
iteration.
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RE: Brownian Control and the Test
No, I didn't set my system up as a control system. I did the
following:
1. define the chemical's concentration c as position dependent:
c = f (x, y) if in two-space
2. define the particle's radius as concentration dependent:
r = g (c)
3. compute the particle's next position change d based on
Einstein's formula: d = k / sqrt (r) with a suitably chosen k; note
that this disregards that in Einstein's formula d is an _average_
value only; since I do not know d's probability density function, I
cannot simulate how it would vary.
4. make the particle move a distance d in a randomly chosen
direction yielding new x and y coordinates.
5. loop to 1.
OK, this is quite analogous to the E. coli situation, although the
effect will be much smaller because of using the absolute concentration
to determine r rather than the rate of change of concentration. This
wouldn't be a position control system, but a concentration control
system. When r gets larger, d gets smaller, so if you make r increase
when the particle is in a smaller concentration, the particle won't move
as far as when the concentration is larger. This gives you the needed
bias on the random walk.
You could put a reference signal in just by changing eq., 2 to
2. r = g(c0 - c), which shows the reference value explicitly.
This control system would move itself toward a position where c = c0,
assuming you choose the function g so the feedback is negative. The loop
gain would depend on what you put into the function g. As I see control
systems, equation 2 describes the active control system itself, and the
other functions describe the environment.
I'll bet that if you used some natural phenomenon, like a particle that
absorbed the substance and swelled as a consequence, you'd find that
when all is said and done the loop _power_ gain would have to be less
than 1. The argument would be the same one used to show that Maxwell's
Demon can't work without an external source of energy. The absorption of
the substance by the particle would alter the local concentration, so
yoy have the same general problem of disturbing the concentration by
observing it.
To get a significant power gain, you'd have to postulate that c affects
r without a change in r having any reverse effect on c. That's what
makes equation 2 into the equation of an active control system.
I do see the difficulty in applying the test as the system was first
described. However, you can still disturb the system by changing T or
eta in the Einstein equation, and determining the effect on c with and
without the dependence of r on c. This doesn't require arbitrarily
moving the particle to a new concentration.
Step 1: set g to zero (no effect of c on r)
Step 2. Change T by some amount.
Step 3. Measure how the concentration at the particle's position
changes, via equation 3 and 1.
Step 4. Reset the system, use a nonzero function for g, and repeat steps
2 and 3.
Step 4. Compare the behavior of the concentration at the particle's
position under the two conditions.
So it does look possible to apply the Test.
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Best to all,
Bill P.