Stability,information, economics

[From Rick Marken (940210.0930)]

Martin Taylor (940209 14:00) --

Very interesting post. It makes me realize that I should say a
couple more things about the stability factors that I reported.

You say:

I computed the gain of Rick's model systems from the
reported stability factors, on the assumption that the stability factor
was all based on control

stability factor, S = [var(dt)+var(c)]/var(t-c)

First, I should have emphasized that the stability factors that I reported
are ratios of VARIANCES. You noticed this but Bill Powers didn't; Bill
thought I was reporting the ratio of standard deviations. So the S values
I reported are not as impressive as Bill thought. The 15.8 for the
"pursuit" stability is only about 4 (not 16) standard deviations better
than no control -- highly significant but not incredible. It could
be improved by slowing dt and by using var(c*) -- the variance
of the reference for the cursor -- rather than the actual variance
of the cursor in the calculation of expected variance; but, of course,
we don't know var(c*) until we have a good model, The S values for
the compensatory tracking are only about 1.14 standard deviations
better than no control (sqrt(2)). But these stability factors are
very much lower than they should be because the expected variance
does not include var(c*). A better estimate of the stability
factor for the compensatory track would have been based on a formula
like [var(dc)+ var(m)+var(c)]/var(c), taking var(c) as an estimate of
var(c*).

I should mention that the stability factor analysis was done
just to give me a first, rough idea of what was going on. As I
continue working on this two-level experiment, my goal will be to
develop a model that matches the subject's behavior in detail. That
is, the time course of both the cursor and mouse movements
should be the same for subject and model. I am working on this
part of the model now, as time permits (and it doesn't permit much).

Bill Leach (940209.10:39) --

Me:

There is no information [in the perception of the cursor] about anything
other than the cursor movement itself.

Bill:

I still believe that the subject will perceive a great deal more about
the disturbing function than just the cursor's position.

Yes. That's the way it seems. But it's not true. Remember, in a
compensatory tracking task, all you are seeing is the perception (p(t))
of the cursor and p(t) = o(t) + d(t) (output and disturbance combined).
You can't see either of the two components of p(t) unless you stop
doing anything (o(t) becomes a constant = zero) but then there is
no control.

It is becoming abundantly obvious to me that I have to somehow run these
experiments.

Yes, that would help. Order the Demo Disk from Dag. Seeing is believing --
or disbelieving -- well, at least it's seeing.

Jim Dundon (940209.1850) --

As I see transmission lines they are loop
segments that have no purpose except to function in
conjunction with purpose.

Right. Transmission lines are one way of implementing a
functional dependency between variables. When I say in
axiom 3, for example, that p = t(i), I mean that (in the
case of the nervous system) the perceptual signal is
functionally dependent on some environmental variable.
The t() function is a complex transmission network, moving
sensory data about environmental input to another part
of the nervous system where it turns up as a perceptual
signal.

But the functional dependencies that I listed in the axioms
and observations of PCT are not always implemented as
transmission lines. They ARE so implemented in the nervous
system -- which is a signal carrying system -- but they
are NOT so implemented in the external world part of a
control loop (not usually, anyway). For example, the
functional dependance of mouse movement on the forces
generated by my hand is not mediated by a transmission
line; it is mediated by whatever it is about the world
that converts force into motion.

It's the functional dependencies that are important in
understanding a control loop -- not how they are actually
implemented (transmission lines, physical interactions,
etc).

Would axioms of PCT be usable in an economic loop?

Of course. Economics (to the extent that I understand it)
is the result of interactions between purposeful systems. Like
all social phenomena, it should be possible to understand the
mass phenomenon that we call "the economy" a lot better once we
understand the workings of the components (people) that make
it happen.

One beautiful example of the applicaiton of PCT to an
ecomonic phenomenon is described in the American Behavioral
Science Issue that I edited. It's a paper by Bill Williams
on the Geffen paradox -- the fact that, under certain
circumstances, an increase in the price of a commodity (bread)
will lead to an INCREASE in the amount of that commodity that
is purchased. PCT handles this (observed) violation of one
of the basic "laws" of economics with ease;. In fact, give the
assumption that people are control systems, there turns out
to be no other possible result -- raising the price of
bread MUST lead to increased purchases of bread: it's all
a matter of people controlling their perceptions.

Best

Rick