[From Bill Powers (2010.05.22.-0920 MDT)]
Martin Taylor 2010.05.21.23.17 –
Rick Marken: Your derivation
produces results that are quite different than mine (using a linear
approximation) and Bill’s (using functional notation).In the derivations
with which I am familiar, d is not a function of I^-1(qo) (the organism
function) in a control loop.
MMT: By definition, qi = E(qo) + d, or qi - E(qo) = d
but I(qi) = qo, so I^-1(qo) = qi
I^-1(qo) - E(qo) = d
(I hope that makes Bill happier, since I put d on th right-hand side this
time. It makes no difference to me whether I write a = b or b = a, but
since it matters to Bill, I am playing nice)
If you don’t see why I don’t like solving for independent variables, I
guess I can’t persuade you that it’s misleading. In algebra, of course,
there is no such thing as dependent or independent variables; all
equalities work both ways. And if you want to deduce what the value of an
independent variable must have been (in the immediate past – there’s no
way to predict the future state), then it’s meaningful to solve an
equation for it.
I would interpret your equation above (boldfaced) to mean that given the
value of d, you know the value that [I^-1(qo) - E(qo)] must have. The
problem here is that d has an arbitrary value, not dependent on any other
variable in the equations. So basically, you’re solving for all cases
with a constant disturbance. You may or may not be able to deduce the
permissible values of qo from that condition.
Consider the equation
P= QA
This can also be written
Q = P/A.
So by varying either P or A, the equation seems to say, you can alter
Q. If you’re thinking only of the abstract mathematical
manipulations, this will not bother you. But if you’re also aware of the
physical situation being represented by the equations, you may know that
Q stands for “Quantity of matter, or mass,” P stands for
“Pushing force”, and A stands for Acceleration. In that case
you won’t won’t consider varying “either” P or A, since
Quantity of matter (Mass) determines the ratio of Force to Acceleration.
If you vary P, A will automatically vary at the same time, just enough to
keep M constant. That connection is not shown in the equation.
This is the sort of thing my old instructor meant when he said it’s
important to keep the equations tied to the physical situation when you
represent real systems mathematically.
In your equation, you might assume that you can vary I^-1(qo) (by
changing some of the parameters of I) while leaving E(qo) alone, and thus
change d. There is nothing in the equation to say you can’t do that. The
reason you can’t do it is hidden in the physical relationships being
described by the equation.
RM: Bill’s derivations are on
pp. 145-146 of LCS I; for the closed-loop case he gets (using your
notation):
qo = 1/E (r - d)
MMT: This makes no sense, since in most analyses that are shown on
CSGnet, E() is the identity function, which would give
qo = 1/(r-d)
BP: No, it would give qo = r - d. The notation E^-1 applies to the
function, not the variables in it. If the function were qo = SQRT(r - d),
the inverse of the square root function would give qo = (r - d)^2, not
1/SQRT(r-d). The notation is perfectly arbitrary, of course: it couLd
also be written “inv E(r-d)”. To be less misleading, it should
probably be written (E^-1)(r-d) or (inv E)(r-d). You first do the
operation in the first parenthesis, and find the function that is the
inverse of E. Then you apply that function to (r - d). Sin^-1(x) means
“the angle whose sine is x,” not 1/sin(x). You would write
the
-1
latter expression as sin(x)^-1, that is, sin(x )
. Only the identity function is its own
inverse.
The equation Rick cites from LCS1 is a little embarrassing because it
needs a better explanation than I gave it. It starts with making the
approximation of infinite loop gain in equation 7, then manipulates the
result, which is always very risky. The starred values were supposed to
indicate the ideal case, but I see this wasn’t explained clearly.
RM (I think): The question is
whether one who observesan S-R relationship (the relationship between d
and qo) will conclude that this relationship reflects F(), the organism
function, or 1/E(), the inverse of the feedback function. Obviously,
conventional researchers take an observed relationship between d and qo
to be a reflection of F().
MMT: So far, I’m with you. This is indeed the question.
BP: I see Martin didn’t notice, or has been seduced into making the same
mistake. E^-1(x) is not 1/E(x). That reciprocal would be written
[E(x)]^-1, to be perfectly clear.
RM: But if the system under
study is closed loop, the the observed relationship between d and qo
actually reflects 1/E(). That’s the behavioral illusion.
MMT: This would be true if control were perfect.
Yes, and that is why the loop gain must be high to use the approximations
Rick is using. Organisms have high loop gain in most circumstances. But
not all – the iris reflex, I seem to recall, has a loop gain of around
7.
MMT: But when control is less
perfect, then changes in I() do matter, as in the case of your cyclist,
who probably will slow down in slippery conditions to compensate for
changes in E().
BP: Yes, I was going to say that I hope Rick’s response to sensing an
incipient skid is not to put the brakes on harder. To avoid skidding, the
cyclist must reduce the braking force.This says that a higher-order
skid-control system (r = 0) must be reducing the reference level for
deceleration, because the deceleration control system, given the same
reference deceleration as for dry pavement, would try to achieve the
higher deceleration by braking harder when the tires began to
slip.
MMT: Now let’s consider another
experimental situation, in which qo has no influence on qi, so that qo =
I(d). This is the typical psychophysical experiment, in which qo is a
report after a trial as to whether a tone was in the first or the second
interval of the trial.
RM: A person doesn’t become open loop when they are in a psychophysical
experiment.
MMT: True. A person doesn’t become open loop in a psychophysical
experiment. But the relation betwenn the presentation on trial N and the
response choice on trial N is most definitely open-loop. We thrashed all
this out a year or more ago.
No, you declared yourself satisfied with your own arguments. As far as I
know, nobody else was satisfied with them. You may have a point of some
kind, but so far you haven’t made it. I disagreed with your basic
analysis of the situation, not with your algebra. Rick’s arguments, I
thought, weren’t pertinent; there’s no reason the person can’t control
one variable while not controlling another. Rick’s objections had to do
with controlling for following the instructions, which we assume in any
case. I came up with a model showing what the possible controlled
variable might be, but evidently I was the only one who understood
it.
RM: My “Power Law”
paper
(
http://www.mindreadings.com/BehavioralIllusion.pdf) shows how the
closed loop might work in a magnitude estimation experiment.
MMT: Yes, I’ve referenced this before, several times. But even there,
there is no way that the subject’s saying “27” can affect the
loudness of the tone to which the subject has assigned that number. The
(several) loops are closed elsewhere.
You’re right, of course. It can’t affect the pitch or the time of
occurrance, either. However, it can affect the relationship between the
tone and the utterance, and that may be what the subject is
controlling.
Here is an abstract that has more than a little relevance to this subject
(copied from a PDF file using Foxit reader):
···
===========================================================================
American Psychologist, 43(3), 151-160
Pavlovian Conditioning
It’s Not What You Think It Is
Robert A. Rescorla
University of Pennsylvania
Abstract: Current thinking about Pavlovian conditioning differs
substantially from that of 20 years ago. Yet the changes that have taken
place remain poorly
appreciated by psychologists generally. Traditional descriptions of
conditioning as the acquired ability of one stimulus to evoke the
original response to another because of their pairing are shown to be
inadequate. They fail to characterize adequately the circumstances
producing learning, the content of that learning, or the manner in which
that learning influences performance. Instead, conditioning is now
described as the learning of relations among events so as to allow the
organism to represent its environment. Within this framework, the
study of Pavlovian conditioning continues to be an intellectually active
area, full of new discoveries and information relevant to other areas of
psychology.
============================================================================
Instead of just observing and describing relationships, as you do in your
analysis, Rescorla came (in 1988) to recognize them as controlled
variables in their own right – but without the concept of controlled
variables. I tried to show how the phenomena you described could come
about through control of relationships, but you didn’t object on that
ground; you objected because one of the elements of the relationship
couldn’t be affected by the subject’s response.
RM: And I am currently working
with Bill on a demonstration of the closed-loop nature of behavior in a
choice reaction time experiment.
MMT: Good, but unless your experiment is set up so that the subject’s
choice influences the physical presentation that leads to the choice,
that critical element is open loop.
Are you saying that if the driver of the car turns left when coming to a
T-junction, that choice is open-loop? Surely turning left doesn’t affect
the presentation that led to that choice, so by your reasoning if there
is an invisible force (a strong wind) affecting the left turn, the driver
will simply steer off the road.
A two-element relationship can be controlled when either element can be
disturbed, even if only one of its elements can be affected by the
controller. I think you’re focusing only on the triggering event and the
element that can’t be affected. However, if that element is changed, the
relationship with the other element is disturbed, and the controller can
react to that disturbance at the relationship level.
In some tracking experiments we combine pursuit tracking (target
disturbed) with compensatory tracking (cursor disturbed). The pertinent
relationship between cursor and target is called the distance between
them. Here we have a person reacting to two disturbances, but able to
affect only one of the elements, the cursor position. If we assume
distance, a function of the positions of both target and cursor, to be
the controlled variable, the model fits the performance within 5% or
better RMS, which is roughly a 20-sigma fit.
I think you are too fixed on the element of the relationship that can’t
be affected by the subject, and are thereby missing the relationship that
the subject might be controlling and that is disturbed by the change in
the presentation. Of course the element that can’t be affected by the
behavior (the brightness of the light bulb) is not controlled, but the
relationship that is disturbed is under control (if the proposed
identification of the controlled variable is correct). I’d be more
specific but I’ve forgotten some of the details.
MMT: Now we have the nub of the
issue. Let’s suppose that there was an experiment in which the
experimenter believed that the setup precluded qo having any influence on
qi, but was wrong, and the subject could control (perhaps poorly). Could
an experimenter who understands PCT discover from the experimental data,
without examining the experimental setup, that the S-R (IV-DV) analysis
was wrong? I think the equations above suggest that the data do not allow
it. The discrimination between S-R and perceptual control must be made on
other grounds.
You can always tell the difference if you can directly disturb qo or some
variable in series with it before the controlled variable is reached.
Maybe that is one of the “other grounds” you mention. If the
system is open-loop, the output and variables immediately depending on it
do not affect the input, so disturbances of the output will not change
the behavior of the system at all. If the loop is closed, the input will
be disturbed and the action of the control system will change.
Of course since the feedback connection is in the environment and is
easily observed, it’s much simpler just to see if there’s a connection
between the action and the input variables. You have to identify the
controlled variable first, of course – and that alone settles the
question. As, indeed, both Rick and you agree:
RM: It must always be made on
other grounds, those being tests to determine that a variable is under
control.
MMT: Yes, and that is the essence of Bill’s comment,
and it is a simple answer to Bruce’s original query.
So why was it necessary to go through all those derivations to get to
this point on which we seem to have agreed beforehand?
Because along the way, a lot of other disagreements were exposed to
view.
Best,
Bill P.
