[Martin Taylor 980307 14:47]
Bill Powers (980307.0603 MST)
Your derivation is correct only in the special case where Fe^-1(C - DS) =
Fe^-1(C) - Fe^-1(DS) -- in other words, where Fe^-1 is linear and
single-valued.
Linear, yes.
Linear AND single-valued. To say that the inverse of the input function is
multiple-valued is only to say that p is a function of several variables.
p is a function of qi, which is scalar. An observer external to the
control loop might observe several variables that could be combined
by some function to form qi, but it is qi that is the variable that
matters in the loop. How qi is formed is as irrelevant to the loop as
is the source of the disturbance(s).
A function whose result is multidimensional is not necessarily multiple
valued. A multiple-valued function is one in which the same set of input
values gives rise to more than one set of output values. If the
dimensionality of the input is lower than that of the result, then the
function is necessarily multiple-valued, but if the dimensionality of
the input is at least as high as that of the result, the function may
or may not be multiple-valued.
In the case in question, the function Fe(o), which yields the influence
of the output on the CCEV, has a scalar input and a scalar result.
Nevertheless, Fe^-1() would be multiple-valued if Fe() was non-monotonic
(as in the real world it often is). Does that matter? Yes it does, and I
discussed how, in the context of Fp^-1().
A reliable relationship is found, at least if Fe^-1(constant) is
constant. The experimenter thinks that a relationship has been found
within the individual. Is this not the "behavioural illusion?"
Yes.
In many cases, the effect of Fe^-1(C)
is _not_ constant, so the relationship between DS and qo is measurable,
and perhaps strong, but not reliable (e.g. correlations may be 0.3 rather
than 0.95).
Please explain what you mean by a relationship that is strong but not
reliable.
More or less what you have frequently talked about. "Reliable" means you
can be pretty sure it will be there every time. "Strong" means it is there
a lot of the time. Here, I'm looking at what an S-R psychologist would
be likely to see and to report.
I think you're arguing verbally where you need a mathematical argument. It
is impossible for the experimenter to directly determine DS; any attempt to
force DS directly to have a particular value will break the control loop.
I don't see that. DS is _not_ qi. To say that fixing DS forces qi to a
particular value is the same as saying that fixing r forces e to a
particular value. qi = DS + Fe(o), remember?
You're right. The experimenter _can_ arbitrarily vary DS. What breaks the
control loop is assuming qi = constant. Both qi and qo are functions of DS.
Approximations should be made only AFTER all equations have been solved
exactly. Otherwise, you can arrive at spurious conclusions. In this case,
you conclude that o can vary when qi remains constant (and r is constant).
Indeed, o _must_ vary if qi and r are constant and DS varies. Likewise,
o must vary if qi is nearly constant and r is constant and DS varies.
You aren't saying anything that influences the argument.
Remember that DS is not literally a variable; it is a computed difference
between two variables, and can't just be set to a particular value without
arbitrarily setting the two variables (qi and qo) to specific values. But
if you force qi and qo to specific values, you don't have a control system
any more.
I don't suppose you remember your posting of a couple of days ago, about
how a function of two inputs and one output could have an infinity of
value pairs of its two inputs for any given value of its output?
That has nothing to do with this case. The "two variables" were the
independent (of each other) input signals that were combined to produce the
perceptual signal. Here we have two variables that are in different places
in the same loop, and their relative values are set by the system equations.
Here we have two variables, o and DS, each of which is affected by the
actions of a different control system. One control system attempts to
set DS to a desired value, the other attempts to set qi to a desired
value. Both can achieve their goal, since one can affect qi by varying o,
and the other has no disturbance to affect the setting of DS.
Incidentally, I _love_ the way you go from a personal assertion to
"Remember that..."
Here we have a simple equation that not only describes the behavioural
illusion and gives a reason for it, but also suggests, without appealing
to the notion of "noise" or random variation, why S-R experimental
results can be erratic.
The equation is exactly the one I do use, and have always used, to describe
the behavioral illusion. I don't see why you think you're saying anything new.
Why did you think the reference to the behavioural illusion was supposed
to be new? But I wasn't aware that you used this equation to show why
S-R experimental results can be erratic.
Sorry, but your explanation for the "erratic" results in the absence of
noise or random variations doesn't make any sense.
Oh, you aren't.
If the variations are
regular, and there is no noise, we can write equations that explain the
output exactly.
Indeed we can.
Your simple equation doesn't explain why, in a mostly
proportional system, the output and disturbance still don't correlate
highly with the controlled variable even though they correlate highly with
each other.
"A mostly proportional system" is key, here. We almost never talk about
mostly proportional systems. Usually we talk about systems in which the
output function is mostly an integrator. Even when it isn't, the effect
of the output on qi usually is, as when the output is a force and the
controlled perception is of a position.
I don't know how the correlations work with a mostly proportional system.
Do you have demo examples?
I do know that in a system that is nearly an integrator between the error
signal and qi, the correlation between perception and DS (or o) must
be low when the loop control ratio is high (and I've been preparing a
message about why this is).
It is not an insight; it is a mistake.
Well, if you know that, perhaps you could present a reason why. Is it
the assumption of linearity in the inverse of the environmental feedback
function?
It is the assumption of linearity AND the assumption that the inverse of
the input function is single-valued AND the premature assumption that qi is
a constant.
What I think you are saying is that my original claim
+Here we have a simple equation that not only describes the behavioural
+illusion and gives a reason for it, but also suggests, without appealing
+to the notion of "noise" or random variation, why S-R experimental
+results can be erratic.
is true if the control loop is the simplest possible one with an
integrator output function, but is not true if the loop is more complicated.
In other words, the relation between Stimulus and response would be
reliable in S-R experiments in which (unknown to the S-R experimenter)
the control loop has non-linear, multiple-valued functions, even though
they are unreliable for the simple control loop to which the analysis
applies rigorously.
You could be right, I suppose.
Let's rephrase the claim.
An S-R experiment in which the "stimulus" is a disturbance to the
controlled perception in a simple control loop with linear components
and an output or environemtal feedback function that includes an integrator
will give erratic results. It is not known whether erratic results will be
obtained if the control system is more complicated.
How's that?
Martin