Testing Simultaneous Hypotheses About CV

[From Rick Marken (970521.1350 PDT)]

Some time ago, Bill Powers (970507.0500 MST) said:

I propose that Rick and Bruce...put their heads together and
develop a rigorous statistical analysis for both of these cases
[predicting responses from disturbance when the controlled
variable is not known oir predicting disturbance from responses
when controlled variable is known -- RM].

Since Bruce hasn't been pestering me to work this problem with him,
I thought I would pester Bruce to work on a different (but similar)
problem with me. The problem is one I ran into long ago while developing
the "Mind Reading" demo -- what is now called "The Test
for the Controlled Variable" in the Java demos. I just ran into it again
recently while trying to develop an improved version of the "Hierarchy
of Behavior and Control" demo.

The problem occurs when one tries to Test more than two hypotheses about
a controlled variable at the same time. In the "Mind Reading"
demo I was testing five hypotheses (just three in the Java version)
at the same time. The hypotheses are that the position of each of the
objects on the screen is under control. I test all hypotheses
simultaneously by appying a _different_ time varying disturbance to the
position of each object. The "Mind Reading" program concludes that the
object being controlled is the one with the _lowest_ correlation
between position (the hypothetical controlled variable) and
disturbance.

This approach to testing several simultaneous hypotheses about the
controlled variable works pretty well. The problem is that sometimes
the correlation between position and disturbance will be lowest for
one of the _uncontrolled_ objects; it looks like an object's position is
controlled when it is _not_. These spuriously low correlations result
from the fact that the disturbances to each of the hypothetical
controlled objects are _correlated with each other_ to some extent!
So occasionally the mouse movements that are negatively correlated with
the disturbance to the controlled object are even _more_ negatively
correlated with the disturbance to another object.

This problem would not arise if the disturbances were _not_ correlated
with each other. It's easy to select two disturbance waveforms that are
not correlated with each other -- a sine and cosine wave, for example.
But I think (and this is where a statistician would help)
that it is impossible to select more than 2 disturbances such
that the correlation between every disturbance and every other
disturbance is 0.0. That is, given three disturbance waveforms,
d1(t), d2(t) and d3(t), I think it is impossible to have it be true that
the correlations between d1(t) and d2(t), d1(t) and d3(t) and d2(t) and
d3(t) are _all_ 0.0.

If this is true, then one way to improve the accuracy of Tests of
more than two simultaneous hypotheses about controlled variables is
by "statistically" removing the covariation between disturbances that
might influence the results of these Tests. This is where Bruce
Abbott comes in. What I would like is a way to determine the correlation
between d1 and cv1 (controlled variable 1) while
factoring out any covariation between d1 and d2 that could
contribute to the existence of a spuriously low correlation between
d2 and cv2 (giving the impression that cv2 rather than cv1 is under
control). To do this, I think it would also be necessary to factor
out any correlation between d2 and m (mouse movements), d2 and cv2
and cv2 and cv1.

I think what I want to compute is called a "partial correlation"
between, say, d1 and cv1 with d2 (and all other disturbances and
other possible contributors to the corrlation between disturbance
and controlled variables) "partialled out". I think I can get
"running values" of all the relevant correlations; what I need is
the formula that will give me the partial correlations (between
d1 and cv1, d2 and cv2, etc), presumably as a function of the
relevant correlations. If I had such a method available I think I could
extend the capability of my "Mind Reading" program so that it
could test not only for object position but also for variables like
"distance between objects". I had problems testing for these more
complex variables _while_ testing for position because the disturbances
to these more complex variables tend to be _highly correlated_ with the
disturbances to position. If I could compute partial correlation I might
be able to include these more complex variables in the set of variables
that the subject could control.

Best

Rick

···

--

Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: marken@leonardo.net
http://www.leonardo.net/Marken

[From
[From Bruce gregory (970521.1710 EDT)]

Rick Marken (970521.1350 PDT)

The problem is that sometimes
the correlation between position and disturbance will be lowest for
one of the _uncontrolled_ objects; it looks like an object's position is
controlled when it is _not_. These spuriously low correlations result
from the fact that the disturbances to each of the hypothetical
controlled objects are _correlated with each other_ to some extent!

Does this happen because the disturbances are sine waves that
differ only in phase?

The Other Bruce

[From Stefan Balke (970522.1000 CET)]

Rick Marken (970521.1350 PDT) --

The problem occurs when one tries to Test more than two hypotheses about
a controlled variable at the same time. In the "Mind Reading"
demo I was testing five hypotheses (just three in the Java version)
at the same time. The hypotheses are that the position of each of the
objects on the screen is under control. I test all hypotheses
simultaneously by appying a _different_ time varying disturbance to the
position of each object. The "Mind Reading" program concludes that the
object being controlled is the one with the _lowest_ correlation
between position (the hypothetical controlled variable) and
disturbance.

Rick, here are some, sometimes not perfectly connected musings about the Test.

At first one silly question, is it intended that the computer models human
behavior as it is in the little man demo? I (as a human) never compute
correlations or even partial correlations. I observe distances, movements,
covariations and patterns, but I don't compute and compare exact correlations.

I guess that the computer is not intended to act as a human being here, but
what else is the purpose here. To show, that the computer is able to detect,
what variable is under control? I run the demo and it worked. The computer
detected what variable I was controlling. The person watching me was not
able to detect the right variable, because he didn't know about my reference
variable and the disturbances, which I counteracted. The computer was cleary
in advantage here, because he knew about the disturbances.

What does this show: If I want to do the Test (to find out what variable is
under control), I need informations about both, the actions and the
disturbances. But the outside observer didn't know anything about the
disturbances. The outside observer was also unable to build up own
hypotheses because he couldn't apply his disturbances to test my reactions.

Isn't it extremly important to know about the disturbances and the actions
to draw a conclusion about the controlled variable. If this is true another
problem arises. An actor perceives a disturbance and evalutes the meaning of
the disturbance along with his personal memories, which are also unknown to
the observer. So the observer can't be sure whether the observed result - no
influence from the disturbance to the controlled variable - is the result of
good control or whether the disturbance simply didn't affect the controled
variable.

Could it be an idea to build up a game, where the player tests by applying
disturbances to some related variables what variable the computer has under
control? Than the player could conclude from the computer counteractions to
the players input what variable the computer controls.

Of course, control sometimes looks like magic, you do something to a
variable where you expect a reaction, but nothing happens with this
variable. That means that we are looking for the invisible :-), the zero
correlation. That is the big difference to the conventional research, they
control for not receiving zero correlations by doing significance tests.

Of course, we are also looking for the perfect correlation between the
disturbance and the counteraction. What do you think, could it be possible
that in a hierachical control system the effective counteraction is divided
in several actions, e.g. speaking louder, using special words and walking
away simultaneously? In that case it would be hard to see a -1.0 correlation
in one dimension.

Best, Stefan

[From Richard Kennaway (970522.1029 BST)]

[From Rick Marken (970521.1350 PDT)]
This problem would not arise if the disturbances were _not_ correlated
with each other. It's easy to select two disturbance waveforms that are
not correlated with each other -- a sine and cosine wave, for example.
But I think (and this is where a statistician would help)
that it is impossible to select more than 2 disturbances such
that the correlation between every disturbance and every other
disturbance is 0.0. That is, given three disturbance waveforms,
d1(t), d2(t) and d3(t), I think it is impossible to have it be true that
the correlations between d1(t) and d2(t), d1(t) and d3(t) and d2(t) and
d3(t) are _all_ 0.0.

It is possible. Any two sine waves of different frequency have zero
correlation in the long run. Therefore, any set of sine waves of different
frequencies will have all pairwise correlations zero.

However, the closer together in frequency they are, the longer a window you
have to look at to get a correlation close to zero. The window for two
frequencies f1 and f2 needs to be proportional to 1/(f1 - f2). For the
mind reading demo you need the frequencies to be small enough that the
subject can control against the disturbances, but large enough for the
correlations between disturbances to be low over the length of time within
which you want to make a prediction. Supposing that the available range of
frequencies is (substitute your own figures here) 0.3 to 1.5 Hz, and taking
into account that you can have two sine waves at each frequency, 90 degrees
out of phase, the five-object demo could use the waveforms

        sin( 0.3t )
        cos( 0.3t )
        sin( 0.9t + 180 degrees )
        cos( 0.9t + 180 degrees )
        sin( 1.5t + 45 degrees )

The phase offsets in the last three are to make sure that in the initial
second or so you don't have a high correlation.

But the statistical analysis you ask for, to deal with non-zero
correlations between disturbances, would also be useful.

-- Richard Kennaway, jrk@sys.uea.ac.uk, http://www.sys.uea.ac.uk/~jrk/
   School of Information Systems, Univ. of East Anglia, Norwich, U.K.

[Hans Blom, 970522]

(Rick Marken (970521.1350 PDT))

The problem occurs when one tries to Test more than two hypotheses
about a controlled variable at the same time.

Not only then, but especially then...

The "Mind Reading" program concludes that the object being
controlled is the one with the _lowest_ correlation between
position (the hypothetical controlled variable) and disturbance.

The point is, I guess, that you _want_ the correlation, but that you
can only _compute_ its estimate. I guess you use something like a
running average, including a forgetting factor. Such a number gives
you the correlation plus or minus some noise. The noise is smaller
the longer you average (i.e. the smaller the forgetting factor), but
large if you want a _rapid_ calculation. There is nothing you can do
about that, alas. There is only so much information in a finite
sequence of samples.

On the one hand you want a rapid calculation, because then you can
rapidly show which of the targets is controlled when the subject
switches his/her choice. On the other hand you want a calculation
over an infinite period, because that will give the most reliable
estimate of the true correlation, and thus the most reliable
indication of which target is controlled. So you're stuck with some
optimum.

When there are N targets of which one is tracked, you'll need to
compute N correlations. Since these are all noisy, chances are that
one of the spurious ones ocasionally is larger than the "true" one.
This chance becomes larger if N increases; the situation with N=2 is
easiest and N=100 would be nigh to impossible. The "noise" in the
calculation is why we're usually satisfied with confidence intervals:
we can only be right most of the time.

So there is no infallible method. If you want to go into the
direction of correctness rather than speedy detection, you may
experience long delays. And those are clearly unacceptable if your
subject rapidly switches from controlling one target to another.

That may suggest a solution: forbid target switching! :slight_smile:

Sorry to be of no help.

Greetings,

Hans

[Hans Blom, 970522b]

(Stefan Balke (970522.1000 CET))

... I (as a human) never compute correlations or even partial
correlations. I observe distances, movements, covariations and
patterns, but I don't compute and compare exact correlations.

There isn't that much difference between a correlation and a
covariation, is there? Or do you mean something deeper?

I run the demo and it worked. The computer detected what variable I
was controlling. The person watching me was not able to detect the
right variable, because he didn't know about my reference variable
and the disturbances, which I counteracted. The computer was cleary
in advantage here, because he knew about the disturbances.

Does your program know about the disturbances, Rick?

What does this show: If I want to do the Test (to find out what
variable is under control), I need informations about both, the
actions and the disturbances.

It is extremely important -- essential, I would say -- to know about
the _actions_, but not necessarily about the disturbances. If the
environment function is a constant gain -- as I assume it is in
Rick's program -- a correlation (computed over a sufficiently long
time) between action and perception close to 1 will identify the
controlled target.

What the correlation (or a covariation) tests is whether there is a
consistent relationship between the action (the mouse movement) and
an ensuing perception (the position change of a target), I suppose.
Am I correct, Rick? That is, at least, how I would do it. Reference
levels play no role. Anyway, the computer cannot directly know what
my reference is; it can only operate on the assumption that I want to
track one of the targets.

The outside observer was also unable to build up own hypotheses
because he couldn't apply his disturbances to test my reactions.

It is my guess that the outside observer cannot know which target is
controlled because _he_ does not control the mouse and thus does not
know/feel the actions. Knowledge of the disturbances is not essential
if they are random or otherwise cancel out.

Thanks, Stefan, for pointing out what the person operating the mouse
perceives more than an innocent bystander.

Greetings,

Hans

[From Rick Marken (970522.0950 PDT)]

Richard Kennaway (970522.1029 BST) --

Thanks for that post! It was incredibly helpful. I think I should
have known some of this stuff (after all, I studied auditory perception
as a grad student -- but that was pretty long ago;-))
I am also impressed by the fact that you understood exactly what
my problem is; to keep all disturbances orthogonal while keeping
them in a controllable frequency range. Either I explained my
problem more clearly than I thought or you are one smart dude
(I'm pretty sure it's the latter;-).

Me:

The problem is that sometimes the correlation between position and
disturbance will be lowest for one of the _uncontrolled_ objects;

Bruce Gregory (970521.1710 EDT) --

Does this happen because the disturbances are sine waves that
differ only in phase?

After thinking about this a bit more carefully I realized that
the specific problem I mention (low correlation between position
and disturbance for uncontrolled objects) is probably a result of the
fact that the subject in the "Mind Reading" demo is _continuously
changing his reference_ for the position the controlled object.

If the reference for the controlled object were always constant,
then the correlation between position and disturbance for an
uncontrolled object would always be exactly proportional to the
correlation between the disturbance to that object and the
disturbance to the controlled object. This is because the output
waveform (mouse movement) would be almost exactly opposite to the
disturbance to the controlled object. So the output waveform would
cancel exactly the variance in the uncontrolled object's position
that corresponds to the proportion of that object's disturbance that
matches the controlled object's disturbance (if you understood that,
please let me know what I said;-))

So I don't think my problem with "spuriously low correlations" in
the "Mind Reading" demo can be cured by partial correlations
or enforced orthogonality of disturbances. These spuriouisly low
correlations result (I think) mainly from coincidenatal matches
between the _reference signal waveform_ and disturbances to uncontrolled
variables. What is needed is a way to estimate, on
the fly, the subject's reference waveform -- where the subject
intends the object to be at any instant. Anyone who can figure
out how to do THAT gets the first PhD in PCT;-)

Stefan Balke (970522.1000 CET)--

Rick, here are some, sometimes not perfectly connected musings about > the Test.

is it intended that the computer models human behavior as it is
in the little man demo?

Not really. The goal of The Test (when done by human or computer)
is to identify a variable or variables that are being controlled by the
controller. In the Mind Reading demo, the computer knows that,
if the subject follows instructions, the position of one of the
objects on the screen is controlled. No other hypotheses (besides
"object position") are considered by the computer.

I (as a human) never compute correlations or even partial
correlations. I observe distances, movements, covariations and
patterns, but I don't compute and compare exact correlations.

Right. I think that you do notice relationships between variables
and that's how you can tell that a variable might be under control.
I don't think formal computation (of things like correlations) is
necessary in order to do the Test. You can notice, for example,
that no matter which way you turn, the car behind you remains
behind you; based on this finding, it is reasonable to conclude
that the person in that car is probably controlling for keeping
you in sight (you're being followed). That's what the informal
Test will be like in most cases.

I guess that the computer is not intended to act as a human being
here, but what else is the purpose here.

The purpose of the Mind Reading demo is to show that you cannot tell
what a person is "doing" (intending to accomplish) by just looking at
what he is "doing" (his visible behavior). I created a special situation
where you can clearly see what a person is "doing"
(their visable behavior -- moving three squares) but you can't see what
the person is "doing" (what result he intends to produce--
moving only _one_ of the three squares). In order to determine
what the person in this demo is doing (controlling) you have
to Test to determine which square is under control.

What does this show: If I want to do the Test (to find out what
variable is under control), I need informations about both, the
actions and the disturbances.

Right! But you can't know what a disturbance might be until you have
guessed what the controlled variable might be. So the first thing you
need to know is what variable _might_ be under control. Then you
manipulate variables that _would be disturbances_ if this variable
were controlled. If the disturbances have far less effect on this
variable that expected (if the variable were not under control)
then you go on to make sure that it is the actions of the system
that are preventing these disturbances from having their effect.

Isn't it extremly important to know about the disturbances and
the actions to draw a conclusion about the controlled variable.

Yes. But, again, you can't just "see" disturbances. Disturbances
are variables in the environment (like the direction in which your
car is moving). A variable like the direction of a car is a disturbance
if the driver behind you is controlling for the
perception of "following" you. In that case, your changes in
direction require that the driver behind you take actions that
compensate for these changes -- in order to maintain his perception
of "following". So I would say what is most important in drawing
conclusions about a control variable is formulting a _clear description_
of what that variable might be!

An actor perceives a disturbance and evalutes the meaning of
the disturbance along with his personal memories, which are
also unknown to the observer. So the observer can't be sure
whether the observed result - no influence from the disturbance
to the controlled variable - is the result of good control or
whether the disturbance simply didn't affect the controled
variable.

I don't know if I understand your example. But it is possible that
lack (or apparent lack) of effect of a disturbance to a possible
controlled variable might be the result of something other than
controlling. For example, your changes in direction may have no
effect on the relatinoship between you and the car behind you; it _looks
like_ these disturbances have no effect; the person behind
you seems to be controlling for following. But this may be a
coincidence; the person behind you may just happen to be going
where you are going.

This is why, when we Test, we also make sure that the system must
be able to perceive the suspected controlled variable. In the
example of following, if the car behind is really controlling for
following, the ability to follow should be lost if the person behind can
no longer see you. If you dodge out of sight, the following driver's
ability to control for following should be lost. If the
car continues on its merry way, the apparent control was a coincidence;
the driver was not controlling for following you. If, however, the
driver tries to recover his perception of you, then
you can be more confident that he is controlling for following you.

Could it be an idea to build up a game, where the player tests by
applying disturbances to some related variables what variable the
computer has under control?

Yes. This is what you do in the "Detection of purpose" demo. You
use the mouse to apply disturbances to the position of all
objects (under the hypothesis that one of the objects is controlling
it's position). Now you (the observer) are the Tester; you apply
disturbances and look for "lack of effect" of your disturbances on
the hypothetical controlled variable (two-dimensional position of
square).

Best

Rick

···

--

Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: marken@leonardo.net
http://www.leonardo.net/Marken

[From Rick Marken (970522.1100 PDT)]

Hans Blom (970522) --

Sorry to be of no help.

It's a correctable problem. All you have to do is stop controlling
for MCT and start controlling for PCT. Then you would not only be
of help, you would also be learning how living systems actually
work. It's your choice.

Hans Blom (970522b)

Does your [Mind Reading] program know about the disturbances, Rick?

Of course. Otherwise, the only way to determine the object the
subject is controlling is by reading his mind;-)

It is extremely important -- essential, I would say -- to know
about the _actions_, but not necessarily about the disturbances.

What the correlation (or a covariation) tests is whether there
is a consistent relationship between the action (the mouse
movement) and an ensuing perception (the position change of a
target), I suppose. Am I correct, Rick?

No. You are wrong on all counts. It is never essential to know about
actions in order to do The Test. It is essential to know about the
disturbance variable and the (hypothetical) controlled variable.

The correlation I compute in the Mind Reading demo does _not_
test whether there is a consistent relationship between action and
perception; in fact, the Mind Reading program has been cleverly designed
so that the action/perception relationship is always _changing_ in order
to make it impossible for the observer to
identify the controlled object in terms of superficial
characteristics of the behavior of the objects on the screen, such
as their positional variance (if the action/percpeption relationship
were constant the controlled object could be identified as the one whose
2-D position varied the least) or their location on the
screen (if the action/percpetion relationship were constant the
controlled object could be identified as the one that stayed
closest to the center of the screen).

It is my guess that the outside observer cannot know which target
is controlled because _he_ does not control the mouse and thus
does not know/feel the actions.

But an outside observer CAN know which target is controlled even
when he does not control the mouse (and how could the subject
control the target if someone else were controlling the mouse?).
The observer can know which target is controlled by doing The Test.
In the Mind Reading demo, the computer is the outside observer who knows
which target is controlled. In the "Detection of purpose"
demo, _you_ can be the outside observer who knows which target is
controlled.

Knowledge of the disturbances is not essential if they are random
or otherwise cancel out.

This, I'm afraid, is nonsense.

Best

Rick

···

--

Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: marken@leonardo.net
http://www.leonardo.net/Marken

[Hans Blom, 970528]

(Rick Marken (970522.1100 PDT))

Sorry to be of no help.

It's a correctable problem. All you have to do is stop controlling
for MCT and start controlling for PCT. Then you would not only be of
help, you would also be learning how living systems actually work.
It's your choice.

Well, thanks Rick. I know it's my choice and I exercise it. And it's
no problem at all. If it were, I might want to correct it. But it
seems I don't. So I guess that despite all your efforts I'll never be
able to learn how living systems actually work :-).

Does your [Mind Reading] program know about the disturbances, Rick?

Of course. Otherwise, the only way to determine the object the
subject is controlling is by reading his mind;-)

The "only way"? How did you establish that there is no other way?

Care to explain what the program does, and how? And maybe even in a
way so that I, lacking the knowledge of how living systems actually
work, might understand it? If so, I might be able to demonstrate that
there is another way. So far, you've kept me guessing about how you
do it. With the "side effect" of me thinking how _I_ would do it.
Differently, it seems...

Greetings,

Hans

[From Rick Marken (970528.0920 PDT)]

Hans Blom (970528) --

So I guess that despite all your efforts I'll never be
able to learn how living systems actually work :-).

Never say "never".

Care to explain what the [Mind Reading] program does, and how?

It continuously computes the correlation between X and Y disturbance
and X and Y cursor movement for each object. The object with the
smallest (by a criterion amount) disturbance-cursor correlation
(in both X and Y dimension) is taken to be the one that is currently
under control.

The computer algorithm can be seen by clicking on "ThreeTrack.java." at
http://www.leonardo.net/Marken/demos.html and taking at look at
the source code.

Best

Rick

···

--

Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: marken@leonardo.net
http://www.leonardo.net/Marken