Illusion and Loops (was Beyond the Fringe)

[rsmarken]

[From Rick Marken (2010.05.22.1430)]

Martin Taylor (2010.05.21.23.17) --

Rick Marken (2010.05.19.2300)--

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)

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)

You must be thinking of E(), then, as the function relating the
disturbance to the controlled variable. I thought that you had been
equating E() to the feedback function. In terms of the equation on p.
139 in LCS I, which says:

qi = g(qo)+h(qd)

I assumed that, in our current notation, this maps to

qi = E(qo) + d

with h() as the identity operator transforming the disturbance
variable into qi. The behavioral illusion is then the appearance that
an observed relationship between qo and d reflects the system
function,

qo = F(d)

rather than the feedback function, as in

qo= 1/E(r-d)

The question is whether one who observes an 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().

So far, I'm with you. This is indeed the question.

Great.

�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.

This would be true if control were perfect.

I don't think so. At least it isn't true in my simulation. The
behavioral illusion, where the relationship between d and qo reflects
only the feedback function, not the system function, occurs for both
very low and very high settings of gain in a simple integral control
model.So it occurs whether control is good or poor.

Let's do the simple analysis
with all the functions being the identity function excepot for the output
gain G.

qi = qo + d
qo = Ge = G(r-p) = G(r -qi) = G(r - qo - d)

qo/G = r - d - qo
qo(1+G)/G = r - d

qo = G(r-d)/(1+G)

The factor G/(1+G) is the sensitivity of qo to changes in d, or the
"compliance" of the system. That compliance, when E() is the identity
function, is purely a property of I(), since in this case I() == G().

I'm better at computer simulations than math. I have written a
simulation of a linear control system (simple integrator) with a
linear feedback function. You can vary the gain (G) of the control
system and the coefficient of the feedback function (E). The model is
very simple. The control system model is:

qo := qo + G(r-qi)

The environment function is:

qi := E(qo) + d

I've run this simulation using different values for G and E while
manipulating d (the IV) and measuring the resulting qo (DV). In every
case, there is a linear relationship between d and qo with a slope
equal to -1/E, the negative inverse of the feedback function. It is
never equal to G, no matter what the value of G is (for G>0).

I'm going to work on improving this spreadsheet and then I'll
distribute it. It's a simple and (I think) pretty clear demonstration
of the behavioral illusion (for linear systems; I might also try a
non-linear system). The simulation shows that the observed
relationship between d and qo, when studying a closed loop system,
reflects_only_ the feedback connection from qo to qi, which is E (in
the form -1/E). The system function, G, that goes from qi to qo, is
completely invisible.

A person doesn't become open loop when they are in a psychophysical
experiment.

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.

But we apparently got nowhere, which I expect will continue now.
Nevertheless, I'll give it another try. The "presentation" in a
psychophysical experiment is an environmental vaiable, like a tone,
that is independent of the actions of the subject. It is therefore a
disturbance, d. The response is an output, qo. There is never a
connection between qo and d in a control loop. The loop is closed due
to the effect of qo on qi (the controlled variable) where qi is
affected by both qo (closing the loop) and d.

In a psychophysical experiment, qi is usually a relationship between d
(the stimulus) and qo (the response) that the subject is asked to
maintain (control). For example, the subject is asked to say "yes"
when a tone is present and "no" when it's not. The disturbance
variable, d, values are "tone" or "no tone"; the output (qo) variable
values are"yes" and "no". qi is the logical relationship between d and
qo. The subject is asked to keep the following logical relationship
"true" (control for "true"): if d="tone then qo = "yes"; if d = "no
tone" then qo = "no".

My "Power Law" paper (http://www.mindreadings.com/BehavioralIllusion.pdf)
shows how the closed loop might work in a magnitude estimation experiment.

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.

Of course not. The subject's response, qo, affects the state of the
controlled variable (a relationship between the stimulus, d, and the
response, qo). The control system is closed loop through the
controlled variable, not through the disturbance.

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.

This "critical element" is always open loop. There is never a direct
effect of qo on d in a control loop.

I'll try to get the behavioral illusion spreadsheet ready soon.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com
www.mindreadings.com

[Bruce_Gregory6]

[From Bruce Gregory (2010.05.22.1802 EDT)]

[From Rick Marken (2010.05.22.1430)]

But we apparently got nowhere, which I expect will continue now.
Nevertheless, I’ll give it another try. The “presentation” in a
psychophysical experiment is an environmental vaiable, like a tone,
that is independent of the actions of the subject. It is therefore a
disturbance, d. The response is an output, qo. There is never a
connection between qo and d in a control loop. The loop is closed due
to the effect of qo on qi (the controlled variable) where qi is
affected by both qo (closing the loop) and d.

This is very helpful. What benighted psychologists think of as information, those in the know call disturbances. The world, in general, is a disturbance. I can see that. I would be perfectly content if there were no information to disturb my control. I know you well enough to feel confident that the same is true for you.

Namaste,

Bruce

[Bill_Powers1]

[From Bill Powers (2010.05.22.2000 MDT)]

Rick Marken (2010.05.22.1430) –

Martin Taylor
(2010.05.21.23.17) –

Rick Marken (2010.05.19.2300)–

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)

RM: You must be thinking of E(), then, as the function relating the

disturbance to the controlled variable. I thought that you had been

equating E() to the feedback function. In terms of the equation on
p.

139 in LCS I, which says:

qi = g(qo)+h(qd)

I assumed that, in our current notation, this maps to

qi = E(qo) + d

with h() as the identity operator transforming the disturbance

variable into qi.

BP: in my derivation I was using h() for the function connecting qi to
qo. Can we settle on this notation below, which is, I think, a reasonable
compromise among the three of us:

     h(d) 

•    f(qi)    
  

  d ---------->qi ----------->qo

^

+|

···

---

e(qo)

With this notation,

(1) qi = h(d) + e(qo)

(2) qo = f(qi)

Solving for qi using both equations:

qi = h(d) + e[f(qi)]

That’s as far as we can go without knowing the forms of the functions. qi
appears on both sides of the equal sign so it can’t in general be
separated as a single variable by algebraic manipulations.

We can go farther, as Martin and I both did, if we replace the functions
with multiplying constants with suitable names. I suggest these names for
constants:

h(d) becomes Kd*d

f(qi) becomes Kf*qi (f now refers to the forward path through the
organism)

e(qo) becomes Ke*qo (e for environment)

Now the equations are

(1) qi = Kdd + Keqo

(2) qo = Kf*qi

These are algebraic formulae which represent the steady state solutions
of the differential equations that describe the dynamics of the
system.

We need a digression here to deal with a problem in Rick’s
simulation.

Rick says: The behavioral
illusion, where the relationship between d and qo reflects only the
feedback function, not the system function, occurs for both

very low and very high settings of gain in a simple integral control

model.So it occurs whether control is good or poor.

There are no integrators in the above steady-state algebraic equations.
The steady-state gain of an integrator is infinite no matter what
multiplier is used; varying the multiplier of a simple integrator only
changes how long it takes the system to correct an error within some
small epsilon. To simulate a system with an integrating output we
generally use an output function in the form of a leaky integrator, so it
will have finite gain (and also so it will be like a real system in which
integrations are never perfect):
qo := qo + slowing*(gain * error - qo)dt
For a constant error signal, qo will come to a steady state with the
value of gain * error. This is the finite gain we would use in the above
equations.
If you use the simpler form,
qi := qo + gainerrordt,
this becomes a perfect integrator which will make the error get smaller
and smaller but never quite get to zero. The apparent gain (the fraction
qo/error) will keep getting larger the longer you let the simulation run,
because e keeps getting smaller while qo approaches an asymptote. If you
use the pure integrator, therefore, you will always get the perfect
behavioral illusion if you wait long enough, no matter what value you
give to the gain or to dt. There will be computational problems if you
make the gain too large relative to dt, but that’s just an artifact due
to simulating a continuous analog system on a digital computer. At a
value of gain that is not too large, error correction will take only one
or two iterations.
Using the leaky integrator in the output function makes the gain into the
true steady-state gain, and you can use its value in the above system of
equations. Then you will find that the behavioral illusion predicts
correctly only for large values of the loop gain, KfKe.
Now equations (1) and (2) can be solved by substitution for qi and qo,
which will both turn out to be functions of d alone. Both qi and qo are
dependent variables completely determined by the value of the
independent variable d. For small values of loop gain, say less than 1,
which is Kf*Ke in equations 1 and 2, the behavior illusion will be shown
very poorly; variations in the forward gain Kf will have an appreciable
effect on qo, so qo is not predictable completely from the inverse of the
feedback function. This is also shown by the fact that for low loop gain,
the input quantity qi is not kept solidly at the reference level, which
means that Kf inside the organism does influence the overall relationship
from d to qo.

In a control-system model with all the functions shown, the forward gain
that we are calling Kf will turn out to be the product of the input gain
and the output gain, assuming a comparator with unity gain. Kf will be
KiKo, and the loop gain will be KiKf*Ke.

I hope we’re all on the same page now. On with the music, maybe with no
DC al Coda this time.

Best,

Bill P.

[MartinT]

[Martin Taylor 2010.05.23.09.19]

This must be some kind of special day, when I find myself agreeing with
Rick rather more than with Bill!

[From Rick Marken (2010.05.22.1430)]

Martin Taylor (2010.05.21.23.17) --

Rick Marken (2010.05.19.2300)--

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)

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)

You must be thinking of E(), then, as the function relating the
disturbance to the controlled variable. I thought that you had been
equating E() to the feedback function.

It doesn’t matter, since Bill said you had misinterpreted the inverse
function as the reciprocal, so what you wrote was wrong anyway.
However, I had expected my diagram to make clear that E( ) was exactly
the relationship between qo and the influence of qo on qi, which in the
simple analyses usually shown on CSGnet is ordinarily taken as the
identity function. Here’s the diagram again.

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.

This would be true if control were perfect.

I probably should have said “good”, rather than “perfect”. When control
is perfect, the structure of the internal system doesn’t get reflected
at all in the output, since the output is determined entirely by E(go)
= d. But you had used the word “reflects” and it is true that if there
is any control at all the relation between d and qo “reflects” (is
influenced by) E( ). But if the control ratio (rms(d )/rms(qi)) is
small (control is poor), then the relationship between d and qo also
“reflects” P( ) and G( ).

I don't think so. At least it isn't true in my simulation. The
behavioral illusion, where the relationship between d and qo reflects
only the feedback function, not the system function, occurs for both
very low and very high settings of gain in a simple integral control
model.So it occurs whether control is good or poor.

Let's do the simple analysis
with all the functions being the identity function excepot for the output
gain G.
qi = qo + d
qo = Ge = G(r-p) = G(r -qi) = G(r - qo - d)
qo/G = r - d - qo
qo(1+G)/G = r - d
qo = G(r-d)/(1+G)
The factor G/(1+G) is the sensitivity of qo to changes in d, or the
"compliance" of the system. That compliance, when E() is the identity
function, is purely a property of I(), since in this case I() == G().

I'm better at computer simulations than math. I have written a
simulation of a linear control system (simple integrator) with a
linear feedback function. You can vary the gain (G) of the control
system and the coefficient of the feedback function (E). The model is
very simple. The control system model is:
qo := qo + G(r-qi)
The environment function is:
qi := E(qo) + d
I've run this simulation using different values for G and E while
manipulating d (the IV) and measuring the resulting qo (DV). In every
case, there is a linear relationship between d and qo with a slope
equal to -1/E, the negative inverse of the feedback function. It is
never equal to G, no matter what the value of G is (for G>0).

We are talking about different things. When G changes, the compliance
changes, but there’s no change in the fact that eventually, after
infinite time, with an integrating output function, the influence of qo
will exactly compensate for d, and what you say here will hold true.

Look, however, at the other end of the time scale, and consider a
stepwise change in d after a long period over which d had not changed.
At the next instant, qo will be what it was before the step, and qi
will be equal to the change in d. The relation between qo and d is a
function of time, and for the period during the exponential approach to
perfect control, the properties of the internal system do matter. For
an integrator output function, G is a measure of the integration time
constant, and if all the other functions are linear, the approach to
perfect control will be an exponential decay. But if, say, p = log(qi)
or p = (qi)^2, then the approach to perfect control will not be the
simple exponential decay that you would see if p = qi and E( ) is
linear. The whole of I( ) will influence the shape of the curve qo(t).

[RM earlier] A person doesn’t become open loop when they
are in a psychophysical experiment.

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.

But we apparently got nowhere, which I expect will continue now.
Nevertheless, I'll give it another try. The "presentation" in a
psychophysical experiment is an environmental vaiable, like a tone,
that is independent of the actions of the subject. It is therefore a
disturbance, d. The response is an output, qo. There is never a
connection between qo and d in a control loop. The loop is closed due
to the effect of qo on qi (the controlled variable) where qi is
affected by both qo (closing the loop) and d.So far, we are in agreement.

So far, we are in agreement, apart from “but we apparently got nowhere”.

I don’t think we got nowhere in the previous thread. After it was
agreed that the output of every individual pathway in a control loop is
a simple function of its input, the discussion moved on to questions of
how the loops controlling the different controlled perceptions
interact. I don’t think anyone maintained that the perception created
by the presentation could be influenced by the subsequent response
output by the subject. We may not have come to full agreement on the
precise interrelationship of the control loops involved in making the
response, but most of that is speculation, anyway.

Anyway, before going on, I want to reiterate that the perception being
judged is not among the controlled perceptions of the experiment. In a
2 alternative forced choice study on the ability of a person to detect
a tone, the perception being judged is the perception of the tone. When
the presentation is Xdb, as a function of X how likely is it that the
subject will perceive “tone”? As you point out in the next paragraph,
the perception of “tone” or “no tone” is a disturbance to a controlled
perception. In itself, the perception of “tone” or “no tome” is not
controlled. If the subject had a knob to change X (as in Bekesy
audiometry) so as to alter the perceptibility of the tone, it would be
a controlled perception. But in most psychophysical studies, it isn’t.

In a psychophysical experiment, qi is usually a relationship between d
(the stimulus) and qo (the response) that the subject is asked to
maintain (control). For example, the subject is asked to say "yes"
when a tone is present and "no" when it's not. The disturbance
variable, d, values are "tone" or "no tone"; the output (qo) variable
values are"yes" and "no". qi is the logical relationship between d and
qo. The subject is asked to keep the following logical relationship
"true" (control for "true"): if d="tone then qo = "yes"; if d = "no
tone" then qo = "no".

Yes, we all agreed to this in the earlier thread. In fact, I think I
was the one who introduced this way of looking at it, in my first model
of how the several levels of control loop in an experiment relate to
one another, the one that used the layered protocol structure to relate
the user’s control structure to that of the experimenter.

The thread became very quickly mainly about the details of how the
control structure for generating the response was organized, not about
the relation between the sensory input and the tone-like-ness of the
perception of the input.

[RM earlier] My "Power Law" paper ()
shows how the closed loop might work in a magnitude estimation experiment.
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.

Of course not. The subject's response, qo, affects the state of the
controlled variable (a relationship between the stimulus, d, and the
response, qo). The control system is closed loop through the
controlled variable, not through the disturbance.

Precisely my point. You are dealing with a different variable,
controlling the perception of the match of the response to the
perceptual value obtained from the presentation, not the perception of
the presentation.

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.

This "critical element" is always open loop. There is never a direct
effect of qo on d in a control loop.

Again, you do make my point.

A larger point here is that in a hierarchical control system, a
perception at one level can and often does become a disturbance at a
higher level. Not all perceptions are controlled, and at any higher
level, some of the contributing lower-level perceptions may well not be
controlled. This is the situation we are talking about in a
psychophysical experiment. You choose to emphasize the controlled
perceptions, and I have no issue with your doing so. It’s important to
get that part right, as your analysis of the power-law experiments
demonstrates.

Where I have a problem is in your dismissal of the fact (which you
acknowledge to be a fact) that the perception of interest to the
psychophysicist is uncontrolled, and is influenced only by the
disturbance introduced by the experimenter. Even in the power-law case,
the question of interest is still how the magnitude of the sensory
input influences the magnitude of the perception. Stevens thought that
numbers were linearly perceived, and came to the power-law result. You
showed that a proper consideration of the generation of the response
led to a different conclusion. But no matter how the response was
generated, the perception of magnitude was and is uncontrolled.

Martin

···

http://www.mindreadings.com/BehavioralIllusion.pdf

[rsmarken]

[From Rick Marken (2010.05.23.1030)]

Bill Powers (2010.05.22.2000 MDT)--

BP: in my derivation I was using h() for the function
connecting qi to qo.

You must mean "connecting d to qi", right. No biggie, you get it right
in the next section.

I suggest these names for constants:

h(d) becomes Kd*d

f(qi) becomes Kf*qi (f now refers to the forward path
through the organism)

e(qo) becomes Ke*qo (e for environment)

OK, this is the notation I use in the attached spreadsheet, with Kd=1 always.

Rick says: The behavioral illusion, where the relationship
between d and qo reflects only the feedback function, not
the system function, occurs for both very low and very
high settings of gain in a simple integral control
model.So it occurs whether control is good or poor.

There are no integrators in the above steady-state
algebraic equations. The steady-state gain of an
integrator is infinite no matter what multiplier is
used; varying the multiplier of a simple integrator only
changes how long it takes the system to correct an error
within some small epsilon. To simulate
a system with an integrating output we generally use an
output function in the form of a leaky integrator, so it will
have finite gain (and also so it will be like a real system
in which integrations are never perfect):

qo := qo + slowing*(gain * error - qo)*dt

You are such a smarty pants;-) Absolutely right. I made the
integration leaky in the simulation I now use in the attached
spreadsheet and, indeed, the lower the gain, the less closely the
relationship between qo and d matches -1/Ke, the inverse of the
feedback function. So the poorer the control the less of a behavioral
illusion there is.

But I think you will see from the spreadsheet that even when control
is very poor, the observed relationship between qo and d never comes
close to the system function, Kf, either. It just gets closer to a
horizontal line, indicating no relationship between qo and d, which is
what you expect if control is poor; the actor is not compensating for
disturbances to the controlled variable, qi.

Here's how to use the spreadsheet. First, make sure that the
calculation options are set to "Iteration" with about 1000 iterations
so that it stabilizes completely. The calculation settings should go
with the spreadsheet but I don't know if this is the case for sure.

In the upper left you will see cells that allow you to set values for
Ke (the feedback function coefficient) and Kf (the system function
coefficent). Both functions are linear in this simulation. The default
value of Kf that comes with the spreadsheet gives fairly good control,
as indicated by the stability and RMS values in rows 17 and 18.
Increase Kf a bit (to, say, 10) and control gets much better.

The graph on the right shows the effect of changes in Kf on the fit of
the observed relationship between d and qo (the black dots) to the
relationship predicted by the behavioral illusion (-1/Ke), indicated
by the red line. Notice that the fit of the dots to the red line
increases as Kf (and thus control ability, as indicated by the
increase in stability and decrease in RMS error) increases (note what
happens to the fit when Kf goes from the default value to 10, for
example).

So you are right (as usual); the behavioral illusion (in the sense of
the relationship between qo and d reflecting the feedback function,
rather than the system function) lessens as control decreases. But
control has to get pretty poor before this happens, far poorer, I
would guess, than even the control we see in the typical psychological
experiment.

But you can also see that, even when control is poor and the observed
relationship between qo and d does not come very close to -1/Ke, the
observed relationship between qo and d does not come close to the
system function, Kf (the blue line on the graph). This can be seen
even more clearly in the tab labeled Non-Linear f. In this sheet the
system function is non-linear: qo is a quadratic rather than a linear
function of error. The default settings in that sheet give pretty poor
control, so the observed relationship between qo and d is not very
close to -1/Ke, but it's still linear, unlike the actual non-linear
system function, shown by the blue line.

So my conclusion, based on this little exercise, is that the observed
relationship between qo and d (such as that observed in conventional
psychology experiments) is never a reflection of the system function
when the system under study happens to be closed loop. And this is
true no matter how well or how poorly the system controls. The system
function of a closed loop system is simply invisible to a conventional
experiment where an environmental variable (d or IV) is manipulated
and a behavioral output (qo or DV) is measured. But the degree to
which the observed relationship between qo and d reflects -1/Ke (which
is the "official" definition of the behavioral illusion) depends on
how well the system controls the variable that is affected by both qo
and d, the controlled variable.

Does this seem like a fair conclusion?

Best

Rick

IllusionSim.xls (25.5 KB)

···

--
Richard S. Marken PhD
rsmarken@gmail.com
www.mindreadings.com

Content-Type: application/vnd.ms-excel; name="IllusionSim.xls"
Content-Disposition: attachment; filename="IllusionSim.xls"
X-Attachment-Id: f_g9k4iptb0

[Bruce_Gregory6]

[From Bruce Gregory (2010.0523.1428 EDT)]

[Martin Taylor 2010.05.23.09.19]

MT to RM: Where I have a problem is in your dismissal of the fact (which you
acknowledge to be a fact) that the perception of interest to the
psychophysicist is uncontrolled, and is influenced only by the
disturbance introduced by the experimenter. Even in the power-law case,
the question of interest is still how the magnitude of the sensory
input influences the magnitude of the perception. Stevens thought that
numbers were linearly perceived, and came to the power-law result. You
showed that a proper consideration of the generation of the response
led to a different conclusion. But no matter how the response was
generated, the perception of magnitude was and is uncontrolled.

BG: I thought Rick was quite clear that actions generated by the experimenter are disturbances to perceptions controlled by the subject. Uncontrolled perceptions play no role in this description, as far as I can tell.

Bruce

[Bill_Powers1]

[From Bill Powers (2010.05.23.1225 MDT)]

Rick Marken (2010.05.23.1030) --

RM: I made the integration leaky in the simulation I now use in the attached
spreadsheet and, indeed, the lower the gain, the less closely the
relationship between qo and d matches -1/Ke, the inverse of the
feedback function. So the poorer the control the less of a behavioral
illusion there is.

But I think you will see from the spreadsheet that even when control
is very poor, the observed relationship between qo and d never comes
close to the system function, Kf, either. It just gets closer to a
horizontal line, indicating no relationship between qo and d, which is
what you expect if control is poor; the actor is not compensating for
disturbances to the controlled variable, qi.

BP: This depends on how you make the control worse. You're doing it by reducing the forward gain. But suppose you do it by reducing the feedback factor instead -- just reduce it to zero, to make this easy. Then adjust Kf, the forward factor, to the right number to get the same output as before, so we are observing the same relationship of d to qo as before. Everything (except the time constant) will look about the way it was with feedback control, until you start playing with disturbances. If you use the same output function as in the control system, you'll find that when Kf is adjusted to give the same qo as before, the time constant increases greatly. If the observed time constant was 1 second when the loop gain was 100, you'll find that removing the feedback and adjusting Kf to get the same steady-state forward characteristics makes the time constant 100 seconds. To put that the other way, if the open-loop system has a natural time constant of 1 second, adding the feedback and raising the loop gain to 100 will reduce the system time constant to 0.01 second. Feedback systems are much faster than open-loop systems made of the same components. H.S. Black published that fact in about 1934.

Add a disturbance to the effect of the controller on qo:

qo = Kf*qi + d2 (d2 is a new disturbance applied to the output quantity along with the effect from the controller)

The truly open-loop system won't alter its output so as to keep qo the same when d2 is applied. The control system will.

And of course with no feedback, the loop gain is zero and the relationship between d and qo will exactly reflect the forward function inside the controlling organism.

Best,

Bill P.

[rsmarken]

[From Rick Marken (2010.05.23.1410)]

Bill Powers (2010.05.23.1225 MDT)--

Rick Marken (2010.05.23.1030) --

RM: But I think you will see from the spreadsheet that even
when control is very poor, the observed relationship between qo
and d never comes close to the system function, Kf, either....

BP: This depends on how you make the control worse. You're
doing it by reducing the forward gain. But suppose you do it by
reducing the feedback factor instead -- just reduce it to zero, to
make this easy... Everything (except the time constant) will
look about the way it was with feedback
control

This is great. Thanks. Yes, when I reduce control by reducing Kf the
form of the observed relationship between d and qo looks more and more
like the system function. This is clearest in the non-linear case; as
Kf is reduced to zero the observed relationship between d and qo looks
more and more like the (negative of) the quadratic system function.
Very cool.

I think this is very important for me to note when I discuss the
problems of studying control systems using conventional methodology.
In fact, conventional methodology can tell us something about the
system to the extent that we can set up the experiment so that the
subject's responses have little or no effect on the controlled
variable (the variable that the subject is instructed to control for
in the experiment). I imagine there are such experiments but I don't
think they is very common.

Martin seems to think that psychophysical experiments are open-loop
but I don't see it; even in detection experiments where subjects are
not given "feedback" (told whether their answer was right or wrong)
they are still able to perceive the effect of their answer on the
relationship between that answer and what they perceived (which is
what they are asked to control for in a detection task). I have done
tracking experiments where I have broken (or changed) the connection
between response and input. In these experiments the subject is
responding to qi open loop.I think Pavlov's experiment where he
measured salivation (going into a fistula instead of the mouth) in
response to food in the mouth is another example. Other examples of
"open loop" experiments would be most welcome.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com
www.mindreadings.com

[Bruce_Gregory6]

[From Bruce Gregory (2010.05.23.1721 EDT)]

[From Rick Marken (2010.05.23.1410)]

I think this is very important for me to note when I discuss the
problems of studying control systems using conventional methodology.
In fact, conventional methodology can tell us something about the
system to the extent that we can set up the experiment so that the
subject’s responses have little or no effect on the controlled
variable (the variable that the subject is instructed to control for
in the experiment). I imagine there are such experiments but I don’t
think they is very common.

BG: They is more common than you think. I cannot recall a verbal learning task where the subjects responses controlled the “disturbance.” If the experimenter eliminated correct responses from the deck, the subject would be influencing the disturbances, but not necessarily in a helpful way. Perhaps you have evidence otherwise.

Namaste,

Bruce

[Bill_Powers1]

[From Bill Powers (20120.05.23.1725 MDT)]

Rick Marken (2010.05.23.1410) --

BP earlier: This depends on how you make the control worse. You're
doing it by reducing the forward gain. But suppose you do it by
reducing the feedback factor instead -- just reduce it to zero, to
make this easy... Everything (except the time constant) will
look about the way it was with feedback
control

RM: This is great. Thanks. Yes, when I reduce control by reducing Kf the
form of the observed relationship between d and qo looks more and more
like the system function. This is clearest in the non-linear case; as
Kf is reduced to zero the observed relationship between d and qo looks
more and more like the (negative of) the quadratic system function.
Very cool.

BP: Good, I was pretty sure it would work that way.

RM: Martin seems to think that psychophysical experiments are open-loop
but I don't see it; even in detection experiments where subjects are
not given "feedback" (told whether their answer was right or wrong)
they are still able to perceive the effect of their answer on the
relationship between that answer and what they perceived (which is
what they are asked to control for in a detection task).

BP: The problems with this subject come from the ambiguous idea of experiments being closed-loop. Every single relationship in every combination is closed-loop? Of course not. Some of the relationships, such as the relationship between error and output, are clearly open loop, aren't they? Or between qi and p? Those relationships don't contain feedback connections: they're just input-output relationships, as in qo = Kf*qi. No closed loop there.

Martin is trying to find a way to investigate such open-loop connections which characterize part of a control loop. We do that, too. For example, in my tracking experiment, one of the relationships the analysis measures is the gain of a leaky integrator in the output function. Another is the amount of velocity perceptual signal obtained from the actual rate of change of the cursor-target distance. Those are constants that belong to just one open loop component of the control system.

The bone of contention between Martin and me is that I claim the Schouten experiment involves control of a relationship, so that a disturbance which alters an uncontrolled element of the relationship can be counteracted by the control system which alters the other, controllable, element. The data can be explained in that way, which suggests some different interpretations. But I'm not going to get back into that argument; my analysis still sits there in the archives, I presume, and I'm content to let others judge its merits. This way, by the time someone discovers my fatal error that calls for an apology to Martin, I'll be dead, so I never will have to apologize. Ha, ha to you, Martin.

Best,

Bill P.

[MartinT]

[Martin Taylor 2010.05.23.23.28]

[From Bill Powers (20120.05.23.1725 MDT)]

The bone of contention between Martin and me is that I claim the
Schouten experiment involves control of a relationship, so that a
disturbance which alters an uncontrolled element of the relationship
can be counteracted by the control system which alters the other,
controllable, element.

Why do you call it a bone of contention? It’s been agreed since at
least Feb 14, 2009. In fact, I’m not sure I wasn’t the one who pointed
it out initially, since it was part of my original diagram showing my
multi-level control system proposal for what goes on in the experiment.

>[From Rick Marken (2010.05.23.1410)]

>Martin seems to think that psychophysical experiments are open-loop

I’m really puzzled as you why you persist in saying things like that.
One would think you simply never read anything I wrote. Are you trying
to create an imaginary “Martin” persona that you can cast into the
wilderness of “those who do not iunderstand PCT”?

Martin

[rsmarken]

[From Rick Marken (2010.05.23.2200)]

Bruce Gregory (2010.05.23.1721 EDT)--

Rick Marken (2010.05.23.1410)

RM: In fact, conventional methodology can tell us something
about the system to the extent that we can set up the
experiment so that the subject's responses have little or no
effect on the controlled variable (the variable that the subject is
instructed to control for in the experiment). I imagine there are
such experiments but I don't think they is very common.

BG: They is more common than you think.

If you get an outfit you can be a cowboy too.

I cannot recall a verbal learning task where the subjects
responses controlled the "disturbance."

There is no psychological experiment in which subjects control the
disturbance. A disturbance is, by definition, an independent influence
on a controlled variable; in an experiment it is an independent
variable. The subjects' responses in an experiment are actions that
prevent these disturbances from moving the controlled variable from
its reference state. In a verbal learning task, such as a free recall
task, the subject is instructed to control for saying words that are
the same as those that we in a previously presented list. The
disturbance to this variable is the "learning set" of words itself.
The responses that aim to bring this controlled variable to the
reference state are the words spoken by the subject. How well a
subject controls this variable depends on how well he or she can store
and retrieve from memory the words that were in the learning set.

Best

Rick

···

--
Richard S. Marken PhD
rsmarken@gmail.com
www.mindreadings.com

[Bruce_Gregory6]

[From Bruce Gregory (2010.05.24.0650 EDT)]

[Martin Taylor 2010.05.23.23.28]

>[From Rick Marken (2010.05.23.1410)]

>Martin seems to think that psychophysical experiments are open-loop

I’m really puzzled as you why you persist in saying things like that.
One would think you simply never read anything I wrote. Are you trying
to create an imaginary “Martin” persona that you can cast into the
wilderness of “those who do not iunderstand PCT”?

BG: I think your tests for the controlled variable demonstrate that this is indeed the case. The set of “those who understand PCT” has very few members. Any effort to enlarge this number invariably encounters strong resistance. This resistance is often coupled with expressions of disdain and contempt. Since this has been the case for decades, it seems unlikely to change in our lifetimes. That is unfortunate, but it definitely appears to be the nature of the beast.

Namaste,

Bruce

[rsmarken]

[From Rick Marken (2010.05.24.1250)]

Martin Taylor (2010.05.23.23.28)–

Rick Marken (2010.05.23.1410)–

Martin seems to think that psychophysical experiments are
open-loop

I’m really puzzled as you why you persist in saying things
like that. One would think you simply never read anything I
wrote.

Maybe I am wrong. But it had sounded a lot like you have been saying that the relationship between stimulus and response in a psychophysical task tells you something about the subject’s perceptual function because the relationship of interest to the experimenter in such experiments is open loop. For example, just yesterday you [Martin Taylor 2010.05.23.09.19] said this:

Where I have a problem is in your dismissal of the fact (which you acknowledge to be a fact) that the perception of interest to the psychophysicist is uncontrolled, and is influenced only by the disturbance introduced by the experimenter. Even in the power-law case, the question of interest is still how the magnitude of the sensory input influences the magnitude of the perception. Stevens thought that numbers were linearly perceived, and came to the power-law result. You showed that a proper consideration of the generation of the response led to a different conclusion. But no matter how the response was generated, the perception of magnitude was and is uncontrolled.

It’s kind of hard to see what you are getting at here. Maybe I can get at it better with a question: Are you agreeing that a psychophysical task is a closed-loop control task and that, therefore, the observed relationship between the physical stimulus (disturbance) and output (numerical response in the magnitude estimation example) tells us virtually nothing about how the magnitude of sensory input (qi) influences the magnitude of the perception? If that’s the case then I stand corrected.

Best

Rick

···

–
Richard S. Marken PhD
rsmarken@gmail.com
www.mindreadings.com

[Bill_Powers1]

[From Bill Powers (2010.05.24.1312 MDT)]

Martin Taylor 2010.05.23.23.28 –

The bone of contention between
Martin and me is that I claim the Schouten experiment involves control of
a relationship, so that a disturbance which alters an uncontrolled
element of the relationship can be counteracted by the control system
which alters the other, controllable, element.

Why do you call it a bone of contention? It’s been agreed since at least
Feb 14, 2009. In fact, I’m not sure I wasn’t the one who pointed it out
initially, since it was part of my original diagram showing my
multi-level control system proposal for what goes on in the
experiment.

Here’s the figure from Feb. 14, 2009.

The subject in this diagram is not controlling any relationship that I
can see. The subject’s controlled perception does not receive any input
from the state of the button, though there is input from the state of the
presentation (labeled “Observe” in red).

The subject does not control the presentation, as is indicated; however,
the subject could control the relationship between button and
presentation if the lower “Perc” function received information
about the state of the button. Then the “interpret” arrow would
be based on the relationship, which would be disturbed when the state of
the presentation changed, and restored when the state of the button
changed. The lower-order perception in the diagram would then be p =
(Button XOR Presentation) and the reference state would be
FALSE.

Is this the diagram you were referring to?

Best,

Bill P.

[MartinT]

[Martin Taylor 2010.05.25.11.17]

This has to be quick, as I am packing for my plane to Europe this
evening.

[From Bill Powers (2010.05.24.1312 MDT)]

Martin Taylor 2010.05.23.23.28 –

The bone of contention
between
Martin and me is that I claim the Schouten experiment involves control
of
a relationship, so that a disturbance which alters an uncontrolled
element of the relationship can be counteracted by the control system
which alters the other, controllable, element.

Why do you call it a bone of contention? It’s been agreed since at
least
Feb 14, 2009. In fact, I’m not sure I wasn’t the one who pointed it out
initially, since it was part of my original diagram showing my
multi-level control system proposal for what goes on in the
experiment.

Here’s the figure from Feb. 14, 2009.

The subject in this diagram is not controlling any relationship that I
can see. The subject’s controlled perception does not receive any input
from the state of the button, though there is input from the state of
the
presentation (labeled “Observe” in red).

Yes, this is the figure.

“The subject’s controlled perception does not receive any input from
the state of the button”. That was the area of contention, not whether
the subject was controlling a relationship perception, of which one
input was the perception of the “stimulus” array.

I considered that the state of the button was a perception controlled
at a much lower level in the output side of the relationship control
unit, whereas you considered it to be a direct input to the
relationship control unit. I considered the relationship to be between
the perception of the “stimulus” and an imagined answer taken from a
repertoire of permissible answers (shown in the diagram as a control
unit “Formulate response”). The imagined answer that gave the best
match was provided as a reference value to whatever control systems
actually fed the answer back to the experimenter, whether button,
voice, or whatever. You preferred the direct link to the perception of
the button state, I thought (and think) that to be less probable than
the setting of a reference value for the output control hierarchy. The
disagreement is shown in the paired diagram I reproduced and reproduce
again here.

Eventually, I decided in my own mind that there was no data to judge
which output mechanism was to be preferred, and there was no point in
continuing the discussion. But right from that initial diagram, it was
clear that we both assumed that what the subject controlled was a
relationship between the perception of the stimulus and the answer to
be presented to the experimenter. I think that should still be clear,
despite that apparently we retain the same different preferences for
the organization of the output processes.

Martin

[MartinT]

[Martin Taylor 2010.05.25.11.34]

[From Rick Marken (2010.05.24.1250)]

Martin Taylor (2010.05.23.23.28)–

Rick Marken (2010.05.23.1410)–

Martin seems to think that psychophysical experiments are

open-loop

I’m really puzzled as you why you persist in saying things

like that. One would think you simply never read anything I

wrote.

Maybe I am wrong. But it had sounded a lot like you have been saying
that the relationship between stimulus and response in a psychophysical
task tells you something about the subject’s perceptual function
because the relationship of interest to the experimenter in such
experiments is open loop.

That’s what I am saying, but the puzzle is how you get from there to
“Martin seems to think that psychophysical experiments are open-loop”.
It’s a real puzzle how you get from the notion that input-output
pathways derive their output from their input to the assertion that I
think the experiment is open loop.

For example, just yesterday you [Martin Taylor
2010.05.23.09.19] said this:

Where I have a problem is in your dismissal of the
fact (which you acknowledge to be a fact) that the perception of
interest to the psychophysicist is uncontrolled, and is influenced only
by the disturbance introduced by the experimenter. Even in the
power-law case, the question of interest is still how the magnitude of
the sensory input influences the magnitude of the perception. Stevens
thought that numbers were linearly perceived, and came to the power-law
result. You showed that a proper consideration of the generation of the
response led to a different conclusion. But no matter how the response
was generated, the perception of magnitude was and is uncontrolled.

It’s kind of hard to see what you are getting at here. Maybe I can get
at it better with a question: Are you agreeing that a psychophysical
task is a closed-loop control task and that, therefore, the observed
relationship between the physical stimulus (disturbance) and output
(numerical response in the magnitude estimation example) tells us
virtually nothing about how the magnitude of sensory input (qi)
influences the magnitude of the perception? If that’s the case then I
stand corrected.

Of course that’s not what I am saying. I’m saying that X, where X is
the magnitude of the perception of the stimulus, is the disturbance to
the controlled perception of the relation between the magnitude
perception and the subject’s choice of response. X = P(qi) at a lower
perceptual level. The experimenter’s stimulus is not itself the
disturbance to the relationship control unit.

It’s a real puzzle how you get from the notion that input-output
pathways derive their output from their input to the assertion that I
think the experiment is open loop. When you run your simulations of
tracking stiudies, don’t you use mathematics that assumes the output of
one function to depend only on its input, say p = P(qi)? Why don’t you
assert that you and Bill think that tracking is open-loop, if you use
such equations?

Martin

Martin

[rsmarken]

[From Rick Marken (2010.05.25.0950)]

Martin Taylor (2010.05.25.11.34)–

I’m saying that X, where X is
the magnitude of the perception of the stimulus, is the disturbance to
the controlled perception of the relation between the magnitude
perception and the subject’s choice of response. X = P(qi) at a lower
perceptual level. The experimenter’s stimulus is not itself the
disturbance to the relationship control unit.

I still don’t get it. You seem to agree that subjects in a psychophysical experiment are controlling a relationship between their perception of the stimulus and a perception of their own response. But then you say that the stimulus is not itself the disturbance to this relationship perception? How could that be? And if it’s true that the stimulus is not a disturbance, why is that not the case in any experiment.

The stimulus is what the experimenter manipulates in an experiment, the IV. PCT suggests that this stimulus must be a disturbance (d) to a controlled variable, the perception that the subject is asked (via the instructions) to control (using their response, qo, the DV) in the experiment.

If the subjects in conventional experiments are controlling a variable, such as the relationship between perceived aspects of the IV and DV, then the relationship between IV (d) and DV (qo) is characterized by the “behavioral illusion” equation, qo = -1/E(d) This is why I have argued that the relationship between IV (stimulus) and response (DV) in typical psychological experiments (including, of course, psychophysical experiments) reflects characteristics of the environment rather than of the subjects themselves.

It looks to me like you are trying to exempt psychophysical experiments from this analysis by suggesting that the stimulus (IV) in these experiments is not a disturbance to a controlled variable. If such an exemption existed then psychophysicists would, indeed, be in the happy position of being able to see, in observed relationships between stimuli (IV) and responses (DV), characteristics of the system, such as its perceptual function, rather than characteristics of the environmental feedback path from response to controlled variable, as in all those other psychological experiments.

I think that subjects in psychophysical experiments are controlling a perception of the relationship between stimulus and response. But you could easily test this using Bill’s suggestion of disturbing qo. If the response to the stimulus is open loop then there will be no resistance to the disturbance.

Anyway, have a nice trip to Europe.

Best

Rick

···

–
Richard S. Marken PhD
rsmarken@gmail.com
www.mindreadings.com

[MartinT]

[Martin Taylor 2010.05.25.13.11]

This is probably my last before shutting off my computer.

[From Rick Marken (2010.05.25.0950)]

Martin Taylor
(2010.05.25.11.34)–

I’m saying that X, where X is
the magnitude of the perception of the stimulus, is the disturbance to
the controlled perception of the relation between the magnitude
perception and the subject’s choice of response. X = P(qi) at a lower
perceptual level. The experimenter’s stimulus is not itself the
disturbance to the relationship control unit.

I still don’t get it. You seem to agree that subjects in a
psychophysical experiment are controlling a relationship between their
perception of the stimulus and a perception of their own response.

No, I tried to make crystal clear that they are NOT " controlling a
relationship between their perception of the stimulus and a perception
of their own response" but they ARE controlling a relationship between
their perception of the magnitude (identity, some property) of the
stimulus and a perception of their own response

But then you say that the stimulus is not itself the disturbance
to this relationship perception? How could that be?

Because the disturbance is the output of a function to which the
stimulus is the input. The question of interest to the psychophysicist
is some property of this function.

The stimulus is what the experimenter manipulates in an experiment, the
IV. PCT suggests that this stimulus must be a disturbance (d) to a
controlled variable,

Nothing in PCT says that the stimulus as perceived by the experimenter
(e.g. the brightness of a light or the intensity of a tone) must be the
actual disturbance. All PCT says is that if some action is correlated
to the presentation of the stimulus, it is probable that the stimulus
contributes to a disturbance to some controlled variable and that the
action is likely to be an input to the environmental feedback path, the
output of which contributes to the controlled variable.

I think that subjects in psychophysical experiments are
controlling a perception of the relationship between stimulus and
response.

Between some property of the stimulus, generated by a perceptual
function inside the subject, and some property of the response (e.g. it
doesn’t matter how hard the response button is predded, in most
psychophysical experiments).

But you could easily test this using Bill’s suggestion of
disturbing qo. If the response to the stimulus is open loop then there
will be no resistance to the disturbance.

There will, of course, be no effect of varying qo on the properties of
the preceding stimulus – unless you really do think that time-travel
to the past should be incorporated into PCT. Varying qo will affect the
compensation against the disturbance to whatever variables are being
controlled. But the immediately preceding stimulus (and it
predecessors) are not among these variables.

Anyway, have a nice trip to Europe.

Thanks. I may or may not check in from time to time over the next 3
weeks.

Martin

[Bill_Powers1]

[From Bill Powers (2010.05.25.1035 MDT)]

Martin Taylor 2010.05.25.11.17 –

Here’s the figure from Feb. 14, 2009.

BP earlier: The subject in this diagram is not controlling any
relationship that I can see. The subject’s controlled perception does not
receive any input from the state of the button, though there is input
from the state of the presentation (labeled “Observe” in
red).

MMT: Yes, this is the figure.

“The subject’s controlled perception does not receive any input from
the state of the button”. That was the area of contention, not
whether the subject was controlling a relationship perception, of which
one input was the perception of the “stimulus” array.

I considered that the state of the button was a perception controlled at
a much lower level in the output side of the relationship control unit,
whereas you considered it to be a direct input to the relationship
control unit.

BP: Take a look at the diagram you cite in this post:

Mine was a simplified version of yours. To convert yours to mine, all you
have to do is delete the imagination connection and run an arrow from the
sensor in your lower-order control system to the input of the next level,
as shown below (the arrows to and from E in my side of the diagram were
somehow reversed by one of us):

You show your higher-order system controlling a relationship between the
presentation (as sensed at a lower level) and an imagined
“answer”, where mine uses the present-time perception of the
“answer” that would be generated by actually pressing the
button. As in all diagrams of hierarchical control, a copy of the
perception at the lower order is sent to the higher order of control.
What you call the “match” between presentation and answer I
would call the reference signal specifying the relationship between
the (lower-level) perceptions of the presentation and the button
press.

MMT: I considered the
relationship to be between the perception of the “stimulus” and
an imagined answer taken from a repertoire of permissible answers (shown
in the diagram as a control unit “Formulate response”). The
imagined answer that gave the best match was provided as a reference
value to whatever control systems actually fed the answer back to the
experimenter, whether button, voice, or whatever.

BP: Your way of modeling this would give the wrong result if the
reference signal supposedly calling for a perception of the button being
depressed failed to have that effect – if, for example, the person
didn’t push quite hard enough to make the button hit bottom. The imagined
button would still be perceived as pressed. By using my added connection
instead, failure of the button to go down would lead to an error in the
relationship so (perhaps, depending on details) the subject would try
again, or increase the reference output to make the lower system push
harder. Same sort of problem if the finger hit the wrong button.

I can see there being an intermediate phase in which the imagined answer
is selected, then the switch is thrown to send the result as a reference
signal to the lower level. In my version of this arrangement (in B:CP)
the output of the higher system is not connected to the lower system at
same time it is being imagined, because that would cause the lower system
to execute wrong behaviors while the right answer was being
“formulated.” Formulation is essentiallly finding the answer
which is perceived to be related to the presentation perception in the
way specified by the highest reference signal. Unless this answer is hit
upon on the first try, the process of checking the relationship for error
would cause the lower system to act. That’s why I put a switch
there.

After initial formulations, the connections would not need to be imagined
any more; they become permanent. The button press becomes the only action
tried.

Your proposal has the main disadvantage of all compute-then-execute
models. Since only the imagined output is part of the controlled
variable, the control system will get a picture of the state of the
controlled quantity that easily can be different from the external
reality. It will think it has done the task while the experimenter
observes that it has not done it.

You preferred the direct link to
the perception of the button state, I thought (and think) that to be less
probable than the setting of a reference value for the output control
hierarchy. The disagreement is shown in the paired diagram I reproduced
and reproduce again here.

I left out the intermediate functions, which was lazy of me. But you
forgot that lower-level controlled perceptions are also copied to the
inputs of higher systems.

Eventually, I decided in my own
mind that there was no data to judge which output mechanism was to be
preferred, and there was no point in continuing the discussion. But right
from that initial diagram, it was clear that we both assumed that what
the subject controlled was a relationship between the perception of the
stimulus and the answer to be presented to the experimenter. I think that
should still be clear, despite that apparently we retain the same
different preferences for the organization of the output
processes.

We still agree about that. I will even let you have a
“formulation” phase while the task is being learned. But once
it’s learned, I still think that the real button-press and the
presentation are the elements of the controlled relationship.

So where does that leave us?

Best,

Bill P.