Tracking task

[From Rupert Young (2014.05.31 17.00)]

Hi Rick,

Looking at the tracking task I notice that although the correlation

C-M is poor it does look if the cursor trace goes up and down when
does the mouse. Has the correlation between the change in the
mouse value and the change in the cursor value been computed? In
other words, is there a good correlation between the change in
output and the change in input? What would be the significance if
this were the case?

![bbfdbeca.png|622x453](upload://xxalez1zJePIpeRYxDZsxUAk5j5.png)
···

-- Regards,
Rupert

[From Rick Marken (2014.05.31.1315)]

bbfdbeca.png

···

Rupert Young (2014.05.31 17.00)–

Hi Rick,



Looking at the tracking task I notice that although the correlation

C-M is poor it does look if the cursor trace goes up and down when
does the mouse. Has the correlation between the change in the
mouse value and the change in the cursor value been computed? In
other words, is there a good correlation between the change in
output and the change in input?

RM: This was the first question I addressed when I started doing research on what we now call PCT. The finding of a lack of relationship between input and output in the compensatory tracking task was so surprising to me (and my colleagues, to whom I demonstrated it, on an Apple II computer!) that I felt like I had to see whether there was some other aspect of the input that was related to (and might be causing) outputs. Since there were many possibilities (such as change in input value, as you suggest, or an earlier value of the input, etc etc) I developed the “repeated disturbance” test that is now available as another net demo at:

http://www.mindreadings.com/ControlDemo/Cause.html

In this task, the same disturbance occurs in the first and second half of the experiment. Since output (mouse position) is highly correlated with disturbance value I knew that repeating the disturbance would lead to repeated output. I then reasoned that if there was something about the input that was related to the output, when the outputs are the same on both trials the inputs should be the same as well. But the results of this “repeated disturbance” demo shows that the inputs are not the same when the outputs on repeated trials are the same. The correlation between outputs when the disturbance is repeated is always greater then .99 while the correlation between inputs is rarely greater than .3 and it’s often close to 0. So there appears to be nothing about the input that is related to the output that keeps the input under control.

RY: What would be the significance if

this were the case?

RM: It almost was the case. It turns out that the correlation between inputs in the repeated disturbance case is nearly 1.0 for a noiseless control system. So if people were noiseless control systems the results of my repeated disturbance demo would not have been quite as dramatic. But fortunately, there is noise in living control systems and this “drowns out” the very small variance in the repeated inputs that is the same on the repeated trials.

I think the whole point of these demonstrations of the lack of relationship between input and output in control is just to try to encourage psychologists to drop the cause-effect model of behavior and start studying control from a control theory (PCT) perspective, which means trying to figure out (using “the test”) the inputs that are controlled by (rather than control) behavior. This is all discussed in Marken, R. S.
and Horth, B. (2011) When Causality Does Not Imply Correlation More Spadework at the Foundations of
Scientific Psychology, Psychological
Reports
, 108, 1-12.

By the way, I put up a new, javascript version of my “Behavioral Illusion” demo up at

http://www.mindreadings.com/ControlDemo/Illusion.html

I’d be interested in hearing what people think of it. I think it might make the point better than the previous (java) version.

Best regards

Rick


-- Regards,
Rupert


Richard S. Marken PhD
www.mindreadings.com
It is difficult to get a man to understand something, when his salary depends upon his not understanding it. – Upton Sinclair

[Martin Taylor 2014.05.31.17.09]

Rupert, imagine an extreme case. Suppose that the loop transport lag

were measured in hours rather than milliseconds. The cursor position
is the sum of the disturbance and the mouse. If, in this extreme
case, the mouse were to move 1 mm, no matter what the disturbance
did, the cursor would move 1 mm more in the mouse direction than
would be caused by the disturbance movement, and that extra movement
would not be compensated until hours later. You would see a
correlation between rapid (quicker than hours-long) movements of the
mouse and movements of the cursor, regardless of what the
disturbance does.
Take the other extreme, with a loop transport lag of zero. In this
case, an arbitrary movement of the mouse would, like a change in the
disturbance, be immediately compensated, and in fact you wouldn’t
see it at all, except for the fact that integration takes time (the
output stage is usually taken to be an integrator). In the real case, there is a finite loop transport lag. In many of
Bill’s and my experiments, our results were best fitted by loop
transport lags of around 18 ms. This means that changes in the mouse
are indeed reflected momentarily (and it’s a short moment) in the
movement of the cursor, so that if you differentiate the cursor
trace and the mouse trace, they will correlate more than will the
simple values. Another way of saying this is that the loop gain at high frequencies
is less than at low frequencies, as it must be, because if the loop
gain is greater than unity at a frequency at which the period is
half a loop delay, the loop will go into sustained oscillation. That
means that rapid disturbance changes are less well controlled than
slow ones, and that more of the cursor signal is due to the mouse at
high frequencies than at low. So, in the correlations, you see the
effect of cursor changes more than of cursor values.
So the simple answer to your question is: “loop transport lag”.
Martin

···

[From Rupert Young (2014.05.31 17.00)]

  Hi Rick,



  Looking at the tracking task I notice that although the

correlation C-M is poor it does look if the cursor trace goes up
and down when does the mouse. Has the correlation between the
change in the mouse value and the change in the cursor value
been computed? In other words, is there a good correlation between
the change in output and the change in input? What would be the
significance if this were the case?

[From Rick Marken (2014.05.31.1810)]

···

Martin Taylor (2014.05.31.17.09)–

RY: Has the correlation between the change in the mouse value and the change in the cursor value been computed? In other words, is there a good correlation between the change in output and the change in input? What would be the significance if this were the case?
MT: So the simple answer to your question is: “loop transport lag”.

RM: Which of Rupert’s questions is that an answer to? It doesn’t see to be an answer to any of them.

Best

Rick


Richard S. Marken PhD
www.mindreadings.com
It is difficult to get a man to understand something, when his salary depends upon his not understanding it. – Upton Sinclair

[Martin Taylor 2014.05.31.22.54]

I guess Rupert will have to decide whether I answered his question,

which I understood to be about whether there was a reason he saw a
relationship between changes in mouse and cursor movements when
there is nearly no relationship between their values. Is it
significant? I think I answered that. Is there a good correlation? I
think I answered that. Has it been computed? I don’t know whether it
has, but it has been addressed analytically, which might count as an
answer to that question.
Martin

···

[From Rick Marken (2014.05.31.1810)]

            Martin Taylor

(2014.05.31.17.09)–

              RY: Has the correlation between

the change in the mouse value and the change in
the cursor value been computed? In other words, is
there a good correlation between the change in output
and the change in input? What would be the
significance if this were the case?
MT: So the simple answer to your question is: “loop
transport lag”.

          RM: Which of Rupert's questions is that an answer to?

It doesn’t see to be an answer to any of them.

Well, I am almost with you; I am playing devil’s advocate to get
things clear in my own mind. You say But I am having difficulty
distinguishing Ss and Rs as you use the same colours for each. I
can’t see which black squares are S1 and which red squares are S2.
How do I tell which is which?
Not quite sure what you mean here. Do you mean the correlation is
100%? Wouldn’t noise just make a small difference
Going back to the basic tracking for a moment. What are C and M? C
is the difference between cursor and target, and M is the absolute
position of the mouse? Then in S-R terms then the S is C, but the R
would be how much you move the mouse, I’ll call it dM. So, then
shouldn’t the useful correlation be dC-dM rather than C-M. And if I
look at the graph and estimate those for the first few datapoints
(making up the units) I get dC -
+10,+10,+20,+5,+5,0,+10,+10,+10,+10,+20,-10 and for dM -
+10,+10,+10,+8,+5,0,+8,+8,+8,+8,+10,-10, which looks like good
correlation.
Btw,
in
Cause.html
.

ehccbbid.jpg

bbfdbeca.png

···

[From Rupert Young
(2014.06.01 13.00)]

(Rick Marken (2014.05.31.1315)
… I developed the “repeated
disturbance” test that is now available as another net
demo at:

http://www.mindreadings.com/ControlDemo/Cause.html

            In this task, the same disturbance occurs in the

first and second half of the experiment. Since output
(mouse position) is highly correlated with disturbance
value I knew that repeating the disturbance would lead
to repeated output. I then reasoned that if there was
something about the input that was related to the
output, when the outputs are the same on both trials the
inputs should be the same as well. But the results of
this “repeated disturbance” demo shows that the inputs
are not the same when the outputs on repeated trials
are the same.

"
Note
that the variations in S during the first (black squares) and
second (red squares) periods of the experiment are not identical
for both subject or model."

              RY: What would be

the significance if this were the case?

            RM: It almost _was_ the case. It turns out that the

correlation between inputs in the repeated disturbance
case is nearly 1.0 for a noiseless control system. So if
people were noiseless control systems the results of my
repeated disturbance demo would not have been quite as
dramatic. But fortunately, there is noise in living
control systems and this “drowns out” the very small
variance in the repeated inputs that is the same on the
repeated trials.

            The correlation between outputs when the disturbance

is repeated is always greater then .99 while the
correlation between inputs is rarely greater than .3 and
it’s often close to 0. So there appears to be nothing
about the input that is related to the output that keeps
the input under control.

  Typo

“subjust”
In
BasicTrack.html you say “Press the :Basic Tracking Experiment”
button ", but I see no such
button

Regards,
Rupert

I’m not sure I am understanding you. You’d see the cursor moving,
but moving the mouse would have no effect.
Do you mean there is a lag due to the processing time of the system?
Or due to the non-zero resolution of the visual system? Are you
saying that there is a correlation?
Perhaps there is a simple answer to my question, in that C = f(M) +
f(D), so by definition C does not correlate with f(M)?

···

[From Rupert Young (2014.06.01 17.00)]

([Martin Taylor 2014.05.31.17.09)

  Rupert, imagine an extreme case. Suppose that the loop transport

lag were measured in hours rather than milliseconds. The cursor
position is the sum of the disturbance and the mouse. If, in this
extreme case, the mouse were to move 1 mm, no matter what the
disturbance did, the cursor would move 1 mm more in the mouse
direction than would be caused by the disturbance movement, and
that extra movement would not be compensated until hours later.
You would see a correlation between rapid (quicker than
hours-long) movements of the mouse and movements of the cursor,
regardless of what the disturbance does.

  Take the other extreme, with a loop transport lag of

zero. In this case, an arbitrary movement of the mouse would, like
a change in the disturbance, be immediately compensated, and in
fact you wouldn’t see it at all, except for the fact that
integration takes time (the output stage is usually taken to be an
integrator).

  In the real case, there is a finite loop transport lag. In many of

Bill’s and my experiments, our results were best fitted by loop
transport lags of around 18 ms. This means that changes in the
mouse are indeed reflected momentarily (and it’s a short moment)
in the movement of the cursor, so that if you differentiate the
cursor trace and the mouse trace, they will correlate more than
will the simple values.

Regards,
Rupert

[From Rupert Young (2014.05.31 17.00)]

    Hi Rick,



    Looking at the tracking task I notice that although the

correlation C-M is poor it does look if the cursor trace goes up
and down when does the mouse. Has the correlation between the
change in the mouse value and the change in the cursor value
been computed? In other words, is there a good correlation
between the change in output and the change in input? What would
be the significance if this were the case?

[From Rick Marken (2014.06.06.1010)]

Rupert Young (2014.06.01 17.00)--

RY: Perhaps there is a simple answer to my question, in that C = f(M) + f(D), so by definition C does not correlate with f(M)?

RM: Bingo!!
Best
Rick

···

--
Richard S. Marken PhD
<http://www.mindreadings.com>www.mindreadings.com

It is difficult to get a man to <http://en.wikiquote.org/wiki/Understand&gt;understand something, when his salary depends upon his not understanding it. -- Upton Sinclair

[From Rick Marken (2014.06.01.1500)]

···

Rupert Young (2014.05.31 17.00)–

Looking at the tracking task I notice that although the correlation

C-M is poor it does look if the cursor trace goes up and down when
does the mouse. Has the correlation between the change in the
mouse value and the change in the cursor value been computed? In
other words, is there a good correlation between the change in
output and the change in input? What would be the significance if
this were the case?

RM: Hi Rupert. I’m going back to your original post here because I actually looked at the correlation between change in output (dM/dt) and input (dC/dt) for some of my tracking data and found that the correlation can be quite high, indeed: on the order of .95. I don’t know why I never looked at this relationship before but your informal observation seems to be correct; the derivative of the input (controlled variable; C in this case) in a control task can be highly correlated with the derivative of the output that keeps that variable under control (I say "can be because I’ve just tried it with some other data and the correlation was only .35 but I’m not sure that analysis was correct so, for now, I’ll assume that the correlation can be quite high; I’m sure that it is positive). So your question about the significance of this fact (it can now be considered a fact) is quite relevant.

A smart mathematician (I’m looking at you Richard Kennaway) could probably figure this out analytically but I did some other analyses of the data and my conclusion is that the high negative correlation between dC/dt and dM/dt simply reflects the instantaneous compensating effect of the output on the controlled variable, C. I conclude this for two reasons. One is that I found that the correlation between the change in the disturbance, dD/dt and the change in the input, dC/dt, was close to 0.0. This suggests that the change in output, dM/dt, was completely compensating for the effect that a change in the disturbance would have had on the input variable if the controller had not been doing anything. I also looked at the “lagged” correlations between dC/dt and dM/dt to see if the correlation between these variables reflected a delayed response (dM/dt) to a change in the input (dC/dt). If this were the case then the correlatoin between dC/dt and dM/dt should be higher if we look at the correlation between values of dC/dt that slightly precede the values of dM/dt. In fact, the correlation between dC/dt and dM/dt decreases significantly, from .95 to .31, when dC/dt leads dM/dt by just one sample period. So the correlatoin between dC/dt and dM/dt is not a reflection of a delayed response to a change in input.

So, again, my conclusion about the meaning of the high correlation between dC/dt and dM/dt is that it reflects the instantaneous disturbance-countering effect of the output (M) on the controlled variable (C). This interpretation is also supported by the fact that the correlation between dC/dt and dM/dt is positive and highest when there is no lag between the variables, which suggest that the correlation between dC/dt and dM/dt reflects the instantaneous effect of output (M) on input (C) rather than a compensating effect of output on a disturbance produced change in input. But it would be nice if someone else could replicate these results and/or provide and analytic (mathematical) explanation.

Best

Rick


Richard S. Marken PhD
www.mindreadings.com
It is difficult to get a man to understand something, when his salary depends upon his not understanding it. – Upton Sinclair

[From Rick Marken (2014.06.01.1715)]

···

Rick Marken (2014.06.01.1500)

RM: I just re-did the analysis correctly and the correlation between dC/dt and dM/dt is on the order of .95. So there is definitely a strong positive relationship between the derivatives of input and output in a tracking task. And the fact that it is positive and large suggests that is the effect of the disturbance compensating output on the controlled input; it is the output that is keeping disturbances from having any effect on the controlled input.

Best

Rick


Richard S. Marken PhD
www.mindreadings.com
It is difficult to get a man to understand something, when his salary depends upon his not understanding it. – Upton Sinclair

Rupert Young (2014.05.31 17.00)–

Looking at the tracking task I notice that although the correlation

C-M is poor it does look if the cursor trace goes up and down when
does the mouse. Has the correlation between the change in the
mouse value and the change in the cursor value been computed? In
other words, is there a good correlation between the change in
output and the change in input? What would be the significance if
this were the case?

RM: the derivative of the input (controlled variable; C in this case) in a control task can be highly correlated with the derivative of the output that keeps that variable under control (I say "can be because I’ve just tried it with some other data and the correlation was only .35 but I’m not sure that analysis was correct so, for now, I’ll assume that the correlation can be quite high; I’m sure that it is positive).

[From Rupert Young (2014.05.31 17.00)]

Is it positive correlation; in your previous email you said negative?

My original concern was that a SR person looking at the plots might say that the S is the change in input and the R the change in output, and I wouldn't be able to argue against that. Your findings seem to support that, at first glance. But thinking about the disturbance as a step function, when it first has a sudden effect, of 50 units say, to the left, then the input (the difference between cursor and target; which is also the change in input in this case) is actually equal to the disturbance. Then as mouse movement comes into play the input will reduce over a few seconds and the output will increase (disturbance stays the same), until it is also 50 (to the right). The rates at which the input and output change would be the same, so we would expect (negative?) correlation. They are the same because the it is only the output which has an effect on the input, as the disturbance is constant (50). That is, the change in input is "caused" by the change in output, not the other way around (if you didn't move the mouse the input would stay at 50).

Hmm, not sure where I am going with this. Perhaps the point I am trying to make is that it is not incorrect to say that the output is a function of input, but the SR view neglects that input is also a function of output.
Regards,
Rupert

···

On 02/06/2014 01:15, Richard Marken wrote:

[From Rick Marken (2014.06.01.1715)]

Rick Marken (2014.06.01.1500)

Rupert Young (2014.05.31 17.00)--

Looking at the tracking task I notice that although the correlation C-M is poor it does look if the cursor trace goes up and down when does the mouse. Has the correlation between the _change_ in the mouse value and the _change_ in the cursor value been computed? In other words, is there a good correlation between the change in output and the change in input? What would be the significance if this were the case?

RM: the derivative of the input (controlled variable; C in this case) in a control task can be highly correlated with the derivative of the output that keeps that variable under control (I say "can be because I've just tried it with some other data and the correlation was only .35 but I'm not sure that analysis was correct so, for now, I'll assume that the correlation can be quite high; I'm sure that it is positive).

RM: I just re-did the analysis correctly and the correlation between dC/dt and dM/dt is on the order of .95. So there is definitely a strong positive relationship between the derivatives of input and output in a tracking task. And the fact that it is positive and large suggests that is the effect of the disturbance compensating output on the controlled input; it is the output that is keeping disturbances from having any effect on the controlled input.
Best
Rick

--
Richard S. Marken PhD
<http://www.mindreadings.com>> www.mindreadings.com

It is difficult to get a man to <http://en.wikiquote.org/wiki/Understand&gt;understand something, when his salary depends upon his not understanding it. -- Upton Sinclair

[From Rick Marken (2014.06.02.0825)]

Rupert Young (2014.05.31 17.00)--

RY: Is it positive correlation; in your previous email you said negative?

RM. I'm pretty sure it's positive but if someone else would check it that would be nice. I'll check it myself again this evening if I get a chance. But I did a lot of checking yesterday. I'd be very surprised if the correlation between dC/dt and dM/dt were actually negative.>

RY: My original concern was that a SR person looking at the plots might say that the S is the change in input and the R the change in output, and I wouldn't be able to argue against that.

RM: I agree that if the relationship between dC/dt and dM/dt were negative it would look a lot like SR. Fortunately it's not (or so it seems now).

RY: Your findings seem to support that, at first glance.

RM: If the dC/dt - dM/dt correlation is positive then this rules out the SR interpretation; it would imply a positive feedback situation if the correlation reflected the causal connection from S to R.

RY: But thinking about the disturbance as a step function, when it first has a sudden effect, of 50 units say, to the left, then the input (the difference between cursor and target; which is also the change in input in this case) is actually equal to the disturbance. Then as mouse movement comes into play the input will reduce over a few seconds and the output will increase (disturbance stays the same), until it is also 50 (to the right). The rates at which the input and output change would be the same, so we would expect (negative?) correlation.

RM: No, positive correlation; the output and input are changing in the same direction after the initial disturbance-produced change in input.

RY: They are the same because the it is only the output which has an effect on the input, as the disturbance is constant (50). That is, the change in input is "caused" by the change in output, not the other way around (if you didn't move the mouse the input would stay at 50).

RM: There you go! And this effect of output on input is picked up even while the disturbance is having an effect, as in the tracking task with smoothly varying disturbance.>

RY: Hmm, not sure where I am going with this. Perhaps the point I am trying to make is that it is not incorrect to say that the output is a function of input, but the SR view neglects that input is also a function of output.

RM: Yes, that's the important point. The rest is commentary.
RM: When there is a closed loop negative feedback relationship between system input and output understanding the behavior of the system means determining what aspects of the input are under control (using some version of the test for the controlled variable), not how inputs relate to outputs (by studying input-output relationships).
Best
Rick

···

--
Richard S. Marken PhD
<http://www.mindreadings.com>www.mindreadings.com

It is difficult to get a man to <http://en.wikiquote.org/wiki/Understand&gt;understand something, when his salary depends upon his not understanding it. -- Upton Sinclair

I may be misunderstanding what is negative correlation. I’d thought
it meant that the absolute values are the same, but one set are
negative, i.e. 10,5,10 and -10,-5,-10 have negative correlation. Is
that not right?
Aren’t they changing in opposite directions? Input goes from 50 to
0, and output from 0 to 50.

···

[From Rupert Young (2014.06.02 17.00)]

(Rick Marken (2014.06.02.0825)]

Rupert Young (2014.05.31 17.00)–

              RY: Is it positive correlation; in your previous email

you said negative?

              RY: But thinking about the disturbance as a step

function, when it first has a sudden effect, of 50
units say, to the left, then the input (the difference
between cursor and target; which is also the change in
input in this case) is actually equal to the
disturbance. Then as mouse movement comes into play
the input will reduce over a few seconds and the
output will increase (disturbance stays the same),
until it is also 50 (to the right). The rates at which
the input and output change would be the same, so we
would expect (negative?) correlation.

          RM: No, positive correlation; the output and input are

changing in the same direction after the initial
disturbance-produced change in input.

Regards,
Rupert

[From Rick Marken (2014.06.02.1530)]

···

Rupert Young (2014.06.02 17.00)

RY: I may be misunderstanding what is negative correlation. I'd thought

it meant that the absolute values are the same, but one set are
negative, i.e. 10,5,10 and -10,-5,-10 have negative correlation. Is
that not right?

RM: No. A negative correlation exists when increases in the value of one variable are associated with decreases in the value of the other. Negative numbers have nothing to do with it; either variable can go negative or not. The “negative” in negative correlation refers to the slope of the best fitting straight line that fits the relationship between the two variables. A positive correlation exists when the line slopes up: increases in one variable are associated with increases in the other. This is what we see when we look at the correlation between dC/dt and dM/dt. Increases in the rate of change in the cursor (dC/dt) are associated with increases in the rate of change in the mouse(dM/dt). Since we know from physical considerations that cursor movement doesn’t affect mouse movement we can be pretty sure that the positive correlation between dC/dt and dM/dt results from the effect of mouse movement on the cursor.

RY: Aren't they changing in opposite directions? Input goes from 50 to

0, and output from 0 to 50.

RM: The step disturbance changes the input (cursor); lets say it moves it to the left of the target (reference) point. Now the subject (control system) moves the cursor back to the target. So all the movement of the cursor is caused by movement of the mouse; the rate of cursor movement (dC/dt) at any instant is directly proportional to the rate of mouse movement (dM/dt). So there will be a perfect positive correlation between dC/dt and dM/dt.

RM: We find the same positive correlation between dC/dt and dM/dt in the continuous disturbance case because the observed rate of cursor movement is caused by the mouse movements. These are the mouse movements that are “compensating” for the disturbance so dM/dt should be negatively related to dD/dt and, indeed, I’ve found that it is. But apparently dM/dt is larger than dD/dt so what we are seeing, as dC/dt, is the “excess” of adding dM/dt to dD/dt, which is mainly dM/dt.

RM: Actually, by looking at the derivatives of the variables in a tracking task it is easy to see that control is not a process of “reacting” to disturbance produced changes in input. If this were the case there would, indeed, be a negative correlation between dC/dt and dM/dt, especially if it were computed as a lagged correlation with dM/dt following dC/dt. But we find no such negative correlation between dC/dt and dM/dt, even when we do it as a lagged correlation. What is actually happening when a variable is under control is that the compensating actions of the control system are literally simultaneous with effects of the disturbance. This is why I think it’s really best to think of control actions as protecting a controlled variable from disturbance.

Best

Rick


Richard S. Marken PhD
www.mindreadings.com
It is difficult to get a man to understand something, when his salary depends upon his not understanding it. – Upton Sinclair

          RM: No, positive correlation; the output and input are

changing in the same direction after the initial
disturbance-produced change in input.

[Martin Taylor 2014.06.01.19.18]

I guess you aren't understanding me! I was setting up an extreme

case in which moving the mouse would immediately influence the
cursor, but that the loop transport lag would be very long, so that
the feedback effect would be delayed by hours.
Although in the long term, control would be established, at least
for very slowly moving disturbances, any more rapid mouse changes
would be closely correlated with cursor changes.
Yes. That has been the result whenever the simple “leaky-integral”
control loop has been used to model simple human tracking such as
you asked about. It usually seems to be about 18 msec (at least for
Bill and me). Physically, every connection in the loop MUST involve
some transport lag, but that’s probably not where most of the lag
occurs. It’s probably mostly processing time. The leaky integral
also imposes its own delay, but that’s not included in the transport
lag. The leaky integral just determines how fast the effect of a
change dies away. Transport lag determines how long it is before the
effect of the change begins to dies away.
That’s not right. There’s no such definition, and C must correlate
with both. Here’s a geometrical way of looking at why this is. If you are
visually inclined, as I am, this way of looking at it makes for much
clarity. If you aren’t, I guess it won’t help. But it’s worth a try.
If X = Y+Z, where X, Y, and Z are sequences of samples of
variables, X will be correlated with both Y and Z unless Y = -Z + k
(k is a constant), in which case X = k is constant. The degree of
correlation of X with each of the other two depends on their
relative range of variation and the correlation between them. For
example, if Y and Z are uncorrelated and have equal range of
variation, X is correlated 0.707 = 1/sqrt(2) with each.
-------- more detail ------
More generally, any sequence of N values identifies a point in
N-dimensional space, or a vector from zero to the point. Two such
sequences represent two vectors. Since any three points in space
define a plane, these two vectors lie in a plane. By doing the
algebra, you can prove that the cosine of the angle between them is
the correlation between the two sequences.
The difference between two vectors defines another vector that is
also in the same plane. So, if X = Y-Z, X is a vector that connects
the points defined by Y and Z. The points defined by Y, Z, and zero
are the corners of a triangle, whose sides are the vectors X, Y, and
Z. Hence, if one knows the lengths of Y and Z (square root of the
sum of squares of the sequence values y1, y2 … yN, and similarly
for Z) together with the correlation between them, one can find the
length of X and its correlation with Y and with Z using the formulae
(from Wikipedia "Solution of Triangles):
Wiki quote**
end Wiki quote*
In correlational terms using X, Y, Z as above, and taking “x”, “y”,
and “z” to be their lengths (or total energies if X, Y, and Z
represent signal values):
x = sqrt(y^2 + z^2 - 2yzcorr(y,z))
corr(x,z) = (x^2+z^2-y^2)/2(xz)
corr(x,y) = (x^2+y^2-z^2)/2(x
y)
You can scale the lengths by dividing by the number of samples if
you want to use sequences extended indefinitely over time (using
power instead of total energy).
If Y and Z are uncorrelated and of the same length, the triangle is
a right-isosceles triangle. Scaling the lengths of Y and Z so that y
= z = 1, corr(x,z) = (2+1-1)/2(sqrt(2)) = 1/sqrt(2) = 0.707
-----------end extra detail--------
Now consider the control situation about which you asked. C = M+D corresponds to X = Y+Z above, but for the vectorial analysis
the triangle was described with X = Y-Z, so let’s put V = -D and say
C = M-V.
If control is very good, M and D have almost the same length and are
correlated nearly -1, so M and V are correlated nearly +1. Putting
these into the correlation formula in the “extra detail” section
above, we get
C = sqrt(M^2+V^2 - 2MV(1-eps)) where eps is a positive number much
less than 1.
Since M and V are almost the same length, C ~ Msqrt(2eps), which
is very small compared to M (or D), where ~ means “almost equal to”.
corr(C,M) = (C^2 + M^2 - V^2)/2(CM)
Since M and V are almost the same,
corr(C,M) ~ -corr(C,D) ~ C^2/2(C
M) = C/2M = sqrt(eps/2)
For the trial in the diagram you showed in [From Rupert Young
(2014.05.31 17.00)], eps = 0.03, which gives corr
corr(C,M) = -corr(C,D) ~ 0.12
corr(C,D) is shown as -0.16, which, given the single-digit accuracy
of eps, is quite close.
But corr(C,M) is shown as 0.38, which is too big. Why is this?
The answer is implicit in the actual traces displayed. Sqrt(eps/2)
is a minimum value, which assumes that there is no autonomous
(noise) mouse motion. But in the traces shown, there is a lot of
variation in M that has no counterpart in D, and that variation,
which is reflected in C, is what caused you to ask your question in
the first place. This extra variation is additional to the variation
that opposes the disturbance, which means M^2 > D^2 (or V^2).
Looking back at the correlation formula, corr(C,M) was taken to be
nearly (C^2)/2(C*M), but that was under the assumption that M^2 =
D^2. Since M^2 > V^2, the difference is an added term that
increases corr(C,M).
The added variation in the displayed traces is clearly not only a
consequence of transport lag or of the delay induced by integration.
It’s too irregular for that, but they do have an effect that is
included in the correlation between cursor and mouse. As I said
earlier [Martin Taylor 2014.05.31.17.09]: “…rapid disturbance
changes are less well controlled than slow ones, and … more of the
cursor signal is due to the mouse at high frequencies than at low.
So, in the correlations, you see the effect of cursor changes more
than of cursor values.”
Martin

b345e1dc09f20fdefdea469f09167892.png

334de1ea38b615839e4ee6b65ee1b103.png

ac4f4495bfcd38f5fd3d55aea9c97277.png

b5cee465082129fc4a2cc9b250d70106.png

1ed7216d89d00b3e0966e8af5ec64426.png

···
  I'm not sure I am understanding you. You'd see the cursor moving,

but moving the mouse would have no effect.

    Take the other extreme, with a loop transport lag

of zero. In this case, an arbitrary movement of the mouse would,
like a change in the disturbance, be immediately compensated,
and in fact you wouldn’t see it at all, except for the fact that
integration takes time (the output stage is usually taken to be
an integrator).

    In the real case, there is a finite loop transport lag. In many

of Bill’s and my experiments, our results were best fitted by
loop transport lags of around 18 ms. This means that changes in
the mouse are indeed reflected momentarily (and it’s a short
moment) in the movement of the cursor, so that if you
differentiate the cursor trace and the mouse trace, they will
correlate more than will the simple values.

  Do you mean there is a lag due to the processing time of the

system?

  Or due to the non-zero resolution of the visual

system? Are you saying that there is a correlation?

  Perhaps there is a simple answer to my question, in that C = f(M)
  • f(D), so by definition C does not correlate with f(M)?

Two sides

and the included angle given (SAS)
Here the lengths of sides and the angle between these
sides are known. The third side can be determined from the law of
cosines:

Now we use law of cosines to find the second angle:

Finally,

    [From Rupert Young (2014.06.01

17.00)]

([Martin Taylor 2014.05.31.17.09)

    Rupert, imagine an extreme case. Suppose that the loop transport

lag were measured in hours rather than milliseconds. The cursor
position is the sum of the disturbance and the mouse. If, in
this extreme case, the mouse were to move 1 mm, no matter what
the disturbance did, the cursor would move 1 mm more in the
mouse direction than would be caused by the disturbance
movement, and that extra movement would not be compensated until
hours later. You would see a correlation between rapid (quicker
than hours-long) movements of the mouse and movements of the
cursor, regardless of what the disturbance does.

[From Rupert Young (2014.05.31 17.00)]

      Hi Rick,



      Looking at the tracking task I notice that although the

correlation C-M is poor it does look if the cursor trace goes
up and down when does the mouse. Has the correlation between
the change in the mouse value and the change in the cursor
value been computed? In other words, is there a good
correlation between the change in output and the change in
input? What would be the significance if this were the case?

I believe that is the case with my example numbers. However, what I
probably should have said was that “the change in absolute values
are the same, but one set are negative compared to the other”.
Thanks for the clarification.
It still looks to me as if they are changing in opposite directions.
I must be missing something.
Regards,
Rupert

···

[From Rupert Young (2014.06.09 14.30)]

(Rick Marken (2014.06.02.1530)]

Rupert Young (2014.06.02 17.00)

            RY: I may be misunderstanding what is negative

correlation. I’d thought it meant that the absolute
values are the same, but one set are negative, i.e.
10,5,10 and -10,-5,-10 have negative correlation. Is
that not right?

          RM: No. A negative correlation exists when increases in

the value of one variable are associated with decreases in
the value of the other.

        RM: No, positive correlation; the

output and input are changing in the same direction after
the initial disturbance-produced change in input.

            RY: Aren't they changing in opposite directions? Input

goes from 50 to 0, and output from 0 to 50.

          RM: The step disturbance changes the input (cursor);

lets say it moves it to the left of the target (reference)
point. Now the subject (control system) moves the cursor
back to the target. So all the movement of the cursor is
caused by movement of the mouse; the rate of cursor
movement (dC/dt) at any instant is directly proportional
to the rate of mouse movement (dM/dt). So there will be a
perfect positive correlation between dC/dt and dM/dt.

[From Rick Marken (2014.06.09.0845)]

···

Rupert Young (2014.06.09 14.30)]

RY: It still looks to me as if they are changing in opposite directions.

I must be missing something.

RM: Why not check it out for yourself. It would be good to have confirmation (or rejection) of this. I went to your perceptual robotics site (which looks great and I will point to from my site) and it looks like you have the software capability to produce your own tracking task. Once you’ve got the tracking task set up it should be easy to look at the correlation between the rate of change in output (dM/dt) and controlled variable (dC/dt in a compensatory task; d(T-C)/dt in a pursuit task, where T is target position). But it makes sense to me that the correlation would be positive; after all, the output has a positive effect on the controlled variable.

            RY: Aren't they changing in opposite directions? Input

goes from 50 to 0, and output from 0 to 50.

          RM: The step disturbance changes the input (cursor);

lets say it moves it to the left of the target (reference)
point. Now the subject (control system) moves the cursor
back to the target. So all the movement of the cursor is
caused by movement of the mouse; the rate of cursor
movement (dC/dt) at any instant is directly proportional
to the rate of mouse movement (dM/dt). So there will be a
perfect positive correlation between dC/dt and dM/dt.

Best

Rick


Richard S. Marken PhD
www.mindreadings.com

[Martin Taylor 2014.06.09.13.25]

Or you might use the Flexible Tracker I just posted. There are lots

of things you could check from the spreadsheets it produces.
Martin

···

[From Rick Marken (2014.06.09.0845)]

Rupert Young (2014.06.09 14.30)]

                            RY: Aren't they changing in opposite

directions? Input goes from 50 to 0, and
output from 0 to 50.

                        RM: The step disturbance changes the

input (cursor); lets say it moves it to the
left of the target (reference) point. Now
the subject (control system) moves the
cursor back to the target. So all the
movement of the cursor is caused by movement
of the mouse; the rate of cursor movement
(dC/dt) at any instant is directly
proportional to the rate of mouse movement
(dM/dt). So there will be a perfect positive
correlation between dC/dt and dM/dt.

            RY: It still looks to me as if they are changing in

opposite directions. I must be missing something.

      RM: Why not check it out for yourself. It would be good to

have confirmation (or rejection) of this. I went to your
perceptual robotics site (which looks great and I will point
to from my site) and it looks like you have the software
capability to produce your own tracking task. Once you’ve got
the tracking task set up it should be easy to look at the
correlation between the rate of change in output (dM/dt) and
controlled variable (dC/dt in a compensatory task; d(T-C)/dt
in a pursuit task, where T is target position). But it makes
sense to me that the correlation would be positive; after all,
the output has a positive effect on the controlled variable.

Well, that probably won’t help if I don’t understand what
correlation is. But I think I realise where the (my) confusion is.
And that is that we were talking about different things when we were
talking about direction. I was talking about the direction of the
change in values that I had expressed, that is one set is going down
(50 to 0) and they other going up (0 to 50). Whereas, you were
talking about how they were changing with respect to the plane of
the task, in that both the input and output move to the right, after
the disturbance.
I think my issue is resolved by taking account of the zero axis, as
being at the target point, so the values should have been -50 to 0
and 0 to 50; the change of both is positive.

···

[From Rupert Young (2014.06.09 20.30)]

(Rick Marken (2014.06.09.0845)]

Rupert Young (2014.06.09 14.30)]

                            RY: Aren't they changing in opposite

directions? Input goes from 50 to 0, and
output from 0 to 50.

                        RM: The step disturbance changes the

input (cursor); lets say it moves it to the
left of the target (reference) point. Now
the subject (control system) moves the
cursor back to the target. So all the
movement of the cursor is caused by movement
of the mouse; the rate of cursor movement
(dC/dt) at any instant is directly
proportional to the rate of mouse movement
(dM/dt). So there will be a perfect positive
correlation between dC/dt and dM/dt.

            RY: It still looks to me as if they are changing in

opposite directions. I must be missing something.

      RM: Why not check it out for yourself. It would be good to

have confirmation (or rejection) of this. I went to your
perceptual robotics site (which looks great and I will point
to from my site) and it looks like you have the software
capability to produce your own tracking task. Once you’ve got
the tracking task set up it should be easy to look at the
correlation between the rate of change in output (dM/dt) and
controlled variable (dC/dt in a compensatory task; d(T-C)/dt
in a pursuit task, where T is target position). But it makes
sense to me that the correlation would be positive; after all,
the output has a positive effect on the controlled variable.

Regards,
Rupert