A PCT approach to the "Power Law"

[Martin Taylor 2016.08.07.11.57]

This message is a start on trying to answer Alex's question of July

6, rather than continuing to teach Rick the difference between a
formal parameter and something that can actually be measured. As a
reminder, here is Alex’s original message.

    Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Let's start with whether the 2/3 (or 1/3) power law is a muscular

output or a perceptual input phenomenon. We can tease these apart by
a modification of Powers’s “Circle-Square” demonstration (p145ff and
Demo 9.1 in LCS III). In this demo you move the mouse to make a dot
that draws a white trace on the screen trace a red square presented
as a reference shape on the screen. But the linkage between mouse
and dot positions is such that the mouse actually has to trace a
circle in order to draw the square.

In the Powers demo, the subject moves the mouse very slowly so as to

make the square trace accurate (and so as not to perceive the
circle, at least not consciously). We are interested in curvature,
and the speed with which moving objects like a tracing finger go
around curves of different radius of curvature, so we can’t insist
on that kind of accuracy. We want people just to draw a shape. So
let us have the perceptual radius of curvature of the shape they
draw be different from the radius of curvature that the mouse
follows. Show an ellipse as a reference trajectory on the screen and
scale the x and/or y of the mouse movement to the cursor movement on
screen. For example:

<img alt="" src="cid:part2.6905B1FD.F42340DE@mmtaylor.net" height="151" width="334">

The subjects should be able to do this quite freely and smoothly if

they aren’t told to be as accurate as possible. If the power law has
a perceptual basis, the speed should be slowest around 3 o’clock and
9 o’clock, and fastest at 12 and 6. If it is muscular, the opposite
should happen. Of course, one would want to make sure the ellipses
were oriented differently on different trials, not always at right
angles to each other, and probably not usually with similar ratios
of long to short axes. If a power law holds, does it hold for speed
around the mouse trajectory or for speed around the screen
trajectory?

If this simple experiment shows either that the power law effect is

totally muscular or totally perceptual, further investigations,
possibly even using modifications of this are easy to imagine.

Martin

(Attachment SqareCircleEllipses.jpg is missing)

HI Martin, that is a great test! Does anyone know if it is possible to get the data output from the existing software as a spreadsheet?
Warren

(Attachment SqareCircleEllipses1.jpg is missing)

···

On Tue, Aug 9, 2016 at 6:06 AM, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2016.08.07.11.57]

This message is a start on trying to answer Alex's question of July

6, rather than continuing to teach Rick the difference between a
formal parameter and something that can actually be measured. As a
reminder, here is Alex’s original message.

    Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Let's start with whether the 2/3 (or 1/3) power law is a muscular

output or a perceptual input phenomenon. We can tease these apart by
a modification of Powers’s “Circle-Square” demonstration (p145ff and
Demo 9.1 in LCS III). In this demo you move the mouse to make a dot
that draws a white trace on the screen trace a red square presented
as a reference shape on the screen. But the linkage between mouse
and dot positions is such that the mouse actually has to trace a
circle in order to draw the square.

In the Powers demo, the subject moves the mouse very slowly so as to

make the square trace accurate (and so as not to perceive the
circle, at least not consciously). We are interested in curvature,
and the speed with which moving objects like a tracing finger go
around curves of different radius of curvature, so we can’t insist
on that kind of accuracy. We want people just to draw a shape. So
let us have the perceptual radius of curvature of the shape they
draw be different from the radius of curvature that the mouse
follows. Show an ellipse as a reference trajectory on the screen and
scale the x and/or y of the mouse movement to the cursor movement on
screen. For example:

The subjects should be able to do this quite freely and smoothly if

they aren’t told to be as accurate as possible. If the power law has
a perceptual basis, the speed should be slowest around 3 o’clock and
9 o’clock, and fastest at 12 and 6. If it is muscular, the opposite
should happen. Of course, one would want to make sure the ellipses
were oriented differently on different trials, not always at right
angles to each other, and probably not usually with similar ratios
of long to short axes. If a power law holds, does it hold for speed
around the mouse trajectory or for speed around the screen
trajectory?

If this simple experiment shows either that the power law effect is

totally muscular or totally perceptual, further investigations,
possibly even using modifications of this are easy to imagine.

Martin

Dr Warren Mansell
Reader in Clinical Psychology

School of Health Sciences
2nd Floor Zochonis Building
University of Manchester
Oxford Road
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Website: http://www.psych-sci.manchester.ac.uk/staff/131406

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - Principles-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check www.pctweb.org for further information on Perceptual Control Theory

(Attachment SqareCircleEllipses2.jpg is missing)

···

[From Rick Marken (2016.08.09.0830)]

Martin Taylor (2016.08.07.11.57)–

MT: This message is a start on trying to answer Alex's question of July

6, rather than continuing to teach Rick the difference between a
formal parameter and something that can actually be measured. As a
reminder, here is Alex’s original message.

    Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

MT: Let's start with whether the 2/3 (or 1/3) power law is a muscular

output or a perceptual input phenomenon. We can tease these apart by
a modification of Powers’s “Circle-Square” demonstration (p145ff and
Demo 9.1 in LCS III). In this demo you move the mouse to make a dot
that draws a white trace on the screen trace a red square presented
as a reference shape on the screen. But the linkage between mouse
and dot positions is such that the mouse actually has to trace a
circle in order to draw the square.

MT: In the Powers demo, the subject moves the mouse very slowly so as to

make the square trace accurate (and so as not to perceive the
circle, at least not consciously). We are interested in curvature,
and the speed with which moving objects like a tracing finger go
around curves of different radius of curvature, so we can’t insist
on that kind of accuracy. We want people just to draw a shape. So
let us have the perceptual radius of curvature of the shape they
draw be different from the radius of curvature that the mouse
follows. Show an ellipse as a reference trajectory on the screen and
scale the x and/or y of the mouse movement to the cursor movement on
screen. For example:

MT: The subjects should be able to do this quite freely and smoothly if

they aren’t told to be as accurate as possible. If the power law has
a perceptual basis, the speed should be slowest around 3 o’clock and
9 o’clock, and fastest at 12 and 6. If it is muscular, the opposite
should happen.

RM: The speed of what? Screen movement or mouse movement? And why do you predict this? How is this derived from the power law? And how do you measure speed at a particular location on the curve? It’s great to see you proposing an actual empirical test of the power law. And I would like to carry it out. But I need a little more detail about exactly what you predict, and why (how is this prediction derived from the power law).

Best

Rick

Of course, one would want to make sure the ellipses

were oriented differently on different trials, not always at right
angles to each other

and probably not usually with similar ratios

of long to short axes. If a power law holds, does it hold for speed
around the mouse trajectory or for speed around the screen
trajectory?

If this simple experiment shows either that the power law effect is

totally muscular or totally perceptual, further investigations,
possibly even using modifications of this are easy to imagine.

Martin


Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From Rick Marken (2016.08.09.0850)]

Rick Marken (2016.08.09.0830)]

RM: The speed of what? Screen movement or mouse movement? And why do you predict this? How is this derived from the power law? And how do you measure speed at a particular location on the curve? It's great to see you proposing an actual empirical test of the power law. And I would like to carry it out. But I need a little more detail about exactly what you predict, and why (how is this prediction derived from the power law).

RM: The best way to show what you predict and why you make that prediction is to make a diagram of your model showing how outputs (mouse movements) are related to perceptual inputs (cursor movements) according to the power law. You have verbally described a model that is an alternative to my model inasmuch as you say it predicts a different result than mine. So I would like to know exactly how your model works, how it is different from mine and exactly how your model (and presumably the subjects also) would behave differently than is predicted by my model. In particular, I want to know precisely what we would measure to distinguish the behavior of your model (and, presumably, also that of the subjects) from the behavior of mine (which I believe will be exactly like that of the subjects in your proposed experiment)?
Best
Rick

···

--
Richard S. Marken
"The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves." -- William T. Powers

[Martin Taylor 2016.08.09.12.36]

Yes.

Because if the power law holds for mouse movement, the sharpest

curve is at the top and bottom of the trajectory, but if it holds
for perception of the track, the sharpest curve is at the left and
right ends.

It's not derived from the power law. It asks whether the power law

holds for one or the other extreme possibility, or might perhaps
hold for some function of both (I hope the last is not the case). Or
it might not hold at all because something about the experiment
removes the critical factor on which the power law depends.

SpeedFormula.jpg

That's not what it is. Lots of people have done that. Some find a

1/3 power law, some a 1/4 power law and some find no power law at
all. But I don’t think there are enough studies to allow us to guess
from those difference what perception(s) is/are being controlled.
Are the the important controlled perceptions the same or different
when the results vary?

What this is, is a simple test. If (and only if) the power law in

some form holds either for mouse speed or for track speed, one could
then modify the experiment in various ways, looking for conditions
that lead to the three different possibilities (personally, I expect
that the viscosity effect will prove to be continuous, very high
viscosity leading to a near-zero power, very low viscosity to a
power near 1/3; but as I have said, my guesses are worth what you
pay for them.)

One caveat that carries a hint as to the controlled variable. One of

the power-law studies (I forget which one right at this moment)
showed that there is actually a lag that affects the computation of
the power law, and separately (maybe it’s in a different study)
there’s a difference in how fast the speed changes when going from a
sharp to a wide curve as compared to vice-versa. I expect that lag
to show up here, too. It’s a computational complication, but not one
that should affect the main result if either track perception or
mouse movement perception is the key power-law variable.

Don't forget to vary the absolute and relative orientations of the

ellipses between trials.

<img alt="" src="cid:part4.6D27F098.1427E5AB@mmtaylor.net" height="149" width="334">

And neither actually has to be an ellipse. I just drew ellipses

because it’s easy to draw them.

I don't predict at this point, though I have some intuition about

how the experiment might turn out. I ask Alex’s question, since the
answer has to start with whether we can find out what perception or
perceptions is/are being controlled that lead to the power law. One
possibility is the perception of the track to be followed, another
is some perception relating to the interaction of the muscle
movements with the track surface, which would have to be tested for
in a following experiment, while a third possibility is “neither of
the above” because the power law doesn’t appear for this experiment.
In the last case, we would have to search for something about this
situation that distinguishes it from the conditions in which the
data do show a power law.

When we have some clue as to what perception(s) is/are being

controlled, then and only then should we try to model and simulate
the various studies.

Martin

PS. Just FYI, when Bill programmed the basic software for my 1994

sleep studies, he included a 2D tracking study at Tom Bourbon’s
behest. But he didn’t include a model for that test as he did for
the other five, because he said he was not sure how to do it in a
way that would lead to results he could believe in. I have a feeling
that in addressing Alex’s question properly we might resolve Bill’s
dilemma.

SqareCircleEllipsesMoment.jpg

(Attachment SqareCircleEllipsesSkew.jpg is missing)

···

[From Rick Marken (2016.08.09.0830)]

            Martin Taylor

(2016.08.07.11.57)–

            MT: This message is a start on trying to answer Alex's

question of July 6, rather than continuing to teach Rick
the difference between a formal parameter and something
that can actually be measured. As a reminder, here is
Alex’s original message.

                Any

ideas why or how “the control of perception” may
give rise to this power law constraining geometry
and kinematics in humans, and now in fruit fly
larvae?

http://biorxiv.org/content/early/2016/07/05/062166

            MT: Let's start with whether the 2/3 (or 1/3) power law

is a muscular output or a perceptual input phenomenon.
We can tease these apart by a modification of Powers’s
“Circle-Square” demonstration (p145ff and Demo 9.1 in
LCS III). In this demo you move the mouse to make a dot
that draws a white trace on the screen trace a red
square presented as a reference shape on the screen. But
the linkage between mouse and dot positions is such that
the mouse actually has to trace a circle in order to
draw the square.

            MT: In the Powers demo, the subject moves the mouse very

slowly so as to make the square trace accurate (and so
as not to perceive the circle, at least not
consciously). We are interested in curvature, and the
speed with which moving objects like a tracing finger go
around curves of different radius of curvature, so we
can’t insist on that kind of accuracy. We want people
just to draw a shape. So let us have the perceptual
radius of curvature of the shape they draw be different
from the radius of curvature that the mouse follows.
Show an ellipse as a reference trajectory on the screen
and scale the x and/or y of the mouse movement to the
cursor movement on screen. For example:

            MT: The subjects should be able to do this quite freely

and smoothly if they aren’t told to be as accurate as
possible. If the power law has a perceptual basis, the
speed should be slowest around 3 o’clock and 9 o’clock,
and fastest at 12 and 6. If it is muscular, the opposite
should happen.

          RM: The speed of what? Screen movement or mouse

movement?

And why do you predict this?

How is this derived from the power law?

          And how do you measure speed at a particular location

on the curve?

          It's great to see you proposing an actual empirical

test of the power law.

And I would like to carry it out.

          But I need a little more detail about exactly what you

predict, and why (how is this prediction derived from the
power law).

Best

Rick

            Of course, one would

want to make sure the ellipses were oriented differently
on different trials, not always at right angles to each
other

            and probably not

usually with similar ratios of long to short axes. If a
power law holds, does it hold for speed around the mouse
trajectory or for speed around the screen trajectory?

            If this simple experiment shows either that the power

law effect is totally muscular or totally perceptual,
further investigations, possibly even using
modifications of this are easy to imagine.

                Martin


Richard S. Marken

                                    "The childhood of the human

race is far from over. We
have a long way to go before
most people will understand that
what they do for
others is just as important to
their well-being as what they do
for
themselves." – William T.
Powers

[From Rick Marken (2016.08.09.1255)]

Re A PCT approach to the Powe.jpg

SpeedFormula.jpg

SqareCircleEllipsesSkew.jpg

···

Martin Taylor (2016.08.09.12.36)–

MT: Because if the power law holds for mouse movement, the sharpest

curve is at the top and bottom of the trajectory, but if it holds
for perception of the track, the sharpest curve is at the left and
right ends.

RM: This implies that you predict (for some reason that is not entirely clear) that the power law will hold for either mouse movements or cursor movements but not both. Is this correct?

Best

Rick


Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

            MT: The subjects should be able to do this quite freely

and smoothly if they aren’t told to be as accurate as
possible. If the power law has a perceptual basis, the
speed should be slowest around 3 o’clock and 9 o’clock,
and fastest at 12 and 6. If it is muscular, the opposite
should happen.

RM: And why do you predict this?

[Martin Taylor 2016.08.09.17.02]

[From Rick Marken (2016.08.09.1255)]

No.

I know that logic has no relevance for you, but even you should be

able to figure out that " ‘If X then Y’ and ‘If P then Q’ " does not
imply that one f X and P but not both must be true.

Martin
···

Martin Taylor (2016.08.09.12.36)–

            MT: Because if the power law holds for mouse

movement, the sharpest curve is at the top and bottom of
the trajectory, but if it holds for perception of the
track, the sharpest curve is at the left and right ends.

          RM: This implies that you predict (for some reason that

is not entirely clear) that the power law will hold for
either mouse movements or cursor movements but not both.
Is this correct?

                          MT:

The subjects should be able to do this
quite freely and smoothly if they aren’t
told to be as accurate as possible. If the
power law has a perceptual basis, the
speed should be slowest around 3 o’clock
and 9 o’clock, and fastest at 12 and 6. If
it is muscular, the opposite should
happen.

RM: And why do you predict this?

[From Rick Marken (2016.08.09.1430)]

···

Martin Taylor (2016.08.09.17.02)

MT: No.

MT: I know that logic has no relevance for you

RM: Now, now, Martin. No need to be insulting.

RM: So if the power law holds for mouse movement, you’ll find one result of your experiment; if it holds for perception of the track, you’ll find a different result. So what will you find if the power law holds for both?

Best

Mr. Spock


Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

            MT: Because if the power law holds for mouse

movement, the sharpest curve is at the top and bottom of
the trajectory, but if it holds for perception of the
track, the sharpest curve is at the left and right ends.

          RM: This implies that you predict (for some reason that

is not entirely clear) that the power law will hold for
either mouse movements or cursor movements but not both.
Is this correct?

RM: And why do you predict this?

Martin Taylor 2016.08.09.17.57)

[From Rick Marken (2016.08.09.1430)]

An error in my mathematics. That’s always a distinct possibility.

Martin
···

Martin Taylor (2016.08.09.17.02)

MT: No.

            MT: I know that logic has no relevance for you

RM: Now, now, Martin. No need to be insulting.

          RM: So if the power law holds for mouse movement,

you’ll find one result of your experiment; if it holds
for perception of the track, you’ll find a different
result. So what will you find if the power law holds for
both?

                          MT: Because if the power law holds

for mouse movement, the sharpest curve is
at the top and bottom of the trajectory,
but if it holds for perception of the
track, the sharpest curve is at the left
and right ends.

                        RM: This implies that you predict (for

some reason that is not entirely clear) that
the power law will hold for either mouse
movements or cursor movements but not both.
Is this correct?

                                      RM: And why do you predict

this?

[From Rick Marken (2016.08.09.1550)]

···

Martin Taylor 2016.08.09.17.57)–

MT: An error in my mathematics. That’s always a distinct possibility.

RM: Again, this sounds to me like you are are saying that the power law cannot hold for both the perception of the track and for the mouse movement. If it could – if the power law could hold for both – then there would be a result of your experiment that would show this to be the case; it would not be an error in your mathematics.

          RM: So if the power law holds for mouse movement,

you’ll find one result of your experiment; if it holds
for perception of the track, you’ll find a different
result. So what will you find if the power law holds for
both?

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Martin Taylor 2016.08.11.17.21]

[From Rick Marken (2016.08.10.1900)]

That intuition is easy enough to test. You don't even need real

data. Here is a spreadsheet with an example pair of velocities that
might be obtained in a run of my experiment when the subject’s motor
velocity follows the V=R1/3 power law.

![SqareCircleEllipsesMoment.jpg|334x149](upload://o83E6x8fyzVkDzTszDMkJaodMSr.jpeg)

For this run, the perceptual reference ellipse (the "screen target

trajectory") is x = 1.5cos
θ, y = 0.666…sin
θ. In order to trace this figure exactly, the mouse would have to
follow what I call in the spreadsheet the “motor ellipse” x =
0.75cos
θ, y=1.333…sin
θ. Theta is the rotation angle counterclockwise from due East (the
positive x axis, so the positive y axis is 90 degrees).

On this imaginary trial, the subject's mouse velocity is made to

conform exactly to V = R1/3 . But you could change the
velocity however you want. The ratio of the velocities at any moment
is given by the ratio of the radii for that value of
θ.

![SqCircGraph_1.jpg|319x211](upload://pB3r1PuGJgR0Mli4z2BioT7VLiK.jpeg)

But, as is necessarily the case, the corresponding perceptual

velocity does not.

![SqCircGraph_2.jpg|418x210](upload://u1OKgQZF0N0yNltlNeYVe7mF1ul.jpeg)

If you care to check the spreadsheet, you may find errors. I hope

not. One thing that is not an error is that both
θ and the derived “dt” must be the same for both ellipses, but the
“R” value for any particular
θ is specific to each ellipse. The calculations are made at 1 degree
intervals of
θ.

You can try changing the ellipse eccentricities by altering cells E2

and F2 for the motor ellipse and R2 and S2 for the perceptual
ellipse. The graphs will change accordingly. Just don’t make the
eccentricity zero, because the spreadsheet will blow up, there being
no variation in variables used in divisions.

Martin

SquareCircleEllipsesMotions.xlsx (239 KB)

···

            MT: In my suggested

experiment, to which motion does the “statistical
artifact” apply? Both perceptual and motor ellipses have
a curved movement, but at different phases (the same
kind of variation you used in your bimanual rotation
demo, which I confess to having had in the back of my
mind when I considered the experiment). Do you expect
the same power law relationship to apply to both if it
applies to either? Apparently you do. At least you asked
a question that presupposed that it might [From Rick
Marken (2016.08.09.1430)].

          RM: The statistical artifact will apply to both the

perceptual and motor ellipses. And if they are both
ellipses, as you expert, then the same (or very similar)
power law relationship (in terms of estimates of b) will
be found for both. But I think it should be possible to
develop a demo where the estimates are quite different.

[From Rick Marken (2016.08.11.1830)]

image293.png

image295.png

···

Martin Taylor (2016.08.11.17.21)–

MT: That intuition is easy enough to test. You don't even need real

data. Here is a spreadsheet with an example pair of velocities that
might be obtained in a run of my experiment when the subject’s motor
velocity follows the V=R1/3 power law…

RM: Thanks for doing this Martin.

MT: On this imaginary trial, the subject's mouse velocity is made to

conform exactly to V = R1/3 . But you could change the
velocity however you want. The ratio of the velocities at any moment
is given by the ratio of the radii for that value of
θ.

MT: But, as is necessarily the case, the corresponding perceptual

velocity does not.

MT: If you care to check the spreadsheet, you may find errors.

RM: I put both ellipses into my spreadsheet and did a log-log regression of log® on log (V) for both the perceptual and motor ellipses. In both cases the regression coefficient was .33 and the R^2 was 1.0. This is the result I would expect for this type of elliptical movement. You would have found the coefficients for motor and perceptual movement to be different if you made the perceptual movement pattern an ellipse and the motor movement pattern a random pattern, or vice versa. The movement pattern below has a power coefficient of .29 and an R^2 value of only .52. If you have used that as the motor movement you would have found the power coefficient for the perceptual movement to be quite different from the motor movement.

RM: The title of this thread is “A PCT approach to the “Power Law””. But I haven’t seen your (or Bruce;s or anyone’s) PCT model of the power law; nor have I seen a diagram of how you think the power law fits into PCT. I think I would understand your (and Bruce’s and everyone else’s) position better is you could show me your PCT model of the power law or a diagrammatic description of how you think the power law fits into PCT.

RM: I’ve shown my model of how I think the power law fits into PCT. Apparently you and Bruce (and everyone else) thinks that that model has it all wrong. That’s fine. But I would better understand what is wrong with my model if you would show me the correct model. After all, this is a discussion group about PCT so how about bringing some PCT into the discussion. I think that would help everyone, including those who are neither math not multivariate statistics mavens, understand what the argument is about. Again, here’s my model:

RM: How about re-drawing it correctly.

          RM: The statistical artifact will apply to both the

perceptual and motor ellipses. And if they are both
ellipses, as you expert, then the same (or very similar)
power law relationship (in terms of estimates of b) will
be found for both. But I think it should be possible to
develop a demo where the estimates are quite different.

Best regards

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

Martin Taylor 2016.08.11.22.57

[Martin Taylor 2016.08.11.17.21]

[From Rick Marken (2016.08.10.1900)]

  That intuition is easy enough to test. You don't even need real

data. Here is a spreadsheet with an example pair of velocities
that might be obtained in a run of my experiment when the
subject’s motor velocity follows the V=R1/3 power law.

  ![SqareCircleEllipsesMoment.jpg|334x149](upload://o83E6x8fyzVkDzTszDMkJaodMSr.jpeg)



  For this run, the perceptual reference ellipse (the "screen target

trajectory") is x = 1.5cos
θ, y = 0.666…sin
θ. In order to trace this figure exactly, the mouse would have to
follow what I call in the spreadsheet the “motor ellipse” x =
0.75cos
θ, y=1.333…sin
θ. Theta is the rotation angle counterclockwise from due East (the
positive x axis, so the positive y axis is 90 degrees).

  On this imaginary trial, the subject's mouse velocity is made to

conform exactly to V = R1/3 . But you could change the
velocity however you want. The ratio of the velocities at any
moment is given by the ratio of the radii for that value of
θ.

  ![SqCircGraph_1.jpg|319x211](upload://pB3r1PuGJgR0Mli4z2BioT7VLiK.jpeg)



  But, as is necessarily the case, the corresponding perceptual

velocity does not.

  ![SqCircGraph_2.jpg|418x210](upload://u1OKgQZF0N0yNltlNeYVe7mF1ul.jpeg)



  If you care to check the spreadsheet, you may find errors. I hope

not. One thing that is not an error is that both
θ and the derived “dt” must be the same for both ellipses, but the
“R” value for any particular
θ is specific to each ellipse. The calculations are made at 1
degree intervals of
θ.

  You can try changing the ellipse eccentricities by altering cells

E2 and F2 for the motor ellipse and R2 and S2 for the perceptual
ellipse. The graphs will change accordingly. Just don’t make the
eccentricity zero, because the spreadsheet will blow up, there
being no variation in variables used in divisions.

  Martin
I did find an error, which meant that the graphs referred not to

ellipses, but to some weird kind of “inside-out” ellipses, because I
had curvature where I meant to have radius of curvature. It makes no
difference to the first graph, but the one for the perceptual
velocity becomes:

![SqrCircGraph_2.jpg|425x212](upload://eDuQQSzjKj5fBqlBltmGBDOYzKj.jpeg)

With real data one wouldn't expect such clean curves, but the

synthetic velocity data were chosen to make one of them exactly
proportional to R1/3, and because the two figures (which could have
been any closed curves for which the radius from centre was a
single-valued function of
θ) were orthogonal ellipses, both graphs are straight lines.

[From Rick Marken (2016.08.11.1830)]
RM: I put both ellipses into my spreadsheet
and did a log-log regression of log® on log (V) for both the
perceptual and motor ellipses. In both cases the regression
coefficient was .33 and the R^2 was 1.0

Yes, that's what you will get if you totally ignore the conditions

of the experiment. To help ensure that you did not ignore them, I
reposted the diagram explaining the experimental relation between
the position of the mouse and the on-screen cursor at a given moment
(and do so again here by requoting the original message). But you
ignored it all the same.

It can not have been easy to ignore the experimental conditions,

since I re-emphasized the point in text as well: “* One thing that
is not an error is that both θ and the derived “dt” must be the
same for both ellipses, but the “R” value for any particular θ is
specific to each ellipse* .” In spite of this, and in spite of
the diagram that explains why this is so, you went ahead and used a
different
θ and hence a different dt for the two ellipses, making each
velocity independent of the other.

But I suppose this is totally in line with the care you usually

devote to your readings of my postings. I half-expected you to post
a response along the lines of what you did post, but truly hoped
that reposting the diagram and backing it up in the text would have
made it harder for you to miss the point.

Martin
···

              MT: In my

suggested experiment, to which motion does the
“statistical artifact” apply? Both perceptual and
motor ellipses have a curved movement, but at
different phases (the same kind of variation you used
in your bimanual rotation demo, which I confess to
having had in the back of my mind when I considered
the experiment). Do you expect the same power law
relationship to apply to both if it applies to either?
Apparently you do. At least you asked a question that
presupposed that it might [From Rick Marken
(2016.08.09.1430)].

            RM: The statistical artifact will apply to both the

perceptual and motor ellipses. And if they are both
ellipses, as you expert, then the same (or very similar)
power law relationship (in terms of estimates of b) will
be found for both. But I think it should be possible to
develop a demo where the estimates are quite different.

[From Rick Marken (2016.08.11.2245)]

Martin Taylor 2016.08.11.22.57

  RM: I put both ellipses into my spreadsheet

and did a log-log regression of . log® on log (V) for both the
perceptual and motor ellipses. In both cases the regression
coefficient was .33 and the R^2 was 1.0

MT: Yes, that's what you will get if you totally ignore the conditions

of the experiment. To help ensure that you did not ignore them, I
reposted the diagram explaining the experimental relation between
the position of the mouse and the on-screen cursor at a given moment
(and do so again here by requoting the original message). But you
ignored it all the same.

RM: Could you please post a diagram of your PCT explanation of the power law. Without that I have no idea why your findings from this demo are evidence against my PCT explanation of the power law. Such a diagram would also help me understand what it was about the experimental conditions that I missed and why this was a crucial flaw in my analysis. Indeed I don’t even know why (or even whether) my results caused you to get so upset. I thought they would be what you wanted to see. But I was just guessing because without your model I have no idea what your model predicts or why it predicts . So please post your model that shows how the power law is involved in movement control.

Thanks

Best

Rick

···
It can not have been easy to ignore the experimental conditions,

since I re-emphasized the point in text as well: “* One thing that
is not an error is that both θ and the derived “dt” must be the
same for both ellipses, but the “R” value for any particular θ is
specific to each ellipse* .” In spite of this, and in spite of
the diagram that explains why this is so, you went ahead and used a
different
θ and hence a different dt for the two ellipses, making each
velocity independent of the other.

But I suppose this is totally in line with the care you usually

devote to your readings of my postings. I half-expected you to post
a response along the lines of what you did post, but truly hoped
that reposting the diagram and backing it up in the text would have
made it harder for you to miss the point.

Martin


Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Martin Taylor 2016.08.12.12.41]

[From Rick Marken (2016.08.11.2245)]

Martin Taylor 2016.08.11.22.57

          RM: I put both ellipses into my

spreadsheet and did a log-log regression of . log® on
log (V) for both the perceptual and motor ellipses. In
both cases the regression coefficient was .33 and the R^2
was 1.0

        MT: Yes, that's what you will get if you totally ignore the

conditions of the experiment. To help ensure that you did
not ignore them, I reposted the diagram explaining the
experimental relation between the position of the mouse and
the on-screen cursor at a given moment (and do so again here
by requoting the original message). But you ignored it all
the same.

      RM: Could you please post a diagram of your PCT explanation

of the power law. Without that I have no idea why your
findings from this demo are evidence against my PCT
explanation of the power law.

Why do you think they are presented as evidence against your model?

Your model is totally irrelevant to the question addressed by this
experiment, which is a first step toward finding the controlled
perception(s) that might be responsible for the power law findings
and non-findings.

You yourself posted the evidence that says your model is premature.

You said that you have not looked for the controlled variable or
variables. You have no idea whether x and y perceptions even exist,
let alone are controlled perceptions in fish, fly larvae, etc.etc.
My suggested experiment is simply a first attempt to tease out
whether the observed power law is associated with perception of the
track or with the muscular outputs needed if the perception of being
on track is to be controlled. If that experiment yields a definitive
answer, it would then be a guide when we look further for what
actually is being controlled.

      Such a diagram would also help me understand what it was

about the experimental conditions that I missed and why this
was a crucial flaw in my analysis.

I refer you to the first part of Chapter 9 in LCS III.

Nothing further is required, unless you don't know the principle

behind the Square-Circle demo. I figured that some readers might not
know what was going on there, so I explained it in a diagram that I
have now posted three times. I’ll do it one more time:

![SqareCircleEllipsesMoment.jpg|334x149](upload://o83E6x8fyzVkDzTszDMkJaodMSr.jpeg)

If the mouse is at the point marked by the small circle on the mouse

trajectory plot, the on screen cursor is at the point marked by the
small circle on the screen target plot. No matter where the mouse
point is in the 2-D space, the cursor on screen is at a place mapped
by the software – in this case an angle-dependent ratio of
distances to a central point. There’s no need for either trajectory
to be an ellipse, or anything like an ellipse. All you need is a
continuous mapping of the mouse point to the screen cursor point.

Do you now see what you missed in your analysis by violating the

experimental condition that the cursor position is completely
determined by the mouse position?

Martin

[From Rick Marken (2016.08.12. 1145)]

SqareCircleEllipsesMoment.jpg

···

Martin Taylor (2016.08.12.12.41)–

      RM: Could you please post a diagram of your PCT explanation

of the power law. Without that I have no idea why your
findings from this demo are evidence against my PCT
explanation of the power law.

MT: Why do you think they are presented as evidence against your model?

Your model is totally irrelevant to the question addressed by this
experiment, which is a first step toward finding the controlled
perception(s) that might be responsible for the power law findings
and non-findings.

RM: If, indeed, your experiment is a “first step toward finding the controlled perception(s) that might be responsible for the power law findings and non-findings” then it must be testing a hypothesis about what the controlled variable is. And if you do have a hypothesis about what the controlled variable is, you must have some concept of how this variable is related to the observation of the power law. That “concept” is the model I would like to see.

MT: You yourself posted the evidence that says your model is premature.

RM: All models are “premature” until they are thoroughly tested. A model is a proposed explanation of an observation. That’s what my model is. Now my model has to be tested. And I have been testing it quite a bit and it’s working quite well to explain the data I’m seeing in the literature on the power law.

MT: My suggested experiment is simply a first attempt to tease out

whether the observed power law is associated with perception of the
track or with the muscular outputs needed if the perception of being
on track is to be controlled.

RM: That doesn’t sound much like a test for the controlled variable. What is your hypothesis about the variable being controlled; and how is the power law related to that variable?

MT: If that experiment yields a definitive

answer, it would then be a guide when we look further for what
actually is being controlled.

RM: OK, so the experiment is not a test to determine a controlled variable that might be the basis of the power law. It’s to test to see whether the observed power law is “associated” with perception of the track or with the muscular outputs. So what I need is a diagram showing how you think the power law is associated with perception of the track or the muscular output that produces the track. You must have a model in your head of how this works or you couldn’t know whether or not your experiment tests it.

      RM: Such a diagram would also help me understand what it was

about the experimental conditions that I missed and why this
was a crucial flaw in my analysis.

MT: I refer you to the first part of Chapter 9 in LCS III.

MT: Nothing further is required, unless you don't know the principle

behind the Square-Circle demo.

RM: Actually, I know the principle behind the square-circle demo quite well and I don’t see what it has to do with the power law. It has to do with the fact that the outputs that produce a controlled result can look nothing like the controlled result itself since that result is a simultaneous result of outputs and disturbances. My model of the power law is completely consistent with the results of the square-circle demo because the movements produced by my model are a controlled result of the combined effects of output and disturbance.

RM: So I don’t need to know how the square-circle demo works. I need to know why you think your version of the square-circle experiment has anything to do with showing how the power law fits into movement control. And in order to know this I need a diagram showing how you think the power law is associated with perception of the track or the muscular output that produces the track.

MT: If the mouse is at the point marked by the small circle on the mouse

trajectory plot, the on screen cursor is at the point marked by the
small circle on the screen target plot…

MT: Do you now see what you missed in your analysis by violating the

experimental condition that the cursor position is completely
determined by the mouse position?

RM: No, I’m afraid not. I do not know what this experiment is about because I don’t know what model it is testing. I have to see a diagram showing how you think the power law is associated with perception of the track or the muscular output that produces the track. The square-circle demo very clearly tests a model – PCT – and the PCT model would produce the same results as a person doing the demo. So I know what the circle-square demo tests because I know the model it tests. I don’t know what your version of the square-circle demo tests because I don’t know the model it tests.

RM: So please show me the model. It would make this discussion much more productive.

Best

Rick


Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Martin Taylor 2016.08.12.16.42]

[From Rick Marken (2016.08.12. 1145)]

True. Like all such questions, the first step is a refinement of the

domain. In 20 questions “Is it animal, vegetable or mineral” is
often a first question. My “first question” is “Is the controlled
perception responsible for the power law when it is obeyed some
perception related to the track, or is it some perception related to
the forces used in following the track?”

First thing first. I actually have a raft of possibilities in mind,

for each answer that the experiment might give, but it would be
stupid to wast time on trying to model control systems for them
before we know even which domain to investigate further.

Not at all. A control model is premature if it presupposes

controlling a variable that is not even known to be perceived. In
respect of your model that presupposes that x and y are
independently perceived in all the wide range of cases, I ask again,
have you evidence that a fly larva separately perceives x and y in
the plane in which it is moving? Or even that a person does when
drawing ellipses in air or under water?

By the way, do you now understand why your analysis of the

experiment was wrong? You say you understand how the square-circle
demo works, so I suppose you do, but it would be nice to know for
sure. It would help this discussion be a bit more productive.

Martin
···

Martin Taylor (2016.08.12.12.41)–

                    RM: Could you please post a diagram of your

PCT explanation of the power law. Without that I
have no idea why your findings from this demo
are evidence against my PCT explanation of the
power law.

            MT: Why do you think they are presented as

evidence against your model? Your model is totally
irrelevant to the question addressed by this experiment,
which is a first step toward finding the controlled
perception(s) that might be responsible for the power
law findings and non-findings.

          RM: If, indeed, your experiment is a "first step toward

finding the controlled perception(s) that might be
responsible for the power law findings and non-findings"
then it must be testing a hypothesis about what the
controlled variable is.

          And if you do have a hypothesis about what the

controlled variable is, you must have some concept of how
this variable is related to the observation of the power
law. That “concept” is the model I would like to see.

            MT: You yourself

posted the evidence that says your model is premature.

          RM: All models are "premature" until they are

thoroughly tested.

RM: I’ve shown my model of how I think the power law fits into PCT. Apparently you and Bruce (and everyone else) thinks that that model has it all wrong. That’s fine. But I would better understand what is wrong with my model if you would show me the correct model. After all, this is a discussion group about PCT so how about bringing some PCT into the discussion. I think that would help everyone, including those who are neither math not multivariate statistics mavens, understand what the argument is about. Again, here’s my model:

image00410.png
RM: How about re-drawing it correctly.

HB : You are right. How about re-drawing it correctly ??? It’s total confussion…. As you are…

There’s a lot to be redrawn here including your knowledge and understanding of PCT. You used here spoiiled Bill’s diagram which we saw earlier that you understand under name of RCT (Rick’s Control Theory). So your wrong approach probably caused so many troubles, as usual. It’s useless to implement Power Law to PCT, beacuse you used wrong assumptions about PCT.

As I understand you see some »Movement Control« of Artist and there is some »Controlled variable« in environment of the LCS (generaly speaking) and so on…. Do I understand iit right ? Some kind of RCT ? And there are two different qi (probably input quantities) x and y which should represent what ? The amount of input to left and right eye ? And there are two different perceptual signals carrying »x and y effects« of the »movement control« in environment of artist ? So if I understand right with left eye you are perceiving x and with right eye y dimension of »movement control« (or vica verse). And each eye (input function) has it’s own output function ???

How can you determine on the first level of control which signals will carry information about x and which signals about y ? It’s just »intensity control« on the first level. How can intensity control distinguish between what is drawn into environment of LCS ?

You think that YOUR MODEL shows how physical quantities in outer environment are selectivelly  transformed into perceptual signal and afterwards into other brain signals ? And it seems that your asumption is that every eye works for itself with it’s own output. People are not Chameleons. Your model shows your fantasy Rick and your perfect ignorance about physiology.

If I understand right you are saying that in perceptual signal coming from eyes (input function) we can clearly distinguish between x intensity control and y intensity control ? And mybe later you will introduce new control unit with z-intensity control. Can you show us exactly how this look like ? You can help yourself with B:CP and other books. But please don’t use your imagination because it’s incredible. Why didn’t you follow Asimov and write science fiction ?

What you are demanding from Martin is for me science fiction. He had to do work for you and connect »Power Law« to PCT and prove that »Movement is Control« and that input function (eyes) can distnguish between x and y perceptual signal and probably some z perceptual signal if everything is happening in 3-dimensional space. Is this what you want from him ? Whatever you are doing it doesn’t feet in PCT. It’s your imagination. And I must say a bad one.

You want the right model. O.K. here are the basic premisis for the right model :

Bill P (B:CP) :

  1.   All sensory endings act to convert the magnitude of some physical interaction into the magnitude of a neural current (with or without significant emphasis of rates of change). Coverversely, all sensory information available to more central parts of the brain must first exist in the form of these primary neural currents.
    
  2.   The perceptual signal from a touch receptor does not reflect whether the cause is an electrical current, a touch, or a chemical poisonng, or whether a touch acuurs to the left or right of the exact receptor location.
    
  3.   There is no information in any one first order visual signal to indicate the origin of the light the input function absorbs : the source can be fluorescence inside the eyeball or an exploding star hundred million years removed in space and time, with no change in the character of the perceptual signal.
    
  4.   All information contained in first order perceptual signal is therefore information about what is happening to the associated input functions and about nothing else.
    
  5.   The fact that a muscle acts as a whole on its attachments to the skeleton serves as a natural way to group these control systems. All the comparator neurons that send error signals to the muscle can be grouped into a comparator emitting an error current measured by summing all the neural currents going to the same muscle**. A large set reference signals enters this COMPOSITE SUBTRACTOR, as does large set of perceptual feedback signals.** The characteristics of the **composite control system** should be expressed in terms of average neural currents summed over the parallel pathways of each kind.
    
  6.   All behavioral acts are produced by the actions of first order control systems, these systems, therefore are interposed between more central parts of the nervous system and the environment.
    

And this is model that feets to the »facts« above :

image00318.png

I told you many times Rick that you have to read all Bills books carefully before you start shaming PCT. So go on and read Bill’s work and than come again on CSGnet forum with better explanation or make your own forum for »behavioral or movement control« analysis and perception on first level that distinguish »controlled variables« in environment of LCS with separate control units perceiving each dimension indipendently including »private« output. How this happens probably only you know.Â

Best,

Boris

Best,

Boris

Best regards

Rick

···

Richard S. Marken

“The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves.” – William T. Powers

RM :

But I was just guessing because without your model I have no idea what your model predicts or why it predicts . So please post your model that shows how the power law is involved in movement control.

HB :

There is no PCT model that »Power Law« can be involved in »movement control« because there is no movement or behavior control in PCT. If it is involved in »movement control« than something is wrong.

It’s just your imagination Rick. Your understanding of PCT is wrong and consequently many things that are binding to it. The model which you are using (I assume it’s Bill’s, though you are talking about your model) is misinterpreted. And thus you are misleding and confusing the whole forum again.

HB :

Alex clearly indicated that you are »bullshiting« as you didn’t want to listen to him what’s wrong with your »Power Law« thinking. You have real talent to make experts angry. I see it also in your discussions with Bruce Abbott and Martin…… Why don’t you withdraw in such a moments aand think it for yourself or try reading Bill’s books to be sure that your understanding is more clear.

You are simplifying too much and changing your mind too often Rick. It’s terrible to look at it and I’m sorry for all those people here who had to watch your exhibiton. Specialy I’m sorry for Powers ladies who beleive you 120 %. And you are still manipulating with them, because of long friendship. Science have nothing to do with friendship and thus with PCT. It’s hard to persuade them to check what you are saying with experts. I must congratulate Bruce Abbott for what he done.

I hope Powers Ladies will follow the example of Bruce. But it’s their forum and it’s their choice if they want to watch all your nonsense (bullshit).

HB : The main problem I see Rick is that you don’t understand PCT so you don’t understand anything what is springing form PCT diagram or what is connected to it. I’ve been telling you this for years. And you stil persist at the paradigm that »behavior is control« or that you want a diagram that involves »movement control«. There is no such thing in PCT.

I’m guessing that your misunderstanding of PCT caused your missunderstanding of connection of »Power Law« to PCT. So I think that you have to change you view about PCT and you’ll probably have no problems with understanding everything.Â

Best,

Boris

Thanks

Best

Rick

···

It can not have been easy to ignore the experimental conditions, since I re-emphasized the point in text as well: “One thing that is not an error is that both θ and the derived “dt” must be the same for both ellipses, but the “R” value for any particular θ is specific to each ellipse.” In spite of this, and in spite of the diagram that explains why this is so, you went ahead and used a different θ and hence a different dt for the two ellipses, making each velocity independent of the other.

But I suppose this is totally in line with the care you usually devote to your readings of my postings. I half-expected you to post a response along the lines of what you did post, but truly hoped that reposting the diagram and backing it up in the text would have made it harder for you to miss the point.

Martin

Richard S. Marken

“The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves.” – William T. Powers