Looking at the Power Law through Laputasian Glasses

[MartinT]

[Martin Taylor 2016.08.20.17.33]

[Martin Taylor 2016.08.16.13.13]

I don't know why I forgot to include the punch line. Scroll down to

find it, twice.

[From Rick Marken(2016.08.19.2100)]

  I suppose that if you ignore the relevant mathematics and say it's

irrelevant when it is presented to you in outline and also when it
is given in excruciating detail, I suppose you could say that. You
shouldn’t, but there you go. Nothing anyone but you can do
anything about it.

  For the few who still might be interested, here's both overview

and detail approaches rolled into one.

  Point 1. No description of shape can depend on anything but

measures of space. Time and velocity cannot enter into them.

  ----- detail follows----



  There is a mathematical description of curvature in a Euclidean (x

vs. y) space. Curvature is defined as 1/R, where R is a radius of
curvature (the radius of an “osculating circle”, the circle
equivalent of a tangent straight line).

  In the following formulae, "s" means distance along the curve, and

(x, y) is a location in the space. the differential ds in terms of
x and y is given by ds = (dx² + dy²) ^1/2
from basic Euclidean geometry. One can do as is done in the
Wikipedia article on Curvature, divide through this equation by ds
and define an “intermediate variable” or “formal variable” which I
will call “w”. The definition of “w” is

  ![spaceVelocityFormula.jpg|227x47](upload://6wzRiXObihJUCWY17TCM3z60XNF.jpeg)



  The basic expression for curvature (it's based on computing the

tangent vector acceleration, which we don’t need to go into. If
you re interested, you could look up the actual derivation) is

  ![curvatureFormulaByS.jpg|245x98](upload://4dWi0tOjX84Gux9EM2qYyNbfrXy.jpeg)



  so we can write



   ---- end detail----



  ![curvatureFormulaBySwithW.jpg|274x74](upload://qA0NXjXzDJhHwOMBlBHsI0JfsJM.jpeg)



  If we do as Rick does, call the numerator of this fraction "D" and

transpose, we get

  w<sup>3</sup> = D*R



  or



  w = D<sup>1/3</sup>*R<sup>1/3</sup>

It should be evident that this equation has the form X = X*Y/Y,

because it can be written w = w*R^1/3/R^1/3

  Notice that nothing at all in this derivation has any

relationship to time or velocity. Those should not and do not
enter into any formula that is purely about the shape of a curve.
And yet we arrive at a formula for “w” that Rick calls a velocity.
Not only that, but Rick claims “w” to be the velocity that the
“power law researchers” measure in their experiments.

  Point 2. Right at the start of this curious set of exchanges, we

pointed out to Rick that he had made a very excusable mistake,
figuring that correcting the mistake would end that particular
discussion, but we were sadly mistaken. The reason that the
mistake is easy to make is that the derivatives in the formula
have been represented in the “dotty” Newtonian notation, and are
often taken to be derivatives with respect to time.

  ----detail follows----



  The problem with this is that the formulae do work if you use time

derivatives, but they do so if and only if you prespecify a
velocity profile along the track of the curve. You can set V as a
function of t (time) or of s (distance along the curve), because
they convert into each other, but you have to set it before you do
any other manipulations. It can be anything, provided that it
defines a one-to-one mapping between t and s.

   V(t)<sub>t= t0</sub> = (ds/dt)<sub>t=t0

  </sub>or<sub>

  </sub>V(s)<sub>s=s0</sub> = (ds/dt)<sub>s=s0</sub>



  If you integrate V(t) after time t0 or V(s) after point s0, you

get a mapping of t onto s. For each moment in time there is a
corresponding point along the curve. Those values depends on the
arbitrarily defined V function of t or s, So let’s see (once
again) what we arrive at when we use this arbitrary mapping of t
onto s, starting with the formulae when you take the dot notation
as signifying differentiation with respect to time.

  The first formula, which defined our intervening variable "w" now

is

  ![SpeedFormulaWithV.jpg|234x50](upload://nVxWZ2LlIEzcRXlxGYQCpCcPOSn.jpeg)



  The curvature formula becomes



  ![curvatureFormulaLeibnitz.jpg|245x98](upload://miDgbNUrNeBHkHCd47n5FGE61ev.jpeg)



  It is perhaps worth noticing at this point that it would be

formally correct to multiply top and bottom by (dt)³ to
avoid having time mentioned in a formula for a description of
space, but we won’t do that because we want to see whether the
formula as written is correct on the assumption that we have
specified a velocity function of either time or distance along the
curve. What we will do is note that in calculus generally, dx/dy =
dx/dzdz/dy and d²x/dy² = d²x/dz(dz/dy)² ,
and use these equalities to show that the equation is correct by
deriving it from the second equation above, the basic formula for
curvature, repeated here.

  <img alt="" src="cid:part2.96276A58.A501B022@mmtaylor.net" height="98" width="245">



  Using the equivalences just mentioned, we can write this as

  ![curvatureFormulaBySandT.jpg|391x98](upload://xjyG6ZI8JEiWWRkrmdIZxMdrfod.jpeg)

  And cancelling out the ds's we get the formula above

  <img alt="" src="cid:part5.3DDA0004.2CCE9B17@mmtaylor.net" height="98" width="245">



  Q.E.D.



  Provided we have chosen some arbitrary velocity in advance as a

function of time or space, so that ds/dt has a value, the equation
for curvature with the time derivatives is correct.

  ----end detail---



  It is very easy to ignore the requirement to specify an arbitrary

V(t) or V(s) and think that the two equations, one for the
velocity and one for the curvature, are independent. That’s
exactly what Rick did when he noticed that the denominator of this
formula is actually V³ . He thought that he could then
do as we did above for the formal variable “w”, which has no
relation to velocity. So we follow Rick and do it here, for V, but
now we do it knowing that we pre-set V. Rick calls the numerator
of the fraction “D”, so the equation becomes

  1/R = D/V<sup>3</sup>



  which immediately gives Rick's favourite equation for

determining(!) the already arbitrarily specified V.

  V = D<sup>1/3</sup>*R<sup>1/3</sup>



  Notice here that V is the same as the "w" variable we found

earlier, except that the equations were based on time derivatives
instead of distance derivatives. They are incorporated in the “D”
variable.

As with the "w" variable, it should be evident that this equation

has the form X = XY/Y, since it can be rewritten as V = VR1/3/R^1/3 .
You can’t use the one to find X, and you can’t use the other to find
V.

  The equation is true, as is any tautology, but we

can’t use it to determine V because in order to create the
equation we had to create the arbitrary V function before we
started. The equation tautologically has V on both sides, as the
proof of correctness of the time-based formula demonstrates. Alex
pointed this out as soon as Rick introduced this so-called
equation, this tautology, but to no avail.

  In other words, none of it, since you continue to claim that the

arbitrarily preset variable “V” in the equation above is what
people measure. They don’t. They measure how fast people and other
organisms move along a curving track. Quite a different thing.

          Not in the slightest. It

says that your so-called test is a cheat because it uses
reference trajectories whose speed along the curve is defined by
the power law. When you use real data, as you seem to do below,
you get what Alex and many other have found. And what Alex would
like to have explained.

          No, there's no implied

relationship between the shape and the speed. There may well be
a relationship in practice, as Alex said when he asked the
original question. His question was why this relation is found
in practice when there is no analytically determined reason for
it. One can, in principle, choose to go fast around the sharp
curves and go slow when there is a flat section. Apparently
people don’t usually do that, and I assume that your data
contribute to the mass of evidence that this is so.

          That's good. Without

knowing what you actually calculate, I can’t judge whether these
betas are what is usually computed, but I’ll take your word that
they are. I wonder whether your apparatus had some equivalent of
viscosity?

  The problem is that your sine waves are sine waves with as a

function of time. The squiggle shapes have no connection with
time, so your comment (and your spreadsheet and your analysis)
fails to make the connection. You don’t need different
disturbances, you need different disturbance velocities .
As I have suggested several times, there are several easy ways to
do this without changing the shapes of either the target squiggle
or the disturbances. You shouldn’t need any new instructions how
to do it.

  Well, you are saying the same thing in different words. Yes you

must, and no you don’t, or haven’t yet.

  You have not, so far as I know, ever demonstrated that your model

produces the power law when the along-track velocity of the target
squiggle does not.

  I agree that's what you *should* do. I keep asking you do do

it. I don’t mind if you refuse to do it. I do mind that you keep
claiming that you have done it. It confuses the readership, and
I’d prefer that the CSGnet readership were treated to honest PCT
discussions rather than ones that depend on a mathematical
misunderstanding, however easy that misunderstanding is to make.

  Martin

Martin

···

Martin Taylor 2016.08.19.14.00)–

              MT: But where is the velocity? It's the

along-track velocity of the reference squiggle that is
critical, not its shape.

            RM: You say it's "critical". Yet the model accounts

for the data without it.

            So as the electrified Dylan said to the fan who

yelled “Judas”, “I don’t believe you”.

              MT: But I ask for

the Nth time, where is the along-track speed variation
in the reference trace in your spreadsheet?

RM: Where is it in your model? As I said,
there is no need to put an along-track speed variation
in the reference trace into the PCT model in order to
account for the data (power law). The reference trace is
actually a trajectory – variation over time – not
just a shape, by the way. So a regular old PCT model
with fixed or variable reference accounts for all of the
power law data with which I am familiar.

              MT: In the one you

distributed, you use a reference track that, by its
construction from x and y variations that are composed
of sine-waves over time, imposes a 1/3 power law on
the cursor velocity.

              RM:  This

implies that you accept my explanation of the power
law.

                  RM: Your

statement is also consistent with my explanation
of the power law because it implies that the power
coefficient that is observed in power law studies
depends on the movement pattern (trajectory)
produced.

                  And, indeed,

it does, as can be seen by pressing the “Scribble”
button over and over (which should work on your
Mac; just the “Data collection” macro doesn’t work
on the Mac) and seeing the different estimates of
beta for the different scribbles in the upper
right corner. Just now I got estimates of beta for
the different scribbles that ranged from .25 to
.38. The average for 10 trials was .31.

              MT: Bruce says

that even the disturbances are made the same way, so
that the output of any decent control system must
conform to the same power law. It’s forced by the
construction, and is not a property of the movement
control system.

            RM: It looks to me like you are complaining about the

fact that the model accounts for the data. But maybe
you are saying that somehow the fit to the data is
forced by my choice of disturbances. Perhaps, like
Bruce, you think the use of sine wave disturbances has
something to do with it. So I’d be happy to use
different disturbances; let me know what you would like
me to use. But remember Fourier’s theorem; every
disturbance waveform can be represented as the sum of
sine waves of different phases and frequencies.

              MT: To test the

model you MUST use a reference track in which the
along-track velocity is independent of the local
curvature.

            RM: No, to test the model you must compare the

model’s behavior to the behavior of the systems whose
behavior you are trying to understand.

            If the model doesn't fit the behavior, perhaps it's

because you haven’t done what you say (used a reference
track in which the along-track velocity is independent
of the local curvature). But my model does fit the data
so your emphatic “MUST” is simply not true.

              MT: It's easy to

do. Here’s anothe method to add to the ones suggested
previously. You could have a “squiggle-like” column
listing some arbitrarily varying local velocity in x
and in y, and integrate that to produce the reference
squiggle. If the cursor speed conforms to the 1/3
power law when the reference speed does not, then you
would really have a significant advance in answering
Alex’s question.

            RM: This is not the way I test models. I test models

against actual observations, not against what someone
says should be but hasn’t yet been observed.

[rsmarken]

[From Rick Marken (2016.08.20.1445)]

···

Martin Taylor (2016.08.16.13.13)–

MT: For the few who still might be interested, here's both overview and

detail approaches rolled into one.

RM: There are apparently several people who are interested. I think you could make this a lot clearer to the on-lookers if you would just describe your theory of why the power law is observed. I think it would be easier for people to understand why the PCT explanation of the power law is wrong if you would describe the explanation of the power law that you think is correct.

Best

Rick

Point 1. No description of shape can depend on anything but measures

of space. Time and velocity cannot enter into them.

----- detail follows----



There is a mathematical description of curvature in a Euclidean (x

vs. y) space. Curvature is defined as 1/R, where R is a radius of
curvature (the radius of an “osculating circle”, the circle
equivalent of a tangent straight line).

In the following formulae, "s" means distance along the curve, and

(x, y) is a location in the space. the differential ds in terms of x
and y is given by ds = (dx² + dy²)^1/2 from
basic Euclidean geometry. One can do as is done in the Wikipedia
article on Curvature, divide through this equation by ds and define
an “intermediate variable” or “formal variable” which I will call
“w”. The definition of “w” is

The basic expression for curvature (it's based on computing the

tangent vector acceleration, which we don’t need to go into. If you
re interested, you could look up the actual derivation) is

so we can write



 ---- end detail----







If we do as Rick does, call the numerator of this fraction "D" and

transpose, we get

w<sup>3</sup> = D*R



or



w = D<sup>1/3</sup>*R<sup>1/3</sup>



Notice that nothing at all in this derivation has any relationship

to time or velocity. Those should not and do not enter into any
formula that is purely about the shape of a curve. And yet we arrive
at a formula for “w” that Rick calls a velocity. Not only that, but
Rick claims “w” to be the velocity that the “power law researchers”
measure in their experiments.

Point 2. Right at the start of this curious set of exchanges, we

pointed out to Rick that he had made a very excusable mistake,
figuring that correcting the mistake would end that particular
discussion, but we were sadly mistaken. The reason that the mistake
is easy to make is that the derivatives in the formula have been
represented in the “dotty” Newtonian notation, and are often taken
to be derivatives with respect to time.

----detail follows----



The problem with this is that the formulae do work if you use time

derivatives, but they do so if and only if you prespecify a velocity
profile along the track of the curve. You can set V as a function of
t (time) or of s (distance along the curve), because they convert
into each other, but you have to set it before you do any other
manipulations. It can be anything, provided that it defines a
one-to-one mapping between t and s.

 V(t)<sub>t= t0</sub> = (ds/dt)<sub>t=t0

</sub>or<sub>

</sub>V(s)<sub>s=s0</sub> = (ds/dt)<sub>s=s0</sub>



If you integrate V(t) after time t0 or V(s) after point s0, you get

a mapping of t onto s. For each moment in time there is a
corresponding point along the curve. Those values depends on the
arbitrarily defined V function of t or s, So let’s see (once again)
what we arrive at when we use this arbitrary mapping of t onto s,
starting with the formulae when you take the dot notation as
signifying differentiation with respect to time.

The first formula, which defined our intervening variable "w" now is





The curvature formula becomes







It is perhaps worth noticing at this point that it would be formally

correct to multiply top and bottom by (dt)³ to avoid
having time mentioned in a formula for a description of space, but
we won’t do that because we want to see whether the formula as
written is correct on the assumption that we have specified a
velocity function of either time or distance along the curve. What
we will do is note that in calculus generally, dx/dy = dx/dzdz/dy
and d²x/dy² = d²x/dz(dz/dy)² ,
and use these equalities to show that the equation is correct by
deriving it from the second equation above, the basic formula for
curvature, repeated here.

Using the equivalences just mentioned, we can write this as



And cancelling out the ds's we get the formula above





Q.E.D.



Provided we have chosen some arbitrary velocity in advance as a

function of time or space, so that ds/dt has a value, the equation
for curvature with the time derivatives is correct.

----end detail---



It is very easy to ignore the requirement to specify an arbitrary

V(t) or V(s) and think that the two equations, one for the velocity
and one for the curvature, are independent. That’s exactly what Rick
did when he noticed that the denominator of this formula is actually
V³ . He thought that he could then do as we did above for
the formal variable “w”, which has no relation to velocity. So we
follow Rick and do it here, for V, but now we do it knowing that we
pre-set V. Rick calls the numerator of the fraction “D”, so the
equation becomes

1/R = D/V<sup>3</sup>



which immediately gives Rick's favourite equation for determining(!)

the already arbitrarily specified V.

V = D<sup>1/3</sup>*R<sup>1/3</sup>



Notice here that V is the same as the "w" variable we found earlier,

except that the equations were based on time derivatives instead of
distance derivatives. They are incorporated in the “D” variable.

The equation is true, but we can't use it to determine V because in

order to create the equation we had to create the arbitrary V
function before we started. The equation tautologically has V on
both sides, as the proof of correctness of the time-based formula
demonstrates. Alex pointed this out as soon as Rick introduced this
so-called equation, this tautology, but to no avail.

In other words, none of it, since you continue to claim that the

arbitrarily preset variable “V” in the equation above is what people
measure. They don’t. They measure how fast people and other
organisms move along a curving track. Quite a different thing.

      Not in the slightest. It

says that your so-called test is a cheat because it uses reference
trajectories whose speed along the curve is defined by the power
law. When you use real data, as you seem to do below, you get what
Alex and many other have found. And what Alex would like to have
explained.

      No, there's no implied

relationship between the shape and the speed. There may well be a
relationship in practice, as Alex said when he asked the original
question. His question was why this relation is found in practice
when there is no analytically determined reason for it. One can,
in principle, choose to go fast around the sharp curves and go
slow when there is a flat section. Apparently people don’t usually
do that, and I assume that your data contribute to the mass of
evidence that this is so.

      That's good. Without

knowing what you actually calculate, I can’t judge whether these
betas are what is usually computed, but I’ll take your word that
they are. I wonder whether your apparatus had some equivalent of
viscosity?

The problem is that your sine waves are sine waves with as a

function of time. The squiggle shapes have no connection with time,
so your comment (and your spreadsheet and your analysis) fails to
make the connection. You don’t need different disturbances, you need
different disturbance velocities . As I have suggested
several times, there are several easy ways to do this without
changing the shapes of either the target squiggle or the
disturbances. You shouldn’t need any new instructions how to do it.

Well, you are saying the same thing in different words. Yes you

must, and no you don’t, or haven’t yet.

You have not, so far as I know, ever demonstrated that your model

produces the power law when the along-track velocity of the target
squiggle does not.

I agree that's what you *should* do. I keep asking you do do

it. I don’t mind if you refuse to do it. I do mind that you keep
claiming that you have done it. It confuses the readership, and I’d
prefer that the CSGnet readership were treated to honest PCT
discussions rather than ones that depend on a mathematical
misunderstanding, however easy that misunderstanding is to make.

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

          So as the electrified Dylan said to the fan who yelled

“Judas”, “I don’t believe you”.

            MT: But I ask for

the Nth time, where is the along-track speed variation
in the reference trace in your spreadsheet?

RM: Where is it in your model? As I said, there
is no need to put an along-track speed variation in the
reference trace into the PCT model in order to account for
the data (power law). The reference trace is actually a
trajectory – variation over time – not just a shape, by
the way. So a regular old PCT model with fixed or variable
reference accounts for all of the power law data with
which I am familiar.

            MT: In the one you

distributed, you use a reference track that, by its
construction from x and y variations that are composed
of sine-waves over time, imposes a 1/3 power law on the
cursor velocity.

            RM:  This

implies that you accept my explanation of the power law.

                RM: Your

statement is also consistent with my explanation of
the power law because it implies that the power
coefficient that is observed in power law studies
depends on the movement pattern (trajectory)
produced.

                And, indeed, it

does, as can be seen by pressing the “Scribble”
button over and over (which should work on your Mac;
just the “Data collection” macro doesn’t work on the
Mac) and seeing the different estimates of beta for
the different scribbles in the upper right corner.
Just now I got estimates of beta for the different
scribbles that ranged from .25 to .38. The average
for 10 trials was .31.

            MT: Bruce says that

even the disturbances are made the same way, so that the
output of any decent control system must conform to the
same power law. It’s forced by the construction, and is
not a property of the movement control system.

          RM: It looks to me like you are complaining about the

fact that the model accounts for the data. But maybe you
are saying that somehow the fit to the data is forced by
my choice of disturbances. Perhaps, like Bruce, you think
the use of sine wave disturbances has something to do with
it. So I’d be happy to use different disturbances; let me
know what you would like me to use. But remember Fourier’s
theorem; every disturbance waveform can be represented as
the sum of sine waves of different phases and frequencies.

            MT: To test the

model you MUST use a reference track in which the
along-track velocity is independent of the local
curvature.

          RM: No, to test the model you must compare the model's

behavior to the behavior of the systems whose behavior you
are trying to understand.

          If the model doesn't fit the behavior, perhaps it's

because you haven’t done what you say (used a reference
track in which the along-track velocity is independent of
the local curvature). But my model does fit the data so
your emphatic “MUST” is simply not true.

            MT: It's easy to do.

Here’s anothe method to add to the ones suggested
previously. You could have a “squiggle-like” column
listing some arbitrarily varying local velocity in x and
in y, and integrate that to produce the reference
squiggle. If the cursor speed conforms to the 1/3 power
law when the reference speed does not, then you would
really have a significant advance in answering Alex’s
question.

          RM: This is not the way I test models. I test models

against actual observations, not against what someone says
should be but hasn’t yet been observed.

[MartinT]

[Martin Taylor 2016.08.21.00.04]

I really wonder why it is so difficult to get across the idea that I

think the first step in finding a PCT explanation of the power law
is to find the controlled variable (or variables). Well, maybe I
don’t wonder very much, because I know from bitter experience that
you simply don’t read most of what I write when it questions
something you have decided is part of your religion. You usually are the first to say that the search for the controlled
variable is the primary, if not the only, proper focus of PCT
research. Why is it that when I say (as I always have) that for a
specific question about what is the best PCT explanation of an
observed effect, one should look for the controlled variable before
trying to build a model. So I have no model and will not have one
before there is a reasonable proposal for what variable is being
controlled.
I proposed an experiment to start the search, which Alex told me
privately he intends to do when he returns from a leave of absence.
Your response to that proposal for a start on a search for the
controlled variable is to call me and Bruce “opponents of PCT”. Sad.
Really sad.
When you have a “model” without having any idea as to the controlled
variable, that’s not good PCT. And when that model is shown by
various methods not to demonstrate what it is claimed to
demonstrate, it’s not good science to continue the same claims
without showing why the criticisms are wrong. Not once have you done
that. You haven’t even said anything to indicate that you understand
what the criticisms are!
All you have done in support of your model is to show time and time
again that the formula for curvature is self-consistent, and to
falsely accuse Bruce and me of being S-R theorists, that we believe
in control of output, that we are opponents of PCT. No. We are
opponents of the deluded belief that because one had an idea, that
idea must be right, no matter what evidence is brought to bear
against it.
Even if those calumnies were true, it would still not be good
science to ignore the evidence that your model is irrelevant to the
problem at issue. Even enemies (which we are not) can have accurate
ideas about your weak points. You refuse even do the trivially
simple tests of your model for which you already have the test
machinery. Instead of doing the experiment, you say that the result
would be irrelevant and cast slurs on those who present the
evidence. Not good science. Nor, I think good pubic relations.
Martin

···

On 2016/08/20 5:44 PM, Richard Marken
wrote:

[From Rick Marken (2016.08.20.1445)]

Martin Taylor (2016.08.16.13.13)–

            MT: For the few who still might be interested, here's

both overview and detail approaches rolled into one.

          RM: There are apparently several people who are

interested. I think you could make this a lot clearer to
the on-lookers if you would just describe your theory of
why the power law is observed. I think it would be easier
for people to understand why the PCT explanation of the
power law is wrong if you would describe the explanation
of the power law that you think is correct.

[From Rick Marken (2016.08.20.1540)]
RM: It’s more than 20 years since Mary wrote that and the
feedforward, information theory, dynamic systems theory and
whatever types (many of them the same people now as then) are
still at it. If Mary was grumpy back then, after only 3 years
of this stuff, imagine how I felt when I wrote my withdrawal
from LCS IV.

      RM: Thanks Dag and Barb. But this is just the way it is for

PCT. There’s no taming its opponents. Those in the
establishment do not suffer revolutions gladly.

Best

Rick

            Point 1. No description of shape can depend on anything

but measures of space. Time and velocity cannot enter
into them.

            ----- detail follows----



            There is a mathematical description of curvature in a

Euclidean (x vs. y) space. Curvature is defined as 1/R,
where R is a radius of curvature (the radius of an
“osculating circle”, the circle equivalent of a tangent
straight line).

            In the following formulae, "s" means distance along the

curve, and (x, y) is a location in the space. the
differential ds in terms of x and y is given by ds = (dx²
+ dy²)^1/2 from basic Euclidean
geometry. One can do as is done in the Wikipedia article
on Curvature, divide through this equation by ds and
define an “intermediate variable” or “formal variable”
which I will call “w”. The definition of “w” is

            The basic expression for curvature (it's based on

computing the tangent vector acceleration, which we
don’t need to go into. If you re interested, you could
look up the actual derivation) is

            so we can write



             ---- end detail----







            If we do as Rick does, call the numerator of this

fraction “D” and transpose, we get

            w<sup>3</sup> = D*R



            or



            w = D<sup>1/3</sup>*R<sup>1/3</sup>



            Notice that nothing at all in this derivation has any

relationship to time or velocity. Those should not and
do not enter into any formula that is purely about the
shape of a curve. And yet we arrive at a formula for “w”
that Rick calls a velocity. Not only that, but Rick
claims “w” to be the velocity that the “power law
researchers” measure in their experiments.

            Point 2. Right at the start of this curious set of

exchanges, we pointed out to Rick that he had made a
very excusable mistake, figuring that correcting the
mistake would end that particular discussion, but we
were sadly mistaken. The reason that the mistake is easy
to make is that the derivatives in the formula have been
represented in the “dotty” Newtonian notation, and are
often taken to be derivatives with respect to time.

            ----detail follows----



            The problem with this is that the formulae do work if

you use time derivatives, but they do so if and only if
you prespecify a velocity profile along the track of the
curve. You can set V as a function of t (time) or of s
(distance along the curve), because they convert into
each other, but you have to set it before you do any
other manipulations. It can be anything, provided that
it defines a one-to-one mapping between t and s.

             V(t)<sub>t= t0</sub> = (ds/dt)<sub>t=t0

            </sub>or<sub>

            </sub>V(s)<sub>s=s0</sub> = (ds/dt)<sub>s=s0</sub>



            If you integrate V(t) after time t0 or V(s) after point

s0, you get a mapping of t onto s. For each moment in
time there is a corresponding point along the curve.
Those values depends on the arbitrarily defined V
function of t or s, So let’s see (once again) what we
arrive at when we use this arbitrary mapping of t onto
s, starting with the formulae when you take the dot
notation as signifying differentiation with respect to
time.

            The first formula, which defined our intervening

variable “w” now is

            The curvature formula becomes







            It is perhaps worth noticing at this point that it would

be formally correct to multiply top and bottom by (dt)³
to avoid having time mentioned in a formula for a
description of space, but we won’t do that because we
want to see whether the formula as written is correct on
the assumption that we have specified a velocity
function of either time or distance along the curve.
What we will do is note that in calculus generally,
dx/dy = dx/dzdz/dy and d²x/dy² =
d²x/dz(dz/dy)² , and use these
equalities to show that the equation is correct by
deriving it from the second equation above, the basic
formula for curvature, repeated here.

            Using the equivalences just mentioned, we can write this

as

            And cancelling out the ds's we get the formula above





            Q.E.D.



            Provided we have chosen some arbitrary velocity in

advance as a function of time or space, so that ds/dt
has a value, the equation for curvature with the time
derivatives is correct.

            ----end detail---



            It is very easy to ignore the requirement to specify an

arbitrary V(t) or V(s) and think that the two equations,
one for the velocity and one for the curvature, are
independent. That’s exactly what Rick did when he
noticed that the denominator of this formula is actually
V³ . He thought that he could then do as we
did above for the formal variable “w”, which has no
relation to velocity. So we follow Rick and do it here,
for V, but now we do it knowing that we pre-set V. Rick
calls the numerator of the fraction “D”, so the equation
becomes

            1/R = D/V<sup>3</sup>



            which immediately gives Rick's favourite equation for

determining(!) the already arbitrarily specified V.

            V = D<sup>1/3</sup>*R<sup>1/3</sup>



            Notice here that V is the same as the "w" variable we

found earlier, except that the equations were based on
time derivatives instead of distance derivatives. They
are incorporated in the “D” variable.

            The equation is true, but we can't use it to determine V

because in order to create the equation we had to create
the arbitrary V function before we started. The equation
tautologically has V on both sides, as the proof of
correctness of the time-based formula demonstrates. Alex
pointed this out as soon as Rick introduced this
so-called equation, this tautology, but to no avail.

                        So as the electrified Dylan said to the

fan who yelled “Judas”, “I don’t believe
you”.

                          MT:

But I ask for the Nth time, where is the
along-track speed variation in the
reference trace in your spreadsheet?

RM: Where is it in your model? As
I said, there is no need to put an
along-track speed variation in the reference
trace into the PCT model in order to account
for the data (power law). The reference
trace is actually a trajectory – variation
over time – not just a shape, by the way.
So a regular old PCT model with fixed or
variable reference accounts for all of the
power law data with which I am familiar.

             In other words, none of it, since you continue

to claim that the arbitrarily preset variable “V” in the
equation above is what people measure. They don’t. They
measure how fast people and other organisms move along a
curving track. Quite a different thing.

                          MT: In

the one you distributed, you use a
reference track that, by its construction
from x and y variations that are composed
of sine-waves over time, imposes a 1/3
power law on the cursor velocity.

                          RM:

This implies that you accept my
explanation of the power law.

                              Not in

the slightest. It says that your so-called test is a
cheat because it uses reference trajectories whose
speed along the curve is defined by the power law.
When you use real data, as you seem to do below, you
get what Alex and many other have found. And what Alex
would like to have explained.

                              RM:

Your statement is also consistent with
my explanation of the power law
because it implies that the power
coefficient that is observed in power
law studies depends on the movement
pattern (trajectory) produced.

                              No,

there’s no implied relationship between the shape and
the speed. There may well be a relationship in
practice, as Alex said when he asked the original
question. His question was why this relation is found
in practice when there is no analytically determined
reason for it. One can, in principle, choose to go
fast around the sharp curves and go slow when there is
a flat section. Apparently people don’t usually do
that, and I assume that your data contribute to the
mass of evidence that this is so.

                              And,

indeed, it does, as can be seen by
pressing the “Scribble” button over
and over (which should work on your
Mac; just the “Data collection” macro
doesn’t work on the Mac) and seeing
the different estimates of beta for
the different scribbles in the upper
right corner. Just now I got estimates
of beta for the different scribbles
that ranged from .25 to .38. The
average for 10 trials was .31.

                              That's

good. Without knowing what you actually calculate, I
can’t judge whether these betas are what is usually
computed, but I’ll take your word that they are. I
wonder whether your apparatus had some equivalent of
viscosity?

                          MT:

Bruce says that even the disturbances are
made the same way, so that the output of
any decent control system must conform to
the same power law. It’s forced by the
construction, and is not a property of the
movement control system.

                        RM: It looks to me like you are

complaining about the fact that the model
accounts for the data. But maybe you are
saying that somehow the fit to the data is
forced by my choice of disturbances.
Perhaps, like Bruce, you think the use of
sine wave disturbances has something to do
with it. So I’d be happy to use different
disturbances; let me know what you would
like me to use. But remember Fourier’s
theorem; every disturbance waveform can be
represented as the sum of sine waves of
different phases and frequencies.

             The problem is that your sine waves are sine

waves with as a function of time. The squiggle shapes
have no connection with time, so your comment (and your
spreadsheet and your analysis) fails to make the
connection. You don’t need different disturbances, you
need different disturbance velocities . As I have
suggested several times, there are several easy ways to
do this without changing the shapes of either the target
squiggle or the disturbances. You shouldn’t need any new
instructions how to do it.

                          MT: To

test the model you MUST use a reference
track in which the along-track velocity is
independent of the local curvature.

                        RM: No, to test the model you must

compare the model’s behavior to the behavior
of the systems whose behavior you are trying
to understand.

             Well, you are saying the same thing in different

words. Yes you must, and no you don’t, or haven’t yet.

                        If the model doesn't fit the behavior,

perhaps it’s because you haven’t done what
you say (used a reference track in which the
along-track velocity is independent of the
local curvature). But my model does fit the
data so your emphatic “MUST” is simply not
true.

             You have not, so far as I know, ever

demonstrated that your model produces the power law when
the along-track velocity of the target squiggle does
not.

                          MT:

It’s easy to do. Here’s anothe method to
add to the ones suggested previously. You
could have a “squiggle-like” column
listing some arbitrarily varying local
velocity in x and in y, and integrate that
to produce the reference squiggle. If the
cursor speed conforms to the 1/3 power law
when the reference speed does not, then
you would really have a significant
advance in answering Alex’s question.

                        RM: This is not the way I test models. I

test models against actual observations, not
against what someone says should be but
hasn’t yet been observed.

             I agree that's what you *should* do. I

keep asking you do do it. I don’t mind if you refuse to
do it. I do mind that you keep claiming that you have
done it. It confuses the readership, and I’d prefer that
the CSGnet readership were treated to honest PCT
discussions rather than ones that depend on a
mathematical misunderstanding, however easy that
misunderstanding is to make.

                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

[rsmarken]

[From Rick Marken (2016.08.20.2320)]

···

Martin Taylor (2016.08.21.00.04)

MT: I really wonder why it is so difficult to get across the idea that I

think the first step in finding a PCT explanation of the power law
is to find the controlled variable (or variables).

RM: Because I don’t know how this search for the controlled variable relates to the phenomenon we are trying to explain: the power law. The first step in a PCT explanation of behavior is to identify a behavior that seems to involve control. And this usually involves noticing that a variable is being kept in a constant or variable state, protected from disturbances that would ordinarily move it from that state. For example, control seems to be involved when a person consistently catches a ball thrown to different locations or remains upright while walking on uneven terrain. So the first step in finding a PCT explanation of any phenomenon is to have a reason to believe that the phenomenon involves control. The next step is to guess what variable(s) is (are) under control. And the next step after that is to test this guess using some version of the TCV.

RM: In the case of the power law, the phenomenon that is observed does not obviously involve control. All we see is a quantitative relationship between measures of the velocity and curvature of a movement trajectory. There is no evidence that either velocity or curvature are being kept in a constant of variable state, protected from disturbances. So I don’t know what finding the controlled variable(s) has to do with explaining the power law. If your model is a PCT model then there must be some controlling that it is explaining. What is the phenomenon that your model explains and how does the power law fit into that model? You should be able to diagram this model without identifying the controlled variable(s); just put q.i’s in the part of the diagram where the to-be-identified controlled variable(s) go.

MT: You usually are the first to say that the search for the controlled

variable is the primary, if not the only, proper focus of PCT
research.

RM: That’s true. But only when you have reason to believe that the phenomenon under study involves control and you have at least one hypothesis about the variable(s) under control. In the baseball catching research we did the TCV because we had reason to believe that catching baseballs was a control process and that the variables controlled were aspects of the optical trajectory of the ball. We actually came in with three different hypotheses about the controlled variable and used the TCV to get it down to one.

MT: So I have no model and will not have one

before there is a reasonable proposal for what variable is being
controlled.

RM: But you do have a model; it is a control model because you assume there is a controlled variable involved. So you should be able to show me a diagram of the model (with the exact controlled variable unspecified) and show me how the power law fits into that model.

MT: I proposed an experiment to start the search, which Alex told me

privately he intends to do when he returns from a leave of absence.

RM: I gave my evaluation of that experiment in another post. I have no idea what you expect to find or how those findings relate to an explanation of the power law.

MT: Your response to that proposal for a start on a search for the

controlled variable is to call me and Bruce “opponents of PCT”. Sad.
Really sad.

RM: No, my response to that proposal was that the goals of the experiment were vague, there was no clear description of the predicted results, why those predictions were made or how those predictions related to explaining the power law.

MT: When you have a "model" without having any idea as to the controlled

variable, that’s not good PCT.

RM: I agree. That’s why my model is good PCT and yours is not. My model has a very good idea as to the controlled variable – it’s the perceived distance from cursor to target in my spreadsheet demo. Your model doesn’t have any idea as to what the controlled variable is; indeed, your model doesn’t even seem to exist.

MT: And when that model is shown by

various methods not to demonstrate what it is claimed to
demonstrate, it’s not good science to continue the same claims
without showing why the criticisms are wrong.

RM:I agree with that too. Fortunately, there is yet to be a demonstration that my model doesn’t demonstrate what it claims to demonstrate: that the power law is a function of the movement trajectory that is produced and has nothing to do with how it is produced.

MT: Not once have you done

that. You haven’t even said anything to indicate that you understand
what the criticisms are!

RM: Actually, I see it as you not understanding that your criticisms are irrelevant to what is demonstrated by the behavior of the model.

MT: All you have done in support of your model is to show time and time

again that the formula for curvature is self-consistent,

RM: No, I have shown that the 1/3 (and 2/3) power law is found for movement trajectories that are produced by output trajectories that are quite different than the resulting movement trajectories. I’ve also shown that the power law that is observed depends on the movement trajectory produced, not on the outputs used to produce it.

MT: and to

falsely accuse Bruce and me of being S-R theorists, that we believe
in control of output, that we are opponents of PCT.

RM: Since I haven’t seen your control model that explains the power law and since everything you’ve said is aimed at showing that measures of velocity and curvature are independent of each other and since you have dismissed my PCT model for no good reason and provided no alternative, it looks to me like you are looking at the power law as an S-R phenomenon and that you are pretty unhappy with a PCT explanation of the phenomenon. So, yes, I believe that you are fundamentally S-R theorists and opponents of PCT. You could change my mind about that if you acted more like a friend of PCT. I think a friend of PCT, one who thought my model of the power law was wrong, would suggest the right way to model it. Neither of you have come close to doing that.

MT: No. We are

opponents of the deluded belief that because one had an idea, that
idea must be right, no matter what evidence is brought to bear
against it.

RM: Great. But rather than just say that you are not an opponent of PCT I would rather you demonstrate it by suggesting the correct way to model the power law using PCT.

Best regards

Rick

[From Rick Marken (2016.08.20.1540)]
RM: It’s more than 20 years since Mary wrote that and the
feedforward, information theory, dynamic systems theory and
whatever types (many of them the same people now as then) are
still at it. If Mary was grumpy back then, after only 3 years
of this stuff, imagine how I felt when I wrote my withdrawal
from LCS IV.

      RM: Thanks Dag and Barb. But this is just the way it is for

PCT. There’s no taming its opponents. Those in the
establishment do not suffer revolutions gladly.

Even if those calumnies were true, it would still not be good

science to ignore the evidence that your model is irrelevant to the
problem at issue. Even enemies (which we are not) can have accurate
ideas about your weak points. You refuse even do the trivially
simple tests of your model for which you already have the test
machinery. Instead of doing the experiment, you say that the result
would be irrelevant and cast slurs on those who present the
evidence. Not good science. Nor, I think good pubic relations.

Martin

–

          RM: There are apparently several people who are

interested. I think you could make this a lot clearer to
the on-lookers if you would just describe your theory of
why the power law is observed. I think it would be easier
for people to understand why the PCT explanation of the
power law is wrong if you would describe the explanation
of the power law that you think is correct.

Best

Rick

            Point 1. No description of shape can depend on anything

but measures of space. Time and velocity cannot enter
into them.

            ----- detail follows----



            There is a mathematical description of curvature in a

Euclidean (x vs. y) space. Curvature is defined as 1/R,
where R is a radius of curvature (the radius of an
“osculating circle”, the circle equivalent of a tangent
straight line).

            In the following formulae, "s" means distance along the

curve, and (x, y) is a location in the space. the
differential ds in terms of x and y is given by ds = (dx²
+ dy²)^1/2 from basic Euclidean
geometry. One can do as is done in the Wikipedia article
on Curvature, divide through this equation by ds and
define an “intermediate variable” or “formal variable”
which I will call “w”. The definition of “w” is

            The basic expression for curvature (it's based on

computing the tangent vector acceleration, which we
don’t need to go into. If you re interested, you could
look up the actual derivation) is

            so we can write



             ---- end detail----







            If we do as Rick does, call the numerator of this

fraction “D” and transpose, we get

            w<sup>3</sup> = D*R



            or



            w = D<sup>1/3</sup>*R<sup>1/3</sup>



            Notice that nothing at all in this derivation has any

relationship to time or velocity. Those should not and
do not enter into any formula that is purely about the
shape of a curve. And yet we arrive at a formula for “w”
that Rick calls a velocity. Not only that, but Rick
claims “w” to be the velocity that the “power law
researchers” measure in their experiments.

            Point 2. Right at the start of this curious set of

exchanges, we pointed out to Rick that he had made a
very excusable mistake, figuring that correcting the
mistake would end that particular discussion, but we
were sadly mistaken. The reason that the mistake is easy
to make is that the derivatives in the formula have been
represented in the “dotty” Newtonian notation, and are
often taken to be derivatives with respect to time.

            ----detail follows----



            The problem with this is that the formulae do work if

you use time derivatives, but they do so if and only if
you prespecify a velocity profile along the track of the
curve. You can set V as a function of t (time) or of s
(distance along the curve), because they convert into
each other, but you have to set it before you do any
other manipulations. It can be anything, provided that
it defines a one-to-one mapping between t and s.

             V(t)<sub>t= t0</sub> = (ds/dt)<sub>t=t0

            </sub>or<sub>

            </sub>V(s)<sub>s=s0</sub> = (ds/dt)<sub>s=s0</sub>



            If you integrate V(t) after time t0 or V(s) after point

s0, you get a mapping of t onto s. For each moment in
time there is a corresponding point along the curve.
Those values depends on the arbitrarily defined V
function of t or s, So let’s see (once again) what we
arrive at when we use this arbitrary mapping of t onto
s, starting with the formulae when you take the dot
notation as signifying differentiation with respect to
time.

            The first formula, which defined our intervening

variable “w” now is

            The curvature formula becomes







            It is perhaps worth noticing at this point that it would

be formally correct to multiply top and bottom by (dt)³
to avoid having time mentioned in a formula for a
description of space, but we won’t do that because we
want to see whether the formula as written is correct on
the assumption that we have specified a velocity
function of either time or distance along the curve.
What we will do is note that in calculus generally,
dx/dy = dx/dzdz/dy and d²x/dy² =
d²x/dz(dz/dy)² , and use these
equalities to show that the equation is correct by
deriving it from the second equation above, the basic
formula for curvature, repeated here.

            Using the equivalences just mentioned, we can write this

as

            And cancelling out the ds's we get the formula above





            Q.E.D.



            Provided we have chosen some arbitrary velocity in

advance as a function of time or space, so that ds/dt
has a value, the equation for curvature with the time
derivatives is correct.

            ----end detail---



            It is very easy to ignore the requirement to specify an

arbitrary V(t) or V(s) and think that the two equations,
one for the velocity and one for the curvature, are
independent. That’s exactly what Rick did when he
noticed that the denominator of this formula is actually
V³ . He thought that he could then do as we
did above for the formal variable “w”, which has no
relation to velocity. So we follow Rick and do it here,
for V, but now we do it knowing that we pre-set V. Rick
calls the numerator of the fraction “D”, so the equation
becomes

            1/R = D/V<sup>3</sup>



            which immediately gives Rick's favourite equation for

determining(!) the already arbitrarily specified V.

            V = D<sup>1/3</sup>*R<sup>1/3</sup>



            Notice here that V is the same as the "w" variable we

found earlier, except that the equations were based on
time derivatives instead of distance derivatives. They
are incorporated in the “D” variable.

            The equation is true, but we can't use it to determine V

because in order to create the equation we had to create
the arbitrary V function before we started. The equation
tautologically has V on both sides, as the proof of
correctness of the time-based formula demonstrates. Alex
pointed this out as soon as Rick introduced this
so-called equation, this tautology, but to no avail.

             In other words, none of it, since you continue

to claim that the arbitrarily preset variable “V” in the
equation above is what people measure. They don’t. They
measure how fast people and other organisms move along a
curving track. Quite a different thing.

                              Not in

the slightest. It says that your so-called test is a
cheat because it uses reference trajectories whose
speed along the curve is defined by the power law.
When you use real data, as you seem to do below, you
get what Alex and many other have found. And what Alex
would like to have explained.

                              No,

there’s no implied relationship between the shape and
the speed. There may well be a relationship in
practice, as Alex said when he asked the original
question. His question was why this relation is found
in practice when there is no analytically determined
reason for it. One can, in principle, choose to go
fast around the sharp curves and go slow when there is
a flat section. Apparently people don’t usually do
that, and I assume that your data contribute to the
mass of evidence that this is so.

                              That's

good. Without knowing what you actually calculate, I
can’t judge whether these betas are what is usually
computed, but I’ll take your word that they are. I
wonder whether your apparatus had some equivalent of
viscosity?

             The problem is that your sine waves are sine

waves with as a function of time. The squiggle shapes
have no connection with time, so your comment (and your
spreadsheet and your analysis) fails to make the
connection. You don’t need different disturbances, you
need different disturbance velocities . As I have
suggested several times, there are several easy ways to
do this without changing the shapes of either the target
squiggle or the disturbances. You shouldn’t need any new
instructions how to do it.

             Well, you are saying the same thing in different

words. Yes you must, and no you don’t, or haven’t yet.

             You have not, so far as I know, ever

demonstrated that your model produces the power law when
the along-track velocity of the target squiggle does
not.

             I agree that's what you *should* do. I

keep asking you do do it. I don’t mind if you refuse to
do it. I do mind that you keep claiming that you have
done it. It confuses the readership, and I’d prefer that
the CSGnet readership were treated to honest PCT
discussions rather than ones that depend on a
mathematical misunderstanding, however easy that
misunderstanding is to make.

                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

                        So as the electrified Dylan said to the

fan who yelled “Judas”, “I don’t believe
you”.

                          MT:

But I ask for the Nth time, where is the
along-track speed variation in the
reference trace in your spreadsheet?

RM: Where is it in your model? As
I said, there is no need to put an
along-track speed variation in the reference
trace into the PCT model in order to account
for the data (power law). The reference
trace is actually a trajectory – variation
over time – not just a shape, by the way.
So a regular old PCT model with fixed or
variable reference accounts for all of the
power law data with which I am familiar.

                          MT: In

the one you distributed, you use a
reference track that, by its construction
from x and y variations that are composed
of sine-waves over time, imposes a 1/3
power law on the cursor velocity.

                          RM:

This implies that you accept my
explanation of the power law.

                              RM:

Your statement is also consistent with
my explanation of the power law
because it implies that the power
coefficient that is observed in power
law studies depends on the movement
pattern (trajectory) produced.

                              And,

indeed, it does, as can be seen by
pressing the “Scribble” button over
and over (which should work on your
Mac; just the “Data collection” macro
doesn’t work on the Mac) and seeing
the different estimates of beta for
the different scribbles in the upper
right corner. Just now I got estimates
of beta for the different scribbles
that ranged from .25 to .38. The
average for 10 trials was .31.

                          MT:

Bruce says that even the disturbances are
made the same way, so that the output of
any decent control system must conform to
the same power law. It’s forced by the
construction, and is not a property of the
movement control system.

                        RM: It looks to me like you are

complaining about the fact that the model
accounts for the data. But maybe you are
saying that somehow the fit to the data is
forced by my choice of disturbances.
Perhaps, like Bruce, you think the use of
sine wave disturbances has something to do
with it. So I’d be happy to use different
disturbances; let me know what you would
like me to use. But remember Fourier’s
theorem; every disturbance waveform can be
represented as the sum of sine waves of
different phases and frequencies.

                          MT: To

test the model you MUST use a reference
track in which the along-track velocity is
independent of the local curvature.

                        RM: No, to test the model you must

compare the model’s behavior to the behavior
of the systems whose behavior you are trying
to understand.

                        If the model doesn't fit the behavior,

perhaps it’s because you haven’t done what
you say (used a reference track in which the
along-track velocity is independent of the
local curvature). But my model does fit the
data so your emphatic “MUST” is simply not
true.

                          MT:

It’s easy to do. Here’s anothe method to
add to the ones suggested previously. You
could have a “squiggle-like” column
listing some arbitrarily varying local
velocity in x and in y, and integrate that
to produce the reference squiggle. If the
cursor speed conforms to the 1/3 power law
when the reference speed does not, then
you would really have a significant
advance in answering Alex’s question.

                        RM: This is not the way I test models. I

test models against actual observations, not
against what someone says should be but
hasn’t yet been observed.

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

[_Boris_Hartman]

RM : I think you and Bruce (and others) are very opposed to my PCT interpretation of the power law because it shows the S-R view of movement production - which you guys try to disguise using the jargon of PCT – is wrong. So prove me wrong and tell me the “correct” PCT explanation of the power law – since you are so emphatic about my PCT model of the power law being wrong.

HB : I think they are not opposed to your PCT explanation as there isn’t any. Your model of PCT is simply wrong. How many times do we have to emphaisze this.

RM: That is the PCT model of arbitrary movement…

HB : This is not a PCT model of arbitrary movement. And correct left side of picture. It’s your Model and it’s wrong, and I told you why. It’s showing »Coontrol of movement«. And you should promote »Control of perception«.

Bruce and Martin are by my assumptions using right model and the only that can be used in PCT, although they don’t emphasize that.

HB : In this (right) model of PCT there is no »Controlled variable« in environment and there is no »movement control«, references are formed inside organism and so on.…. This is cloose to how organisms work.

So you have first to draw a right model and then ask for Bruce’s and Martin’s oppinion. Or whatever you want to do.

Best,

Boris

Best

Rick

···

From: Richard Marken [mailto:rsmarken@gmail.com]
Sent: Saturday, August 20, 2016 6:04 AM
To: csgnet@lists.illinois.edu
Cc: Henry Yin
Subject: Re: Looking at the Power Law through Laputasian Glasses

–

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

[wmansell]

Hi Rick,

I see the pattern of the movements that people subject to the power law as analogous to the squiggles in the rubber band demo or the movement of your modelled fielder to get to a position to catch the ball. In neither of these cases is the movement controlled. The movement emerges as a means to control a perceptual variable in each case, as you well know better than anyone. Don’t you think that the movements consistent with the power law also emerge as a means to control one or more perceptual variables from the perspective of the organism moving? Surely these variables need to be from the organism’s perspective like your optical velocity on the retina models, rather than from a bird’s eye x and y desired location?

I like using your own models as examples when I reply to you because it shows how much respect and admiration I have for your pu
blished work. I would like to see a model from one of us, including you, and me, Martin, Bruce or Alex, that shows that if we use a very likely CV or combinations of CVs that are biologically, physically and perceptually possible, we get the power law emerging under normal circumstances, but at the same time we prove the exceptions. To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it. Note that in every case the movements will be means to control input (as all of us, Bruce, Martin and Alex included), but that doesn’t mean they can’t be measured, as we know that from every tracking task that correlates mouse movement with disturbance and mouse movement with variation in the likely CV in the environment (e.g. Relative cursor-target position).

Does that help at all?

Warren

···

Martin Taylor (2016.08.21.00.04)

MT: I really wonder why it is so difficult to get across the idea that I

think the first step in finding a PCT explanation of the power law
is to find the controlled variable (or variables).

RM: Because I don’t know how this search for the controlled variable relates to the phenomenon we are trying to explain: the power law. The first step in a PCT explanation of behavior is to identify a behavior that seems to involve control. And this usually involves noticing that a variable is being kept in a constant or variable state, protected from disturbances that would ordinarily move it from that state. For example, control seems to be involved when a person consistently catches a ball thrown to different locations or remains upright while walking on uneven terrain. So the first step in finding a PCT explanation of any phenomenon is to have a reason to believe that the phenomenon involves control. The next step is to guess what variable(s) is (are) under control. And the next step after that is to test this guess using some version of the TCV.

RM: In t
he case of the power law, the phenomenon that is observed does not obviously involve control. All we see is a quantitative relationship between measures of the velocity and curvature of a movement trajectory. There is no evidence that either velocity or curvature are being kept in a constant of variable state, protected from disturbances. So I don’t know what finding the controlled variable(s) has to do with explaining the power law. If your model is a PCT model then there must be some controlling that it is explaining. What is the phenomenon that your model explains and how does the power law fit into that model? You should be able to diagram this model without identifying the controlled variable(s); just put q.i’s in the part of the diagram where the to-be-identified controlled variable(s) go.

MT: You usually are the first to say that the search for the controlled
variable is the primary, if not the only, proper focus of PCT
research.

RM: That’s true. But only when you have reason to believe that the phenomenon under study involves control and you have at least one hypothesis about the variable(s) under control. In the baseball catching research we did the TCV because we had reason to believe that catching baseballs was a control process and that the variables controlled were aspects of the optical trajectory of the ball. We actually came in with three different hypotheses about the controlled variable and used the TCV to get it down to one.

MT: So I have no model and will not have one

before there is a reasonable proposal for what variable is being
controlled.

RM: But you do have a model; it is a control model because you assume there is a controlled variable involved. So you should be able to show me a diagram of the model (with the exact controlled variable unspecified) and show me how the power law fits into that model.

MT: I proposed an experiment to start the search, which Alex told me

privately he intends to do when he returns from a leave of absence.

RM: I gave my evaluation of that experiment in another post. I have no idea what you expect to find or how those findings relate to an explanation of the power law.

MT: Your response to that proposal for a start on a search for the

controlled variable is to call me and Bruce “opponents of PCT”. Sad.
Really sad.

RM: No, my response to that proposal was that the goals of the experiment were vague, there was no clear description of the predicted results, why those predictions were made or how those predictions related to explaining the power law.

MT: When you have a "model" without having any idea as to the controlled

variable, that’s not good PCT.

RM: I agree. That’s why my model is good PCT and yours is not. My model has a very good idea as to the controlled variable – it’s the perceived distance from cursor to target in my spreadsheet demo. Your model doesn’t have any idea as to what the controlled variable is; indeed, your model doesn’t even seem to exist.

MT: And when that model is shown by

various methods not to demonstrate what it is claimed to
demonstrate, it’s not good science to continue the same claims
without showing why the criticisms are wrong.

RM:I agree with that too. Fortunately, there is yet to be a demonstration that my model doesn’t demonstrate what it claims to demonstrate: that the power law is a function of the movement trajectory that is produced and has nothing to do with how it is produced.

MT: Not once have you done

that. You haven’t even said anything to indicate that you understand
what the criticisms are!

RM: Actually, I see it as you not understanding that your criticisms are irrelevant to what is demonstrated by the behavior of the model.

MT: All you have done in support of your model is to show time and time

again that the formula for curvature is self-consistent,

RM: No, I have shown that the 1/3 (and 2/3) power law is found for movement trajectories that are produced by output trajectories that are quite different than the resulting movement trajectories. I’ve also shown that the power law that is observed depends on the movement trajectory produced, not on the outputs used to produce it.

MT: and to

falsely accuse Bruce and me of being S-R theorists, that we believe
in control of output, that we are opponents of PCT.

RM: Since I haven’t seen your control model that explains the power law and since everything you’ve said is aimed at showing that measures of velocity and curvature are independent of each other and since you have dismissed my PCT model for no good reason and provided no alternative, it looks to me like you are looking at the power law as an S-R phenomenon and that you are pretty unhappy with a PCT explanation of the phenomenon. So, yes, I believe that you are fundamentally S-R theorists and opponents of PCT. You could change my mind about that if you acted more like a friend of PCT. I think a friend of PCT, one who thought my model of the power law was wrong, would suggest the right way to model it. Neither of you have come close to doing that.

MT: No. We are

opponents of the deluded belief that because one had an idea, that
idea must be right, no matter what evidence is brought to bear
against it.

RM: Great. But rather than just say that you are not an opponent of PCT I would rather you demonstrate it by suggesting the correct way to model the power law using PCT.

Best regards

Rick

[From Rick Marken (2016.08.20.1540)]
RM: It’s more than 20 years since Mary wrote that and the
feedforward, information theory, dynamic systems theory and
whatever types (many of them the same people now as then) are
still at it. If Mary was grumpy back then, after only 3 years
of this stuff, imagine how I felt when I wrote my withdrawal
from LCS IV.

      RM: Thanks Dag and Barb. But this is just the way it is for

PCT. There’s no taming its opponents. Those in the
establishment do not suffer revolutions gladly.

Even if those calumnies were true, it would still not be good

science to ignore the evidence that your model is irrelevant to the
problem at issue. Even enemies (which we are not) can have accurate
ideas about your weak points. You refuse even do the trivially
simple tests of your model for which you already have the test
machinery. Instead of doing the experiment, you say that the result
would be irrelevant and cast slurs on those who present the
evidence. Not good science. Nor, I think good pubic relations.

Martin

–

          RM: There are apparently several people who are

interested. I think you could make this a lot clearer to
the on-lookers if you would just describe your theory of
why the power law is observed. I think it would be easier
for people to understand why the PCT explanation of the
power law is wrong if you would describe the explanation
of the power law that you think is correct.

Best

Rick

            Point 1. No description of shape can depend on anything

but measures of space. Time and velocity cannot enter
into them.

            ----- detail follows----



            There is a mathematical description of curvature in a

Euclidean (x vs. y) space. Curvature is defined as 1/R,
where R is a radius of curvature (the radius of an
“osculating circle”, the circle equivalent of a tangent
straight line).

            In the following formulae, "s" means distance along the

curve, and (x, y) is a location in the space. the
differential ds in terms of x and y is given by ds = (dx²
+ dy²)^1/2 from basic Euclidean
geometry. One can do as is done in the Wikipedia article
on Curvature, divide through this equation by ds and
define an “intermediate variable” or “formal variable”
which I will call “w”. The definition of “w” is

            <mime-attachment.jpg>



            The basic expression for curvature (it's based on

computing the tangent vector acceleration, which we
don’t need to go into. If you re interested, you could
look up the actual derivation) is

            <mime-attachment.jpg>



            so we can write



             ---- end detail----



            <mime-attachment.jpg>



            If we do as Rick does, call the numerator of this

fraction “D” and transpose, we get

            w<sup>3</sup> = D*R



            or



            w = D<sup>1/3</sup>*R<sup>1/3</sup>



            Notice that nothing at all in this derivation has any

relationship to time or velocity. Those should not and
do not enter into any formula that is purely about the
shape of a curve. And yet we arrive at a formula for “w”
that Rick calls a velocity. Not only that, but Rick
claims “w” to be the velocity that the “power law
researchers” measure in their experiments.

            Point 2. Right at the start of this curious set of

exchanges, we pointed out to Rick that he had made a
very excusable mistake, figuring that correcting the
mistake would end that particular discussion, but we
were sadly mistaken. The reason that the mistake is easy
to make is that the derivatives in the formula have been
represented in the “dotty” Newtonian notation, and are
often taken to be derivatives with respect to time.

            ----detail follows----



            The problem with this is that the formulae do work if

you use time derivatives, but they do so if and only if
you prespecify a velocity profile along the track of the
curve. You can set V as a function of t (time) or of s
(distance along the curve), because they convert into
each other, but you have to set it before you do any
other manipulations. It can be anything, provided that
it defines a one-to-one mapping between t and s.

             V(t)<sub>t= t0</sub> = (ds/dt)<sub>t=t0

            </sub>or<sub>

            </sub>V(s)<sub>s=s0</sub> = (ds/dt)<sub>s=s0</sub>



            If you integrate V(t) after time t0 or V(s) after point

s0, you get a mapping of t onto s. For each moment in
time there is a corresponding point along the curve.
Those values depends on the arbitrarily defined V
function of t or s, So let’s see (once again) what we
arrive at when we use this arbitrary mapping of t onto
s, starting with the formulae when you take the dot
notation as signifying differentiation with respect to
time.

            The first formula, which defined our intervening

variable “w” now is

            <mime-attachment.jpg>



            The curvature formula becomes



            <mime-attachment.jpg>



            It is perhaps worth noticing at this point that it would

be formally correct to multiply top and bottom by (dt)³
to avoid having time mentioned in a formula for a
description of space, but we won’t do that because we
want to see whether the formula as written is correct on
the assumption that we have specified a velocity
function of either time or distance along the curve.
What we will do is note that in calculus generally,
dx/dy = dx/dzdz/dy and d²x/dy² =
d²x/dz(dz/dy)² , and use these
equalities to show that the equation is correct by
deriving it from the second equation above, the basic
formula for curvature, repeated here.

            <mime-attachment.jpg>



            Using the equivalences just mentioned, we can write this

as

            <mime-attachment.jpg>

            And cancelling out the ds's we get the formula above

            <mime-attachment.jpg>



            Q.E.D.



            Provided we have chosen some arbitrary velocity in

advance as a function of time or space, so that ds/dt
has a value, the equation for curvature with the time
derivatives is correct.

            ----end detail---



            It is very easy to ignore the requirement to specify an

arbitrary V(t) or V(s) and think that the two equations,
one for the velocity and one for the curvature, are
independent. That’s exactly what Rick did when he
noticed that the denominator of this formula is actually
V³ . He thought that he could then do as we
did above for the formal variable “w”, which has no
relation to velocity. So we follow Rick and do it here,
for V, but now we do it knowing that we pre-set V. Rick
calls the numerator of the fraction “D”, so the equation
becomes

            1/R = D/V<sup>3</sup>



            which immediately gives Rick's favourite equation for

determining(!) the already arbitrarily specified V.

            V = D<sup>1/3</sup>*R<sup>1/3</sup>



            Notice here that V is the same as the "w" variable we

found earlier, except that the equations were based on
time derivatives instead of distance derivatives. They
are incorporated in the “D” variable.

            The equation is true, but we can't use it to determine V

because in order to create the equation we had to create
the arbitrary V function before we started. The equation
tautologically has V on both sides, as the proof of
correctness of the time-based formula demonstrates. Alex
pointed this out as soon as Rick introduced this
so-called equation, this tautology, but to no avail.

             In other words, none of it, since you continue

to claim that the arbitrarily preset variable “V” in the
equation above is what people measure. They don’t. They
measure how fast people and other organisms move along a
curving track. Quite a different thing.

                              Not in

the slightest. It says that your so-called test is a
cheat because it uses reference trajectories whose
speed along the curve is defined by the power law.
When you use real data, as you seem to do below, you
get what Alex and many other have found. And what Alex
would like to have explained.

                              No,

there’s no implied relationship between the shape and
the speed. There may well be a relationship in
practice, as Alex said when he asked the original
question. His question was why this relation is found
in practice when there is no analytically determined
reason for it. One can, in principle, choose to go
fast around the sharp curves and go slow when there is
a flat section. Apparently people don’t usually do
that, and I assume that your data contribute to the
mass of evidence that this is so.

                              That's

good. Without knowing what you actually calculate, I
can’t judge whether these betas are what is usually
computed, but I’ll take your word that they are. I
wonder whether your apparatus had some equivalent of
viscosity?

             The problem is that your sine waves are sine

waves with as a function of time. The squiggle shapes
have no connection with time, so your comment (and your
spreadsheet and your analysis) fails to make the
connection. You don’t need different disturbances, you
need different disturbance velocities . As I have
suggested several times, there are several easy ways to
do this without changing the shapes of either the target
squiggle or the disturbances. You shouldn’t need any new
instructions how to do it.

             Well, you are saying the same thing in different

words. Yes you must, and no you don’t, or haven’t yet.

             You have not, so far as I know, ever

demonstrated that your model produces the power law when
the along-track velocity of the target squiggle does
not.

             I agree that's what you *should* do. I

keep asking you do do it. I don’t mind if you refuse to
do it. I do mind that you keep claiming that you have
done it. It confuses the readership, and I’d prefer that
the CSGnet readership were treated to honest PCT
discussions rather than ones that depend on a
mathematical misunderstanding, however easy that
misunderstanding is to make.

                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

                        So as the electrified Dylan said to the

fan who yelled “Judas”, “I don’t believe
you”.

                          MT:

But I ask for the Nth time, where is the
along-track speed variation in the
reference trace in your spreadsheet?

RM: Where is it in your model? As
I said, there is no need to put an
along-track speed variation in the reference
trace into the PCT model in order to account
for the data (power law). The reference
trace is actually a trajectory – variation
over time – not just a shape, by the way.
So a regular old PCT model with fixed or
variable reference accounts for all of the
power law data with which I am familiar.

                          MT: In

the one you distributed, you use a
reference track that, by its construction
from x and y variations that are composed
of sine-waves over time, imposes a 1/3
power law on the cursor velocity.

                          RM:

This implies that you accept my
explanation of the power law.

                              RM:

Your statement is also consistent with
my explanation of the power law
because it implies that the power
coefficient that is observed in power
law studies depends on the movement
pattern (trajectory) produced.

                              And,

indeed, it does, as can be seen by
pressing the “Scribble” button over
and over (which should work on your
Mac; just the “Data collection” macro
doesn’t work on the Mac) and seeing
the different estimates of beta for
the different scribbles in the upper
right corner. Just now I got estimates
of beta for the different scribbles
that ranged from .25 to .38. The
average for 10 trials was .31.

                          MT:

Bruce says that even the disturbances are
made the same way, so that the output of
any decent control system must conform to
the same power law. It’s forced by the
construction, and is not a property of the
movement control system.

                        RM: It looks to me like you are

complaining about the fact that the model
accounts for the data. But maybe you are
saying that somehow the fit to the data is
forced by my choice of disturbances.
Perhaps, like Bruce, you think the use of
sine wave disturbances has something to do
with it. So I’d be happy to use different
disturbances; let me know what you would
like me to use. But remember Fourier’s
theorem; every disturbance waveform can be
represented as the sum of sine waves of
different phases and frequencies.

                          MT: To

test the model you MUST use a reference
track in which the along-track velocity is
independent of the local curvature.

                        RM: No, to test the model you must

compare the model’s behavior to the behavior
of the systems whose behavior you are trying
to understand.

                        If the model doesn't fit the behavior,

perhaps it’s because you haven’t done what
you say (used a reference track in which the
along-track velocity is independent of the
local curvature). But my model does fit the
data so your emphatic “MUST” is simply not
true.

                          MT:

It’s easy to do. Here’s anothe method to
add to the ones suggested previously. You
could have a “squiggle-like” column
listing some arbitrarily varying local
velocity in x and in y, and integrate that
to produce the reference squiggle. If the
cursor speed conforms to the 1/3 power law
when the reference speed does not, then
you would really have a significant
advance in answering Alex’s question.

                        RM: This is not the way I test models. I

test models against actual observations, not
against what someone says should be but
hasn’t yet been observed.

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

[rsmarken]

[From Rick Marken (2016.08.21.0950)]

PowerLawRegressionwDataCollect&Model1.xlsm (590 KB)

···

On Sun, Aug 21, 2016 at 2:31 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I see the pattern of the movements that people subject to the power law as analogous to the squiggles in the rubber band demo or the movement of your modelled fielder to get to a position to catch the ball. In neither of these cases is the movement controlled.

RM: What does it mean to be “subject to the power law”? How does being “subject to the power law” fit into the PCT model of the “squiggles” in the rubber band demo? I presume you mean the squiggle movements made by S’s finger in response to E’s disturbances to the knot’s position. I made a model that accounts for those movements for our paper on “Control Blindness”. Did I leave something out; the something that makes those movements “subject to the power law”?

WM: The movement emerges as a means to control a perceptual variable in each case, as you well know better than anyone.

RM: Apparently. And I also seem to know better than anyone that this movement “emerges” from the operation of a closed loop control organization. The movements are not “subject to a power law”.

WM: Don’t you think that the movements consistent with the power law also emerge as a means to control one or more perceptual variables from the perspective of the organism moving?

RM: I don’t know what you mean by “movements consistent with the power law”. Is it movements that are fit by a power function, regardless of the value of the best fit power coefficient? Or is it only movements that are fit by the 1/3 or 2/3 power function? My spreadsheet (Now attached again) shows that all control movement trajectories, whether they are controlled variables or the outputs that keep variables under control, follow a power law to some degree (in terms of the R^2 fit of a power function regression analysis). But only certain trajectories follow the 1/3 or 2/3 power law with an R^2 close to .9.

WM: Surely these variables need to be from the organism’s perspective like your optical velocity on the retina models, rather than from a bird’s eye x and y desired location?

RM: Of course they are. It’s control of perception. But when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory. The same way I do it in the attached spreadsheet (see the tab labeled “Regression”.

WM: I like using your own models as examples when I reply to you because it shows how much respect and admiration I have for your published work.

RM: I’d much rather have people understand than respect me.

WM: I would like to see a model from one of us, including you, and me, Martin, Bruce or Alex, that shows that if we use a very likely CV or combinations of CVs that are biologically, physically and perceptually possible, we get the power law emerging under normal circumstances, but at the same time we prove the exceptions.

RM: Me too. But I’ve already provided it (it’s in the attached spreadsheet). Now I’d like to see one from someone else, like Bruce, Martin, Alex or you.

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!

WM: Note that in every case the movements will be means to control input (as all of us, Bruce, Martin and Alex included), but that doesn’t mean they can’t be measured, as we know that from every tracking task that correlates mouse movement with disturbance and mouse movement with variation in the likely CV in the environment (e.g. Relative cursor-target position).

RM: Have you looked at my spreadsheet? It does exactly what you suggest.

WM: Does that help at all?

RM: It helped me know that you may not have gotten a copy of my spreadsheet that collects actual movement data, runs a model simulation at the same time and measures the fit of a power law to the data. So I’m re-attaching it so that you can give it a try. I plan to develop a better version of this demo that makes it even clearer that the so called “power law” is simply a property of movement trajectories and has nothing to do with how those trajectories was produced.

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

[Abbott]

[From Bruce Abbott (2016.08.21.1310 EDT)]

Well put, Warren, well put.

Bruce

···

From: Warren Mansell [mailto:wmansell@gmail.com]
Sent: Sunday, August 21, 2016 5:31 AM
To: csgnet@lists.illinois.edu
Subject: Re: Looking at the Power Law through Laputasian Glasses

Hi Rick,

I see the pattern of the movements that people subject to the power law as analogous to the squiggles in the rubber band demo or the movement of your modelled fielder to get to a position to catch the ball. In neither of these cases is the movement controlled. The movement emerges as a means to control a perceptual variable in each case, as you well know better than anyone. Don’t you think that the movements consistent with the power law also emerge as a means to control one or more perceptual variables from the perspective of the organism moving? Surely these variables need to be from the organism’s perspective like your optical velocity on the retina models, rather than from a bird’s eye x and y desired location?

I like using your own models as examples when I reply to you because it shows how much respect and admiration I have for your pu blished work. I would like to see a model from one of us, including you, and me, Martin, Bruce or Alex, that shows that if we use a very likely CV or combinations of CVs that are biologically, physically and perceptually possible, we get the power law emerging under normal circumstances, but at the same time we prove the exceptions. To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it. Note that in every case the movements will be means to control input (as all of us, Bruce, Martin and Alex included), but that doesn’t mean they can’t be measured, as we know that from every tracking task that correlates mouse movement with disturbance and mouse movement with variation in the likely CV in the environment (e.g. Relative cursor-target position).

Does that help at all?

Warren

On 21 Aug 2016, a t 07:25, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.20.2320)]

Martin Taylor (2016.08.21.00.04)

RM: There are apparently several people who are interested. I think you could make this a lot clearer to the on-lookers if you would just describe your theory of why the power law is observed. I think it would be easier for people to understand why the PCT explanation of the power law is wrong if you would describe the explanation of the power law that you think is correct.

MT: I really wonder why it is so difficult to get across the idea that I think the first step in finding a PCT explanation of the power law is to find the controlled variable (or variables).

RM: Because I don’t know how this search for the controlled variable relates to the phenomenon we are trying to explain: the power law. The first step in a PCT explanation of behavior is to identify a behavior that seems to involve control. And this usually involves noticing that a variable is being kept in a constant or variable state, protected from disturbances that would ordinarily move it from that state. For example, control seems to be involved when a person consistently catches a ball thrown to different locations or remains upright while walking on uneven terrain. So the first step in finding a PCT explanation of any phenomenon is to have a reason to believe that the phenomenon involves control. The next step is to guess what variable(s) is (are) under control. And the next step after that is to test this guess using some version of the TCV.

RM: In t he case of the power law, the phenomenon that is observed does not obviously involve control. All we see is a quantitative relationship between measures of the velocity and curvature of a movement trajectory. There is no evidence that either velocity or curvature are being kept in a constant of variable state, protected from disturbances. So I don’t know what finding the controlled variable(s) has to do with explaining the power law. If your model is a PCT model then there must be some controlling that it is explaining. What is the phenomenon that your model explains and how does the power law fit into that model? You should be able to diagram this model without identifying the controlled variable(s); just put q.i’s in the part of the diagram where the to-be-identified controlled variable(s) go.

MT: You usually are the first to say that the search for the controlled variable is the primary, if not the only, proper focus of PCT research.

RM: That’s true. But only when you have reason to believe that the phenomenon under study involves control and you have at least one hypothesis about the variable(s) under control. In the baseball catching research we did the TCV because we had reason to believe that catching baseballs was a control process and that the variables controlled were aspects of the optical trajectory of the ball. We actually came in with three different hypotheses about the controlled variable and used the TCV to get it down to one.

MT: So I have no model and will not have one before there is a reasonable proposal for what variable is being controlled.

RM: But you do have a model; it is a control model because you assume there is a controlled variable involved. So you should be able to show me a diagram of the model (with the exact controlled variable unspecified) and show me how the power law fits into that model.

MT: I proposed an experiment to start the search, which Alex told me privately he intends to do when he returns from a leave of absence.

RM: I gave my evaluation of that experiment in another post. I have no idea what you expect to find or how those findings relate to an explanation of the power law.

MT: Your response to that proposal for a start on a search for the controlled variable is to call me and Bruce “opponents of PCT”. Sad. Really sad.

RM: No, my response to that proposal was that the goals of the experiment were vague, there was no clear description of the predicted results, why those predictions were made or how those predictions related to explaining the power law.

MT: When you have a “model” without having any idea as to the controlled variable, that’s not good PCT.

RM: I agree. That’s why my model is good PCT and yours is not. My model has a very good idea as to the controlled variable – it’s the perceived distance from cursor to target in my spreadsheet demo. Your model doesn’t have any idea as to what the controlled variable is; indeed, your model doesn’t even seem to exist.

MT: And when that model is shown by various methods not to demonstrate what it is claimed to demonstrate, it’s not good science to continue the same claims without showing why the criticisms are wrong.

RM:I agree with that too. Fortunately, there is yet to be a demonstration that my model doesn’t demonstrate what it claims to demonstrate: that the power law is a function of the movement trajectory that is produced and has nothing to do with how it is produced.

MT: Not once have you done that. You haven’t even said anything to indicate that you understand what the criticisms are!

RM: Actually, I see it as you not understanding that your criticisms are irrelevant to what is demonstrated by the behavior of the model.

MT: All you have done in support of your model is to show time and time again that the formula for curvature is self-consistent,

RM: No, I have shown that the 1/3 (and 2/3) power law is found for movement trajectories that are produced by output trajectories that are quite different than the resulting movement trajectories. I’ve also shown that the power law that is observed depends on the movement trajectory produced, not on the outputs used to produce it.

MT: and to falsely accuse Bruce and me of being S-R theorists, that we believe in control of output, that we are opponents of PCT.

RM: Since I haven’t seen your control model that explains the power law and since everything you’ve said is aimed at showing that measures of velocity and curvature are independent of each other and since you have dismissed my PCT model for no good reason and provided no alternative, it looks to me like you are looking at the power law as an S-R phenomenon and that you are pretty unhappy with a PCT explanation of the phenomenon. So, yes, I believe that you are fundamentally S-R theorists and opponents of PCT. You could change my mind about that if you acted more like a friend of PCT. I think a friend of PCT, one who thought my model of the power law was wrong, would suggest the right way to model it. Neither of you have come close to doing that.

MT: No. We are opponents of the deluded belief that because one had an idea, that idea must be right, no matter what evidence is brought to bear against it.

RM: Great. But rather than just say that you are not an opponent of PCT I would rather you demonstrate it by suggesting the correct way to model the power law using PCT.

Best regards

Rick

[From Rick Marken (2016.08.20.1540)]

RM: It’s more than 20 years since Mary wrote that and the feedforward, information theory, dynamic systems theory and whatever types (many of them the same people now as then) are still at it. If Mary was grumpy back then, after only 3 years of this stuff, imagine how I felt when I wrote my withdrawal from LCS IV.

RM: Thanks Dag and Barb. But this is just the way it is for PCT. There’s no taming its opponents. Those in the establishment do not suffer revolutions gladly.

Even if those calumnies were true, it would still not be good science to ignore the evidence that your model is irrelevant to the problem at issue. Even enemies (which we are not) can have accurate ideas about your weak points. You refuse even do the trivially simple tests of your model for which you already have the test machinery. Instead of doing the experiment, you say that the result would be irrelevant and cast slurs on those who present the evidence. Not good science. Nor, I think good pubic relations.

Martin

Best

Rick

Point 1. No description of shape can depend on anything but measures of space. Time and velocity cannot enter into them.

----- detail follows----

There is a mathematical description of curvature in a Euclidean (x vs. y) space. Curvature is defined as 1/R, where R is a radius of curvature (the radius of an “osculating circle”, the circle equivalent of a tangent straight line).

In the following formulae, “s” means distance along the curve, and (x, y) is a location in the space. the differential ds in terms of x and y is given by ds = (dx² + dy²)^1/2 from basic Euclidean geometry. One can do as is done in the Wikipedia article on Curvature, divide through this equation by ds and define an “intermediate variable” or “formal variable” which I will call “w”. The definition of “w” is
<mime-attachment.jpg>

The basic expression for curvature (it’s based on computing the tangent vector acceleration, which we don’t need to go into. If you re interested, you could look up the actual derivation) is

<mime-attachment.jpg>

so we can write

---- end detail----

<mime-attachment.jpg>

If we do as Rick does, call the numerator of this fraction “D” and transpose, we get

w³ = D*R

or

w = D^1/3*R^1/3

Notice that nothing at all in this derivation has any relationship to time or velocity. Those should not and do not enter into any formula that is purely about the shape of a curve. And yet we arrive at a formula for “w” that Rick calls a velocity. Not only that, but Rick claims “w” to be the velocity that the “power law researchers” measure in their experiments.

Point 2. Right at the start of this curious set of exchanges, we pointed out to Rick that he had made a very excusable mistake, figuring that correcting the mistake would end that particular discussion, but we were sadly mistaken. The reason that the mistake is easy to make is that the derivatives in the formula have been represented in the “dotty” Newtonian notation, and are often taken to be derivatives with respect to time.

----detail follows----

The problem with this is that the formulae do work if you use time derivatives, but they do so if and only if you prespecify a velocity profile along the track of the curve. You can set V as a function of t (time) or of s (distance along the curve), because they convert into each other, but you have to set it before you do any other manipulations. It can be anything, provided that it defines a one-to-one mapping between t and s.

V(t)_{t= t0} = (ds/dt)_t=t0
or
V(s)_s=s0 = (ds/dt)_s=s0

If you integrate V(t) after time t0 or V(s) after point s0, you get a mapping of t onto s. For each moment in time there is a corresponding point along the curve. Those values depends on the arbitrarily defined V function of t or s, So let’s see (once again) what we arrive at when we use this arbitrary mapping of t onto s, starting with the formulae when you take the dot notation as signifying differentiation with respect to time.

The first formula, which defined our intervening variable “w” now is
<mime-attachment.jpg>

The curvature formula becomes

<mime-attachment.jpg>

It is perhaps worth noticing at this point that it would be formally correct to multiply top and bottom by (dt)³ to avoid having time mentioned in a formula for a description of space, but we won’t do that because we want to see whether the formula as written is correct on the assumption that we have specified a velocity function of either time or distance along the curve. What we will do is note that in calculus generally, dx/dy = dx/dzdz/dy and d²x/dy² = d²x/dz(dz/dy)², and use these equalities to show that the equation is correct by deriving it from the second equation above, the basic formula for curvature, repeated here.

<mime-attachment.jpg>

Using the equivalences just mentioned, we can write this as
<mime-attachment.jpg>
And cancelling out the ds’s we get the formula above
<mime-attachment.jpg>

Q.E.D.

Provided we have chosen some arbitrary velocity in advance as a function of time or space, so that ds/dt has a value, the equation for curvature with the time derivatives is correct.

----end detail—

It is very easy to ignore the requirement to specify an arbitrary V(t) or V(s) and think that the two equations, one for the velocity and one for the curvature, are independent. That’s exactly what Rick did when he noticed that the denominator of this formula is actually V³. He thought that he could then do as we did above for the formal variable “w”, which has no relation to velocity. So we follow Rick and do it here, for V, but now we do it knowing that we pre-set V. Rick calls the numerator of the fraction “D”, so the equation becomes

1/R = D/V³

which immediately gives Rick’s favourite equation for determining(!) the already arbitrarily specified V.

V = D^1/3*R^1/3

Notice here that V is the same as the “w” variable we found earlier, except that the equations were based on time derivatives instead of distance derivatives. They are incorporated in the “D” variable.

The equation is true, but we can’t use it to determine V because in order to create the equation we had to create the arbitrary V function before we started. The equation tautologically has V on both sides, as the proof of correctness of the time-based formula demonstrates. Alex pointed this out as soon as Rick introduced this so-called equation, this tautology, but to no avail.

So as the electrified Dylan said to the fan who yelled “Judas”, “I don’t believe you”.

MT: But I ask for the Nth time, where is the along-track speed variation in the reference trace in your spreadsheet?

RM: Where is it in your model? As I said, there is no need to put an along-track speed variation in the reference trace into the PCT model in order to account for the data (power law). The reference trace is actually a trajectory – variation over time – not just a shape, by the way. So a regular old PCT model with fixed or variable reference accounts for all of the power law data with which I am familiar.

In other words, none of it, since you continue to claim that the arbitrarily preset variable “V” in the equation above is what people measure. They don’t. They measure how fast people and other organisms move along a curving track. Quite a different thing.

MT: In the one you distributed, you use a reference track that, by its construction from x and y variations that are composed of sine-waves over time, imposes a 1/3 power law on the cursor velocity.

RM: This implies that you accept my explanation of the power law.

Not in the slightest. It says that your so-called test is a cheat because it uses reference trajectories whose speed along the curve is defined by the power law. When you use real data, as you seem to do below, you get what Alex and many other have found. And what Alex would like to have explained.

RM: Your statement is also consistent with my explanation of the power law because it implies that the power coefficient that is observed in power law studies depends on the movement pattern (trajectory) produced.

No, there’s no implied relationship between the shape and the speed. There may well be a relationship in practice, as Alex said when he asked the original question. His question was why this relation is found in practice when there is no analytically determined reason for it. One can, in principle, choose to go fast around the sharp curves and go slow when there is a flat section. Apparently people don’t usually do that, and I assume that your data contribute to the mass of evidence that this is so.

And, indeed, it does, as can be seen by pressing the “Scribble” button over and over (which should work on your Mac; just the “Data collection” macro doesn’t work on the Mac) and seeing the different estimates of beta for the different scribbles in the upper right corner. Just now I got estimates of beta for the different scribbles that ranged from .25 to .38. The average for 10 trials was .31.

That’s good. Without knowing what you actually calculate, I can’t judge whether these betas are what is usually computed, but I’ll take your word that they are. I wonder whether your apparatus had some equivalent of viscosity?

MT: Bruce says that even the disturbances are made the same way, so that the output of any decent control system must conform to the same power law. It’s forced by the construction, and is not a property of the movement control system.

RM: It looks to me like you are complaining about the fact that the model accounts for the data. But maybe you are saying that somehow the fit to the data is forced by my choice of disturbances. Perhaps, like Bruce, you think the use of sine wave disturbances has something to do with it. So I’d be happy to use different disturbances; let me know what you would like me to use. But remember Fourier’s theorem; every disturbance waveform can be represented as the sum of sine waves of different phases and frequencies.

The problem is that your sine waves are sine waves with as a function of time. The squiggle shapes have no connection with time, so your comment (and your spreadsheet and your analysis) fails to make the connection. You don’t need different disturbances, you need different disturbance velocities. As I have suggested several times, there are several easy ways to do this without changing the shapes of either the target squiggle or the disturbances. You shouldn’t need any new instructions how to do it.

MT: To test the model you MUST use a reference track in which the along-track velocity is independent of the local curvature.

RM: No, to test the model you must compare the model’s behavior to the behavior of the systems whose behavior you are trying to understand.

Well, you are saying the same thing in different words. Yes you must, and no you don’t, or haven’t yet.

If the model doesn’t fit the behavior, perhaps it’s because you haven’t done what you say (used a reference track in which the along-track velocity is independent of the local curvature). But my model does fit the data so your emphatic “MUST” is simply not true.

You have not, so far as I know, ever demonstrated that your model produces the power law when the along-track velocity of the target squiggle does not.

MT: It’s easy to do. Here’s anothe method to add to the ones suggested previously. You could have a “squiggle-like” column listing some arbitrarily varying local velocity in x and in y, and integrate that to produce the reference squiggle. If the cursor speed conforms to the 1/3 power law when the reference speed does not, then you would really have a significant advance in answering Alex’s question.

RM: This is not the way I test models. I test models against actual observations, not against what someone says should be but hasn’t yet been observed.

I agree that’s what you should do. I keep asking you do do it. I don’t mind if you refuse to do it. I do mind that you keep claiming that you have done it. It confuses the readership, and I’d prefer that the CSGnet readership were treated to honest PCT discussions rather than ones that depend on a mathematical misunderstanding, however easy that misunderstanding is to make.

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

–

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

[wmansell]

Hi Rick, see below…

[From Rick Marken (2016.08.21.0950)]

I was using ‘subject’ as a verb in that sentence ‘movements that researchers choose subject to the power law’ not that are subject to the power law.

See above. I see the ‘power law’ not a a law of physics but as a pattern that has been detected whatever its origins.

I am not sure they do show that - not all control movement trajectories - because how do we know that there are not CVs in the organism for which these movements might be the means to control they we haven’t hypothesised about yet? They certainly don’t need to be CV
s to do with movement because movement is the means to achieve them, not the CV itself. The CV might be ‘high intensity of food chemical’ or ‘low use of energy’ or ‘vertical balance of the body’… for example.

That’s correct for measuring the movement, but not for testing the CV. You know that Rick. If we think hierarchically the CV at the top of the hierarchy for the movements of my fingers typing this reply has nothing to do with movement and everything to do with the perceptions I want to experience by doing so.

I do think that I underst
and your published work and o do think that you think that I understand them too. I just don’t understand your approach to this question.

That is very fair dues. It is a shame we can’t all find an objective way to agree yet!

But I think to do that the CV cannot be movement itself. The CV has to be something that requires movement to be successfully controlled for the organism.

See above.

I haven’t got to my PC with Excel on yet, I think I mentioned that won’t be until Wednesday. Do you want mcevoy hold off making mire comments before then? If you can reassure me that your CV is not movement then I am with you…

Warren

···

On Sun, Aug 21, 2016 at 2:31 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I see the pattern of the movements that people subject to the power law as analogous to the squiggles in the rubber band demo or the movement of your modelled fielder to get to a position to catch the ball. In neither of these cases is the movement controlled.

RM: What does it mean to be “subject to the power law”? How does being “subject to the power law” fit into the PCT model of the “squiggles” in the rubber band demo? I presume you mean the squiggle movements made by S’s finger in response to E’s disturbances to the knot’s position. I made a model that accounts for those movements for our paper on “Control Blindness”. Did I leave something out; the something that makes those movements “subject to the power law”?

WM: The movement emerges as a means to control a perceptual variable in each case, as you well know better than anyone.

RM: Apparently. And I also seem to know better than anyone that this movement “emerges” from the operation of a closed loop control organization. The movements are not “subject to a power law”.

WM: Don’t you think that the movements consistent with the power law also emerge as a means to control one or more perceptual variables from the perspective of the organism moving?

RM: I don’t know what you mean by “movements consistent with the power law”. Is it movements that are fit by a power function, regardless of the value of the best fit power coefficient? Or is it only movements that are fit by the 1/3 or 2/3 power function? My spreadsheet (Now attached again) shows that all control movement trajectories, whether they are controlled variables or the outputs that keep variables under control, follow a power law to some degree (in terms of the R^2 fit of a power function regression analysis). But only certain trajectories follow the 1/3 or 2/3 power law with an R^2 close to .9.

WM: Surely these variables need to be from the organism’s perspective like your optical velocity on the retina models, rather than from a bird’s eye x and y desired location?

RM: Of course they are. It’s control of perception. But when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory. The same way I do it in the attached spreadsheet (see the tab labeled “Regression”.

WM: I like using your own models as examples when I reply to you because it shows how much respect and admiration I have for your published work.

RM: I’d much rather have people understand than respect me.

WM: I would like to see a model from one of us, including you, and me, Martin, Bruce or Alex, that shows that if we use a very likely CV or combinations of CVs that are biologically, physically and perceptually possible, we get the power law emerging under normal circumstances, but at the same time we prove the exceptions.

RM: Me too. But I’ve already provided it (it’s in the attached spreadsheet). Now I’d like to see one from someone else, like Bruce, Martin, Alex or you.

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!

WM: Note that in every case the movements will be means to control input (as all of us, Bruce, Martin and Alex included), but that doesn’t mean they can’t be measured, as we know that from every tracking task that correlates mouse movement with disturbance and mouse movement with variation in the likely CV in the environment (e.g. Relative cursor-target position).

RM: Have you looked at my spreadsheet? It does exactly what you suggest.

WM: Does that help at all?

RM: It helped me know that you may not have gotten a copy of my spreadsheet that collects actual movement data, runs a model simulation at the same time and measures the fit of a power law to the data. So I’m re-attaching it so that you can give it a try. I plan to develop a better version of this demo that makes it even clearer that the so called “power law” is simply a property of movement trajectories and has nothing to do with how those trajectories was produced.

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

[FredNickols]

[From Fred Nickols (2016.08.21.1501 ET)]

A couple of things.

Where can I read up on the Power Law as it is being discussed here. What I found on Wikipedia seems to be an exercise in statistics (as yet unproven).

And for Warren: Why can’t movement be a CV? It seems to me that I could establish a reference signal for the movement of my arm and then have the movement of my arm match that reference signal? Am I missing something here?

Fred Nickols

···

From: Warren Mansell [mailto:wmansell@gmail.com]
Sent: Sunday, August 21, 2016 2:43 PM
To: csgnet@lists.illinois.edu
Subject: Re: Looking at the Power Law through Laputasian Glasses

Hi Rick, see below…

On 21 Aug 2016, at 17:53, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.21.0950)]

On Sun, Aug 21, 2016 at 2:31 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I see the pattern of the movements that people subject to the power law as analogous to the squiggles in the rubber band demo or the movement of your modelled fielder to get to a position to catch the ball. In neither of these cases is the movement controlled.

RM: What does it mean to be “subject to the power law”? How does being “subject to the power law” fit into the PCT model of the “squiggles” in the rubber band demo? I presume you mean the squiggle movements made by S’s finger in response to E’s disturbances to the knot’s position. I made a model that accounts for those movements for our paper on “Control Blindness”. Did I leave something out; the something that makes those movements “subject to the power law”?

I was using ‘subject’ as a verb in that sentence ‘movements that researchers choose subject to the power law’ not that are subject to the power law.

WM: The movement emerges as a means to control a perceptual variable in each case, as you well know better than anyone.

RM: Apparently. And I also seem to know better than anyone that this movement “emerges” from the operation of a closed loop control organization. The movements are not “subject to a power law”.

See above. I see the ‘power law’ not a a law of physics but as a pattern that has been detected whatever its origins.

WM: Don’t you think that the movements consistent with the power law also emerge as a means to control one or more perceptual variables from the perspective of the organism moving?

RM: I don’t know what you mean by “movements consistent with the power law”. Is it movements that are fit by a power function, regardless of the value of the best fit power coefficient? Or is it only movements that are fit by the 1/3 or 2/3 power function? My spreadsheet (Now attached again) shows that all control movement trajectories, whether they are controlled variables or the outputs that keep variables under control, follow a power law to some degree (in terms of the R^2 fit of a power function regression analysis). But only certain trajectories follow the 1/3 or 2/3 power law with an R^2 close to .9.

I am not sure they do show that - not all control movement trajectories - because how do we know that there are not CVs in the organism for which these movements might be the means to control they we haven’t hypothesised about yet? They certainly don’t need to be CV s to do with movement because movement is the means to achieve them, not the CV itself. The CV might be ‘high intensity of food chemical’ or ‘low use of energy’ or ‘vertical balance of the body’… for example.

WM: Surely these variables need to be from the organism’s perspective like your optical velocity on the retina models, rather than from a bird’s eye x and y desired location?

RM: Of course they are. It’s control of perception. But when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory. The same way I do it in the attached spreadsheet (see the tab labeled “Regression”.

That’s correct for measuring the movement, but not for testing the CV. You know that Rick. If we think hierarchically the CV at the top of the hierarchy for the movements of my fingers typing this reply has nothing to do with movement and everything to do with the perceptions I want to experience by doing so.

WM: I like using your own models as examples when I reply to you because it shows how much respect and admiration I have for your published work.

RM: I’d much rather have people understand than respect me.

I do think that I underst and your published work and o do think that you think that I understand them too. I just don’t understand your approach to this question.

WM: I would like to see a model from one of us, including you, and me, Martin, Bruce or Alex, that shows that if we use a very likely CV or combinations of CVs that are biologically, physically and perceptually possible, we get the power law emerging under normal circumstances, but at the same time we prove the exceptions.

RM: Me too. But I’ve already provided it (it’s in the attached spreadsheet). Now I’d like to see one from someone else, like Bruce, Martin, Alex or you.

That is very fair dues. It is a shame we can’t all find an objective way to agree yet!

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!

But I think to do that the CV cannot be movement itself. The CV has to be something that requires movement to be successfully controlled for the organism.

WM: Note that in every case the movements will be means to control input (as all of us, Bruce, Martin and Alex included), but that doesn’t mean they can’t be measured, as we know that from every tracking task that correlates mouse movement with disturbance and mouse movement with variation in the likely CV in the environment (e.g. Relative cursor-target position).

RM: Have you looked at my spreadsheet? It does exactly what you suggest.

See above.

WM: Does that help at all?

RM: It helped me know that you may not have gotten a copy of my spreadsheet that collects actual movement data, runs a model simulation at the same time and measures the fit of a power law to the data. So I’m re-attaching it so that you can give it a try. I plan to develop a better version of this demo that makes it even clearer that the so called “power law” is simply a property of movement trajectories and has nothing to do with how those trajectories was produced.

I haven’t got to my PC with Excel on yet, I think I mentioned that won’t be until Wednesday. Do you want mcevoy hold off making mire comments before then? If you can reassure me that your CV is not movement then I am with you…

Warren

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

<PowerLawRegressionwDataCollect&Model.xlsm>

[MartinT]

[From
Fred Nickols (2016.08.21.1501 ET)]

        A

couple of things.

        Where

can I read up on the Power Law as it is being discussed
here. What I found on Wikipedia seems to be an exercise in
statistics (as yet unproven).

···

From:
Warren Mansell [mailto:wmansell@gmail.com]
Sent: Sunday, August 21, 2016 2:43 PM
To: csgnet@lists.illinois.edu
Subject: Re: Looking at the Power Law through
Laputasian Glasses

          Hi Rick,

see below…

          On 21 Aug 2016, at 17:53, Richard Marken <rsmarken@gmail.com              >

wrote:

              [From Rick Marken

(2016.08.21.0950)]

                  On Sun, Aug 21, 2016 at 2:31

AM, Warren Mansell <wmansell@gmail.com >
wrote:

                      WM: I see the pattern of

the movements that people subject to the power
law as analogous to the squiggles in the
rubber band demo or the movement of your
modelled fielder to get to a position to catch
the ball. In neither of these cases is the
movement controlled.

                    RM: What does it mean to be

“subject to the power law”? How does being
“subject to the power law” fit into the PCT
model of the “squiggles” in the rubber band
demo? I presume you mean the squiggle movements
made by S’s finger in response to E’s
disturbances to the knot’s position. I made a
model that accounts for those movements for our
paper on “Control Blindness”. Did I leave
something out; the something that makes those
movements “subject to the power law”?

          I was using 'subject' as a verb in that

sentence ‘movements that researchers choose subject to
the power law’ not that are subject to the power law.

                      WM: The movement emerges as

a means to control a perceptual variable in
each case, as you well know better than
anyone.

                  RM: Apparently. And I also seem

to know better than anyone that this movement
“emerges” from the operation of a closed loop
control organization. The movements are not
“subject to a power law”.

          See above. I see the 'power law' not a

a law of physics but as a pattern that has been detected
whatever its origins.

                      WM: Don't you think that

the movements consistent with the power law
also emerge as a means to control one or more
perceptual variables from the perspective of
the organism moving?

                  RM: I don't know what you mean

by “movements consistent with the power law”. Is
it movements that are fit by a power function,
regardless of the value of the best fit power
coefficient? Or is it only movements that are fit
by the 1/3 or 2/3 power function? My spreadsheet
(Now attached again) shows that all control
movement trajectories, whether they are controlled
variables or the outputs that keep variables under
control, follow a power law to some degree (in
terms of the R^2 fit of a power function
regression analysis). But only certain
trajectories follow the 1/3 or 2/3 power law with
an R^2 close to .9.

          I am not sure they do show that - not

all control movement trajectories - because how do we know
that there are not CVs in the organism for which these
movements might be the means to control they we haven’t
hypothesised about yet? They certainly don’t need to be CV
s to do with movement because movement is the means to
achieve them, not the CV itself. The CV might be ‘high
intensity of food chemical’ or ‘low use of energy’ or
‘vertical balance of the body’… for example.

                      WM: Surely these variables

need to be from the organism’s perspective
like your optical velocity on the retina
models, rather than from a bird’s eye x and y
desired location?

                  RM: Of course they are. It's

control of perception. But when researchers
determine the fit of the power law to the movement
trajectory they do it by measuring curvature and
velocity from the observed variations in x,y
positions of the movement trajectory. The same
way I do it in the attached spreadsheet (see the
tab labeled “Regression”.

          That's correct for measuring the

movement, but not for testing the CV. You know that Rick.
If we think hierarchically the CV at the top of the
hierarchy for the movements of my fingers typing this
reply has nothing to do with movement and everything to do
with the perceptions I want to experience by doing so.

                      WM: I like using your own

models as examples when I reply to you because
it shows how much respect and admiration I
have for your published work.

                  RM: I'd much rather have people

understand than respect me.

          I do think that I underst and your

published work and o do think that you think that I
understand them too. I just don’t understand your approach
to this question.

                      WM: I would like to see a

model from one of us, including you, and me,
Martin, Bruce or Alex, that shows that if we
use a very likely CV or combinations of CVs
that are biologically, physically and
perceptually possible, we get the power law
emerging under normal circumstances, but at
the same time we prove the exceptions.

                  RM: Me too. But I've already

provided it (it’s in the attached spreadsheet).
Now I’d like to see one from someone else, like
Bruce, Martin, Alex or you.

          That is very fair dues. It is a shame

we can’t all find an objective way to agree yet!

                      WM: To me, this doesn't

require rearranging the equations but simply
running the hypothetical control systems,
collecting movement data and doing the
existing formula on it.

                  RM: That's exactly what my

spreadsheet does!!!

          But I think to do that the CV cannot be

movement itself. The CV has to be something that requires
movement to be successfully controlled for the organism.

                      WM: Note that in every case

the movements will be means to control input
(as all of us, Bruce, Martin and Alex
included), but that doesn’t mean they can’t be
measured, as we know that from every tracking
task that correlates mouse movement with
disturbance and mouse movement with variation
in the likely CV in the environment (e.g.
Relative cursor-target position).

                  RM: Have you looked at my

spreadsheet? It does exactly what you suggest.

See above.

WM: Does that help at all?

                  RM: It helped me know that you

may not have gotten a copy of my spreadsheet that
collects actual movement data, runs a model
simulation at the same time and measures the fit
of a power law to the data. So I’m re-attaching it
so that you can give it a try. I plan to develop a
better version of this demo that makes it even
clearer that the so called “power law” is simply a
property of movement trajectories and has nothing
to do with how those trajectories was produced.

          I haven't got to my PC with Excel on

yet, I think I mentioned that won’t be until Wednesday. Do
you want mcevoy hold off making mire comments before then?
If you can reassure me that your CV is not movement then I
am with you…

Warren

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

<PowerLawRegressionwDataCollect&Model.xlsm>

[rsmarken]

[From Rick Marken (2016.08.21.1700)]

···

On Sun, Aug 21, 2016 at 11:43 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I am not sure they do show that

RM: Neither is anyone else so join the party.

WM: That’s correct for measuring the movement, but not for testing the CV. You know that Rick.

RM: I don’t think I know what you think I know. If we were testing for the variable controlled in movement control we would certainly formulate our hypotheses about the controlled variable in terms of measures of aspects of the observed variations in the x,y positions of the movement itself. That’s what I do in the Mind Reading demo (http://www.mindreadings.com/ControlDemo/Mindread.html) to determine which avatar is being intentionally moved (whose movement is being controlled). It’s also what I did in the object interception research.

WM: I do think that I underst
and your published work and o do think that you think that I understand them too. I just don’t understand your approach to this question.

RM: I can see now that it’s a little more difficult to understand than I thought. I hope you’ll get it eventually.

WM: But I think to do that the CV cannot be movement itself.

RM: I don’t know why not. But it’s not the movement that is controlled in my model anyway; it’s the distance from cursor to target.

WM: …If you can reassure me that your CV is not movement then I am with you…

RM: Yes, I can assure you that the CV in my model is not movement.

Best

Rick

–

RM: My spreadsheet (Now attached again) shows that all control movement trajectories, whether they are controlled variables or the outputs that keep variables under control, follow a power law to some degree…

RM: …when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory.

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!

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

[wmansell]

This is good, getting somewhere even before I can run the demo! I get they the CV is the distance from the cursor to the target, but then it becomes crucial how the movement of the target is programmed. If the target movement already corresponds to the power law then the decks are stacked in favour of finding it in the movement of the cursor aren’t they? In the examples of the human squiggle and the larva, I don’t think they are tracking a moving target are they? I am thinking that the Crowd Demo is a better (but not perfect) analogue because the CV is closeness to static points via various obstacles. If the movement of those agents corresponded to the power law it would be more interesting because no patterned movement is built in to the environment or CV. It truly emerges. If you think this is worthwhile I know a man who can convert video to x y coordinates by time…
.

Warren

···

On Sun, Aug 21, 2016 at 11:43 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I am not sure they do show that

RM: Neither is anyone else so join the party.

WM: That’s correct for measuring the movement, but not for testing the CV. You know that Rick.

RM: I don’t think I know what you think I know. If we were testing for the variable controlled in movement control we would certainly formulate our hypotheses about the controlled variable in terms of measures of aspects of the observed variations in the x,y positions of the movement itself. That’s what I do in the Mind Reading demo (http://www.mindreadings.com/ControlDemo/Mindread.html) to determine which avatar is being intentionally moved (whose movement is being controlled). It’s also what I did in the object interception research.

WM: I do think that I underst
and your published work and o do think that you think that I understand them too. I just don’t understand your approach to this question.

RM: I can see now that it’s a little more difficult to understand than I thought. I hope you’ll get it eventually.

WM: But I think to do that the CV cannot be movement itself.

RM: I don’t know why not. But it’s not the movement that is controlled in my model anyway; it’s the distance from cursor to target.

WM: …If you can reassure me that your CV is not movement then I am with you…

RM: Yes, I can assure you that the CV in my model is not movement.

Best

Rick

RM: My spreadsheet (Now attached again) shows that all control movement trajectories, whether they ar
e controlled variables or the outputs that keep variables under control, follow a power law to some degree…

RM: …when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory.

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!

Richard S. Marken&nbs
p;

“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

[rsmarken]

[From Rick Marken (2016.08.22.1000)]

···

On Mon, Aug 22, 2016 at 1:45 AM, Warren Mansell wmansell@gmail.com wrote:

WM: This is good, getting somewhere even before I can run the demo! I get they the CV is the distance from the cursor to the target, but then it becomes crucial how the movement of the target is programmed.

RM: Why?

WM: If the target movement already corresponds to the power law then the decks are stacked in favour of finding it in the movement of the cursor aren’t they?

RM: Yes, that’s what I have been saying. It’s because the observed fit of a power law to the relationship between curvature and velocity in a movement trajectory depends completely on the form of the trajectory; it has nothing to do with how the movement trajectory was produced.

WM: In the examples of the human squiggle and the larva, I don’t think they are tracking a moving target are they?

RM: The larva is tracking, the human squigglers aren’t. But it makes no difference how the movement is produced, the power relationship that fits the data depends only on the form of the movement trajectory, not on how it was produced.

WM: I am thinking that the Crowd Demo is a better (but not perfect) analogue because the CV is closeness to static points via various obstacles.

RM: The results for the Crowd trajectories would be the same as that for any movement trajectories. The fit of a power function to the data depends on the form of the particular movement trajectory, not on how it was produced.

WM: If the movement of those agents corresponded to the power law it would be more interesting because no patterned movement is built in to the environment or CV. It truly emerges.

RM: It doesn’t emerge by magic. The best fit power function relating curvature to velocity for any of the Crowd trajectories will depend on the shape of the individual trajectory. Trajectories where the variable D contributes little to the variance in velocity will result in a good fit of the 1/3 and 2/3 power function. When D does contributes a lot of variance to velocity the power coefficient will deviate from 1/3 or 2/3 (depending on how velocity and curvature are measured) and the R^2 will be very low.

RM: Actually, my current plan is to have the computer create different random squiggle trajectories, measure the power law for each squiggle and then use each squiggle as the target in a 2 D tracking task. The prediction is that the power law found for the tracking data will be very close to being the same as it was for the corresponding computer produced squiggle. This, and the fact that, because of disturbances, the outputs that keep the cursor on target are very different than the resulting tracking movement, will show that the power relationship found for different movement trajectories depends on the form of the trajectory and not on how it was produced.

WM: If you think this is worthwhile I know a man who can convert video to x y coordinates by time…
.

RM: Sure, that would be a great.

Best

Rick

Warren

On 22 Aug 2016, at 00:59, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.21.1700)]–
Richard S. Marken&nbs
p;

“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

–
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

On Sun, Aug 21, 2016 at 11:43 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I am not sure they do show that

RM: Neither is anyone else so join the party.

WM: That’s correct for measuring the movement, but not for testing the CV. You know that Rick.

RM: I don’t think I know what you think I know. If we were testing for the variable controlled in movement control we would certainly formulate our hypotheses about the controlled variable in terms of measures of aspects of the observed variations in the x,y positions of the movement itself. That’s what I do in the Mind Reading demo (http://www.mindreadings.com/ControlDemo/Mindread.html) to determine which avatar is being intentionally moved (whose movement is being controlled). It’s also what I did in the object interception research.

WM: I do think that I underst
and your published work and o do think that you think that I understand them too. I just don’t understand your approach to this question.

RM: I can see now that it’s a little more difficult to understand than I thought. I hope you’ll get it eventually.

WM: But I think to do that the CV cannot be movement itself.

RM: I don’t know why not. But it’s not the movement that is controlled in my model anyway; it’s the distance from cursor to target.

WM: …If you can reassure me that your CV is not movement then I am with you…

RM: Yes, I can assure you that the CV in my model is not movement.

Best

Rick

RM: My spreadsheet (Now attached again) shows that all control movement trajectories, whether they ar
e controlled variables or the outputs that keep variables under control, follow a power law to some degree…

RM: …when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory.

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!

[wmansell]

Hi Rick, I’m confused again! I guess I just don’t believe that movement trajectories exist by themselves, like a mathematical entity. I think that they have to be formed by some kind of controlling agent which must itself be formed from simpler, non-movement trajectory components. But that’s just my belief. It is my reductionism coming in

···

On Mon, Aug 22, 2016 at 1:45 AM, Warren Mansell wmansell@gmail.com wrote:

WM: This is good, getting somewhere even before I can run the demo! I get they the CV is the distance from the cursor to the target, but then it becomes crucial how the movement of the target is programmed.

RM: Why?

WM: If the target movement already corresponds to the power law then the decks are stacked in favour of finding it in the movement of the cursor aren’t they?

RM: Yes, that’s what I have been saying. It’s because the observed fit of a power law to the relationship between curvature and velocity in a movement trajectory depends completely on the form of the trajectory; it has nothing to do with how the movement trajectory was produced.

WM: In the examples of the human squiggle and the larva, I don’t think they are tracking a moving target are they?

RM: The larva is tracking, the human squigglers aren’t. But it makes no difference how the movement is produced, the power relationship that fits the data depends only on the form of the movement trajectory, not on how it was produced.

WM: I am thinking that the Crowd Demo is a better (but not perfect) analogue because the CV is closeness to static points via various obstacles.

RM: The results for the Crowd trajectories would be the same as that for any movement trajectories. The fit of a power function to the data depends on the form of the particular movement trajectory, not on how it was produced.

WM: If the movement of those agents corresponded to the power law it would be more interesting because no patterned movement is built in to the environment or CV. It truly emerges.

RM: It doesn’t emerge by magic. The best fit power function relating curvature to velocity for any of the Crowd trajectories will depend on the shape of the individual trajectory. Trajectories where the variable D contributes little to the variance in velocity will result in a good fit of the 1/3 and 2/3 power function. When D does contributes a lot of variance to velocity the power coefficient will deviate from 1/3 or 2/3 (depending on how velocity and curvature are measured) and the R^2 will be very low.

RM: Act
ually, my current plan is to have the computer create different random squiggle trajectories, measure the power law for each squiggle and then use each squiggle as the target in a 2 D tracking task. The prediction is that the power law found for the tracking data will be very close to being the same as it was for the corresponding computer produced squiggle. This, and the fact that, because of disturbances, the outputs that keep the cursor on target are very different than the resulting tracking movement, will show that the power relationship found for different movement trajectories depends on the form of the trajectory and not on how it was produced.

WM: If you think this is worthwhile I know a man who can convert video to x y coordinates by time…
.

RM: Sure, that would be a great.

Best

Rick

Warren

On 22 Aug 2016, at 00:59, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.21.1700)]–
Richard S. Marken&nbs
p;

“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

–
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

On Sun, Aug 21, 2016 at 11:43 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I am not sure they do show that

RM: Neither is anyone else so join the party.

WM: That’s correct for measuring the movement, but not for testing the CV. You know that Rick.

RM: I don’t think I know what you think I know. If we were testing for the variable controlled in movement control we would certainly formulate our hypotheses about the controlled variable in terms of measures of aspects of the observed variations in the x,y positions of the movement itself. That’s what I do in the Mind Reading demo (http://www.mindreadings.com/ControlDemo/Mindread.html) to determine which avatar is being intentionally moved (whose movement is being controlled). It’s also what I did in the object interception research.

WM: I do think that I underst
and your published work and o do think that you think that I understand them too. I just don’t understand your approach to this question.

RM: I can see now that it’s a little more difficult to understand than I thought. I hope you’ll get it eventually.

WM: But I think to do that the CV cannot be movement itself.

RM: I don’t know why not. But it’s not the movement that is controlled in my model anyway; it’s the distance from cursor to target.

WM: …If you can reassure me that your CV is not movement then I am with you…

RM: Yes, I can assure you that the CV in my model is not movement.

Best

Rick

RM: My spreadsheet (Now attached again) shows that all control movement trajectories, whether they ar
e controlled variables or the outputs that keep variables under control, follow a power law to some degree…

RM: …when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory.

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!

[rsmarken]

[From Rick Marken (2016.08.221810)]

···

On Mon, Aug 22, 2016 at 4:42 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I’m confused again! I guess I just don’t believe that movement trajectories exist by themselves, like a mathematical entity.

RM: I’m confused too. What is a movement trajectory that exists by itself?

WM: I think that they have to be formed by some kind of controlling agent

RM: Why a controlling agent? The movement trajectory of a thrown baseball or of a pendulum or of a planet is not formed by a controlling agent and yet, they are movement trajectories.

WM: which must itself be formed from simpler, non-movement trajectory components. But that’s just my belief. It is my reductionism coming in

RM: It seems more like animism than reductionism. You seem to see all movement as being produced purposefully (by a controlling agent) rather than by non-purposefully (by causal forces). Some movement trajectories are, indeed, produced purposefully; some are not. This distinction, by the way, is not made by power law researchers;it would be the first thing to be determined by a PCT researcher. PCT researchers would first determine whether the the observed movement trajectory seems to be produced on purpose (controlled) or not.

RM: My analysis of movement trajectories predicts that the same power law will be found for the same movement trajectory that is produced purposefully (controlled) or not (caused). That would certainly show that you can’t tell how a movement trajectory was produced by looking at the power law coefficient for that trajectory.

Best

Rick

On 22 Aug 2016, at 18:00, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.22.1000)]

–
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

On Mon, Aug 22, 2016 at 1:45 AM, Warren Mansell wmansell@gmail.com wrote:

WM: This is good, getting somewhere even before I can run the demo! I get they the CV is the distance from the cursor to the target, but then it becomes crucial how the movement of the target is programmed.

RM: Why?

WM: If the target movement already corresponds to the power law then the decks are stacked in favour of finding it in the movement of the cursor aren’t they?

RM: Yes, that’s what I have been saying. It’s because the observed fit of a power law to the relationship between curvature and velocity in a movement trajectory depends completely on the form of the trajectory; it has nothing to do with how the movement trajectory was produced.

WM: In the examples of the human squiggle and the larva, I don’t think they are tracking a moving target are they?

RM: The larva is tracking, the human squigglers aren’t. But it makes no difference how the movement is produced, the power relationship that fits the data depends only on the form of the movement trajectory, not on how it was produced.

WM: I am thinking that the Crowd Demo is a better (but not perfect) analogue because the CV is closeness to static points via various obstacles.

RM: The results for the Crowd trajectories would be the same as that for any movement trajectories. The fit of a power function to the data depends on the form of the particular movement trajectory, not on how it was produced.

WM: If the movement of those agents corresponded to the power law it would be more interesting because no patterned movement is built in to the environment or CV. It truly emerges.

RM: It doesn’t emerge by magic. The best fit power function relating curvature to velocity for any of the Crowd trajectories will depend on the shape of the individual trajectory. Trajectories where the variable D contributes little to the variance in velocity will result in a good fit of the 1/3 and 2/3 power function. When D does contributes a lot of variance to velocity the power coefficient will deviate from 1/3 or 2/3 (depending on how velocity and curvature are measured) and the R^2 will be very low.

RM: Act
ually, my current plan is to have the computer create different random squiggle trajectories, measure the power law for each squiggle and then use each squiggle as the target in a 2 D tracking task. The prediction is that the power law found for the tracking data will be very close to being the same as it was for the corresponding computer produced squiggle. This, and the fact that, because of disturbances, the outputs that keep the cursor on target are very different than the resulting tracking movement, will show that the power relationship found for different movement trajectories depends on the form of the trajectory and not on how it was produced.

WM: If you think this is worthwhile I know a man who can convert video to x y coordinates by time…
.

RM: Sure, that would be a great.

Best

Rick

Warren

On 22 Aug 2016, at 00:59, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.21.1700)]–
Richard S. Marken&nbs
p;

“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

–
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

On Sun, Aug 21, 2016 at 11:43 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I am not sure they do show that

RM: Neither is anyone else so join the party.

WM: That’s correct for measuring the movement, but not for testing the CV. You know that Rick.

RM: I don’t think I know what you think I know. If we were testing for the variable controlled in movement control we would certainly formulate our hypotheses about the controlled variable in terms of measures of aspects of the observed variations in the x,y positions of the movement itself. That’s what I do in the Mind Reading demo (http://www.mindreadings.com/ControlDemo/Mindread.html) to determine which avatar is being intentionally moved (whose movement is being controlled). It’s also what I did in the object interception research.

WM: I do think that I underst
and your published work and o do think that you think that I understand them too. I just don’t understand your approach to this question.

RM: I can see now that it’s a little more difficult to understand than I thought. I hope you’ll get it eventually.

WM: But I think to do that the CV cannot be movement itself.

RM: I don’t know why not. But it’s not the movement that is controlled in my model anyway; it’s the distance from cursor to target.

WM: …If you can reassure me that your CV is not movement then I am with you…

RM: Yes, I can assure you that the CV in my model is not movement.

Best

Rick

RM: My spreadsheet (Now attached again) shows that all control movement trajectories, whether they ar
e controlled variables or the outputs that keep variables under control, follow a power law to some degree…

RM: …when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory.

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!

[wmansell]

So by that logic, are you saying that the reason that the behaviour of living things appears to obey the power law is because the living things are tracking the movements of non-living things that obey the power law, presumably by virtue of the physics of the world. That would be a very exciting idea. To test it, we could put a ball on a huge platform on springs and randomly put force at different positions of the platform over time. The movements of the ball could be recorded. If these conform with the power law then that would be strong evidence that the power law has nothing to do with control by living things. Has no one done this?
Warren

···

On Tue, Aug 23, 2016 at 2:08 AM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.221810)]

–

On Mon, Aug 22, 2016 at 4:42 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I’m confused again! I guess I just don’t believe that movement trajectories exist by themselves, like a mathematical entity.

RM: I’m confused too. What is a movement trajectory that exists by itself?

WM: I think that they have to be formed by some kind of controlling agent

RM: Why a controlling agent? The movement trajectory of a thrown baseball or of a pendulum or of a planet is not formed by a controlling agent and yet, they are movement trajectories.

WM: which must itself be formed from simpler, non-movement trajectory components. But that’s just my belief. It is my reductionism coming in

RM: It seems more like animism than reductionism. You seem to see all movement as being produced purposefully (by a controlling agent) rather than by non-purposefully (by causal forces). Some movement trajectories are, indeed, produced purposefully; some are not. This distinction, by the way, is not made by power law researchers;it would be the first thing to be determined by a PCT researcher. PCT researchers would first determine whether the the observed movement trajectory seems to be produced on purpose (controlled) or not.

RM: My analysis of movement trajectories predicts that the same power law will be found for the same movement trajectory that is produced purposefully (controlled) or not (caused). That would certainly show that you can’t tell how a movement trajectory was produced by looking at the power law coefficient for that trajectory.

Best

Rick

On 22 Aug 2016, at 18:00, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.22.1000)]

–
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

WM: This is good, getting somewhere even before I can run the demo! I get they the CV is the distance from the cursor to the target, but then it becomes crucial how the movement of the target is programmed.

RM: Why?

WM: If the target movement already corresponds to the power law then the decks are stacked in favour of finding it in the movement of the cursor aren’t they?

RM: Yes, that’s what I have been saying. It’s because the observed fit of a power law to the relationship between curvature and velocity in a movement trajectory depends completely on the form of the trajectory; it has nothing to do with how the movement trajectory was produced.

WM: In the examples of the human squiggle and the larva, I don’t think they are tracking a moving target are they?

RM: The larva is tracking, the human squigglers aren’t. But it makes no difference how the movement is produced, the power relationship that fits the data depends only on the form of the movement trajectory, not on how it was produced.

WM: I am thinking that the Crowd Demo is a better (but not perfect) analogue because the CV is closeness to static points via various obstacles.

RM: The results for the Crowd trajectories would be the same as that for any movement trajectories. The fit of a power function to the data depends on the form of the particular movement trajectory, not on how it was produced.

WM: If the movement of those agents corresponded to the power law it would be more interesting because no patterned movement is built in to the environment or CV. It truly emerges.

RM: It doesn’t emerge by magic. The best fit power function relating curvature to velocity for any of the Crowd trajectories will depend on the shape of the individual trajectory. Trajectories where the variable D contributes little to the variance in velocity will result in a good fit of the 1/3 and 2/3 power function. When D does contributes a lot of variance to velocity the power coefficient will deviate from 1/3 or 2/3 (depending on how velocity and curvature are measured) and the R^2 will be very low.

On Mon, Aug 22, 2016 at 1:45 AM, Warren Mansell wmansell@gmail.com wrote:
RM: Act
ually, my current plan is to have the computer create different random squiggle trajectories, measure the power law for each squiggle and then use each squiggle as the target in a 2 D tracking task. The prediction is that the power law found for the tracking data will be very close to being the same as it was for the corresponding computer produced squiggle. This, and the fact that, because of disturbances, the outputs that keep the cursor on target are very different than the resulting tracking movement, will show that the power relationship found for different movement trajectories depends on the form of the trajectory and not on how it was produced.

WM: If you think this is worthwhile I know a man who can convert video to x y coordinates by time…
.

RM: Sure, that would be a great.

Best

Rick

Warren

On 22 Aug 2016, at 00:59, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.21.1700)]–
Richard S. Marken&nbs
p;

“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

–
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

On Sun, Aug 21, 2016 at 11:43 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I am not sure they do show that

RM: Neither is anyone else so join the party.

WM: That’s correct for measuring the movement, but not for testing the CV. You know that Rick.

RM: I don’t think I know what you think I know. If we were testing for the variable controlled in movement control we would certainly formulate our hypotheses about the controlled variable in terms of measures of aspects of the observed variations in the x,y positions of the movement itself. That’s what I do in the Mind Reading demo (http://www.mindreadings.com/ControlDemo/Mindread.html) to determine which avatar is being intentionally moved (whose movement is being controlled). It’s also what I did in the object interception research.

WM: I do think that I underst
and your published work and o do think that you think that I understand them too. I just don’t understand your approach to this question.

RM: I can see now that it’s a little more difficult to understand than I thought. I hope you’ll get it eventually.

WM: But I think to do that the CV cannot be movement itself.

RM: I don’t know why not. But it’s not the movement that is controlled in my model anyway; it’s the distance from cursor to target.

WM: …If you can reassure me that your CV is not movement then I am with you…

RM: Yes, I can assure you that the CV in my model is not movement.

Best

Rick

RM: My spreadsheet (Now attached again) shows that all control movement trajectories, whether they ar
e controlled variables or the outputs that keep variables under control, follow a power law to some degree…

RM: …when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory.

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!

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

[rsmarken]

[From Rick Marken (2016.08.23.1545)]

···

On Mon, Aug 22, 2016 at 10:23 PM, Warren Mansell wmansell@gmail.com wrote:

WM: So by that logic, are you saying that the reason that the behaviour of living things appears to obey the power law is because the living things are tracking the movements of non-living things that obey the power law,

RM: No, I’m saying that the power law that is observed for any movement trajectory (produced by any means) is a statistical artifact that results from failure to include the variable D in the regression analysis relating the movement’s curvature (measured as R or V) to its velocity (measured as V or A). I will explain this further after I reply to Bruce A., who seems to be getting all upset about being ignored.

Best

Rick

presumably by virtue of the physics of the world. That would be a very exciting idea. To test it, we could put a ball on a huge platform on springs and randomly put force at different positions of the platform over time. The movements of the ball could be recorded. If these conform with the power law then that would be strong evidence that the power law has nothing to do with control by living things. Has no one done this?
Warren

–
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

On Tue, Aug 23, 2016 at 2:08 AM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.221810)]

–
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

On Mon, Aug 22, 2016 at 4:42 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I’m confused again! I guess I just don’t believe that movement trajectories exist by themselves, like a mathematical entity.

RM: I’m confused too. What is a movement trajectory that exists by itself?

WM: I think that they have to be formed by some kind of controlling agent

RM: Why a controlling agent? The movement trajectory of a thrown baseball or of a pendulum or of a planet is not formed by a controlling agent and yet, they are movement trajectories.

WM: which must itself be formed from simpler, non-movement trajectory components. But that’s just my belief. It is my reductionism coming in

RM: It seems more like animism than reductionism. You seem to see all movement as being produced purposefully (by a controlling agent) rather than by non-purposefully (by causal forces). Some movement trajectories are, indeed, produced purposefully; some are not. This distinction, by the way, is not made by power law researchers;it would be the first thing to be determined by a PCT researcher. PCT researchers would first determine whether the the observed movement trajectory seems to be produced on purpose (controlled) or not.

RM: My analysis of movement trajectories predicts that the same power law will be found for the same movement trajectory that is produced purposefully (controlled) or not (caused). That would certainly show that you can’t tell how a movement trajectory was produced by looking at the power law coefficient for that trajectory.

Best

Rick

On 22 Aug 2016, at 18:00, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.22.1000)]

–
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

WM: This is good, getting somewhere even before I can run the demo! I get they the CV is the distance from the cursor to the target, but then it becomes crucial how the movement of the target is programmed.

RM: Why?

WM: If the target movement already corresponds to the power law then the decks are stacked in favour of finding it in the movement of the cursor aren’t they?

RM: Yes, that’s what I have been saying. It’s because the observed fit of a power law to the relationship between curvature and velocity in a movement trajectory depends completely on the form of the trajectory; it has nothing to do with how the movement trajectory was produced.

WM: In the examples of the human squiggle and the larva, I don’t think they are tracking a moving target are they?

RM: The larva is tracking, the human squigglers aren’t. But it makes no difference how the movement is produced, the power relationship that fits the data depends only on the form of the movement trajectory, not on how it was produced.

WM: I am thinking that the Crowd Demo is a better (but not perfect) analogue because the CV is closeness to static points via various obstacles.

RM: The results for the Crowd trajectories would be the same as that for any movement trajectories. The fit of a power function to the data depends on the form of the particular movement trajectory, not on how it was produced.

WM: If the movement of those agents corresponded to the power law it would be more interesting because no patterned movement is built in to the environment or CV. It truly emerges.

RM: It doesn’t emerge by magic. The best fit power function relating curvature to velocity for any of the Crowd trajectories will depend on the shape of the individual trajectory. Trajectories where the variable D contributes little to the variance in velocity will result in a good fit of the 1/3 and 2/3 power function. When D does contributes a lot of variance to velocity the power coefficient will deviate from 1/3 or 2/3 (depending on how velocity and curvature are measured) and the R^2 will be very low.

On Mon, Aug 22, 2016 at 1:45 AM, Warren Mansell wmansell@gmail.com wrote:
RM: Act
ually, my current plan is to have the computer create different random squiggle trajectories, measure the power law for each squiggle and then use each squiggle as the target in a 2 D tracking task. The prediction is that the power law found for the tracking data will be very close to being the same as it was for the corresponding computer produced squiggle. This, and the fact that, because of disturbances, the outputs that keep the cursor on target are very different than the resulting tracking movement, will show that the power relationship found for different movement trajectories depends on the form of the trajectory and not on how it was produced.

WM: If you think this is worthwhile I know a man who can convert video to x y coordinates by time…
.

RM: Sure, that would be a great.

Best

Rick

Warren

On 22 Aug 2016, at 00:59, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.21.1700)]–
Richard S. Marken&nbs
p;

“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

–
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

On Sun, Aug 21, 2016 at 11:43 AM, Warren Mansell wmansell@gmail.com wrote:

WM: I am not sure they do show that

RM: Neither is anyone else so join the party.

WM: That’s correct for measuring the movement, but not for testing the CV. You know that Rick.

RM: I don’t think I know what you think I know. If we were testing for the variable controlled in movement control we would certainly formulate our hypotheses about the controlled variable in terms of measures of aspects of the observed variations in the x,y positions of the movement itself. That’s what I do in the Mind Reading demo (http://www.mindreadings.com/ControlDemo/Mindread.html) to determine which avatar is being intentionally moved (whose movement is being controlled). It’s also what I did in the object interception research.

WM: I do think that I underst
and your published work and o do think that you think that I understand them too. I just don’t understand your approach to this question.

RM: I can see now that it’s a little more difficult to understand than I thought. I hope you’ll get it eventually.

WM: But I think to do that the CV cannot be movement itself.

RM: I don’t know why not. But it’s not the movement that is controlled in my model anyway; it’s the distance from cursor to target.

WM: …If you can reassure me that your CV is not movement then I am with you…

RM: Yes, I can assure you that the CV in my model is not movement.

Best

Rick

RM: My spreadsheet (Now attached again) shows that all control movement trajectories, whether they ar
e controlled variables or the outputs that keep variables under control, follow a power law to some degree…

RM: …when researchers determine the fit of the power law to the movement trajectory they do it by measuring curvature and velocity from the observed variations in x,y positions of the movement trajectory.

WM: To me, this doesn’t require rearranging the equations but simply running the hypothetical control systems, collecting movement data and doing the existing formula on it.

RM: That’s exactly what my spreadsheet does!!!