Bogus mathematics, (was Re: L'état de PCT, c'es t moi (was ...))

[bnhpct]

[Bruce Nevin 2018-07-19_09:46:11 ET]

Martin Taylor 2018.07.17.17.16 –

Rick, Martin listed “eight … falsehoods you incorporated in your rebuttal.” You replied “they are not “falsehoods” but the best we could do to understand your criticisms.” That seems to affirm that you did not understand his criticisms very well.Â

I know of two ways to demonstrate understanding, and both of them involve a test of understanding that is akin to the Test for controlled variables. One of the two ways is to apply what is understood. This demonstrates control of the perceptions intended by the words. The other way is to paraphrase in different words and ask if the paraphrase is correct. This is similar to e.g. the Coin Game.

Would it be a fair paraphrase of your to enclose each of the eight pairs (statements in Martin’s list and your rejoinders to them) in this frame?

When you said [quote from Martin’s rebuttal] it appeared to us that you meant [quote from your rebuttal of the rebuttal]. Is that what you intended? If so, [further rebuttal].

You actually did this, in effect, at this point of your reply:

It seems to me, naively, that this is not an accurate paraphrase. I think Martin’s point isÂ

a. that one form of the equation is a generalization across all possible velocities,Â

b. that the other form of the equation can be applied only to particular velocity data from a particular experiment, andÂ

c. that you employed the latter (b) as though it were equivalently (a) a generalization across all possible velocities.

Only Martin can say whether or not I have accurately paraphrased what he wrote. If he affirms that I did, are these paraphrase statements incorrect?

I think you have a kind of important typographical error here:

I think you meant to say “we said that your critique was based on your misunderstanding of those equations.” Is that correct? Are there possibly other misstatements confusing the discussion?

···

Rick Marken 2018-07-17_10:31:31 --​

This dispute seems at last to be converging toward common perceptions of what is in dispute, but I still am not understanding it.

RM: What you are saying is that we made the mistake of taking the dot derivatives in the two Gribble/Ostry equations as being time derivatives.

RM: No,
​​

we said that your critique was based on our misunderstanding of those equations. Specifically that the derivatives in the curvature equation were different from those in the velocity equation. Your claim that these derivatives are different is simply wrong and, thus, invalidates your mathematical critique from the get go.

[Rick Marken 2018-07-17_10:31:31]

[Martin Taylor 2018.07.16.15.12]

            MT: As well you have known for a very long time, I have

insufficient hubris to attempt a model of observed
behaviour before trying the TCV to figure out what
variable(s) might be being controlled during the task. I
have no means to do the TCV needed, so I refrain from
suggesting a model. You are not so inhibited.

          RM: You have to have had some idea of what the

controlled variable might be when people make curved
movements or you wouldn’t know that the power law is
"almost certainly a side-effect in any
of the experiments that find velocity to have a near
power-law relationship to the radius of curvature ",
as you note in your rebuttal. In PCT, a “side-effect” is a
relationship between variables that exists because a
variable is under control but this relationship not part
of the process that results in control of that variable.
For example, the relationship between disturbance and
output in a tracking task is a side effect of controlling
the position of the cursor but is not part of the process
that results in control of cursor position. In order to
know that the power law is, indeed, a side-effect, you had
to have an idea of what variable is under control when
people make curved movements as well as having an idea of
how the instantaneous curvature and velocity of these
movements are related to this variable. This should have
been enough to let you develop a first approximation to a
model of curved movements that would demonstrate why the instantaneous
curvature and velocity of these movements is a side
effect of controlling this variable. The model itself
would have been a basis for the TCV; it would be a test
of the correctness of your hypothesis regarding the
variable under control. So it would not have been
hubris to model the behavior before doing the TCV since
you presumably had to have had the essential components of
the model in mind when you said that the power law is
almost certainly a side effect.Â

            MT: For the record, here are just

eight of the falsehoods you incorporated in your
rebuttal of my comment on the Marken and Shaffer paper
(copied from [Martin Taylor 2018.03.08.23.07]). Despite
having been made aware of their falsity, yet you
continue to repeat some of them on CSGnet. Why do you do
that?

          RM: Because  they are not "falsehoods" but the best we

could do to understand your criticisms.

Â

            ----------begin quote (replacing

references to “you” with references to “they”, and added
numbering)-------

            *                  MT: (1) In the very first paragraph you claim that my

reason for writing a critique was that the idea that
the power law might be a behavioural illusion caused
“consternation”, whereas I made explicit that nothing
in my critique had any bearing on that issue. Indeed,
I finished my critique with the statement that perhaps
the power law is indeed a behavioural illusion, though
M&S sheds no light on that issue.*

          RM: Since, as I noted above, you came up with no

hypothesis about what variable might be controlled, I
dismissed your claims of accepting that the power law is a
behavioral illusion because you gave no evidence of
understanding what a behavioral illusion is.

Â

•              MT: (2) M&S say that my
  

critique of their use of Gribble and Ostry’s equations
is based on my belief that those equations are wrong
or misleading, whereas I pointed out that they are
well known and universally accepted equations for
using observed data to measure the velocity (equation

1. and curvature (equation 2) profiles observed in an
   experiment. Neither Gribble and Ostry nor (so far as I
   know) anyone other than Marken and Shaffer ever
   claimed that the observed velocity was the only
   velocity that could be used to get the correct
   curvature from the equation for R.*

          RM: No, we said that your critique was based on our

misunderstanding of those equations. Specifically that the
derivatives in the curvature equation were different from
those in the velocity equation. Your claim that these
derivatives are different is simply wrong and, thus,
invalidates your mathematical critique from the get go.

Â

•              MT: (3) I never said that the
  

derivation of V = R**^1/3D^1/3** was wrong. I said that since the formula for D was
velocity (V) times a constant in spatial variables,
the equation is not an equation from which one can
determine V. The M&S claim that it is an equation
from which one can determine V is the core of my
critique.*

          RM: And we never said that you said that the derivation

of

•            V
  

= R**^1/3D^1/3**Â * was
wrong. We said that what you said about it not being an
equation that can be used to predict V using linear
regression is wrong. Which it is.Â

•              MT: (4) M&S falsely claim that I argue that
  

“it should have been obvious that X-dot and Y-dot are
derivatives with respect to time in the expression for
V, whereas they are derivatives with respect to space
in the expression for R (p. 5)”. On the contrary, I
devote the first couple of pages of my critique to
showing why, despite the radius of curvature being a
spatial property, nevertheless it is quite proper to
use time derivatives in the formula for R.*

RM: But that’s what you argued, right here:Â

          RM: What you are saying is that we made the mistake of

taking the dot derivatives in the two Gribble/Ostry
equations as being time derivatives. But that was not
mistake. The mistake is all yours.

•              MT: (5) M&S say that because
  

Gribble and Ostry correctly transformed Viviani and
Stucchi’s expression for R using spatial derivatives
into one using time derivatives (a derivation with
which I started my comment), therefore they were
correct to say that ONLY the velocity found in an
experiment can be substituted into the numerator of
the expression for R, whereas both my derivation and
that of Viviani and Stucchi (essentially the same)
makes it crystal clear that this is not true.*

          RM: Well, that would be news to all the power law

researchers who computed velocity and curvature the way I
did in my analyses, using time derivatives.

Â

•              MT: (6) M&S follow this
  

astounding assertion with an couple of paragraphs to
show why the V = R**^1/3D^1/3** equation is correct, implying that my comment claimed
it to be wrong. Early in my comment, however, I wrote:
“They then write their key Eq (6) [V = R**^1/3D^1/3** ],
which is true for any value of V whatever…” Any
implication that my comment claimed the equation to be
incorrect is false.*

          RM: What we showed is that that equation has been used

by others to show what we showed in our paper – that
using only R (curvature) as the predictor in a regression
on V (speed) – will result in an estimate of the power
coefficient of R that deviates from 1/3 by an amount
proportional to the correlation between R and D (radial
velocity).Â

Â

•              MT: (7) Omitted Variable Bias:
  

My comment demonstrated that the finding predicted and
reported by M&S was actually a tautology having no
relation to experimental findings, which will always
produce the result claimed by M&S to be an
experimental result. M&S in the paper and in the
rebuttal treat it as a discovery that can be made only
by careful statistical analysis, and do not
acknowledge the tautology criticism at all.*

          RM: Your demonstration that our findings are a

“tautology” made no sense to us. You made this claim based
on your derivation of an equation for V of the form V = V.
But this is true for any equation. If X = f(Y) then you
can substitute X for the right side of the equation and
write the equation X = X. That’s not a tautology; that’s
just an irrelevant observation.

•              MT: (8) M&S: "At the heart of the criticisms
  

of our paper by Z/M and Taylor is the assumption that
the power law is a result of a direct causal
connection between curvature and speed of movement or
between these variables and the physiological
mechanisms that produce them." I have no idea how this
astonishing statement can be derived from my
exposition of the mathematical and logical flaws in
their paper. My comment is designed to refute exactly
M&S’s claim of my motivation. The comment shows
that there is NO necessary relationship, causal
connection or otherwise, between curvature and speed
of movement.
*

          RM: You were apparently trying to show, mathematically,

that the curvature and velocity of a curved movement are
physically independent, like the disturbance and output in
a tracking task. Since you didn’t speculate about the
controlled variable that might be simultaneously affected
by these two variables I assumed that you were dong this
to justify the assumptions of power law researchers that
these two variables are either causally related or
simultaneously caused by a third variable.Â

Â

--------end quote-------

            MT: I repeat from my last message: *"*                What's the

advantage to you of refusing to deal with scientific
points people bring up about your work?"

          RM: We dealt with your confusing rebuttal as best we

could. There was nothing scientific about it inasmuch as
it was purely mathematical.

Best

Rick

Â

            Well, I guess predictions aren't always wrong,

and I am indeed not surprised.

                Martin

–
Richard S. MarkenÂ

                                  "Perfection

is achieved not when you have
nothing more to add, but when you
have
nothing left to take away.�
  Â
            Â
–Antoine de Saint-Exupery

Best

Rick

Â

                              What's the

advantage to you of refusing to deal
with scientific points people bring up
about your work? In what perception
you control would it create error if
you were to accept normal mathematics
or physics as being valid? When your
work is good, it’s good, but when you
make a mistake, why does it seem so
difficult for you to correct it? In
the curvature paper none of the
criticisms were relevant to a PCT
interpretation, but you make out that
all of them were intended to refute a
“correct PCT analysis” of the
experimental findings. Why?

                              I don't expect an answer to a question

raised, but I wouldn’t be surprised at
an answer to something completely
different.

[rsmarken]

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[rsmarken]

[Rick Marken 2018-07-19_16:57:19]

[Martin Taylor 2018.07.19.14.09]

MT: Rick, could you help me correct my "bogus mathematics" by pointing out by page and paragraph or by equation number specifically where the mathematics in my comment on the Marken and Shaffer paper is "bogus".

RM: The mathematics are fine. It's the conclusions that are wrong. A particularly egregious example is your "proof" that our equation relating V to R and D (V = R^1/3*D^1/3, equation 6 in your paper) is a tautology. You do this by showing (correctly, I assume) that D^1/3 is equivalent to V*(1/(R^1/3))Â

MT: Of course, that is NOT at all what I showed...Since we are talking about my comment and not your rebuttal, I'll use my numbering.

MT: (1) is the standard expression for R in purely spatial variables, those being the x,y coordinates of a place along a curve, and s being the distance along the curve to that point from some arbitrary zero point. Simple physical argument indicates that a description of a spatial quantity (such as curvature or radius of curvature) must be a function of purely spatial variables, and if non-spatial variables are used, for convenience, they must cancel out of the expression actually used for the calculation.

 RM: Does this mean that the formulas we used to calculate R and V (and C and A) from the data are incorrect?>

(2) shows how this cancellation works for the substitution of an arbitrary parameter "z" that is a function of "s". It shows that no matter what z might be, if it has a continuous derivative dz/ds or the inverse ds/dz, the expression for R in (1) can be transformed into the equivalent form in z. Depending on the direction the equivalence is shown, numerator and denominator each have a multiplying factor (ds/dz)3 or (dz/ds)3. These multipliers cancel out, which is why the substitution of z for s (or vice-versa) produces the same result for any z.

RM: What does this have to do with our analysis? That is, how does it relate to the findings of our regression analyses?>

MT: In (3), z is taken to be the time it takes for an object that moves arbitrarily along the curve without stopping or retracing to reach the point at which the derivatives are taken. In this case, the numerator of the expression simplifies to (ds/dt)3 = V3. In this equation and the last equivalence of (2), the denominator is Marken and Shaffer's "cross-correlation correction factor" D. If the argument so far has not made it clear that D is V3*f(x,y,s), equation (7) later demonstrates it explicitly. As is necessarily true from basic physical principles, the explicit calculations demonstrate that the general point mentioned above for an arbitrary parameter z holds also if the parameter is time or velocity. The effects of the added variable (in this case V) cancel out.

RM: So why did our regression analyses work so well? What did we do wrong?
Â

MT: Marken and Shaffer choose to ignore the generality of the parameter substitution and the fact that in their specific substitution of the measured velocities for a single experimental run V3 cancels out from numerator and denominator of the fraction that is the expression for R. Instead, they leave V3 explicitly in the numerator, but hide it in their newly discovered "Cross-correlation correction factor". They then use the "cccf" as though it were independent of V in the rest of their paper.

RM: We didn't ignore this. We knew nothing about it. All we knew was what we found in the reports of research on the power law. And there was nothing in the literature about the "generality of the parameter substitution" of which you speak. And what was, indeed, our newly discovered "cross correlation" variable (D) turns out to be a well known parameter of curved movement: affine velocity.Â
Â

MT: I think this is, to put it mildly, a little different from what Rick said above that I showed.

RM: I really tried to find some relevance of your mathematical analysis to the research we described in our power law paper. But I'm not sure there is any relevance because you don't seem to understand -- or want to understand -- what we did. This is evidenced by what you say at the beginning of your mathematical critique of our work: "Accordingly, they assert that measured values of the power law that depart from 1/3 are in error because they omit consideration of D". In fact, we never "asserted" this. What we demonstrated is that measured values of the power law coefficient will depart from 1/3 (for the relationship between R and V and 2/3 for the relationship between C and A) to the extent that the variable D, which power law researchers always omit from the regression analysis, covaries with the curvature variable (R or C) that is included as the predictor variable in the analysis. >

MT: Try again, Rick. I keep hoping to be able to learn something from one of your postings, but I haven't won this lottery jackpot yet.

RM: Sure, I'll try again. But you might have better luck if you would explain, as clearly as possible, how your mathematical analysis relates to our regression analysis of actual data from curved movements.
Â

MT: If I have made a mathematical error in my other comments on Marken and Shaffer, I really would like to know. But you please comment on what I wrote, rather than on something you invented, as you did in this case.

RM: As I said before, I don't think you have made any mathematical errors. I just don't see the relevance of your mathematical analysis to what we actually did with our analysis of actual curved movement data. Did we use the wrong formulas to calculate instantaneous velocity and curvature? Did we do the regression incorrectly? Did we use the wrong variables in the regressions?Â
Best
Rick
 >

···

Martin

so that V = R^1/3*D^1/3 = R^1/3* V*(1/(R^1/3)) which, of course, reduces to V=V.Â
RM: But as I've said, that's true of any equation. The fact that V = R^1/3*D^1/3 can be reduced to V = V doesn't negate the value of knowing that V = R^1/3*D^1/3. This equation analyzes V into its components just as simple one way analysis of variance (ANOVA) analyzes the total variance in scores in an experiment (MS.total) into two components, the variance in scores across (MS.between) and within (MS.within) conditions, so that MS.total = MS.between + MS.within. This is the basic equation of ANOVA.Â
RM: Of course, it's possible to show that MS.total = MS.between + MS.within is a "tautology": MS.total = MS.total. We can do this by noting that MS.within = MS.total - MS.between so that MS.total = MS.between + MS.total - MS.between which, you'll note, reduces to MS.total = MS.total.
RM: But by analyzing MS.total into MS.between and MS.within we can learn some interesting things about the data by computing the two variance components of MS.total and forming the ratio MS.between/MS.within, a ratio known as F (for Sir Ronald Fisher, who invented this analysis method and, as far as I know, never caught flack from anyone about the basic equation of ANOVA being a tautology). Knowing the probability of getting different F ratios in experiments where the independent variable has no effect (the null hypothesis), it is possible to use the F ratio observed in an experiment to decide whether one can reject the null hypothesis with a sufficiently small probability of being wrong.Â
RM: Just as it has proved useful to analyze the total variance in experiments ( MS.total)Â into variance component (MS.between, MS.within and sometimes MS.interaction and MS subjects) it proved useful to us to analyze the variance in the velocity, V, of a curved movement into components, R and D. This analysis produced the equation V = R^1/3*D^1/3. R and D are measures of two different components of the temporal variation in curved movement just as MS.between and MS.within are measures of two different components of the variation in the scores observed in an experiment; R is the variation in curvature and D is the variation in affine velocity.Â
RM: Our equation says that the variation in V for a curved movement will be exactly equal to R^1/3*D^1/3. Linearizing this equation by taking the log of both sides we get log (V) = 1/3*log (R) +1/3*log (D) . This equation shows that if one did a linear regression using the variables log(R) and log(D) as predictors and the variable log(V) as the criterion, the coefficients of the two predictor variables would be exactly 1/3 with an intercept of 0. More importantly, this equation shows that if the variable log (D) isomitted from the regression, the coefficient of log(R) will not necessarily be found to be exactly 1/3 and the intercept will not necessarily be found to be exactly 0. This is where Omitted Variable Bias (OVB) analysis comes in. This analysis makes if possible to predict exactly what a regression analysis will find the coefficient of log(R) to be if log(D) is omitted from the regression.
RM: This finding is important because the "power law" of movement is determined by doing a regression of log (R) on log (V) using the regression equation log (V) = k + b*log(R), omitting the variable log(D). The term "power law" refers to the fact that the results of this regression analysis consistently finds that the power coefficient b is close to 1/3. Our analysis shows that this is a statistical artifact that results from having left the variable log(D) out of the regression analysis. OVB analyiss shows that the amount by which the b coefficient is found to deviate from 1/3 depends on the degree of covariation between the variable included in the regression (log (R)) and the variable omitted from the regression (log(D)). Since both log (R) and log (D) are measured from data (temporal variations in the x,y position of the curved movement) the covariation between these variables is easily calculated and the predicted deviation of the power coefficient, b, from 1/3 can be exactly predicted.Â
RM: The covariation between log (R) and log (D) depends on the nature of the curved movement trajectory itself and has nothing to do with how that movement was generated. It is in this sense that the observed power law is a "behavioral illusion", the illusion being that the relatively consistent observation of an approximately 1/3 power relationship between the curvature (R) and velocity (V) of curved movements seems to reveal something important about how these movements are produced, when it doesn't.Â
RM: So the fact that the equation V = R^1/3*D^1/3 can be reduced to V = V does not negate the value of analyzing V into its components any more than the fact that the equation MS.total = MS.between + MS.within can be reduced to MS.total = MS.total negates the value of analyzing MS.total into its components.Â
RM: There are many other incorrect conclusions in your rebuttal to our paper, Martin. But I think this is enough for now since your "tautology" claim (based on our alleged mathematical mistake) seemed to be central to your argument.Â
BestÂ
Rick

You can do this without referring either to your rebuttal or to the eight falsehoods that I asked you not to try to justify at this point. My question is not about them, but specifically about what in my mathematics you have shown to be bogus. Your previous response did not address this question.

Martin

--

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[wmansell]

Hi guys, putting the maths aside for now. How about this as a solution? The faster you go round a corner, the greater the centripetal force you need to counteract, and the more likely it is that one’s lateral velocity will overcome friction and lead to an unstable trace (a ‘skid’ if you like) . So, if there are limits on the force you can apply, then it makes sense to slow down as you go into a bend and speed up when you come out of it. But of course this slowing down and speeding up requires a force too. So might the most efficient relationship between curvature and velocity be one at which the force required to slow down when coming to a bend (increase in curvature) or speed up when coming out of it (decrease in curvature), is no greater than the force one would need to apply to counteract the centripetal force as one goes round the bend of a specific curvature at that velocity. Expressing this in maths is beyond me at the moment, but I think if this is the reason for the power law then it would be independent of ‘how’ the trace is produced by the person/vehicle/animal because it would be a function of only the curvature of the trace and not the actions used to achieve it. It should also generate some predictions regarding a direct inverse relationship between on-track acceleration and the derivative of curvature (increase or decrease in curvature as one goes round the trace).Â

···

On Fri, Jul 20, 2018 at 12:57 AM, Richard Marken csgnet@lists.illinois.edu wrote:

–

[Rick Marken 2018-07-19_16:57:19]

[Martin Taylor 2018.07.19.14.09]

MT: Of course, that is NOT at all what I showed...Since we are talking about

my comment and not your rebuttal, I’ll use my numbering.

MT: (1) is the standard expression for R in purely spatial variables,

those being the x,y coordinates of a place along a curve, and s
being the distance along the curve to that point from some arbitrary
zero point. Simple physical argument indicates that a description of
a spatial quantity (such as curvature or radius of curvature) must
be a function of purely spatial variables, and if non-spatial
variables are used, for convenience, they must cancel out of the
expression actually used for the calculation.

 RM: Does this mean that the formulas we used to calculate R and V (and C and A) from the data are incorrect?

(2) shows how this cancellation works for the substitution of an

arbitrary parameter “z” that is a function of “s”. It shows that no
matter what z might be, if it has a continuous derivative dz/ds or
the inverse ds/dz, the expression for R in (1) can be transformed
into the equivalent form in z. Depending on the direction the
equivalence is shown, numerator and denominator each have a
multiplying factor (ds/dz)³ or (dz/ds)³ . These
multipliers cancel out, which is why the substitution of z for s (or
vice-versa) produces the same result for any z.

RM: What does this have to do with our analysis? That is, how does it relate to the findings of our regression analyses?

MT: In (3), z is taken to be the time it takes for an object that moves

arbitrarily along the curve without stopping or retracing to reach
the point at which the derivatives are taken. In this case, the
numerator of the expression simplifies to (ds/dt)³ = V³ .
In this equation and the last equivalence of (2), the denominator is
Marken and Shaffer’s “cross-correlation correction factor” D. If the
argument so far has not made it clear that D is V3*f(x,y,s),
equation (7) later demonstrates it explicitly. As is necessarily
true from basic physical principles, the explicit calculations
demonstrate that the general point mentioned above for an arbitrary
parameter z holds also if the parameter is time or velocity. The
effects of the added variable (in this case V) cancel out.

RM: So why did our regression analyses work so well? What did we do wrong?

Â

MT: Marken and Shaffer choose to ignore the generality of the parameter

substitution and the fact that in their specific substitution of the
measured velocities for a single experimental run V³
cancels out from numerator and denominator of the fraction that is
the expression for R. Instead, they leave V³ explicitly
in the numerator, but hide it in their newly discovered
“Cross-correlation correction factor”. They then use the “cccf” as
though it were independent of V in the rest of their paper.

RM: We didn’t ignore this. We knew nothing about it. All we knew was what we found in the reports of research on the power law. And there was nothing in the literature about the “generality of the parameter substitution” of which you speak. And what was, indeed, our newly discovered “cross correlation” variable (D) turns out to be a well known parameter of curved movement: affine velocity.Â

Â

MT: I think this is, to put it mildly, a little different from what Rick

said above that I showed.

RM: I really tried to find some relevance of your mathematical analysis to the research we described in our power law paper. But I’m not sure there is any relevance because you don’t seem to understand – or want to understand – what we did. This is evidenced by what you say at the beginning of your mathematical critique of our work:Â “Accordingly, they assert that measured values of the power law that depart from 1/3 are in error because they omit consideration of D”. In fact, we never “asserted” this. What we demonstrated is that measured values of the power law coefficient will depart from 1/3 (for the relationship between R and V and 2/3 for the relationship between C and A) to the extent that the variable D, which power law researchers always omit from the regression analysis, covaries with the curvature variable (R or C) that is included as the predictor variable in the analysis.Â

MT: Try again, Rick. I keep hoping to be able to learn something from

one of your postings, but I haven’t won this lottery jackpot yet.

RM: Sure, I’ll try again. But you might have better luck if you would explain, as clearly as possible, how your mathematical analysis relates to our regression analysis of actual data from curved movements.

Â

MT: If

I have made a mathematical error in my other comments on Marken and
Shaffer, I really would like to know. But you please comment on what
I wrote, rather than on something you invented, as you did in this
case.

RM: As I said before, I don’t think you have made any mathematical errors. I just don’t see the relevance of your mathematical analysis to what we actually did with our analysis of actual curved movement data. Did we use the wrong formulas to calculate instantaneous velocity and curvature? Did we do the regression incorrectly? Did we use the wrong variables in the regressions?Â

Best

Rick

Â

Martin

–
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

            MT: Rick, could you help me correct my "bogus

mathematics" by pointing out by page and paragraph or by
equation number specifically where the mathematics in my
comment on the Marken and Shaffer paper is “bogus”.

          RM: The mathematics are fine. It's the conclusions that

are wrong. A particularly egregious example is your
“proof” that our equation relating V to R and D (V =
R^1/3D^1/3, equation 6 in your paper) is a tautology.Â
You do this by showing (correctly, I assume) that D^1/3 is
equivalent to V(1/(R^1/3))Â

          so that 
            V

= R^1/3D^1/3 =Â R^1/3
V*(1/(R^1/3))
which, of course, reduces to V=V.Â

            RM:

But as I’ve said, that’s true of any equation. The fact
thatÂ
V
= R^1/3D^1/3 can be reduced to V = V doesn’t negate
the value of knowing thatÂ
V
= R^1/3D^1/3. This equation analyzes V into its
components just as simple one way analysis of
variance (ANOVA) analyzes the total variance in
scores in an experiment (MS.total) into two
components, the variance in scores across
(MS.between) and within (MS.within) conditions, so
that MS.total = MS.between + MS.within. This is the
basic equation of ANOVA.Â

                RM:

Of course, it’s possible to show that MS.total
= MS.between + MS.within is a
“tautology”:Â MS.total = MS.total. We can do this by noting
that MS.within = MS.total - MS.between so thatÂ
MS.total = MS.between + MS.total - MS.between which,
you’ll note, reduces toÂ
MS.total = MS.total.

          RM:  But by analyzing MS.total into MS.between and

MS.within we can learn some interesting things about the
data by computing the two variance components of MS.total
and forming the ratio MS.between/MS.within, a ratio known
as F (for Sir Ronald Fisher, who invented this analysis
method and, as far as I know, never caught flack from
anyone about the basic equation of ANOVA being a
tautology). Knowing the probability of getting different F
ratios in experiments where the independent variable has
no effect (the null hypothesis), it is possible to use the
F ratio observed in an experiment to decide whether one
can reject the null hypothesis with a sufficiently small
probability of being wrong.Â

                                RM:

Just as it has proved useful to
analyze the total variance in
experiments (
MS.total) Â into
variance component ( MS.between,
MS.within and sometimes
MS.interaction and MS subjects) it
proved useful to us to analyze the
variance in the velocity, V, of a
curved movement into components, R
and D. This analysis produced the
equation V = R^1/3*D^1/3. R and D
are measures of two different
components of the temporal
variation in curved movement just
as MS.between and MS.within are
measures of two different
components of the variation in the
scores observed in an experiment;
R is the variation in curvature
and D is the variation in affine
velocity.Â

                                  RM:

Our equation says that the
variation in V for a curved
movement will be exactly equal to
R^1/3D^1/3. Linearizing this
equation by taking the log of both
sides we get log (V) = 1/3log (R)
+1/3*log (D) . This equation shows
that if one did a linear
regression using the variables
log(R) and log(D) as predictors
and the variable log(V) as the
criterion, the coefficients of the
two predictor variables would be
exactly 1/3 with an intercept of
0. More importantly, this equation
shows that if the variable log (D)
isomitted from the regression, the
coefficient of log(R) will not
necessarily be found to be exactly
1/3 and the intercept will not
necessarily be found to be exactly
0. This is where Omitted Variable
Bias (OVB) analysis comes in. This
analysis makes if possible to
predict exactly what a regression
analysis will find the coefficient
of log(R) to be if log(D) is
omitted from the regression.

                                  RM:

This finding is important because
the “power law” of movement is
determined by doing a regression
of log (R) on log (V) using the
regression equation log (V) = k +
b*log(R), omitting the variable
log(D). The term “power law”
refers to the fact that the
results of this regression
analysis consistently finds that
the power coefficient b is close
to 1/3. Our analysis shows that
this is a statistical artifact
that results from having left the
variable log(D) out of the
regression analysis. OVB analyiss
shows that the amount by which the
b coefficient is found to deviate
from 1/3 depends on the degree of
covariation between the variable
included in the regression (log
(R)) and the variable omitted from
the regression (log(D)). Since
both log (R) and log (D) are
measured from data (temporal
variations in the x,y position of
the curved movement) the
covariation between these
variables is easily calculated and
the predicted deviation of the
power coefficient, b, from 1/3 can
be exactly predicted.Â

                                  RM:

The covariation between log (R)
and log (D) depends on the nature
of the curved movement trajectory
itself and has nothing to do with
how that movement was generated.
It is in this sense that the
observed power law is a
“behavioral illusion”, the
illusion being that the relatively
consistent observation of an
approximately 1/3 power
relationship between the curvature
(R) and velocity (V) of curved
movements seems to reveal
something important about how
these movements are produced, when
it doesn’t.Â

                                  RM:

So the fact that the equation V =
R^1/3*D^1/3 can be reduced to V =
V does not negate the value of
analyzing V into its components
any more than the fact that the
equation MS.total
= MS.between + MS.within can be
reduced to MS.total
= MS.total negates the value
of analyzing MS.total into its
components.Â

                                      RM:

There are many other incorrect
conclusions in your rebuttal
to our paper, Martin. But I
think this is enough for now
since your “tautology” claim
(based on our alleged
mathematical mistake) seemed
to be central to your
argument.Â

BestÂ

Rick

            You can do this without referring either to your

rebuttal or to the eight falsehoods that I asked you not
to try to justify at this point. My question is not
about them, but specifically about what in my
mathematics you have shown to be bogus. Your previous
response did not address this question.

            Martin

–
Richard S. MarkenÂ

                                  "Perfection

is achieved not when you have
nothing more to add, but when you
have
nothing left to take away.�
  Â
            Â
–Antoine de Saint-Exupery

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
Â
Tel: +44 (0) 161 275 8589
Â
Website: http://www.psych-sci.manchester.ac.uk/staff/131406
Â
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

[bnhpct]

[Bruce Nevin 2018-07-20_08:47:34 ET]

···

​RM: ​

​

we said that your critique was based on our misunderstanding of those equations.

​MMT: ​

I interpreted them as meaning … that I thought that they misunderstood the equations.Â

                    Rick Marken

2018-07-17_10:31:31 --​

                    This dispute

seems at last to be converging toward common
perceptions of what is in dispute, but I still
am not understanding it.

                RM: What you are saying is that we made the mistake

of taking the dot derivatives in the two
Gribble/Ostry equations as being time derivatives.

RM: No,
​​

              we said that your critique was based on our

misunderstanding of those equations. Specifically that
the derivatives in the curvature equation were
different from those in the velocity equation. Your
claim that these derivatives are different is simply
wrong and, thus, invalidates your mathematical
critique from the get go.

                [Rick Marken

2018-07-17_10:31:31]

[Martin Taylor 2018.07.16.15.12]

                    MT: As well you have known for a very long time,

I have insufficient hubris to attempt a model of
observed behaviour before trying the TCV to
figure out what variable(s) might be being
controlled during the task. I have no means to
do the TCV needed, so I refrain from suggesting
a model. You are not so inhibited.

                  RM: You have to have had some idea of what the

controlled variable might be when people make
curved movements or you wouldn’t know that the
power law is "almost certainly a side-effect in any
of the experiments that find velocity to have a
near power-law relationship to the radius of
curvature ", as you note in your rebuttal.
In PCT, a “side-effect” is a relationship between
variables that exists because a variable is under
control but this relationship not part of the
process that results in control of that variable.
For example, the relationship between disturbance
and output in a tracking task is a side effect of
controlling the position of the cursor but is not
part of the process that results in control of
cursor position. In order to know that the power
law is, indeed, a side-effect, you had to have an
idea of what variable is under control when people
make curved movements as well as having an idea of
how the instantaneous curvature and velocity of
these movements are related to this variable. This
should have been enough to let you develop a first
approximation to a model of curved movements that
would demonstrate why the instantaneous
curvature and velocity of these movements is a
side effect of controlling this variable. The
model itself would have been a basis for the
TCV; it would be a test of the correctness of
your hypothesis regarding the variable under
control. So it would not have been hubris
to model the behavior before doing the TCV since
you presumably had to have had the essential
components of the model in mind when you said that
the power law is almost certainly a side effect.Â

                    MT: For the record, here

are just eight of the falsehoods you
incorporated in your rebuttal of my comment on
the Marken and Shaffer paper (copied from
[Martin Taylor 2018.03.08.23.07]). Despite
having been made aware of their falsity, yet you
continue to repeat some of them on CSGnet. Why
do you do that?

                  RM: Because  they are not "falsehoods" but the

best we could do to understand your criticisms.

Â

                    ----------begin quote

(replacing references to “you” with references
to “they”, and added numbering)-------

                    *                          MT: (1) In the very first paragraph you claim

that my reason for writing a critique was that
the idea that the power law might be a
behavioural illusion caused “consternation”,
whereas I made explicit that nothing in my
critique had any bearing on that issue.
Indeed, I finished my critique with the
statement that perhaps the power law is indeed
a behavioural illusion, though M&S sheds
no light on that issue.*

                  RM: Since, as I noted above, you came up with

no hypothesis about what variable might be
controlled, I dismissed your claims of accepting
that the power law is a behavioral illusion
because you gave no evidence of understanding what
a behavioral illusion is.

Â

•                      MT: (2) M&S say that
  

my critique of their use of Gribble and
Ostry’s equations is based on my belief that
those equations are wrong or misleading,
whereas I pointed out that they are well known
and universally accepted equations for using
observed data to measure the velocity
(equation 1) and curvature (equation 2)
profiles observed in an experiment. Neither
Gribble and Ostry nor (so far as I know)
anyone other than Marken and Shaffer ever
claimed that the observed velocity was the
only velocity that could be used to get the
correct curvature from the equation for R.*

                  RM: No, we said that your critique was based on

our misunderstanding of those equations.
Specifically that the derivatives in the curvature
equation were different from those in the velocity
equation. Your claim that these derivatives are
different is simply wrong and, thus, invalidates
your mathematical critique from the get go.

Â

•                      MT: (3) I never said
  

that the derivation of V = R**^1/3D^1/3** was wrong. I said that since the formula for D
was velocity (V) times a constant in spatial
variables, the equation is not an equation
from which one can determine V. The M&S
claim that it is an equation from which one
can determine V is the core of my critique.*

                  RM: And we never said that you said that the

derivation of * V
= R**^1/3D^1/3**Â * was
wrong. We said that what you said about it not
being an equation that can be used to predict V
using linear regression is wrong. Which it is.Â

•                      MT: (4) M&S falsely claim that I
  

argue that “it should have been obvious that
X-dot and Y-dot are derivatives with respect
to time in the expression for V, whereas they
are derivatives with respect to space in the
expression for R (p. 5)”. On the contrary, I
devote the first couple of pages of my
critique to showing why, despite the radius of
curvature being a spatial property,
nevertheless it is quite proper to use time
derivatives in the formula for R.*

RM: But that’s what you argued, right here:Â

                  RM: What you are saying is that we made the

mistake of taking the dot derivatives in the two
Gribble/Ostry equations as being time derivatives.
But that was not mistake. The mistake is all
yours.

•                      MT: (5) M&S say that
  

because Gribble and Ostry correctly
transformed Viviani and Stucchi’s expression
for R using spatial derivatives into one using
time derivatives (a derivation with which I
started my comment), therefore they were
correct to say that ONLY the velocity found in
an experiment can be substituted into the
numerator of the expression for R, whereas
both my derivation and that of Viviani and
Stucchi (essentially the same) makes it
crystal clear that this is not true.*

                  RM: Well, that would be news to all the power

law researchers who computed velocity and
curvature the way I did in my analyses, using time
derivatives.

Â

•                      MT: (6) M&S follow
  

this astounding assertion with an couple of
paragraphs to show why the V = R**^1/3D^1/3** equation is correct, implying that my comment
claimed it to be wrong. Early in my comment,
however, I wrote: “They then write their key
Eq (6) [V = R**^1/3D^1/3** ],
which is true for any value of V whatever…”
Any implication that my comment claimed the
equation to be incorrect is false.*

                  RM: What we showed is that that equation has

been used by others to show what we showed in our
paper – that using only R (curvature) as the
predictor in a regression on V (speed) – will
result in an estimate of the power coefficient of
R that deviates from 1/3 by an amount proportional
to the correlation between R and D (radial
velocity).Â

Â

•                      MT: (7) Omitted Variable
  

Bias: My comment demonstrated that the finding
predicted and reported by M&S was actually
a tautology having no relation to experimental
findings, which will always produce the result
claimed by M&S to be an experimental
result. M&S in the paper and in the
rebuttal treat it as a discovery that can be
made only by careful statistical analysis, and
do not acknowledge the tautology criticism at
all.*

                  RM: Your demonstration that our findings are a

“tautology” made no sense to us. You made this
claim based on your derivation of an equation for
V of the form V = V. But this is true for any
equation. If X = f(Y) then you can substitute X
for the right side of the equation and write the
equation X = X. That’s not a tautology; that’s
just an irrelevant observation.

•                      MT: (8) M&S: "At the heart of the
  

criticisms of our paper by Z/M and Taylor is
the assumption that the power law is a result
of a direct causal connection between
curvature and speed of movement or between
these variables and the physiological
mechanisms that produce them." I have no idea
how this astonishing statement can be derived
from my exposition of the mathematical and
logical flaws in their paper. My comment is
designed to refute exactly M&S’s claim of
my motivation. The comment shows that there is
NO necessary relationship, causal connection
or otherwise, between curvature and speed of
movement.
*

                  RM: You were apparently trying to show,

mathematically, that the curvature and velocity of
a curved movement are physically independent, like
the disturbance and output in a tracking task.
Since you didn’t speculate about the controlled
variable that might be simultaneously affected by
these two variables I assumed that you were dong
this to justify the assumptions of power law
researchers that these two variables are either
causally related or simultaneously caused by a
third variable.Â

Â

--------end quote-------

                    MT: I repeat from my last message: *"*                        What's

the advantage to you of refusing to deal with
scientific points people bring up about your
work?"

                  RM: We dealt with your confusing rebuttal as

best we could. There was nothing scientific about
it inasmuch as it was purely mathematical.

Best

Rick

Â

                    Well, I guess predictions aren't always

wrong, and I am indeed not surprised.

                        Martin

–
Richard S.
MarkenÂ

                                          "Perfection

is achieved not when you
have nothing more to add,
but when you
have
nothing left to take
away.�
Â
            Â
  --Antoine de
Saint-Exupery

Best

Rick

Â

                                      What's

the advantage to you of
refusing to deal with
scientific points people bring
up about your work? In what
perception you control would
it create error if you were to
accept normal mathematics or
physics as being valid? When
your work is good, it’s good,
but when you make a mistake,
why does it seem so difficult
for you to correct it? In the
curvature paper none of the
criticisms were relevant to a
PCT interpretation, but you
make out that all of them were
intended to refute a “correct
PCT analysis” of the
experimental findings. Why?

                                      I don't expect an answer to a

question raised, but I
wouldn’t be surprised at an
answer to something completely
different.

[rsmarken]

A third variable explanation requires that the cause of movement—the muscle forcces— consistently affects curvature and velocity in such a way that velocity is a power function of curvature. However, this explanation ignores the fact that different muscle forces are required to produce the same movement trajectory on different occasions due to variations in the circumstances that exist each time the movement is produced (Marken,1988). For example, the forces required to move a finger in an elliptical trajectory are different each time the movement is produced due to slight changes in one’s orientation relative to gravity. Therefore, muscle forces will not be consistently related to the curvature and velocity of the movement; the same power relationship between curvature and velocity will be associated with somewhat different muscle forces each time the same movement trajectory is produced. (Marken & Shaffer, 2017).Â

···

[Rick Marken 2018-07-20_18:38:03]

On Fri, Jul 20, 2018 at 1:09 AM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi guys, putting the maths aside for now. How about this as a solution?The faster you go round a corner, the greater the centripetal force you need to counteract, and the more likely it is that one’s lateral velocity will overcome friction and lead to an unstable trace (a ‘skid’ if you like) . So, if there are limits on the force you can apply, then it makes sense to slow down as you go into a bend and speed up when you come out of it.

RM: This implies that the instantaneous velocity of curved movement is a reaction to the curvature through which the movement is taking place. That is, it implies that velocity is a dependent variable and curvature an independent variable. But, in the curved movements studied in power law experiments, both velocity and curvature are actually dependent variables inasmuch as both are a simultaneous result of a “third variable” – the muscle forces used to produce the curved movement. So any observed relationship between curvature and velocity could not possibly be the result of a reaction of one to the other.Â

RM: Moreover, the third variable explanation of the power law is ruled out by the fact that the curved movements under study are a controlled result of muscle forces – the third variable that is the presumed cause of both the curvature and velocity of these movements. We discuss this in the “Correlation and Causality” section of our original paper (https://www.dropbox.com/s/g3tcy8p46c957f7/MarkenShaffer2017.pdf?dl=0). Here’s the relevant passage from that paper:

RM: So by recognizing that curved movements are a controlled result of action, we can see that the power law must be a behavioral illusion in the sense that it cannot possibly be revealing anything about the processes that produce curved movements; this because the processes that produce these movements (muscle forces) are going to be uncorrelated with the movements themselves due to the effects of disturbances to the controlled movement.Â

RM: So there is no math required to see that the power law is a behavioral illusion; all you have to know is that the curved movements observed in power law studies are a controlled result of actions (muscle forces). The math is only used to explain why researchers have consistently found (using regression analysis) that the relationship between the velocity and curvature of curved movements is fit by a power function with coefficient close to 1/3 or 2/3 depending on how velocity and curvature are measured. The math that showed why this consistency is found was very simple; I just observed that the formulas used to compute the velocity (V or A) and curvature (R or C) variables that go into the regression analysis are related to each other such that V = R^1/3D^1/3 and A = C^2/3D^1/3.Â

RM: I was pretty gobsmacked when I saw these results since the power coefficients of curvature in both equations correspond to the power coefficients of the 1/3 and 2/3 power law, 1/3 when velocity and curvature are measured as V and R and 2/3 then these variables are measured as A and C. So I realized right then that the “empirically determined” power law was an artifact of the regression analysis used to determine th power coefficient and that the observed deviation of the empirically determined power law coefficient from it’s “lawful” value (1/3 or 2/3) was a result of omitting the variable D out of the regression analysis.Â

RM: So recognizing that curved movements are a controlled result of action makes it possible to see that the power law is a behavioral illusion in the sense that it appears to show something important about how curved movement is generated, but it doesn’t. And the math shows that the reason for the consistent finding of a power law relationship in curved movements is an artifact of the statistical method (regression) that is used to determine whether or not a power law fits the movement data.Â

RM: I hope you can see that Martin’s math has nothing to do with either of the points made here. His math does not change the fact that the power law is a behavioral illusion. And it has nothing to do with the fact that the computational formulas for the variables that go into the regression analysis used to determine the power law are related asÂ
V = R^1/3D^1/3 and A = C^2/3D^1/3.Â

RM: I’m quite sure that this will be difficult – and possibly impossible – for some people to accept. But that’s the way it is with PCT. It’s revolutionary and people with a vested interest in the status quo – even people who are fans of PCT – will resist it when it leads to conclusions they don’t like. Bill knew this and some of his last words, written as the introduction to what he hoped would be a collection of revolutionary papers on PCT, express a recognition of this fact:Â

BP: The massive opposition from some quarters and
the passive resistance from others came as a disappointing surprise, but
perhaps it shouldn’t have. Science has a social as well as an intellectual
aspect. Scientists are not stupid. They can look at an idea and quickly work
out where it fits in with existing knowledge and where it doesn’t. And
scientists are all too human: when they see that the new idea means their
life’s work could end up mostly in the trash-can, their second reaction is
simply to think “That idea is obviously wrong.” That relieves the
sinking feeling in the pit of the stomach that is the first reaction. Being
wrong about something is unpleasant enough; being wrong about something one has
worked hard to learn and has believed, taught, written about, and researched,
is close to intolerable. All scientists of any talent have had that experience.
The best of them have recognized that their own principles require them to put
those personal reactions aside or at least save them for later. They know that
any such upheaval is going to be important, and ignoring it is not an option.
But those who recognize and embrace a revolution in science are the exception.
Most scientists practice ‘normal science’ within a securely established – and
well-defended – paradigm.

Â

BP: That is what we are up against here, and have
been struggling with since before most of you readers were born. We have spent
that time learning more about this new idea and getting better at explaining
it, but no better at persuading others to change their minds in a serious way
when their career commitments are threatened by it. What we had thought would
be a nutritious and deliciously buttered carrot has proven to function like a
stick. The clearer we have made the idea, the more defenses it has aroused.

Â

BP: We are now facing reality. This is going to be a
revolution whether we like it or not. There are going to be arguments,
screaming and yelling or cool and polite. It’s time to sink or swim.

RM: In other words, revolutions (scientific or otherwise) are not peaceful. But the weapons in the PCT revolution are models and data. These things might hurt some feelings but they leave people intact enough to reorganize, if they want to. Â

BestÂ

Rick

---

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[rsmarken]

[Rick Marken 2018-07-20_20:26:21]

···

On Fri, Jul 20, 2018 at 4:22 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Following on from my last email, can anyone tell me that centripetal force isn’t relevant to the power law?

https://physics.stackexchange.com/questions/294141/what-limits-the-top-speed-in-curves

RM: It’s relevant to the power law in the sense that it is one of the disturbances to curved movements that makes the means of producing those movements uncorrelated with the movements themselves. So these force disturbances are relevant to the power law inasmuch as they demonstrate that the power law is an example of a behavioral illusion.Â

RM: Since centripetal (and, more likely, centrifugal forces) and the muscle forces that compensate for them are invisible it’s hard for see that the curved movements observed in power law research are a controlled result of the muscle forces. That’s why I created the experiment I describe at the beginning of my rebuttal to the rebuttals (https://www.dropbox.com/s/3m51ko4vs1xdult/MarkenShafferReappraisal.pdf?dl=0). In that experiment curved movements were made with a cursor on the computer screen. These cursor movements were a joint result of mouse movements (the analog of muscle forces) and computer generated disturbances (the analog of the force disturbances to curved movements that are made by moving a finger through the air or water).

RM: The experiment shows that the curved cursor movements – the controlled result of mouse movements --Â follow the 1/3 and 2/3 power law but the mouse movements that produce these cursor movements don’t. So this easily repeated experiment shows, sans math, that the power law is an example of a behavioral illusion. The math shows why the power law holds for some movements but not others; it depends on the nature of the trajectory of the curved movements themselves; trajectories where affine velocity (D) is close to being constant will be found to be consistent with the power law; trajectories where affine velocity is not constant and, therefore, somewhat correlated with curvature,will deviate from the power law, possibly by a great deal. So affiine velocity may be a variable that people control when they produce curved movement. Some PCT research aimed at testing this hypothesis was suggested at the end of our rebuttal.Â

BestÂ

Rick

On 20 Jul 2018, at 09:09, Warren Mansell wmansell@gmail.com wrote:

Hi guys, putting the maths aside for now. How about this as a solution? The faster you go round a corner, the greater the centripetal force you need to counteract, and the more likely it is that one’s lateral velocity will overcome friction and lead to an unstable trace (a ‘skid’ if you like) . So, if there are limits on the force you can apply, then it makes sense to slow down as you go into a bend and speed up when you come out of it. But of course this slowing down and speeding up requires a force too. So might the most efficient relationship between curvature and velocity be one at which the force required to slow down when coming to a bend (increase in curvature) or speed up when coming out of it (decrease in curvature), is no greater than the force one would need to apply to counteract the centripetal force as one goes round the bend of a specific curvature at that velocity. Expressing this in maths is beyond me at the moment, but I think if this is the reason for the power law then it would be independent of ‘how’ the trace is produced by the person/vehicle/animal because it would be a function of only the curvature of the trace and not the actions used to achieve it. It should also generate some predictions regarding a direct inverse relationship between on-track acceleration and the derivative of curvature (increase or decrease in curvature as one goes round the trace).Â

–
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

On Fri, Jul 20, 2018 at 12:57 AM, Richard Marken csgnet@lists.illinois.edu wrote:

–
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
Â
Tel: +44 (0) 161 275 8589
Â
Website: http://www.psych-sci.manchester.ac.uk/staff/131406
Â
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

[Rick Marken 2018-07-19_16:57:19]

[Martin Taylor 2018.07.19.14.09]

MT: Of course, that is NOT at all what I showed...Since we are talking about

my comment and not your rebuttal, I’ll use my numbering.

MT: (1) is the standard expression for R in purely spatial variables,

those being the x,y coordinates of a place along a curve, and s
being the distance along the curve to that point from some arbitrary
zero point. Simple physical argument indicates that a description of
a spatial quantity (such as curvature or radius of curvature) must
be a function of purely spatial variables, and if non-spatial
variables are used, for convenience, they must cancel out of the
expression actually used for the calculation.

 RM: Does this mean that the formulas we used to calculate R and V (and C and A) from the data are incorrect?

(2) shows how this cancellation works for the substitution of an

arbitrary parameter “z” that is a function of “s”. It shows that no
matter what z might be, if it has a continuous derivative dz/ds or
the inverse ds/dz, the expression for R in (1) can be transformed
into the equivalent form in z. Depending on the direction the
equivalence is shown, numerator and denominator each have a
multiplying factor (ds/dz)³ or (dz/ds)³ . These
multipliers cancel out, which is why the substitution of z for s (or
vice-versa) produces the same result for any z.

RM: What does this have to do with our analysis? That is, how does it relate to the findings of our regression analyses?

MT: In (3), z is taken to be the time it takes for an object that moves

arbitrarily along the curve without stopping or retracing to reach
the point at which the derivatives are taken. In this case, the
numerator of the expression simplifies to (ds/dt)³ = V³ .
In this equation and the last equivalence of (2), the denominator is
Marken and Shaffer’s “cross-correlation correction factor” D. If the
argument so far has not made it clear that D is V3*f(x,y,s),
equation (7) later demonstrates it explicitly. As is necessarily
true from basic physical principles, the explicit calculations
demonstrate that the general point mentioned above for an arbitrary
parameter z holds also if the parameter is time or velocity. The
effects of the added variable (in this case V) cancel out.

RM: So why did our regression analyses work so well? What did we do wrong?

Â

MT: Marken and Shaffer choose to ignore the generality of the parameter

substitution and the fact that in their specific substitution of the
measured velocities for a single experimental run V³
cancels out from numerator and denominator of the fraction that is
the expression for R. Instead, they leave V³ explicitly
in the numerator, but hide it in their newly discovered
“Cross-correlation correction factor”. They then use the “cccf” as
though it were independent of V in the rest of their paper.

RM: We didn’t ignore this. We knew nothing about it. All we knew was what we found in the reports of research on the power law. And there was nothing in the literature about the “generality of the parameter substitution” of which you speak. And what was, indeed, our newly discovered “cross correlation” variable (D) turns out to be a well known parameter of curved movement: affine velocity.Â

Â

MT: I think this is, to put it mildly, a little different from what Rick

said above that I showed.

RM: I really tried to find some relevance of your mathematical analysis to the research we described in our power law paper. But I’m not sure there is any relevance because you don’t seem to understand – or want to understand – what we did. This is evidenced by what you say at the beginning of your mathematical critique of our work:Â “Accordingly, they assert that measured values of the power law that depart from 1/3 are in error because they omit consideration of D”. In fact, we never “asserted” this. What we demonstrated is that measured values of the power law coefficient will depart from 1/3 (for the relationship between R and V and 2/3 for the relationship between C and A) to the extent that the variable D, which power law researchers always omit from the regression analysis, covaries with the curvature variable (R or C) that is included as the predictor variable in the analysis.Â

MT: Try again, Rick. I keep hoping to be able to learn something from

one of your postings, but I haven’t won this lottery jackpot yet.

RM: Sure, I’ll try again. But you might have better luck if you would explain, as clearly as possible, how your mathematical analysis relates to our regression analysis of actual data from curved movements.

Â

MT: If

I have made a mathematical error in my other comments on Marken and
Shaffer, I really would like to know. But you please comment on what
I wrote, rather than on something you invented, as you did in this
case.

RM: As I said before, I don’t think you have made any mathematical errors. I just don’t see the relevance of your mathematical analysis to what we actually did with our analysis of actual curved movement data. Did we use the wrong formulas to calculate instantaneous velocity and curvature? Did we do the regression incorrectly? Did we use the wrong variables in the regressions?Â

Best

Rick

Â

Martin

–
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

            MT: Rick, could you help me correct my "bogus

mathematics" by pointing out by page and paragraph or by
equation number specifically where the mathematics in my
comment on the Marken and Shaffer paper is “bogus”.

          RM: The mathematics are fine. It's the conclusions that

are wrong. A particularly egregious example is your
“proof” that our equation relating V to R and D (V =
R^1/3D^1/3, equation 6 in your paper) is a tautology.Â
You do this by showing (correctly, I assume) that D^1/3 is
equivalent to V(1/(R^1/3))Â

          so that 
            V

= R^1/3D^1/3 =Â R^1/3
V*(1/(R^1/3))
which, of course, reduces to V=V.Â

            RM:

But as I’ve said, that’s true of any equation. The fact
thatÂ
V
= R^1/3D^1/3 can be reduced to V = V doesn’t negate
the value of knowing thatÂ
V
= R^1/3D^1/3. This equation analyzes V into its
components just as simple one way analysis of
variance (ANOVA) analyzes the total variance in
scores in an experiment (MS.total) into two
components, the variance in scores across
(MS.between) and within (MS.within) conditions, so
that MS.total = MS.between + MS.within. This is the
basic equation of ANOVA.Â

                RM:

Of course, it’s possible to show that MS.total
= MS.between + MS.within is a
“tautology”:Â MS.total = MS.total. We can do this by noting
that MS.within = MS.total - MS.between so thatÂ
MS.total = MS.between + MS.total - MS.between which,
you’ll note, reduces toÂ
MS.total = MS.total.

          RM:  But by analyzing MS.total into MS.between and

MS.within we can learn some interesting things about the
data by computing the two variance components of MS.total
and forming the ratio MS.between/MS.within, a ratio known
as F (for Sir Ronald Fisher, who invented this analysis
method and, as far as I know, never caught flack from
anyone about the basic equation of ANOVA being a
tautology). Knowing the probability of getting different F
ratios in experiments where the independent variable has
no effect (the null hypothesis), it is possible to use the
F ratio observed in an experiment to decide whether one
can reject the null hypothesis with a sufficiently small
probability of being wrong.Â

                                RM:

Just as it has proved useful to
analyze the total variance in
experiments (
MS.total) Â into
variance component ( MS.between,
MS.within and sometimes
MS.interaction and MS subjects) it
proved useful to us to analyze the
variance in the velocity, V, of a
curved movement into components, R
and D. This analysis produced the
equation V = R^1/3*D^1/3. R and D
are measures of two different
components of the temporal
variation in curved movement just
as MS.between and MS.within are
measures of two different
components of the variation in the
scores observed in an experiment;
R is the variation in curvature
and D is the variation in affine
velocity.Â

                                  RM:

Our equation says that the
variation in V for a curved
movement will be exactly equal to
R^1/3D^1/3. Linearizing this
equation by taking the log of both
sides we get log (V) = 1/3log (R)
+1/3*log (D) . This equation shows
that if one did a linear
regression using the variables
log(R) and log(D) as predictors
and the variable log(V) as the
criterion, the coefficients of the
two predictor variables would be
exactly 1/3 with an intercept of
0. More importantly, this equation
shows that if the variable log (D)
isomitted from the regression, the
coefficient of log(R) will not
necessarily be found to be exactly
1/3 and the intercept will not
necessarily be found to be exactly
0. This is where Omitted Variable
Bias (OVB) analysis comes in. This
analysis makes if possible to
predict exactly what a regression
analysis will find the coefficient
of log(R) to be if log(D) is
omitted from the regression.

                                  RM:

This finding is important because
the “power law” of movement is
determined by doing a regression
of log (R) on log (V) using the
regression equation log (V) = k +
b*log(R), omitting the variable
log(D). The term “power law”
refers to the fact that the
results of this regression
analysis consistently finds that
the power coefficient b is close
to 1/3. Our analysis shows that
this is a statistical artifact
that results from having left the
variable log(D) out of the
regression analysis. OVB analyiss
shows that the amount by which the
b coefficient is found to deviate
from 1/3 depends on the degree of
covariation between the variable
included in the regression (log
(R)) and the variable omitted from
the regression (log(D)). Since
both log (R) and log (D) are
measured from data (temporal
variations in the x,y position of
the curved movement) the
covariation between these
variables is easily calculated and
the predicted deviation of the
power coefficient, b, from 1/3 can
be exactly predicted.Â

                                  RM:

The covariation between log (R)
and log (D) depends on the nature
of the curved movement trajectory
itself and has nothing to do with
how that movement was generated.
It is in this sense that the
observed power law is a
“behavioral illusion”, the
illusion being that the relatively
consistent observation of an
approximately 1/3 power
relationship between the curvature
(R) and velocity (V) of curved
movements seems to reveal
something important about how
these movements are produced, when
it doesn’t.Â

                                  RM:

So the fact that the equation V =
R^1/3*D^1/3 can be reduced to V =
V does not negate the value of
analyzing V into its components
any more than the fact that the
equation MS.total
= MS.between + MS.within can be
reduced to MS.total
= MS.total negates the value
of analyzing MS.total into its
components.Â

                                      RM:

There are many other incorrect
conclusions in your rebuttal
to our paper, Martin. But I
think this is enough for now
since your “tautology” claim
(based on our alleged
mathematical mistake) seemed
to be central to your
argument.Â

BestÂ

Rick

            You can do this without referring either to your

rebuttal or to the eight falsehoods that I asked you not
to try to justify at this point. My question is not
about them, but specifically about what in my
mathematics you have shown to be bogus. Your previous
response did not address this question.

            Martin

–
Richard S. MarkenÂ

                                  "Perfection

is achieved not when you have
nothing more to add, but when you
have
nothing left to take away.�
  Â
            Â
–Antoine de Saint-Exupery

[rsmarken]

[Rick Marken 2018-07-21_10:29:50]

WM: Thank you both Rick and Martin for replying to me. I agree with you Rick that the power law is a behavioural illusion to the degree that it doesn’t tell us how muscle movements are produced. I am trying to explain it purely within the physics for this reason.

 RM: I believe we showed that there is no physics explanation of the power law; the power law purely a result of how it is determined -- using regression analysis that omits one of the predictors of velocity. The degree to which the power law is found using this methodology depends completely on the nature of the movement trajectory itself. So a power law will be found for some trajectories that are produced intentionally (like elliptical finger movements) and some that are produced unintentionally (like the movement paths of the pursuers of toy helicopters). And the power law will also not be found for for some trajectories that are produced intentionally (like the mouse movements that produce elliptical cursor movements) and some that are produced unintentionally (like the movement trajectory of the plants).Â

WM: To answer Martin’s point it strikes me that even a lava has the same kind of constraints as a racing car - limits in the force it can apply, a requirement to get to its destination as quickly as possibly, physical constraints that require it to curve its movement at certain points. The fact that a person driving a car and a larva have vastly different locomotion systems in itself seems to indicate that the power law tells us nothing about how the movements are produced, as Rick suggests.

RM: Again, this is simply due to the nature of the trajectories themselves. Movement trajectories that happen to have constant affine velocity will be found to fit the power law perfectly; movement trajectories that happen to have affine velocity that correlate with curvature will not fit the power law at all.Â
Â

WM: However, unlike Rick and Martin, I’d like to try to get to the proof purely through the share physics of these various examples of curved motion rather than through the maths alone. To me, the maths is always an abstraction of a more fundamental relationship.Â

RM: There are two very different "maths" that are being used in this debate; conflating the two by calling them both "maths" may be the heart of the problem here. One kind of maths, the kind used by Martin, is being used to show that an observed power law relationship between velocity and curvature reflects a real, physical relationship between these variables. I called this use of maths "bogus", not because there was any error in the math itself but because these maths are irrelevant to our analysis and, thus, the criticisms of our analysis based on these maths are misguided.
RM: The other kind of maths, the kind used by Marken and Shaffer, are tied intimately to how the data in power law studies is analyzed. We start with the computational formulas that power law researchers use to compute the velocity and curvature at each instant during a curved movement. These computed values are then used in a regression analysis to see if the data are fit by a power law.
RM: It is a simple matter to show that there is a simple mathematical relationship between the formula used to compute velocity and that used to compute curvature. That relationship (linearized by taking the log of both sides) is log (V) = 1/3*log(R) +1/3 * log(D) or log (A) = 2/3*log(C) +1/3 * log(D). So this simple analysis of the way data is collected and analyzed shows that if the variable log(D) is omitted from the regression analysis used to determine the coefficient of the best fitting power function relating curvature to velocity, that estimate will deviate from the "power law" coefficient (1/3 or 2/3) in proportion to the size of the correlation between log (D) and the measure of curvature (log(R) or log(D) in the relationship.Â
RM: So the "proof"Â that the power law is a behavioral illusion already exists; we proved it. The importance of this "proof" is that is shows that by concentrating on the power law per se power law researchers are being distracted from what is actually going on when people make curved movements. What is going on is CONTROL and the focus of research on movement should be on what variables are under control.Â
Â

WM: I do think there can be a two way relationship between curvature and velocity too, because if you head too fast around a corner, you ‘skid’ and therefore the curvature decreases...

RM I think you mean that there can be an actual physical -- not just a mathematical - relationship between these variables. I don't know about "two way" relationship but curvature could be related to velocity as a disturbance is to output in a control system if curvature is an independent variable. And curvature is an independent variable when people race cars around racetracks. Here the curvature of the race track is an independent variable that is a disturbance to a controlled variable -- keeping the car on the track. So assuming that the driver is also controlling for (among other things) going fast, the driver will have to vary how fast she goes through the turn so that she also keeps the car on the track. She will also have to vary how much curvature she gives to her own car when going through the turn. So there are a lot of possible controlled variables involved here and the driver's own curvature is one of them, as it is when moving a finger in an ellipse, but now the curvature the road has been added as a separate disturbance.
Best
Rick
 >

All the bestÂ
WarrenÂ

[Rick Marken 2018-07-20_20:26:21]

WM: Following on from my last email, can anyone tell me that centripetal force isn’t relevant to the power law?

<https://physics.stackexchange.com/questions/294141/what-limits-the-top-speed-in-curves&gt;https://physics.stackexchange.com/questions/294141/what-limits-the-top-speed-in-curves

RM: It's relevant to the power law in the sense that it is one of the disturbances to curved movements that makes the means of producing those movements uncorrelated with the movements themselves. So these force disturbances are relevant to the power law inasmuch as they demonstrate that the power law is an example of a behavioral illusion.Â
RM: Since centripetal (and, more likely, centrifugal forces) and the muscle forces that compensate for them are invisible it's hard for see that the curved movements observed in power law research are a controlled result of the muscle forces. That's why I created the experiment I describe at the beginning of my rebuttal to the rebuttals (<https://www.dropbox.com/s/3m51ko4vs1xdult/MarkenShafferReappraisal.pdf?dl=0&gt;https://www.dropbox.com/s/3m51ko4vs1xdult/MarkenShafferReappraisal.pdf?dl=0\). In that experiment curved movements were made with a cursor on the computer screen. These cursor movements were a joint result of mouse movements (the analog of muscle forces) and computer generated disturbances (the analog of the force disturbances to curved movements that are made by moving a finger through the air or water).
RM: The experiment shows that the curved cursor movements -- the controlled result of mouse movements --Â follow the 1/3 and 2/3 power law but the mouse movements that produce these cursor movements don't. So this easily repeated experiment shows, sans math, that the power law is an example of a behavioral illusion. The math shows why the power law holds for some movements but not others; it depends on the nature of the trajectory of the curved movements themselves; trajectories where affine velocity (D) is close to being constant will be found to be consistent with the power law; trajectories where affine velocity is not constant and, therefore, somewhat correlated with curvature,will deviate from the power law, possibly by a great deal. So affiine velocity may be a variable that people control when they produce curved movement. Some PCT research aimed at testing this hypothesis was suggested at the end of our rebuttal.Â
BestÂ
>>>

Hi guys, putting the maths aside for now. How about this as a solution? The faster you go round a corner, the greater the centripetal force you need to counteract, and the more likely it is that one's lateral velocity will overcome friction and lead to an unstable trace (a 'skid' if you like) . So, if there are limits on the force you can apply, then it makes sense to slow down as you go into a bend and speed up when you come out of it. But of course this slowing down and speeding up requires a force too. So might the most efficient relationship between curvature and velocity be one at which the force required to slow down when coming to a bend (increase in curvature) or speed up when coming out of it (decrease in curvature), is no greater than the force one would need to apply to counteract the centripetal force as one goes round the bend of a specific curvature at that velocity. Expressing this in maths is beyond me at the moment, but I think if this is the reason for the power law then it would be independent of 'how' the trace is produced by the person/vehicle/animal because it would be a function of only the curvature of the trace and not the actions used to achieve it. It should also generate some predictions regarding a direct inverse relationship between on-track acceleration and the derivative of curvature (increase or decrease in curvature as one goes round the trace).Â

[Rick Marken 2018-07-19_16:57:19]

[Martin Taylor 2018.07.19.14.09]

MT: Rick, could you help me correct my "bogus mathematics" by pointing out by page and paragraph or by equation number specifically where the mathematics in my comment on the Marken and Shaffer paper is "bogus".

RM: The mathematics are fine. It's the conclusions that are wrong. A particularly egregious example is your "proof" that our equation relating V to R and D (V = R^1/3*D^1/3, equation 6 in your paper) is a tautology. You do this by showing (correctly, I assume) that D^1/3 is equivalent to V*(1/(R^1/3))Â

MT: Of course, that is NOT at all what I showed...Since we are talking about my comment and not your rebuttal, I'll use my numbering.

MT: (1) is the standard expression for R in purely spatial variables, those being the x,y coordinates of a place along a curve, and s being the distance along the curve to that point from some arbitrary zero point. Simple physical argument indicates that a description of a spatial quantity (such as curvature or radius of curvature) must be a function of purely spatial variables, and if non-spatial variables are used, for convenience, they must cancel out of the expression actually used for the calculation.

 RM: Does this mean that the formulas we used to calculate R and V (and C and A) from the data are incorrect?>>>>>>

(2) shows how this cancellation works for the substitution of an arbitrary parameter "z" that is a function of "s". It shows that no matter what z might be, if it has a continuous derivative dz/ds or the inverse ds/dz, the expression for R in (1) can be transformed into the equivalent form in z. Depending on the direction the equivalence is shown, numerator and denominator each have a multiplying factor (ds/dz)3 or (dz/ds)3. These multipliers cancel out, which is why the substitution of z for s (or vice-versa) produces the same result for any z.

RM: What does this have to do with our analysis? That is, how does it relate to the findings of our regression analyses?>>>>>>

MT: In (3), z is taken to be the time it takes for an object that moves arbitrarily along the curve without stopping or retracing to reach the point at which the derivatives are taken. In this case, the numerator of the expression simplifies to (ds/dt)3 = V3. In this equation and the last equivalence of (2), the denominator is Marken and Shaffer's "cross-correlation correction factor" D. If the argument so far has not made it clear that D is V3*f(x,y,s), equation (7) later demonstrates it explicitly. As is necessarily true from basic physical principles, the explicit calculations demonstrate that the general point mentioned above for an arbitrary parameter z holds also if the parameter is time or velocity. The effects of the added variable (in this case V) cancel out.

RM: So why did our regression analyses work so well? What did we do wrong?

Â

MT: Marken and Shaffer choose to ignore the generality of the parameter substitution and the fact that in their specific substitution of the measured velocities for a single experimental run V3 cancels out from numerator and denominator of the fraction that is the expression for R. Instead, they leave V3 explicitly in the numerator, but hide it in their newly discovered "Cross-correlation correction factor". They then use the "cccf" as though it were independent of V in the rest of their paper.

RM: We didn't ignore this. We knew nothing about it. All we knew was what we found in the reports of research on the power law. And there was nothing in the literature about the "generality of the parameter substitution" of which you speak. And what was, indeed, our newly discovered "cross correlation" variable (D) turns out to be a well known parameter of curved movement: affine velocity.Â

Â

MT: I think this is, to put it mildly, a little different from what Rick said above that I showed.

RM: I really tried to find some relevance of your mathematical analysis to the research we described in our power law paper. But I'm not sure there is any relevance because you don't seem to understand -- or want to understand -- what we did. This is evidenced by what you say at the beginning of your mathematical critique of our work: "Accordingly, they assert that measured values of the power law that depart from 1/3 are in error because they omit consideration of D". In fact, we never "asserted" this. What we demonstrated is that measured values of the power law coefficient will depart from 1/3 (for the relationship between R and V and 2/3 for the relationship between C and A) to the extent that the variable D, which power law researchers always omit from the regression analysis, covaries with the curvature variable (R or C) that is included as the predictor variable in the analysis. >>>>>>

MT: Try again, Rick. I keep hoping to be able to learn something from one of your postings, but I haven't won this lottery jackpot yet.

RM: Sure, I'll try again. But you might have better luck if you would explain, as clearly as possible, how your mathematical analysis relates to our regression analysis of actual data from curved movements.

Â

···

On Fri, Jul 20, 2018 at 11:59 PM, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On 21 Jul 2018, at 04:26, Richard Marken (<mailto:rsmarken@gmail.com>rsmarken@gmail.com via csgnet Mailing List) <<mailto:csgnet@lists.illinois.edu>csgnet@lists.illinois.edu> wrote:

On Fri, Jul 20, 2018 at 4:22 PM, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On 20 Jul 2018, at 09:09, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On Fri, Jul 20, 2018 at 12:57 AM, Richard Marken <<mailto:csgnet@lists.illinois.edu>csgnet@lists.illinois.edu> wrote:

MT: If I have made a mathematical error in my other comments on Marken and Shaffer, I really would like to know. But you please comment on what I wrote, rather than on something you invented, as you did in this case.

RM: As I said before, I don't think you have made any mathematical errors. I just don't see the relevance of your mathematical analysis to what we actually did with our analysis of actual curved movement data. Did we use the wrong formulas to calculate instantaneous velocity and curvature? Did we do the regression incorrectly? Did we use the wrong variables in the regressions?Â
Best
Rick
 >>>>>>

Martin

so that V = R^1/3*D^1/3 = R^1/3* V*(1/(R^1/3)) which, of course, reduces to V=V.Â
RM: But as I've said, that's true of any equation. The fact that V = R^1/3*D^1/3 can be reduced to V = V doesn't negate the value of knowing that V = R^1/3*D^1/3. This equation analyzes V into its components just as simple one way analysis of variance (ANOVA) analyzes the total variance in scores in an experiment (MS.total) into two components, the variance in scores across (MS.between) and within (MS.within) conditions, so that MS.total = MS.between + MS.within. This is the basic equation of ANOVA.Â
RM: Of course, it's possible to show that MS.total = MS.between + MS.within is a "tautology": MS.total = MS.total. We can do this by noting that MS.within = MS.total - MS.between so that MS.total = MS.between + MS.total - MS.between which, you'll note, reduces to MS.total = MS.total.
RM: But by analyzing MS.total into MS.between and MS.within we can learn some interesting things about the data by computing the two variance components of MS.total and forming the ratio MS.between/MS.within, a ratio known as F (for Sir Ronald Fisher, who invented this analysis method and, as far as I know, never caught flack from anyone about the basic equation of ANOVA being a tautology). Knowing the probability of getting different F ratios in experiments where the independent variable has no effect (the null hypothesis), it is possible to use the F ratio observed in an experiment to decide whether one can reject the null hypothesis with a sufficiently small probability of being wrong.Â
RM: Just as it has proved useful to analyze the total variance in experiments ( MS.total)Â into variance component (MS.between, MS.within and sometimes MS.interaction and MS subjects) it proved useful to us to analyze the variance in the velocity, V, of a curved movement into components, R and D. This analysis produced the equation V = R^1/3*D^1/3. R and D are measures of two different components of the temporal variation in curved movement just as MS.between and MS.within are measures of two different components of the variation in the scores observed in an experiment; R is the variation in curvature and D is the variation in affine velocity.Â
RM: Our equation says that the variation in V for a curved movement will be exactly equal to R^1/3*D^1/3. Linearizing this equation by taking the log of both sides we get log (V) = 1/3*log (R) +1/3*log (D) . This equation shows that if one did a linear regression using the variables log(R) and log(D) as predictors and the variable log(V) as the criterion, the coefficients of the two predictor variables would be exactly 1/3 with an intercept of 0. More importantly, this equation shows that if the variable log (D) isomitted from the regression, the coefficient of log(R) will not necessarily be found to be exactly 1/3 and the intercept will not necessarily be found to be exactly 0. This is where Omitted Variable Bias (OVB) analysis comes in. This analysis makes if possible to predict exactly what a regression analysis will find the coefficient of log(R) to be if log(D) is omitted from the regression.
RM: This finding is important because the "power law" of movement is determined by doing a regression of log (R) on log (V) using the regression equation log (V) = k + b*log(R), omitting the variable log(D). The term "power law" refers to the fact that the results of this regression analysis consistently finds that the power coefficient b is close to 1/3. Our analysis shows that this is a statistical artifact that results from having left the variable log(D) out of the regression analysis. OVB analyiss shows that the amount by which the b coefficient is found to deviate from 1/3 depends on the degree of covariation between the variable included in the regression (log (R)) and the variable omitted from the regression (log(D)). Since both log (R) and log (D) are measured from data (temporal variations in the x,y position of the curved movement) the covariation between these variables is easily calculated and the predicted deviation of the power coefficient, b, from 1/3 can be exactly predicted.Â
RM: The covariation between log (R) and log (D) depends on the nature of the curved movement trajectory itself and has nothing to do with how that movement was generated. It is in this sense that the observed power law is a "behavioral illusion", the illusion being that the relatively consistent observation of an approximately 1/3 power relationship between the curvature (R) and velocity (V) of curved movements seems to reveal something important about how these movements are produced, when it doesn't.Â
RM: So the fact that the equation V = R^1/3*D^1/3 can be reduced to V = V does not negate the value of analyzing V into its components any more than the fact that the equation MS.total = MS.between + MS.within can be reduced to MS.total = MS.total negates the value of analyzing MS.total into its components.Â
RM: There are many other incorrect conclusions in your rebuttal to our paper, Martin. But I think this is enough for now since your "tautology" claim (based on our alleged mathematical mistake) seemed to be central to your argument.Â
BestÂ
Rick

You can do this without referring either to your rebuttal or to the eight falsehoods that I asked you not to try to justify at this point. My question is not about them, but specifically about what in my mathematics you have shown to be bogus. Your previous response did not address this question.

Martin

--

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Dr Warren Mansell
Reader in Clinical Psychology
School of Health Sciences
2nd Floor Zochonis Building
University of Manchester
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Manchester M13 9PL
Email: <mailto:warren.mansell@manchester.ac.uk>warren.mansell@manchester.ac.uk
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Tel: +44 (0) 161 275 8589
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Website: <Redirect | Faculty of Biology, Medicine and Health | The University of Manchester

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Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - <http://www.routledge.com/books/details/9780415738781/&gt;Principles\-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check <http://www.pctweb.org>www.pctweb.org for further information on Perceptual Control Theory

--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[rsmarken]

[Rick Marken 2018-07-24_19:54:03]

[Martin Taylor 2018.07.24.13.05]

MT: Rick, Do you intend to respond to my last message of four days ago? Are you going to apologise and acknowledge that my maths is not "bogus" or are you going to explain to me exactly how it is bogus?

RM: I thought I answered this but I'll try again.

RM: I said your math was "bogus" because it is irrelevant to our analysis of the power law. Our analysis is based on the computational formulas for the curvature and velocity variables that go into the regression analysis used to determine whether or not a movement trajectory is fit by the "power law". Your math was based on the fact that the derivatives in the physics equation describing the velocity of a curved trajectory are taken with respect to time while derivatives in the physics equation describing the curvature of a curved trajectory are taken with respect to space. This is irrelevant to our analysis because the derivatives in the computational formulas for both curvature and velocity are taken with respect to time.Â
RM: You used the difference in the derivatives in the physics equations to show that our derivation of the equation relating velocity to curvature and a "cross product variable" was incorrect. But our equation is actually a correct derivation of the relationship between the computational formulas for curvature and velocity, And it's that derived relationship that matters in terms of interpreting the results of the regression analysis.Â
RM: So your math was "bogus" in the sense that it was "deceptive" (one of the synonyms of "bogus"). It was aimed at deceiving the audience, using fancy mathematical operations to make it seem like we made a mathematical error. And the deception has been quite successful. But it doesn't change the fact that our analysis, based on the computational formulas that are actually used by power law researchers to compute the variables used in their regression analysis, shows that all the findings regarding power law relationships between velocity and curvature are "spurious" (another synonym for "bogus").>

MT: We can't move on until you do one or the other. Until then, your replies to Warren since my last message are completely irrelevant to the whole curvature-velocity relationship question.

RM: I really think the best way to deal with this is for you to propose your own explanation of the power law. Just saying that my explanation is wrong is not very interesting.Â
BestÂ
Rick
 >

Martin

[Martin Taylor 2018.07.20.10.54]

...

Anyway, this is all irrelevant to the topic, which is that you asserted in a message to Bruce Nevin that the mathematics in my comment on your original paper was "bogus", a fact that I am unable to corroborate. I asked you to back up that claim by showing me my error(s). So far, all you have said is that you assume my route through equations 1,2,3, and 7 is correct (4, 5, and 6 are copied from your paper, and are irrelevant). So now I must assume that the bogus part of the mathematics lies elsewhere. Please tell me where, and forget about its relevance. I just want to know exactly where the "bogus" appears in my mathematics.

Of course, there is another alternative, which is that you could apologise for the slur and acknowledge that my mathematics is not "bogus", in which case we can talk about relevance if you want. But I think that simply reading my published comment should suffice, if the mathematics is indeed not bogus.

Martin

Martin

[Martin Taylor 2018.07.19.14.09]

MT: Rick, could you help me correct my "bogus mathematics" by pointing out by page and paragraph or by equation number specifically where the mathematics in my comment on the Marken and Shaffer paper is "bogus".

RM: The mathematics are fine. It's the conclusions that are wrong. A particularly egregious example is your "proof" that our equation relating V to R and D (V = R^1/3*D^1/3, equation 6 in your paper) is a tautology. You do this by showing (correctly, I assume) that D^1/3 is equivalent to V*(1/(R^1/3))Â

MT: Of course, that is NOT at all what I showed...Since we are talking about my comment and not your rebuttal, I'll use my numbering.

MT: (1) is the standard expression for R in purely spatial variables, those being the x,y coordinates of a place along a curve, and s being the distance along the curve to that point from some arbitrary zero point. Simple physical argument indicates that a description of a spatial quantity (such as curvature or radius of curvature) must be a function of purely spatial variables, and if non-spatial variables are used, for convenience, they must cancel out of the expression actually used for the calculation.

 RM: Does this mean that the formulas we used to calculate R and V (and C and A) from the data are incorrect?

No.

(2) shows how this cancellation works for the substitution of an arbitrary parameter "z" that is a function of "s". It shows that no matter what z might be, if it has a continuous derivative dz/ds or the inverse ds/dz, the expression for R in (1) can be transformed into the equivalent form in z. Depending on the direction the equivalence is shown, numerator and denominator each have a multiplying factor (ds/dz)3 or (dz/ds)3. These multipliers cancel out, which is why the substitution of z for s (or vice-versa) produces the same result for any z.

RM: What does this have to do with our analysis? That is, how does it relate to the findings of our regression analyses?>>>>>

MT: In (3), z is taken to be the time it takes for an object that moves arbitrarily along the curve without stopping or retracing to reach the point at which the derivatives are taken. In this case, the numerator of the expression simplifies to (ds/dt)3 = V3. In this equation and the last equivalence of (2), the denominator is Marken and Shaffer's "cross-correlation correction factor" D. If the argument so far has not made it clear that D is V3*f(x,y,s), equation (7) later demonstrates it explicitly. As is necessarily true from basic physical principles, the explicit calculations demonstrate that the general point mentioned above for an arbitrary parameter z holds also if the parameter is time or velocity. The effects of the added variable (in this case V) cancel out.

RM: So why did our regression analyses work so well? What did we do wrong?

Â

MT: Marken and Shaffer choose to ignore the generality of the parameter substitution and the fact that in their specific substitution of the measured velocities for a single experimental run V3 cancels out from numerator and denominator of the fraction that is the expression for R. Instead, they leave V3 explicitly in the numerator, but hide it in their newly discovered "Cross-correlation correction factor". They then use the "cccf" as though it were independent of V in the rest of their paper.

RM: We didn't ignore this. We knew nothing about it. All we knew was what we found in the reports of research on the power law. And there was nothing in the literature about the "generality of the parameter substitution" of which you speak. And what was, indeed, our newly discovered "cross correlation" variable (D) turns out to be a well known parameter of curved movement: affine velocity.Â

Â

MT: I think this is, to put it mildly, a little different from what Rick said above that I showed.

RM: I really tried to find some relevance of your mathematical analysis to the research we described in our power law paper. But I'm not sure there is any relevance because you don't seem to understand -- or want to understand -- what we did. This is evidenced by what you say at the beginning of your mathematical critique of our work: "Accordingly, they assert that measured values of the power law that depart from 1/3 are in error because they omit consideration of D". In fact, we never "asserted" this. What we demonstrated is that measured values of the power law coefficient will depart from 1/3 (for the relationship between R and V and 2/3 for the relationship between C and A) to the extent that the variable D, which power law researchers always omit from the regression analysis, covaries with the curvature variable (R or C) that is included as the predictor variable in the analysis. >>>>>

MT: Try again, Rick. I keep hoping to be able to learn something from one of your postings, but I haven't won this lottery jackpot yet.

RM: Sure, I'll try again. But you might have better luck if you would explain, as clearly as possible, how your mathematical analysis relates to our regression analysis of actual data from curved movements.

Â

···

MT: If I have made a mathematical error in my other comments on Marken and Shaffer, I really would like to know. But you please comment on what I wrote, rather than on something you invented, as you did in this case.

RM: As I said before, I don't think you have made any mathematical errors. I just don't see the relevance of your mathematical analysis to what we actually did with our analysis of actual curved movement data. Did we use the wrong formulas to calculate instantaneous velocity and curvature? Did we do the regression incorrectly? Did we use the wrong variables in the regressions?Â
Best
Rick
 >>>>>

Martin

so that V = R^1/3*D^1/3 = R^1/3* V*(1/(R^1/3)) which, of course, reduces to V=V.Â
RM: But as I've said, that's true of any equation. The fact that V = R^1/3*D^1/3 can be reduced to V = V doesn't negate the value of knowing that V = R^1/3*D^1/3. This equation analyzes V into its components just as simple one way analysis of variance (ANOVA) analyzes the total variance in scores in an experiment (MS.total) into two components, the variance in scores across (MS.between) and within (MS.within) conditions, so that MS.total = MS.between + MS.within. This is the basic equation of ANOVA.Â
RM: Of course, it's possible to show that MS.total = MS.between + MS.within is a "tautology": MS.total = MS.total. We can do this by noting that MS.within = MS.total - MS.between so that MS.total = MS.between + MS.total - MS.between which, you'll note, reduces to MS.total = MS.total.
RM: But by analyzing MS.total into MS.between and MS.within we can learn some interesting things about the data by computing the two variance components of MS.total and forming the ratio MS.between/MS.within, a ratio known as F (for Sir Ronald Fisher, who invented this analysis method and, as far as I know, never caught flack from anyone about the basic equation of ANOVA being a tautology). Knowing the probability of getting different F ratios in experiments where the independent variable has no effect (the null hypothesis), it is possible to use the F ratio observed in an experiment to decide whether one can reject the null hypothesis with a sufficiently small probability of being wrong.Â
RM: Just as it has proved useful to analyze the total variance in experiments ( MS.total)Â into variance component (MS.between, MS.within and sometimes MS.interaction and MS subjects) it proved useful to us to analyze the variance in the velocity, V, of a curved movement into components, R and D. This analysis produced the equation V = R^1/3*D^1/3. R and D are measures of two different components of the temporal variation in curved movement just as MS.between and MS.within are measures of two different components of the variation in the scores observed in an experiment; R is the variation in curvature and D is the variation in affine velocity.Â
RM: Our equation says that the variation in V for a curved movement will be exactly equal to R^1/3*D^1/3. Linearizing this equation by taking the log of both sides we get log (V) = 1/3*log (R) +1/3*log (D) . This equation shows that if one did a linear regression using the variables log(R) and log(D) as predictors and the variable log(V) as the criterion, the coefficients of the two predictor variables would be exactly 1/3 with an intercept of 0. More importantly, this equation shows that if the variable log (D) isomitted from the regression, the coefficient of log(R) will not necessarily be found to be exactly 1/3 and the intercept will not necessarily be found to be exactly 0. This is where Omitted Variable Bias (OVB) analysis comes in. This analysis makes if possible to predict exactly what a regression analysis will find the coefficient of log(R) to be if log(D) is omitted from the regression.
RM: This finding is important because the "power law" of movement is determined by doing a regression of log (R) on log (V) using the regression equation log (V) = k + b*log(R), omitting the variable log(D). The term "power law" refers to the fact that the results of this regression analysis consistently finds that the power coefficient b is close to 1/3. Our analysis shows that this is a statistical artifact that results from having left the variable log(D) out of the regression analysis. OVB analyiss shows that the amount by which the b coefficient is found to deviate from 1/3 depends on the degree of covariation between the variable included in the regression (log (R)) and the variable omitted from the regression (log(D)). Since both log (R) and log (D) are measured from data (temporal variations in the x,y position of the curved movement) the covariation between these variables is easily calculated and the predicted deviation of the power coefficient, b, from 1/3 can be exactly predicted.Â
RM: The covariation between log (R) and log (D) depends on the nature of the curved movement trajectory itself and has nothing to do with how that movement was generated. It is in this sense that the observed power law is a "behavioral illusion", the illusion being that the relatively consistent observation of an approximately 1/3 power relationship between the curvature (R) and velocity (V) of curved movements seems to reveal something important about how these movements are produced, when it doesn't.Â
RM: So the fact that the equation V = R^1/3*D^1/3 can be reduced to V = V does not negate the value of analyzing V into its components any more than the fact that the equation MS.total = MS.between + MS.within can be reduced to MS.total = MS.total negates the value of analyzing MS.total into its components.Â
RM: There are many other incorrect conclusions in your rebuttal to our paper, Martin. But I think this is enough for now since your "tautology" claim (based on our alleged mathematical mistake) seemed to be central to your argument.Â
BestÂ
Rick

You can do this without referring either to your rebuttal or to the eight falsehoods that I asked you not to try to justify at this point. My question is not about them, but specifically about what in my mathematics you have shown to be bogus. Your previous response did not address this question.

Martin

--

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[rsmarken]

[Rick Marken 2018-07-25_09:29:22]
WM: Oh, maybe I can carry on!

RM: I should note that these comments from Warren are part of an off line conversation we are having about the poer law. Warren is proposing that the power law is seen in curved movement because centripedal forces created by the movement itself are a disturbance to that movement which are compensated for by the forces that produced the movement. I find this model wanting because centripedal force is not a disturbance becuase it is not independent of the actions that keep the implied controlled variable (movement) under control.Â

WM: I don’t think there is a rule that a disturbance has to be completely independent from the organism is there?

RM: Yes, there is. A disturbance is an effect on a controlled variable that is completely independent of the actions of the control system.Â
Â

WM: Surely my experience of any (randomly occurring) disturbance is contingent on my being in a location near it where it can affect me, for a start?

RM: We generally don't experience the disturbances to the variables we are controlling; we only experience the effects of disturbances to controlled variables. And we experience those effects only when we don't have good control of the controlled variable.Â
Â

WM: And the disturbance is at right angles to the muscle forces that govern velocity, not in the same direction ?

RM: So in order to compensate for these "disturbances" the muscle forces, which are creating these "disturbances" would have to simultaneously be applying forces at right angles to these "disturbances" in order to compensate for them. I think it would be great if you could "carry on" and try to build a model that does this. I think you would learn some very important things about modeling and about PCT.Â
Best
Rick

All the bestÂ
WarrenÂ

Hi Warren

Hi Rick, I don’t see how the power law itself is incompatible with PCT, just how people interpret it.Â

The power law is "compatible" with PCT if it is "interpreted" as an irrelevant side effect of control. The power law is incompatible with PCT if it is interpreted as telling us anything about how the purposeful behavior of living systems works.Â
Â

My starting point is not the power law or PCT but a recognition that the centripetal force around a curve will be proportional to the velocity round that curve and it will require a counteracting force high enough to prevent a ‘skidding’ deviation from the curve.

The main problem here is that centripetal force can't possibly be a disturbance because it is not an independent variable; as you note, it is proportional to the velocity of movement around the curve. Both the velocity of movement around the curve and the degree of curvature of the movement itself are a result of muscle forces. Muscle forces cannot be used to both create a centripetal force disturbance, by producing a movement of a particular instantaneous velocity and curvature, and compensate for it at the same time.Â

But I should probably stop now because I don’t have the time or maths skill to see it through...

The power law is important to me because it is a perfect demonstration of why PCT has had such a tough time becoming mainstream. As Bill said over and over again, it is really revolutionary in the most real way -- it really says that most of the research done in the context of the conventional view of behavior "can be deposited in the wastebasket". The power law is a very specialized area but it represents a huge line of research done by a lot of researchers over many years who don't want their work to be exposed as worthless. So they fight back, aggressively against my analysis. I wish I could fight back with the help of people on CSGNet who are purportedly fans of PCT; I think if I could have gotten help from Abbott, Taylor and you we could have persuaded Alex to start doing PCT research. But it was not to be. So I have to fight back alone (at least on CSGNet). And fight back I will because I loved Bill's work, I loved Bill, and I will not go to my grave without having tried to keep his incredibly important and brilliant revolution going.Â
Best
Rick
 >>>

Hi Warren

Hi Rick, what I want to do is see whether the observations and power law emerges from a more detailed understanding of the physics of the problem, rather than use either observations of data or the power law and work backwards.

Why? You must have some kind of model of how curved movements are produced in mind that would lead you o suspect that physics is involved. What is it?

Â

But I see what you mean in that because the applied force in a particular direction is variable rather than static or obeying a simple law itself like in a non-living system, then there must be something to explain regarding how a living or control system does it.

This doesn't sound much like what I mean. My point is that when you make curved movements in space with your finger you are controlling the position of your finger: the position of you finger (or some variable related to that) is a controlled variable. The main disturbances to this variable are the varying forces applied to it --the gravitational and centripetal forces that are varying because of the variations in the position of the finger relative to the earth and the body. These invisible disturbances are being compensated for by the appropriately varying (and invisible) forces exerted by the muscles. The resulting movement trajectory will show a power law to the extent that the affine velocity of trajectory is close to being constant. So if you move the finger in something close to an ellipse you get a power law; if you move it in the trajectory made by the mouse in my demo in the rebuttal paper you get no power law. That is, the power law has nothing to do with physics or how the movement was produced; it is simply a mathematical property of the nature of curved movement trajectories.Â

Â

My feeling is that the power law doesn’t explain the physiological and perceptual process but merely some relationship between force constraints, but I could be completely wrong, and I’m unlikely to be able to show it myself without time and/or the mathematical expertise....

This suggests that you mental model of what is going on with the power law is similar to that of the "ecological" theorists who are doing this kind of research. They (and you) are succumbing to a behavioral illusion. Save yourself before its too late and you end up like Alex and Martin Taylor and Bruce Abbott and Adam Matic. There is still time!! Actually, maybe you can save them by trying to convince them that they are laboring under an illusion. You could do this by suggesting research they could do to test or the variable(s) people control when making curved movements. Save them from their Gibsonian nightmare and bring them over to PCT heaven!
Best
Rick

[Rick Marken 2018-07-21_10:29:50]

WM: Thank you both Rick and Martin for replying to me. I agree with you Rick that the power law is a behavioural illusion to the degree that it doesn’t tell us how muscle movements are produced. I am trying to explain it purely within the physics for this reason.

 RM: I believe we showed that there is no physics explanation of the power law; the power law purely a result of how it is determined -- using regression analysis that omits one of the predictors of velocity. The degree to which the power law is found using this methodology depends completely on the nature of the movement trajectory itself. So a power law will be found for some trajectories that are produced intentionally (like elliptical finger movements) and some that are produced unintentionally (like the movement paths of the pursuers of toy helicopters). And the power law will also not be found for for some trajectories that are produced intentionally (like the mouse movements that produce elliptical cursor movements) and some that are produced unintentionally (like the movement trajectory of the plants).Â

WM: To answer Martin’s point it strikes me that even a lava has the same kind of constraints as a racing car - limits in the force it can apply, a requirement to get to its destination as quickly as possibly, physical constraints that require it to curve its movement at certain points. The fact that a person driving a car and a larva have vastly different locomotion systems in itself seems to indicate that the power law tells us nothing about how the movements are produced, as Rick suggests.

RM: Again, this is simply due to the nature of the trajectories themselves. Movement trajectories that happen to have constant affine velocity will be found to fit the power law perfectly; movement trajectories that happen to have affine velocity that correlate with curvature will not fit the power law at all.Â

Â

WM: However, unlike Rick and Martin, I’d like to try to get to the proof purely through the share physics of these various examples of curved motion rather than through the maths alone. To me, the maths is always an abstraction of a more fundamental relationship.Â

RM: There are two very different "maths" that are being used in this debate; conflating the two by calling them both "maths" may be the heart of the problem here. One kind of maths, the kind used by Martin, is being used to show that an observed power law relationship between velocity and curvature reflects a real, physical relationship between these variables. I called this use of maths "bogus", not because there was any error in the math itself but because these maths are irrelevant to our analysis and, thus, the criticisms of our analysis based on these maths are misguided.
RM: The other kind of maths, the kind used by Marken and Shaffer, are tied intimately to how the data in power law studies is analyzed. We start with the computational formulas that power law researchers use to compute the velocity and curvature at each instant during a curved movement. These computed values are then used in a regression analysis to see if the data are fit by a power law.
RM: It is a simple matter to show that there is a simple mathematical relationship between the formula used to compute velocity and that used to compute curvature. That relationship (linearized by taking the log of both sides) is log (V) = 1/3*log(R) +1/3 * log(D) or log (A) = 2/3*log(C) +1/3 * log(D). So this simple analysis of the way data is collected and analyzed shows that if the variable log(D) is omitted from the regression analysis used to determine the coefficient of the best fitting power function relating curvature to velocity, that estimate will deviate from the "power law" coefficient (1/3 or 2/3) in proportion to the size of the correlation between log (D) and the measure of curvature (log(R) or log(D) in the relationship.Â
RM: So the "proof"Â that the power law is a behavioral illusion already exists; we proved it. The importance of this "proof" is that is shows that by concentrating on the power law per se power law researchers are being distracted from what is actually going on when people make curved movements. What is going on is CONTROL and the focus of research on movement should be on what variables are under control.Â

Â

WM: I do think there can be a two way relationship between curvature and velocity too, because if you head too fast around a corner, you ‘skid’ and therefore the curvature decreases...

RM I think you mean that there can be an actual physical -- not just a mathematical - relationship between these variables. I don't know about "two way" relationship but curvature could be related to velocity as a disturbance is to output in a control system if curvature is an independent variable. And curvature is an independent variable when people race cars around racetracks. Here the curvature of the race track is an independent variable that is a disturbance to a controlled variable -- keeping the car on the track. So assuming that the driver is also controlling for (among other things) going fast, the driver will have to vary how fast she goes through the turn so that she also keeps the car on the track. She will also have to vary how much curvature she gives to her own car when going through the turn. So there are a lot of possible controlled variables involved here and the driver's own curvature is one of them, as it is when moving a finger in an ellipse, but now the curvature the road has been added as a separate disturbance.
Best
Rick

 >>>>>>>

All the bestÂ
WarrenÂ

[Rick Marken 2018-07-20_20:26:21]

WM: Following on from my last email, can anyone tell me that centripetal force isn’t relevant to the power law?

<https://physics.stackexchange.com/questions/294141/what-limits-the-top-speed-in-curves&gt;https://physics.stackexchange.com/questions/294141/what-limits-the-top-speed-in-curves

RM: It's relevant to the power law in the sense that it is one of the disturbances to curved movements that makes the means of producing those movements uncorrelated with the movements themselves. So these force disturbances are relevant to the power law inasmuch as they demonstrate that the power law is an example of a behavioral illusion.Â
RM: Since centripetal (and, more likely, centrifugal forces) and the muscle forces that compensate for them are invisible it's hard for see that the curved movements observed in power law research are a controlled result of the muscle forces. That's why I created the experiment I describe at the beginning of my rebuttal to the rebuttals (<https://www.dropbox.com/s/3m51ko4vs1xdult/MarkenShafferReappraisal.pdf?dl=0&gt;https://www.dropbox.com/s/3m51ko4vs1xdult/MarkenShafferReappraisal.pdf?dl=0\). In that experiment curved movements were made with a cursor on the computer screen. These cursor movements were a joint result of mouse movements (the analog of muscle forces) and computer generated disturbances (the analog of the force disturbances to curved movements that are made by moving a finger through the air or water).
RM: The experiment shows that the curved cursor movements -- the controlled result of mouse movements --Â follow the 1/3 and 2/3 power law but the mouse movements that produce these cursor movements don't. So this easily repeated experiment shows, sans math, that the power law is an example of a behavioral illusion. The math shows why the power law holds for some movements but not others; it depends on the nature of the trajectory of the curved movements themselves; trajectories where affine velocity (D) is close to being constant will be found to be consistent with the power law; trajectories where affine velocity is not constant and, therefore, somewhat correlated with curvature,will deviate from the power law, possibly by a great deal. So affiine velocity may be a variable that people control when they produce curved movement. Some PCT research aimed at testing this hypothesis was suggested at the end of our rebuttal.Â
BestÂ
>>>>>>>>>

Hi guys, putting the maths aside for now. How about this as a solution? The faster you go round a corner, the greater the centripetal force you need to counteract, and the more likely it is that one's lateral velocity will overcome friction and lead to an unstable trace (a 'skid' if you like) . So, if there are limits on the force you can apply, then it makes sense to slow down as you go into a bend and speed up when you come out of it. But of course this slowing down and speeding up requires a force too. So might the most efficient relationship between curvature and velocity be one at which the force required to slow down when coming to a bend (increase in curvature) or speed up when coming out of it (decrease in curvature), is no greater than the force one would need to apply to counteract the centripetal force as one goes round the bend of a specific curvature at that velocity. Expressing this in maths is beyond me at the moment, but I think if this is the reason for the power law then it would be independent of 'how' the trace is produced by the person/vehicle/animal because it would be a function of only the curvature of the trace and not the actions used to achieve it. It should also generate some predictions regarding a direct inverse relationship between on-track acceleration and the derivative of curvature (increase or decrease in curvature as one goes round the trace).Â

[Rick Marken 2018-07-19_16:57:19]

[Martin Taylor 2018.07.19.14.09]

MT: Rick, could you help me correct my "bogus mathematics" by pointing out by page and paragraph or by equation number specifically where the mathematics in my comment on the Marken and Shaffer paper is "bogus".

RM: The mathematics are fine. It's the conclusions that are wrong. A particularly egregious example is your "proof" that our equation relating V to R and D (V = R^1/3*D^1/3, equation 6 in your paper) is a tautology. You do this by showing (correctly, I assume) that D^1/3 is equivalent to V*(1/(R^1/3))Â

MT: Of course, that is NOT at all what I showed...Since we are talking about my comment and not your rebuttal, I'll use my numbering.

MT: (1) is the standard expression for R in purely spatial variables, those being the x,y coordinates of a place along a curve, and s being the distance along the curve to that point from some arbitrary zero point. Simple physical argument indicates that a description of a spatial quantity (such as curvature or radius of curvature) must be a function of purely spatial variables, and if non-spatial variables are used, for convenience, they must cancel out of the expression actually used for the calculation.

 RM: Does this mean that the formulas we used to calculate R and V (and C and A) from the data are incorrect?>>>>>>>>>>>>

(2) shows how this cancellation works for the substitution of an arbitrary parameter "z" that is a function of "s". It shows that no matter what z might be, if it has a continuous derivative dz/ds or the inverse ds/dz, the expression for R in (1) can be transformed into the equivalent form in z. Depending on the direction the equivalence is shown, numerator and denominator each have a multiplying factor (ds/dz)3 or (dz/ds)3. These multipliers cancel out, which is why the substitution of z for s (or vice-versa) produces the same result for any z.

RM: What does this have to do with our analysis? That is, how does it relate to the findings of our regression analyses?>>>>>>>>>>>>

MT: In (3), z is taken to be the time it takes for an object that moves arbitrarily along the curve without stopping or retracing to reach the point at which the derivatives are taken. In this case, the numerator of the expression simplifies to (ds/dt)3 = V3. In this equation and the last equivalence of (2), the denominator is Marken and Shaffer's "cross-correlation correction factor" D. If the argument so far has not made it clear that D is V3*f(x,y,s), equation (7) later demonstrates it explicitly. As is necessarily true from basic physical principles, the explicit calculations demonstrate that the general point mentioned above for an arbitrary parameter z holds also if the parameter is time or velocity. The effects of the added variable (in this case V) cancel out.

RM: So why did our regression analyses work so well? What did we do wrong?

Â

MT: Marken and Shaffer choose to ignore the generality of the parameter substitution and the fact that in their specific substitution of the measured velocities for a single experimental run V3 cancels out from numerator and denominator of the fraction that is the expression for R. Instead, they leave V3 explicitly in the numerator, but hide it in their newly discovered "Cross-correlation correction factor". They then use the "cccf" as though it were independent of V in the rest of their paper.

RM: We didn't ignore this. We knew nothing about it. All we knew was what we found in the reports of research on the power law. And there was nothing in the literature about the "generality of the parameter substitution" of which you speak. And what was, indeed, our newly discovered "cross correlation" variable (D) turns out to be a well known parameter of curved movement: affine velocity.Â

Â

MT: I think this is, to put it mildly, a little different from what Rick said above that I showed.

RM: I really tried to find some relevance of your mathematical analysis to the research we described in our power law paper. But I'm not sure there is any relevance because you don't seem to understand -- or want to understand -- what we did. This is evidenced by what you say at the beginning of your mathematical critique of our work: "Accordingly, they assert that measured values of the power law that depart from 1/3 are in error because they omit consideration of D". In fact, we never "asserted" this. What we demonstrated is that measured values of the power law coefficient will depart from 1/3 (for the relationship between R and V and 2/3 for the relationship between C and A) to the extent that the variable D, which power law researchers always omit from the regression analysis, covaries with the curvature variable (R or C) that is included as the predictor variable in the analysis. >>>>>>>>>>>>

MT: Try again, Rick. I keep hoping to be able to learn something from one of your postings, but I haven't won this lottery jackpot yet.

RM: Sure, I'll try again. But you might have better luck if you would explain, as clearly as possible, how your mathematical analysis relates to our regression analysis of actual data from curved movements.

Â

···

On Wed, Jul 25, 2018 at 12:59 AM, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On 25 Jul 2018, at 03:07, Richard Marken <<mailto:rsmarken@gmail.com>rsmarken@gmail.com> wrote:

On Tue, Jul 24, 2018 at 1:33 AM, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On 23 Jul 2018, at 17:19, Richard Marken <<mailto:rsmarken@gmail.com>rsmarken@gmail.com> wrote:

On Mon, Jul 23, 2018 at 1:37 AM, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On 21 Jul 2018, at 18:38, Richard Marken (<mailto:rsmarken@gmail.com>rsmarken@gmail.com via csgnet Mailing List) <<mailto:csgnet@lists.illinois.edu>csgnet@lists.illinois.edu> wrote:

On Fri, Jul 20, 2018 at 11:59 PM, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On 21 Jul 2018, at 04:26, Richard Marken (<mailto:rsmarken@gmail.com>rsmarken@gmail.com via csgnet Mailing List) <<mailto:csgnet@lists.illinois.edu>csgnet@lists.illinois.edu> wrote:

On Fri, Jul 20, 2018 at 4:22 PM, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On 20 Jul 2018, at 09:09, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On Fri, Jul 20, 2018 at 12:57 AM, Richard Marken <<mailto:csgnet@lists.illinois.edu>csgnet@lists.illinois.edu> wrote:

MT: If I have made a mathematical error in my other comments on Marken and Shaffer, I really would like to know. But you please comment on what I wrote, rather than on something you invented, as you did in this case.

RM: As I said before, I don't think you have made any mathematical errors. I just don't see the relevance of your mathematical analysis to what we actually did with our analysis of actual curved movement data. Did we use the wrong formulas to calculate instantaneous velocity and curvature? Did we do the regression incorrectly? Did we use the wrong variables in the regressions?Â
Best
Rick
 >>>>>>>>>>>>

Martin

so that V = R^1/3*D^1/3 = R^1/3* V*(1/(R^1/3)) which, of course, reduces to V=V.Â
RM: But as I've said, that's true of any equation. The fact that V = R^1/3*D^1/3 can be reduced to V = V doesn't negate the value of knowing that V = R^1/3*D^1/3. This equation analyzes V into its components just as simple one way analysis of variance (ANOVA) analyzes the total variance in scores in an experiment (MS.total) into two components, the variance in scores across (MS.between) and within (MS.within) conditions, so that MS.total = MS.between + MS.within. This is the basic equation of ANOVA.Â
RM: Of course, it's possible to show that MS.total = MS.between + MS.within is a "tautology": MS.total = MS.total. We can do this by noting that MS.within = MS.total - MS.between so that MS.total = MS.between + MS.total - MS.between which, you'll note, reduces to MS.total = MS.total.
RM: But by analyzing MS.total into MS.between and MS.within we can learn some interesting things about the data by computing the two variance components of MS.total and forming the ratio MS.between/MS.within, a ratio known as F (for Sir Ronald Fisher, who invented this analysis method and, as far as I know, never caught flack from anyone about the basic equation of ANOVA being a tautology). Knowing the probability of getting different F ratios in experiments where the independent variable has no effect (the null hypothesis), it is possible to use the F ratio observed in an experiment to decide whether one can reject the null hypothesis with a sufficiently small probability of being wrong.Â
RM: Just as it has proved useful to analyze the total variance in experiments ( MS.total)Â into variance component (MS.between, MS.within and sometimes MS.interaction and MS subjects) it proved useful to us to analyze the variance in the velocity, V, of a curved movement into components, R and D. This analysis produced the equation V = R^1/3*D^1/3. R and D are measures of two different components of the temporal variation in curved movement just as MS.between and MS.within are measures of two different components of the variation in the scores observed in an experiment; R is the variation in curvature and D is the variation in affine velocity.Â
RM: Our equation says that the variation in V for a curved movement will be exactly equal to R^1/3*D^1/3. Linearizing this equation by taking the log of both sides we get log (V) = 1/3*log (R) +1/3*log (D) . This equation shows that if one did a linear regression using the variables log(R) and log(D) as predictors and the variable log(V) as the criterion, the coefficients of the two predictor variables would be exactly 1/3 with an intercept of 0. More importantly, this equation shows that if the variable log (D) isomitted from the regression, the coefficient of log(R) will not necessarily be found to be exactly 1/3 and the intercept will not necessarily be found to be exactly 0. This is where Omitted Variable Bias (OVB) analysis comes in. This analysis makes if possible to predict exactly what a regression analysis will find the coefficient of log(R) to be if log(D) is omitted from the regression.
RM: This finding is important because the "power law" of movement is determined by doing a regression of log (R) on log (V) using the regression equation log (V) = k + b*log(R), omitting the variable log(D). The term "power law" refers to the fact that the results of this regression analysis consistently finds that the power coefficient b is close to 1/3. Our analysis shows that this is a statistical artifact that results from having left the variable log(D) out of the regression analysis. OVB analyiss shows that the amount by which the b coefficient is found to deviate from 1/3 depends on the degree of covariation between the variable included in the regression (log (R)) and the variable omitted from the regression (log(D)). Since both log (R) and log (D) are measured from data (temporal variations in the x,y position of the curved movement) the covariation between these variables is easily calculated and the predicted deviation of the power coefficient, b, from 1/3 can be exactly predicted.Â
RM: The covariation between log (R) and log (D) depends on the nature of the curved movement trajectory itself and has nothing to do with how that movement was generated. It is in this sense that the observed power law is a "behavioral illusion", the illusion being that the relatively consistent observation of an approximately 1/3 power relationship between the curvature (R) and velocity (V) of curved movements seems to reveal something important about how these movements are produced, when it doesn't.Â
RM: So the fact that the equation V = R^1/3*D^1/3 can be reduced to V = V does not negate the value of analyzing V into its components any more than the fact that the equation MS.total = MS.between + MS.within can be reduced to MS.total = MS.total negates the value of analyzing MS.total into its components.Â
RM: There are many other incorrect conclusions in your rebuttal to our paper, Martin. But I think this is enough for now since your "tautology" claim (based on our alleged mathematical mistake) seemed to be central to your argument.Â
BestÂ
Rick

You can do this without referring either to your rebuttal or to the eight falsehoods that I asked you not to try to justify at this point. My question is not about them, but specifically about what in my mathematics you have shown to be bogus. Your previous response did not address this question.

Martin

--

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Dr Warren Mansell
Reader in Clinical Psychology
School of Health Sciences
2nd Floor Zochonis Building
University of Manchester
Oxford Road
Manchester M13 9PL
Email: <mailto:warren.mansell@manchester.ac.uk>warren.mansell@manchester.ac.uk
Â
Tel: +44 (0) 161 275 8589
Â
Website: <Redirect | Faculty of Biology, Medicine and Health | The University of Manchester

Â

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - <http://www.routledge.com/books/details/9780415738781/&gt;Principles\-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check <http://www.pctweb.org>www.pctweb.org for further information on Perceptual Control Theory

--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[rsmarken]

···

[Rick Marken 2018-07-25_10:09:41]

[Martin Taylor 2018.07.25.09.20]

MT:Â Why do you keep repeating this lie?Â

 RM: Because it’s not a lie.

MT: If you have read ANYTHING I have written on the subject in the last

many months, or even if you had simply read my published critique,
you absolutely know that it is not true, So why say it again and
again?

RM: I have read your stuff very carefully and what I say above is precisely true!

Â

MT: Let’s make the actual criticism clear to all the CSGnet readers.

RM: Yes, let’s!

Â

---

MT: If you have an equation

MT: z cancels out of the equation. It is therefore impossible to say

anything about a relationship between x and z.

 RM: You betcha.Â

MT: This is the case for the equations used by Marken and Shaffer:

RM: This is only one equation and it is not one of the ones used by Marken and Shaffer.Â

Â

MT: Since V³ cancels out of the equation, R can be written as
a function only of s, x, and y, and it is therefore impossible to
say anything about a relationship between V and R.

RM: Yes, that is true for that equation.

Â

MT: In my published comment equations 1, 2, 3, and 7, which Marken has

accepted as correct, all make this same point.Â

---

RM: Right. It is a point that is completely irrelevant to our analysis. Our analysis is based on the computational formulas that are used to compute the values of the instantaneous velocity and curvature variables that go into the regression analysis that is used to determine whether the data are fit by a power law. The computational formulas are as follows:

RM: These are the formulas used to compute the velocity (V) and curvature (R) at each instant during a curved movement. The derivatives are all computed with respect to time. Actually, what is computed is an approximation to the time derivative. For example, the time derivative of movement in the X direction in these equations, X.dot, is computed as [X(t)-X(t-tau)]/tau where X(t) and (X(t-tau) are the recorded positions of the movement in the X dimension at time t and time t-tau, tau being the sampling interval.Â

RM: From equations (2) and (3) we can see that:

and, rearranging terms we get:

RM: This equation describes the mathematical relationship that exists between the computed values of V and R. That relationship is also turns out to be a function of the computed value of another variable, D, which we called the cross product variable but, as we learned later from the power law literature itself, is measure of affine velocity.Â

RM: Note that there is nothing in our maths that looks anything like your equation above.Â

RM: But, again, I know that you are not going to give up on this. So how about doing something more productive, like telling me your explanation of the power law. As I said before, I think you will have a lot more luck convincing me that my explanation of the power law (as a statistical artifact) is wrong if you can tell me what the explanation is that you think is right.Â

Best

Rick

Children are (or were) taught early in their math education about an easily made mistake that allows you to prove 1=2. The proof involves using a fraction x/y, which is legitimate except when x = y = 0 (or infinity), and if the proof is written appropriately, it may not be obvious that you have used 0/0. Marken and Shaffer’s error, though they did not use 0/0 literally, is of this same “concealed” class. It has nothing whatever to do with using time derivatives.

Â

–
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

              MT:

Rick, Do you intend to respond to my last message of
four days ago? Are you going to apologise and
acknowledge that my maths is not “bogus” or are you
going to explain to me exactly how it is bogus?

RM: I thought I answered this but I’ll try again.

          RM:Â  I said your math was "bogus" because it is

irrelevant to our analysis of the power law. Our analysis
is based on the computational formulas for the curvature
and velocity variables that go into the regression
analysis used to determine whether or not a movement
trajectory is fit by the “power law”. Your math was based
on the fact that the derivatives in the physics equation
describing the velocity of a curved trajectory are taken
with respect to time while derivatives
in the physics equation describing the curvature of a
curved trajectory are taken with respect to space.Â

[rsmarken]

[Rick Marken 2018-07-25_21:13:52]

Hi Warren

···

On Wed, Jul 25, 2018 at 10:09 AM, Warren Mansell wmansell@gmail.com wrote:

RM: Hi Rick, sorry but it’s not easy to write a reply to your last email for a few different reasons, so I’ll come back to it another time.

RM: No problem. But I will say that while thinking about this (putting physics in a control model of curved movement) I realized that the law that determines the effect of centripetal force on the controlled variable (say, the movement of a pointing finger) is the feedback function that determines the effect of output (the forces exerted by the muscles) on the controlled variable. The effect of muscle forces on finger movement will, indeed, vary due to changes in these forces. So the effect of output on controlled input will be varying over time due to variations in the amount of centripetal force produced by the movements created by muscle forces and the control loop will automatically compensate for these variations, just as it automatically compensates for variations in disturbances (which are independently caused by variations in the effect of gravity on the finger as the finger moves) and maintains the controlled variable (finger position) in its temporally varying reference state.Â

RM: It would be great to set up such a model. It would be a good way to test whether the physics has anything to do with finding the power law. You could test this by having the model produce curved movements with the physics “on” and with the physics “off”. My guess is that the power law will be found for curves made under both conditions.Â

BestÂ

Rick

–
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[rsmarken]

[Rick Marken 2018-07-28_10:31:30]

[Rick Marken 2018-07-25_21:13:52]
RM: It would be great to set up such a model [a control model of curved movement with physics]. It would be a good way to test whether the physics has anything to do with finding the power law. You could test this by having the model produce curved movements with the physics "on" and with the physics "off". My guess is that the power law will be found for curves made under both conditions.Â

RM: I take it back. It might be fun to build such a model but we already know that the power law will be obtained with the physics on or off because I have already shown that such a model produces a power law even with the physics off. This is shown by my demo at the beginning of our rebuttal to all the rebuttals of our power law paper (<https://www.dropbox.com/s/1of34ctxtd9e4lw/MarkenShafferReappraisal.pdf?dl=0&gt;https://www.dropbox.com/s/1of34ctxtd9e4lw/MarkenShafferReappraisal.pdf?dl=0\).Â
RM: So building a control model to see whether physics has something to do with why the power law is found for curved movements would be a waste of time or, worse, a diversion from doing some actual PCT research on curved movement production. PCT research would be aimed at determining the variables organisms control when they produce curved movements. Our demonstration that the power law is a behavioral illusion was aimed at showing that doing research aimed at trying to explain the power law as though that law reflected something important about how organisms produce curved movement is a fool's errand that divert's research away from the study what PCT shows is essential to understanding purposeful behavior: the variables that are being controlled when we see such behaviors being produced.Â
BestÂ
Rick

···

--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[rsmarken]

···

[Rick Marken 2018-07-29_15:20:10]

[Martin Taylor 2018.07.29,10.41]

MT: Exactly!!!

 RM: Excellent. So it’s settled. V = R^1/3V^1/3. Or, in linear form log (V) = 1/3 log(R) + 1/3*log (D).Â

MT: Since you have carefully read, and mathematically understand my

published equations, why on earth do you say that your equation (3)
is not of the form

     

RM: Because it’s not. It’s of the form:

Â

RM: Which is the form used to calculate R from the data. Your equation is not the equation used to calculate R in power law research.Â

f2(s, x, y) is simply (dx/ds)(d<sup>2</sup>y/ds<sup>2</sup>)-(dy/ds)(d<sup>2</sup>x/ds<sup>2</sup>).

RM: So how do you calculate ds from the data?Â

MT: So, because V³ cancels out, leaving R a simple function
of spatial variables, as normal physics says it must be,

RM: What physics? I think there is only mathematics involved here, the mathematics of curved movement trajectories. In the actual production of curved movements, the physics involved is that which relates muscle and external forces to the resulting curved movement. The curved movement that results is characterized by it’s velocity and curvature at each point in the movement. There is no physical relationship between the curvature and velocity of the movement.Â

Â

MT: Experimenters have a useable velocity profile at hand from which to

compute R, so they use it.

RM: Experimenters don’t use a velocity profile to compute R. The velocity profile of the movement is completely invisible to the regression analysis used to determine the power law. Once you have computed the paired values of V and R (using equations 2 and 3) to be used in the regression they can be entered into the analysis in any order; in a random order if you like. This is further proof that the relationship between V and R has nothing to do with physics.

Â

MT: I have to admit that …Â

RM: All you have to admit is that you have no idea what is involved in determining the power law. And you have no explanation of how organisms produce curved movement. You just don’t want to believe that PCT shows that a whole line of research is based on a mistake. I understand your concern but I just haven’t got time to care.Â

Best

RickÂ

–

          RM:... Our analysis is based on the

computational formulas that are used to compute the values
of the instantaneous velocity and curvature variables that
go into the regression analysis that is used to determine
whether the data are fit by a power law. The computational
formulas are as follows:

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[PHILIP_JERAIR_YERANO]

In what way do you get from these equations:

to this equation:

I see that R = V^3/what looks like some chain rule expression.

Can you please explain how you see V^3 in the denominator?

···

On Sun, Jul 29, 2018 at 3:25 PM, Richard Marken csgnet@lists.illinois.edu wrote:

[Rick Marken 2018-07-29_15:20:10]

[Martin Taylor 2018.07.29,10.41]

MT: Exactly!!!

 RM: Excellent. So it’s settled. V = R^1/3V^1/3. Or, in linear form log (V) = 1/3 log(R) + 1/3*log (D).Â

MT: Since you have carefully read, and mathematically understand my

published equations, why on earth do you say that your equation (3)
is not of the form

     

RM: Because it’s not. It’s of the form:

Â

RM: Which is the form used to calculate R from the data. Your equation is not the equation used to calculate R in power law research.Â

f2(s, x, y) is simply (dx/ds)(d<sup>2</sup>y/ds<sup>2</sup>)-(dy/ds)(d<sup>2</sup>x/ds<sup>2</sup>).

RM: So how do you calculate ds from the data?Â

MT: So, because V³ cancels out, leaving R a simple function
of spatial variables, as normal physics says it must be,

RM: What physics? I think there is only mathematics involved here, the mathematics of curved movement trajectories. In the actual production of curved movements, the physics involved is that which relates muscle and external forces to the resulting curved movement. The curved movement that results is characterized by it’s velocity and curvature at each point in the movement. There is no physical relationship between the curvature and velocity of the movement.Â

Â

MT: Experimenters have a useable velocity profile at hand from which to

compute R, so they use it.

RM: Experimenters don’t use a velocity profile to compute R. The velocity profile of the movement is completely invisible to the regression analysis used to determine the power law. Once you have computed the paired values of V and R (using equations 2 and 3) to be used in the regression they can be entered into the analysis in any order; in a random order if you like. This is further proof that the relationship between V and R has nothing to do with physics.

Â

MT: I have to admit that …Â

RM: All you have to admit is that you have no idea what is involved in determining the power law. And you have no explanation of how organisms produce curved movement. You just don’t want to believe that PCT shows that a whole line of research is based on a mistake. I understand your concern but I just haven’t got time to care.Â

Best

RickÂ

–
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

          RM:... Our analysis is based on the

computational formulas that are used to compute the values
of the instantaneous velocity and curvature variables that
go into the regression analysis that is used to determine
whether the data are fit by a power law. The computational
formulas are as follows:

[PHILIP_JERAIR_YERANO]

I agree with Rick’s assertion that the power law should be stated as A = D^1/3 * C^2/3 unless it can be shown that D = x’y’’ - x’‘y’ may behave like a constant. If D behaves like a constant, then A = k*C^b is the proper expression of the power law. If D does not behave like a constant, then the curved movements produced by living organisms (and perhaps all bodies) follow a power law where the velocity of movement is a power function of both (1) the degree of curvature through which the movement is made and (2) the factor D, which is the cross product between the two vectors (x’, x’‘) and (y’, y’'). Being a cross product, the factor D represents a vector which is orthogonal to the plane of the ellipse.Â
If you simply perform the substitutions of variables, you will find that equation 6 is correct but equation 1 is incorrect.

Intuitively, Rick is proposing a new law of motion - a fact that is relevant to physics in general.  Â

···

On Sun, Jul 29, 2018 at 4:38 PM, PHILIP JERAIR YERANOSIAN pyeranos@ucla.edu wrote:

In what way do you get from these equations:

to this equation:

I see that R = V^3/what looks like some chain rule expression.

Can you please explain how you see V^3 in the denominator?

On Sun, Jul 29, 2018 at 3:25 PM, Richard Marken csgnet@lists.illinois.edu wrote:

[Rick Marken 2018-07-29_15:20:10]

[Martin Taylor 2018.07.29,10.41]

MT: Exactly!!!

 RM: Excellent. So it’s settled. V = R^1/3V^1/3. Or, in linear form log (V) = 1/3 log(R) + 1/3*log (D).Â

MT: Since you have carefully read, and mathematically understand my

published equations, why on earth do you say that your equation (3)
is not of the form

     

RM: Because it’s not. It’s of the form:

Â

RM: Which is the form used to calculate R from the data. Your equation is not the equation used to calculate R in power law research.Â

f2(s, x, y) is simply (dx/ds)(d<sup>2</sup>y/ds<sup>2</sup>)-(dy/ds)(d<sup>2</sup>x/ds<sup>2</sup>).

RM: So how do you calculate ds from the data?Â

MT: So, because V³ cancels out, leaving R a simple function
of spatial variables, as normal physics says it must be,

RM: What physics? I think there is only mathematics involved here, the mathematics of curved movement trajectories. In the actual production of curved movements, the physics involved is that which relates muscle and external forces to the resulting curved movement. The curved movement that results is characterized by it’s velocity and curvature at each point in the movement. There is no physical relationship between the curvature and velocity of the movement.Â

Â

MT: Experimenters have a useable velocity profile at hand from which to

compute R, so they use it.

RM: Experimenters don’t use a velocity profile to compute R. The velocity profile of the movement is completely invisible to the regression analysis used to determine the power law. Once you have computed the paired values of V and R (using equations 2 and 3) to be used in the regression they can be entered into the analysis in any order; in a random order if you like. This is further proof that the relationship between V and R has nothing to do with physics.

Â

MT: I have to admit that …Â

RM: All you have to admit is that you have no idea what is involved in determining the power law. And you have no explanation of how organisms produce curved movement. You just don’t want to believe that PCT shows that a whole line of research is based on a mistake. I understand your concern but I just haven’t got time to care.Â

Best

RickÂ

–
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

          RM:... Our analysis is based on the

computational formulas that are used to compute the values
of the instantaneous velocity and curvature variables that
go into the regression analysis that is used to determine
whether the data are fit by a power law. The computational
formulas are as follows:

[rsmarken]

···

[Rick Marken 2018-08-03_22:07:26]

[Martin Taylor 2018.07.30.01.08]

  On 2018/07/29 7:38 PM, PHILIP JERAIR

YERANOSIAN (pyeranos@ucla.edu via csgnet Mailing List) wrote:

PY: In what way do you get from these equations:

to this equation:

      PY: I see that R = V^3/what looks like some chain rule

expression.

Can you please explain how you see V^3 in the denominator?

MT: You are going the wrong way around, because any expression for R

must be expressed in spatial variables alone. The point you want is
to demonstrate that this is the case for the expression that starts
with velocities and accelerations, which are expressions in space
and time. The usual way to get there is to start from the expression
in s, x, and y as did Viviani and Stucchi (whom Gribble and Ostry
used as their source – Marken and Shaffer used Gribble and Ostry to
justify their equation) and work from there to the equation labelled
(3). But I’ll work it backwards for you.

RM: The math is irrelevant to seeing that the power law is a side effect of control. The power law is precisely analogous to the tangential velocity profiles discovered by Atkeson and Hollerbach and discussed by Bill Powers in the posts from the CSGNet archives that are copied below. Like the power law, tangential velocity profiles are an “invariant” side effect of making controlled movements of a limb. These side effects become a behavioral illusion when they are taken to tell us something about how the observed movement was produced.Â

RM: The mathematics presented by Dennis Shaffer and I shows why the invariance of power law is found for many controlled (and uncontrolled) movements. Powers’ Little Man model of movement production shows why invariant velocity profiles are found for controlled movements. (note in [Bill Powers (931029.0750 MDT)] where Bill says: “The velocity profiles are individually scaled both as to amplitude and duration in order to generate the congruence that the authors found. If you do this for the joint angular velocity profiles in Little Man V2, you will find similar invariances, even though there is nothing computing them” (emphases mine–RM)).

RM: Even if the math used to explain the invariance of the power law were wrong (which it’s not) it is still easy to see that the power law is a side effect of control and, to the extent that it is taken to tell us something about how the observed movement was produced, it is a behavioral illusion.Â

RM: So it is rather ironic that people who are supposed to be experts on PCT and the behavioral illusion have succumbed to that illusion so thoroughly that they are trying to use math to show that an obvious example of a behavioral illusion – the power law – is not an illusion at all.

BestÂ

Rick

=====================================================

[From Bill Powers (931029.0750 MDT)]

Greg Williams (931028) –

Very nice job in picking up the quote on motor schemas. Don’t

forget to include William James – the constancy of ends and the

variability of means. All this shows that the problem has been

known for a very long time (100 years), but that the solution has

eluded a continuing search.

The critical misdirection is contained in

Thus whatever is learned and stored in long-term memory cannot

be a specific set of muscle commands but must represent a more

generic or general set of specifications of how to reach the

desired goal.

Behind the idea of “specifications of how to reach the goal” is

still a picture of direct causality: the specification is for

how to reach the goal, instead of what goal to reach. You can

see this same idea in Atkeson and Hollerbach:

"A strategy for gaining insight into planning and control

processes of the motor system is to look for kinematic

invariances in trajectories of movement. The significance of

straight-line movements is that they imply movement planning at

the hand or object level." (p. 2318)

In the discussion:

"Taken together, shape invariances for path and tangential

velocity profiles indicates that subjects execute only one form

of trajectory between any two targets when not instructed to do

otherwise. The only changes in the trajectory are simple scaling

operations to accomodate different speeds. … Different subjects

use the same tangential velocity profile shape." (pp. 2325-6).

And making the problem even clearer:

"A number of issues remain with regard to these dynamic scaling

results. How are the initial torques for the first movement

generated? If the motor controller has the ability to fashion the

correct torques for one movement, why does it not use this same

ability for all subsequent movements rather than utilize the

dynamic scaling properties? Among the possibilities we are

considering, the first is a generalized motor tape where only one

movement between points need be known if the dynamic components

in Equation 6 are stored separately. … A second possibility is

a modification of tabular approaches (Rabert, 1978) where the

dimensionality and parameter adjustment problem could be reduced

by separate tables for the four components in equation 6." (p.

2327).

The only possibilities being considered are those that involve

open-loop generation of the torques that will produce a pre-

planned trajectory. “Equation 6” is an equation expressing joint

torque as a nonlinear function of scaling factors applied to

torques previously produced for a known movement. As noted above,

the problem is how the torques for the known movement are

generated in the first place. This problem is not solved. What

the authors hope for is that by finding invariants such as

velocity profiles, they will be able to deduce a motor program

that will produce constant results in object space even when

variations in torque are required.

They simply haven’t got far enough into the problem to see that

this quest is hopeless. If they did manage to come up with a

motor program that could realistically create several

trajectories of hand movement between different pairs of points,

they would then have to ask how this can work with different

loads. If they solved that problem, they would have to explain

how the trajectories are produced when the loads are vary

unexpectedly and the muscles progressively fatigue. And then they

would have to explain how the right torques can be produced under

varying loads (such as the varying friction between pencil and

paper) and with fatiguing muscles, when the task is to write the

subject’s name. And then they would have to explain how a subject

can accomplish the same movements under the same uncertain

conditions for the tip of a pointer held in the hand. In truth,

they are extrapolating a long way ahead, and predicting success,

when they have not even found the simplest motor program of all:

that for moving quickly between two points under undisturbed

conditions. Their projected work simply expresses faith that

somehow the require movements in object space can be generated by

a clever enough motor programming device – without ever taking

feedback into account, the feedback both kinesthetic and visual

that is known to exist and that is known to be essential for

skilled performance.

There is an explanation for the observed invariances of velocity

profiles that Atkeson and Hollerbach never consider: these

invariances might simply be the natural outcome of physical

processes of control. There might be no need at all for the motor

program to precompute them. The velocity profiles are

individually scaled both as to amplitude and duration in order to

generate the congruence that the authors found. If you do this

for the joint angular velocity profiles in Little Man V2, you

will find similar invariances, even though there is nothing

computing them. It just happens that when a control system is

given a step-change of reference-signal, the trajectory of the

controlled variable naturally scales up or down so that the

velocity rises and falls along the same generic curve. This is

purely a consequence of the mathematical relationships of control

and the passive dynamical properties of the arm; nothing is

acting to make sure that the trajectory follows any particular

path. The trajectory is a side-effect, not a planned movement.

Evidence of trajectory planning would appear only if the actual

trajectory departed from the one that can be explained as a step-

change in the reference signal of a control system from one fixed

value to another. For example, one can easily move a finger from

one point to another along a semi-circle or an S-shaped curve.

That requires a “program” of velocity or position reference

signals. But it still doesn’t require precomputing torques.

---

The key idea to look for in all these sources is how the authors

propose to account for the forces that create movements. It’s

clear in Atkeson and Hollerbach that the torques are going to be

computed so as to have the required object-space consequences and

that proprioceptive and visual feedback are not considered. All

approaches that propose to use inverse kinematic or inverse

dynamical computations are also attempting to solve the problem

open-loop. In all such approaches, the key idea that is missing

is comparison of the observed consequences with the desired

consequence in real time as the means of producing the required

output signals.

---

Atkeson, C. G., and Hollerbach, J.M.; Kinematic Features of

Unrestrained Vertical Arm Movements. The Journal oif Neuroscience

5, No. 9, pp. 2318-2330. Sept. 1985.

---

Best,

Bill P.

[From Bill Powers (950527.0950 MDT)]

Just got back from seeing our daughter Barbara off in the start of the

Iron Horse bike race, Durango to Silverton. The length is 45 miles, the

total climb over two main passes is 5500 feet (the highest pass, Molas,

is about 11,000 feet). Last year (her first, at age 35) she did it in

4:20; this year she hopes for under 4:00. The pro winning time last year

was 2:10. She should be about halfway right now, starting the four-mile

climb to Coal Bank Pass (2500 foot climb to over 10,000 ft). Go Bara!

---

Rick Marken, Bruce Abbott (continuing) –

===================================================

When you push on a control system, it pushes back.

===================================================

---

RE: trajectories vs. system organization

In a great deal of modern behavioral research, trajectories of movement

are examined in the hope of finding invariants that will reveal secrets

of behavior. This approach ties in with system models that compute

inverse kinematics and dynamics and use motor programs to produce

actions open-loop. These models assume that the path followed by a limb

or the whole body is specified in advance in terms of end-positions and

derivatives during the transition, so the path that is followed reflects

the computations that are going on inside the system.

It is this orientation that explains papers like

Atkeson, C. G. and Hollerback, J.M.(1985); Kinematic features of

unrestrained vertical arm movements. The Journal of Neuroscience 5,

#9, 2318-2330.

In the described experiments, subjects move a hand in the vertical plane

at various prescribed speeds from a starting point to variously located

targets, and the positions are recorded as videos of the positions of

illuminated targets fastened to various parts of the arm and hand.

The authors constructed a tangential-velocity vs time profile of the

wrist movement for various speeds, directions, and distances of

movement. They normalized the profiles to a fixed magnitude, then to a

fixed duration, and found that the curves then had very nearly the same

shape. Using a “similarity” calculation, they quantified the measures of

similarity.

They were then able to compare these normalized tangential velocity

profiles across various directions and amounts of movement and show that

the treated profiles were very close to the same. They conclude:

    Taken together, shape invariance for path and tangential velocity

    profile indicates that subjects execute only one form of trajectory

    between any two targets when not instructed to do otherwise. The

    only changes in trajectory are simple scaling operations to

    accomodate different speeds. Furthermore, subjects use the same

    tangential velocity profile shape to make radically different

    movements, even when the shapes of the paths are not the same in

    extrinsic coordinates. Different subjects use the same tangential

    velocity profile shape.

    … this would be consistent with a simplifying strategy for joint

    torque formation by separation of gravity torques from dynamic

    torques and a uniform scaling of the tangential velocity profile

    … (p. 2325)

    … if the motor controller has the ability to fashion correct

    torques for one movement, why does it not use this same ability for

    all subsequent movements rather than utilize the dynamic scaling

    properties? Among the possibilities we are considering, the first

    is a generalized motor tape where only one movement between points

    must be known if the dynanmic components in equation 6 are stored

    separately…A second possibility is a modification of tabular

    approaches [ref] where the dimensionality and parameter adjustment

    problem could be reduced by separate tables for the four components

    in equation 6. (p. 2326)

This paper was sent to me by Greg Williams as a source of data about

actual hand movements, for comparison with the hand movements generated

by Little Man v. 2, the version using actual arm dynamics for the

external part of the model. The model’s hand movements were, as Greg

will attest, quite close to those shown in this paper, being slightly

curved lines connecting the end-points. Forward and reverse movements

followed somewhat different paths, and by adjustment of model parameters

this difference, too, could be reproduced.

What is interesting is that the fit between the Little Man and the real

data was found without considering tangential velocity profiles or doing

any scaling or normalization. In other words, the invariances noted by

the authors were simply side-effects of the operation of the control

systems of the arm interacting with the dynamics of the physical arm. In

the Little Man there is no trajectory planning, no storage of movement

parameters, no table-lookup facility, no computation of invariant

velocity profiles. The observed behavior is simply a reflection of the

organization of the control system and the physical plant.

The path which Atkeson, Hollerbach (and many others at MIT and

elsewhere) are treading is a blind alley, because no matter how

carefully the observations are made and the invariances are calculated,

there will be no hint of the control-system organization, the SIMPLE

control-system organization, that (I claim) is actually creating the

observed trajectories. No doubt a sufficiently complex trajectory-

control model, with just the right tables of coefficients and velocity

profiles, would ultimately be able to match the behavior. But this line

of investigation, with its underlying assumptions, will never lead to

the far simpler and anatomically correct PCT model.

In terms of the current discussion on the net, the observations made by

the authors were interesting as checks on the model, but were actually

irrelevant to what the control systems were doing. The control systems

(the first two levels of the Little Man model) controlled only three

kinds of variables that underlay the perceptual signals: angular

positions, angular velocities, and angular accelerations. They received

no information about wrist position in laboratory space. They contained

no provision for computing tangential velocities, or for computing

positions of points on the physical arm in space, or for computing

space-time invariants. The behavior of the control systems, in other

words, took place in a proprioceptive perceptual space that no outside

observer could see. In order to translate from this perceptual space

into variables that were observable, the computer program generated the

resulting arm positions and plotted them in a form suitable for visual

inspection. So a side-effect of the actual control process was presented

for comparison with a corresponding side-effect of the real control

process, as visible to an outside observer.

The approach of Atkeson and Hollerbach appears in many guises. We have

already talked about the apparent scaling and normalization of

trajectories seen when two hands move rapidly and simultaneously to

targets at different distances. In operant conditioning experiments, we

have seen how the control of reinforcement by behavior is obscured by

the fact that variations in behavior tend to stabilize reinforcement

rates, thus making reinforcement rate appear to be the independent

variable.

We have also seen a few – a very few, so far – studies in which the

PCT orientation was used, Srinivasan’s being the most recent. What is

the difference? I think the difference is in whether the emphasis is on

seeing the behavior from the behaving systems’s point of view, as best

we can imagine it, and seeing it strictly from the human observer’s

point of view.

From the human observer’s point of view, it seems that we must account

for the detailed movements and physical interactions that are seen to

occur. This leads to trying to find invariances or striking mathematical

regularities of some sort in the observed behaviors. It leads to

imagining an internal system that is producing explicitly what we are

observing; if we observe a trajectory, there must be some generator that

is specifically calculating that trajectory.

But from the behaving system’s point of view, we can consider only the

information that is available to the behaving system; we must look for

our explanations there. The trajectories of movement that result from

the system’s operation are basically side-effects; they are not planned

and they are constant only in a constant environment. Furthermore, they

are unknown to the behaving system and play no part in the production of

behavior. We can deduce from the model of the behaving system what the

observable side-effects would be in a given environment, and so can

compare those side-effects with our external observations of the

behavior. But our explanation of the behavior is not based on those

side-effects.

Most important, when we simply describe behavior as a sequence of

physical happenings and relationships, we have no way of knowing whether

we are describing controlled variables or side-effects. When we see a

fly landing on a ceiling, it is perfectly possible that NOT A SINGLE

ASPECT OF WHAT WE SEE is perceived and controlled by the fly. When we

see the fly extending its legs just prior to landing, the fly may have

no perception of the configuration of its legs; to the fly, all that is

controlled may be two or three joint-angle signals, not even identified

by the fly as representing joint angle. When we see the wings stop

flapping, to the fly all that may be controlled is a sensation of

vibration. When we see the fly’s body making a steep angle with the

surface, the fly may simply be experiencing a visual signal indicating,

as Rick guessed, a gradient of illumination or texture. Not one of the

variables we are observing may ever appear in the ultimate model of the

fly’s internal organization, just as in the Little Man the actual arm

configuration and hand position never appear in the model of the first

two (kinesthetic) levels of control. Once we have the right model, we

can always compute how its operation will appear to an observer who is

focusing on various side-effects of the actions. But the model itself

says nothing about those appearances, and makes no use of them.

---

Best to all,

Bill P.

–
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[amatic]

[From Adam Matic]

My bet is you still don’t know the difference between common uses of trajectory and path, Rick. Other that that, you keep saying mutually conflicting statements, or you keep flipping between two extreme positions on some issues. Your statements reveal confusion, but your are still very confrontational and trying to sound authoritative. That is just silly. You’re on fast track to being a crank scientist. However, given your modus operandi of giving strong statements, then going back on them after a few days (weeks, months?) when you realize your mistake, you just might come back from the dark side. That is probably Martin’s hope too.

RM: The math is irrelevant to seeing that the power law is a side effect of control. The power law is precisely analogous to the tangential velocity profiles discovered by Atkeson and Hollerbach and discussed by Bill Powers in the posts from the CSGNet archives that are copied below. Like the power law, tangential velocity profiles are an “invariant” side effect of making controlled movements of a limb. These side effects become a behavioral illusion when they are taken to tell us something about how the observed movement was produced.Â

AM: Yes, the power law is analogous to bell-shaped velocity profiles. They are not a controlled result (as Bill says), just like power law trajectories are not controlled results (this is my hypothesis). You, on the other hand, claim that power-law trajectories are a controlled result. You also published a model of ellipse tracing which has a 2/3 power law trajectory as the reference trajectory. That is where you are conflicting with yourself - if the trajectory is a side effect then it is not controlled. If it is controlled, then it is telling us that the reference trajectory is also a 2/3 power law trajectory.

AM: The behavioral illusion is one of the issues you have flipped up and down several times, I’ll number it. After publishing two papers saying 1) the power law is behavioral illusion; sometime in May on csgnet you had no problems saying that 2) the “power law is not a behavioral illusion as in Powers(1978)”, then 2) “it is maybe the illusion of control”, then 3) maybe we should just stop telling people about this illusion business; and again back to 1) it is a behavioral illusion.

According to the 1978 paper, a behavioral illusion happens when someone mistakes a correlation between a stimulus and a response for a causal relationship. No one is taking the instantaneous curvatures and velocities of the response to be causally related, those are both properties of the response. There are hypotheses on causal role of the brain in the power law (the reference trajectory is a power law trajectory) and there is the unrelated causal role of the shape of the stimulus and the exponent of the power law (you get different exponents if you draw different shapes).Â

The math you and Shaffer present on the meaning of the power law exponent is based on some serious misunderstanding. The exponent of curvature - angular velocity power law just tells you how much an object traveling along a path is slowing down in corners. If the exponent is 1, the speed is going to be constant all over. If the exponent is lower, such as 2/3, it means the object is going a bit slower in the corners than in the straight parts. If is still lower, it means the object is slowing down a lot in the corners, and speeding up a lot in the straight parts. It is complete nonsense to speak about a “true exponent” (see, no need for math to show your bogus logic).

RM: So it is rather ironic that people who are supposed to be experts on PCT and the behavioral illusion have succumbed to that illusion so thoroughly that they are trying to use math to show that an obvious example of a behavioral illusion – the power law – is not an illusion at all.

AM:

It is ironic that you keep saying “PCT says” and then present your own bogus hypotheses. There is nothing in PCT that says the power law is a behavioral illusion or that the research on the power law is based on a mistake of confusing correlation for a causal relationship. You are saying that. And you are kinda shit at math, as you have been before, but you seem to have been quicker to admit it and did not make grand claims based on your “calculations”. Maybe it was Bill who corrected you when you strayed over to crank territory, now you are just pushing forward.Â

Best,

Adam

···

[Rick Marken 2018-08-03_22:07:26]

[Martin Taylor 2018.07.30.01.08]

  On 2018/07/29 7:38 PM, PHILIP JERAIR

YERANOSIAN (pyeranos@ucla.edu via csgnet Mailing List) wrote:

PY: In what way do you get from these equations:

to this equation:

      PY: I see that R = V^3/what looks like some chain rule

expression.

Can you please explain how you see V^3 in the denominator?

MT: You are going the wrong way around, because any expression for R

must be expressed in spatial variables alone. The point you want is
to demonstrate that this is the case for the expression that starts
with velocities and accelerations, which are expressions in space
and time. The usual way to get there is to start from the expression
in s, x, and y as did Viviani and Stucchi (whom Gribble and Ostry
used as their source – Marken and Shaffer used Gribble and Ostry to
justify their equation) and work from there to the equation labelled
(3). But I’ll work it backwards for you.

RM: The math is irrelevant to seeing that the power law is a side effect of control. The power law is precisely analogous to the tangential velocity profiles discovered by Atkeson and Hollerbach and discussed by Bill Powers in the posts from the CSGNet archives that are copied below. Like the power law, tangential velocity profiles are an “invariant” side effect of making controlled movements of a limb. These side effects become a behavioral illusion when they are taken to tell us something about how the observed movement was produced.Â

RM: The mathematics presented by Dennis Shaffer and I shows why the invariance of power law is found for many controlled (and uncontrolled) movements. Powers’ Little Man model of movement production shows why invariant velocity profiles are found for controlled movements. (note in [Bill Powers (931029.0750 MDT)] where Bill says: “The velocity profiles are individually scaled both as to amplitude and duration in order to generate the congruence that the authors found. If you do this for the joint angular velocity profiles in Little Man V2, you will find similar invariances, even though there is nothing computing them” (emphases mine–RM)).

RM: Even if the math used to explain the invariance of the power law were wrong (which it’s not) it is still easy to see that the power law is a side effect of control and, to the extent that it is taken to tell us something about how the observed movement was produced, it is a behavioral illusion.Â

RM: So it is rather ironic that people who are supposed to be experts on PCT and the behavioral illusion have succumbed to that illusion so thoroughly that they are trying to use math to show that an obvious example of a behavioral illusion – the power law – is not an illusion at all.

BestÂ

Rick

=====================================================

[From Bill Powers (931029.0750 MDT)]

Greg Williams (931028) –

Very nice job in picking up the quote on motor schemas. Don’t

forget to include William James – the constancy of ends and the

variability of means. All this shows that the problem has been

known for a very long time (100 years), but that the solution has

eluded a continuing search.

The critical misdirection is contained in

Thus whatever is learned and stored in long-term memory cannot

be a specific set of muscle commands but must represent a more

generic or general set of specifications of how to reach the

desired goal.

Behind the idea of “specifications of how to reach the goal” is

still a picture of direct causality: the specification is for

how to reach the goal, instead of what goal to reach. You can

see this same idea in Atkeson and Hollerbach:

"A strategy for gaining insight into planning and control

processes of the motor system is to look for kinematic

invariances in trajectories of movement. The significance of

straight-line movements is that they imply movement planning at

the hand or object level." (p. 2318)

In the discussion:

"Taken together, shape invariances for path and tangential

velocity profiles indicates that subjects execute only one form

of trajectory between any two targets when not instructed to do

otherwise. The only changes in the trajectory are simple scaling

operations to accomodate different speeds. … Different subjects

use the same tangential velocity profile shape." (pp. 2325-6).

And making the problem even clearer:

"A number of issues remain with regard to these dynamic scaling

results. How are the initial torques for the first movement

generated? If the motor controller has the ability to fashion the

correct torques for one movement, why does it not use this same

ability for all subsequent movements rather than utilize the

dynamic scaling properties? Among the possibilities we are

considering, the first is a generalized motor tape where only one

movement between points need be known if the dynamic components

in Equation 6 are stored separately. … A second possibility is

a modification of tabular approaches (Rabert, 1978) where the

dimensionality and parameter adjustment problem could be reduced

by separate tables for the four components in equation 6." (p.

2327).

The only possibilities being considered are those that involve

open-loop generation of the torques that will produce a pre-

planned trajectory. “Equation 6” is an equation expressing joint

torque as a nonlinear function of scaling factors applied to

torques previously produced for a known movement. As noted above,

the problem is how the torques for the known movement are

generated in the first place. This problem is not solved. What

the authors hope for is that by finding invariants such as

velocity profiles, they will be able to deduce a motor program

that will produce constant results in object space even when

variations in torque are required.

They simply haven’t got far enough into the problem to see that

this quest is hopeless. If they did manage to come up with a

motor program that could realistically create several

trajectories of hand movement between different pairs of points,

they would then have to ask how this can work with different

loads. If they solved that problem, they would have to explain

how the trajectories are produced when the loads are vary

unexpectedly and the muscles progressively fatigue. And then they

would have to explain how the right torques can be produced under

varying loads (such as the varying friction between pencil and

paper) and with fatiguing muscles, when the task is to write the

subject’s name. And then they would have to explain how a subject

can accomplish the same movements under the same uncertain

conditions for the tip of a pointer held in the hand. In truth,

they are extrapolating a long way ahead, and predicting success,

when they have not even found the simplest motor program of all:

that for moving quickly between two points under undisturbed

conditions. Their projected work simply expresses faith that

somehow the require movements in object space can be generated by

a clever enough motor programming device – without ever taking

feedback into account, the feedback both kinesthetic and visual

that is known to exist and that is known to be essential for

skilled performance.

There is an explanation for the observed invariances of velocity

profiles that Atkeson and Hollerbach never consider: these

invariances might simply be the natural outcome of physical

processes of control. There might be no need at all for the motor

program to precompute them. The velocity profiles are

individually scaled both as to amplitude and duration in order to

generate the congruence that the authors found. If you do this

for the joint angular velocity profiles in Little Man V2, you

will find similar invariances, even though there is nothing

computing them. It just happens that when a control system is

given a step-change of reference-signal, the trajectory of the

controlled variable naturally scales up or down so that the

velocity rises and falls along the same generic curve. This is

purely a consequence of the mathematical relationships of control

and the passive dynamical properties of the arm; nothing is

acting to make sure that the trajectory follows any particular

path. The trajectory is a side-effect, not a planned movement.

Evidence of trajectory planning would appear only if the actual

trajectory departed from the one that can be explained as a step-

change in the reference signal of a control system from one fixed

value to another. For example, one can easily move a finger from

one point to another along a semi-circle or an S-shaped curve.

That requires a “program” of velocity or position reference

signals. But it still doesn’t require precomputing torques.

---

The key idea to look for in all these sources is how the authors

propose to account for the forces that create movements. It’s

clear in Atkeson and Hollerbach that the torques are going to be

computed so as to have the required object-space consequences and

that proprioceptive and visual feedback are not considered. All

approaches that propose to use inverse kinematic or inverse

dynamical computations are also attempting to solve the problem

open-loop. In all such approaches, the key idea that is missing

is comparison of the observed consequences with the desired

consequence in real time as the means of producing the required

output signals.

---

Atkeson, C. G., and Hollerbach, J.M.; Kinematic Features of

Unrestrained Vertical Arm Movements. The Journal oif Neuroscience

5, No. 9, pp. 2318-2330. Sept. 1985.

---

Best,

Bill P.

[From Bill Powers (950527.0950 MDT)]

Just got back from seeing our daughter Barbara off in the start of the

Iron Horse bike race, Durango to Silverton. The length is 45 miles, the

total climb over two main passes is 5500 feet (the highest pass, Molas,

is about 11,000 feet). Last year (her first, at age 35) she did it in

4:20; this year she hopes for under 4:00. The pro winning time last year

was 2:10. She should be about halfway right now, starting the four-mile

climb to Coal Bank Pass (2500 foot climb to over 10,000 ft). Go Bara!

---

Rick Marken, Bruce Abbott (continuing) –

===================================================

When you push on a control system, it pushes back.

===================================================

---

RE: trajectories vs. system organization

In a great deal of modern behavioral research, trajectories of movement

are examined in the hope of finding invariants that will reveal secrets

of behavior. This approach ties in with system models that compute

inverse kinematics and dynamics and use motor programs to produce

actions open-loop. These models assume that the path followed by a limb

or the whole body is specified in advance in terms of end-positions and

derivatives during the transition, so the path that is followed reflects

the computations that are going on inside the system.

It is this orientation that explains papers like

Atkeson, C. G. and Hollerback, J.M.(1985); Kinematic features of

unrestrained vertical arm movements. The Journal of Neuroscience 5,

#9, 2318-2330.

In the described experiments, subjects move a hand in the vertical plane

at various prescribed speeds from a starting point to variously located

targets, and the positions are recorded as videos of the positions of

illuminated targets fastened to various parts of the arm and hand.

The authors constructed a tangential-velocity vs time profile of the

wrist movement for various speeds, directions, and distances of

movement. They normalized the profiles to a fixed magnitude, then to a

fixed duration, and found that the curves then had very nearly the same

shape. Using a “similarity” calculation, they quantified the measures of

similarity.

They were then able to compare these normalized tangential velocity

profiles across various directions and amounts of movement and show that

the treated profiles were very close to the same. They conclude:

    Taken together, shape invariance for path and tangential velocity

    profile indicates that subjects execute only one form of trajectory

    between any two targets when not instructed to do otherwise. The

    only changes in trajectory are simple scaling operations to

    accomodate different speeds. Furthermore, subjects use the same

    tangential velocity profile shape to make radically different

    movements, even when the shapes of the paths are not the same in

    extrinsic coordinates. Different subjects use the same tangential

    velocity profile shape.

    … this would be consistent with a simplifying strategy for joint

    torque formation by separation of gravity torques from dynamic

    torques and a uniform scaling of the tangential velocity profile

    … (p. 2325)

    … if the motor controller has the ability to fashion correct

    torques for one movement, why does it not use this same ability for

    all subsequent movements rather than utilize the dynamic scaling

    properties? Among the possibilities we are considering, the first

    is a generalized motor tape where only one movement between points

    must be known if the dynanmic components in equation 6 are stored

    separately…A second possibility is a modification of tabular

    approaches [ref] where the dimensionality and parameter adjustment

    problem could be reduced by separate tables for the four components

    in equation 6. (p. 2326)

This paper was sent to me by Greg Williams as a source of data about

actual hand movements, for comparison with the hand movements generated

by Little Man v. 2, the version using actual arm dynamics for the

external part of the model. The model’s hand movements were, as Greg

will attest, quite close to those shown in this paper, being slightly

curved lines connecting the end-points. Forward and reverse movements

followed somewhat different paths, and by adjustment of model parameters

this difference, too, could be reproduced.

What is interesting is that the fit between the Little Man and the real

data was found without considering tangential velocity profiles or doing

any scaling or normalization. In other words, the invariances noted by

the authors were simply side-effects of the operation of the control

systems of the arm interacting with the dynamics of the physical arm. In

the Little Man there is no trajectory planning, no storage of movement

parameters, no table-lookup facility, no computation of invariant

velocity profiles. The observed behavior is simply a reflection of the

organization of the control system and the physical plant.

The path which Atkeson, Hollerbach (and many others at MIT and

elsewhere) are treading is a blind alley, because no matter how

carefully the observations are made and the invariances are calculated,

there will be no hint of the control-system organization, the SIMPLE

control-system organization, that (I claim) is actually creating the

observed trajectories. No doubt a sufficiently complex trajectory-

control model, with just the right tables of coefficients and velocity

profiles, would ultimately be able to match the behavior. But this line

of investigation, with its underlying assumptions, will never lead to

the far simpler and anatomically correct PCT model.

In terms of the current discussion on the net, the observations made by

the authors were interesting as checks on the model, but were actually

irrelevant to what the control systems were doing. The control systems

(the first two levels of the Little Man model) controlled only three

kinds of variables that underlay the perceptual signals: angular

positions, angular velocities, and angular accelerations. They received

no information about wrist position in laboratory space. They contained

no provision for computing tangential velocities, or for computing

positions of points on the physical arm in space, or for computing

space-time invariants. The behavior of the control systems, in other

words, took place in a proprioceptive perceptual space that no outside

observer could see. In order to translate from this perceptual space

into variables that were observable, the computer program generated the

resulting arm positions and plotted them in a form suitable for visual

inspection. So a side-effect of the actual control process was presented

for comparison with a corresponding side-effect of the real control

process, as visible to an outside observer.

The approach of Atkeson and Hollerbach appears in many guises. We have

already talked about the apparent scaling and normalization of

trajectories seen when two hands move rapidly and simultaneously to

targets at different distances. In operant conditioning experiments, we

have seen how the control of reinforcement by behavior is obscured by

the fact that variations in behavior tend to stabilize reinforcement

rates, thus making reinforcement rate appear to be the independent

variable.

We have also seen a few – a very few, so far – studies in which the

PCT orientation was used, Srinivasan’s being the most recent. What is

the difference? I think the difference is in whether the emphasis is on

seeing the behavior from the behaving systems’s point of view, as best

we can imagine it, and seeing it strictly from the human observer’s

point of view.

From the human observer’s point of view, it seems that we must account

for the detailed movements and physical interactions that are seen to

occur. This leads to trying to find invariances or striking mathematical

regularities of some sort in the observed behaviors. It leads to

imagining an internal system that is producing explicitly what we are

observing; if we observe a trajectory, there must be some generator that

is specifically calculating that trajectory.

But from the behaving system’s point of view, we can consider only the

information that is available to the behaving system; we must look for

our explanations there. The trajectories of movement that result from

the system’s operation are basically side-effects; they are not planned

and they are constant only in a constant environment. Furthermore, they

are unknown to the behaving system and play no part in the production of

behavior. We can deduce from the model of the behaving system what the

observable side-effects would be in a given environment, and so can

compare those side-effects with our external observations of the

behavior. But our explanation of the behavior is not based on those

side-effects.

Most important, when we simply describe behavior as a sequence of

physical happenings and relationships, we have no way of knowing whether

we are describing controlled variables or side-effects. When we see a

fly landing on a ceiling, it is perfectly possible that NOT A SINGLE

ASPECT OF WHAT WE SEE is perceived and controlled by the fly. When we

see the fly extending its legs just prior to landing, the fly may have

no perception of the configuration of its legs; to the fly, all that is

controlled may be two or three joint-angle signals, not even identified

by the fly as representing joint angle. When we see the wings stop

flapping, to the fly all that may be controlled is a sensation of

vibration. When we see the fly’s body making a steep angle with the

surface, the fly may simply be experiencing a visual signal indicating,

as Rick guessed, a gradient of illumination or texture. Not one of the

variables we are observing may ever appear in the ultimate model of the

fly’s internal organization, just as in the Little Man the actual arm

configuration and hand position never appear in the model of the first

two (kinesthetic) levels of control. Once we have the right model, we

can always compute how its operation will appear to an observer who is

focusing on various side-effects of the actions. But the model itself

says nothing about those appearances, and makes no use of them.

---

Best to all,

Bill P.

–
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[rsmarken]

[Rick Marken 2018-08-05_18:52:37]

[From Adam Matic]

AM: Yes, the power law is analogous to bell-shaped velocity profiles. They are not a controlled result (as BillÂ
says), just like power law trajectories are not controlled results (this is my hypothesis).

RM: It's the power law -- not the power law trajectory -- that is not a controlled result.Â

AM: You, on the other hand, claim that power-law trajectories *are a controlled result*.

RM: No, I claim that it is the time varying position of a purposefully produced movement that is a controlled result.Â

AM: You also published a model of ellipse tracing which has a 2/3 power law trajectory as the reference trajectory.
That is where you are conflicting with yourself - if the trajectory is a side effect then it is not controlled.

RM: It's not the trajectory that is a side effect; it is the power law that is the side effect.

AM: If it is controlled, then it is telling us that the reference trajectory is also a 2/3 power law trajectory.

RM: The model I presented accounts for the behavior in an experiment where a person is asked to make elliptical movements
with a cursor while the two dimensional position of the cursor is being affected by a filtered random two dimensional disturbance.Â
The point of this experiment (and the model that accounts nearly perfectly for the behavior observed in the experiment) was to
show that a movement trajectory can follow the power law (as was the case for the movement trajectory of the cursor) while theÂ
means used to produce this trajectory (the trajectory of the mouse movements) don't. That is, the experimental results (and theÂ
model that accounts for those results) show that the power law of movement tells us nothing about how the power law conformingÂ
movement trajectory was produced. The experiment (and model) were not designed to show why the cursor movement trajectoryÂ
followed a power law.Â

AM: The behavioral illusion is one of the issues you have flipped up and down several times, I'll number it. After publishingÂ
two papers saying 1) the power law is behavioral illusion; sometime in May on csgnet you had no problems saying thatÂ
2) the "power law is not a behavioral illusion as in Powers(1978)", then 2) "it is maybe the illusion of control", then 3) maybeÂ
we should just stop telling people about this illusion business; and again back to 1) it is a behavioral illusion.

RM: I don't see any of those as "flips". The power law is a behavioral illusion to those who take it to say something importantÂ
about how movement is produced. The power law illusion is not the same as the S-R illusion described by Powers (1978)
because it is based on a different side effect of control. I don't know the context of the statement "it is maybe the illusion ofÂ
control" so I have no idea what that statement pertained to. And I might have suggested that it may be wise to "stop telling
people about this illusion business" (if that's what I said) because people who are told that may find it insulting. Â

AM: According to the 1978 paper, a behavioral illusion happens when someone mistakes a correlation between a stimulusÂ
and a response for a causal relationship.

RM: Yes, that is one type of behavioral illusion. Another is the illusion that reinforcement selects behavior. Another is thatÂ
outputs are emitted. And another is the kind seen in both the invariant tangential velocity profiles and the power law ofÂ
movement; the illusion that some invariant mathematical property of behavior tells something about how the behaviorÂ
was produced.
Â

AM: No one is taking the instantaneous curvatures and velocities of the response to be causally related, those are bothÂ
properties of the response. There are hypotheses on causal role of the brain in the power law (the reference trajectoryÂ
is a power law trajectory) and there is the unrelated causal role of the shape of the stimulus and the exponent of theÂ
power law (you get different exponents if you draw different shapes).Â

RM: Yes, those are open loop models that are readily shown to be false when you introduce disturbances to what you callÂ
the "response" (actually, the controlled variable).

AM: The math you and Shaffer present on the meaning of the power law exponent is based on some serious misunderstanding.Â
The exponent of curvature - angular velocity power law just tells you how much an object traveling along a path is slowing downÂ
in corners. If the exponent is 1, the speed is going to be constant all over. If the exponent is lower, such as 2/3, it means theÂ
object is going a bit slower in the corners than in the straight parts. If is still lower, it means the object is slowing down a lot inÂ
the corners, and speeding up a lot in the straight parts. It is complete nonsense to speak about a "true exponent" (see, noÂ
need for math to show your bogus logic).

RM: I don't see how this shows that our math is based on a misunderstanding. Â

AM: It is ironic that you keep saying "PCT says" and then present your own bogus hypotheses.

RM: I actually presented data and a model.Â
Â

AM: There is nothing in PCT that says the power law is a behavioral illusion...

Â
RM: I think we have shown that there is. And we have done it without any insults or ad hominum
attacks.Â
BestÂ
Rick

Best,

Adam

[Rick Marken 2018-08-03_22:07:26]

[Martin Taylor 2018.07.30.01.08]

PY: In what way do you get from these equations:

to this equation:

PY: I see that R = V^3/what looks like some chain rule expression.

Can you please explain how you see V^3 in the denominator?

···

On Sat, Aug 4, 2018 at 9:15 AM Richard Marken <<mailto:csgnet@lists.illinois.edu>csgnet@lists.illinois.edu> wrote:

On 2018/07/29 7:38 PM, PHILIP JERAIR YERANOSIAN (<mailto:pyeranos@ucla.edu>pyeranos@ucla.edu via csgnet Mailing List) wrote:
MT: You are going the wrong way around, because any expression for R must be expressed in spatial variables alone. The point you want is to demonstrate that this is the case for the expression that starts with velocities and accelerations, which are expressions in space and time. The usual way to get there is to start from the expression in s, x, and y as did Viviani and Stucchi (whom Gribble and Ostry used as their source -- Marken and Shaffer used Gribble and Ostry to justify their equation) and work from there to the equation labelled (3). But I'll work it backwards for you.

RM: The math is irrelevant to seeing that the power law is a side effect of control. The power law is precisely analogous to the tangential velocity profiles discovered by Atkeson and Hollerbach and discussed by Bill Powers in the posts from the CSGNet archives that are copied below. Like the power law, tangential velocity profiles are an "invariant" side effect of making controlled movements of a limb. These side effects become a behavioral illusion when they are taken to tell us something about how the observed movement was produced.Â
RM: The mathematics presented by Dennis Shaffer and I shows why the invariance of power law is found for many controlled (and uncontrolled) movements. Powers' Little Man model of movement production shows why invariant velocity profiles are found for controlled movements. (note in [Bill Powers (931029.0750 MDT)] where Bill says: "The velocity profiles are individually scaled both as to amplitude and duration in order to generate the congruence that the authors found. If you do this for the joint angular velocity profiles in Little Man V2, you will find similar invariances, even though there is nothing computing them" (emphases mine--RM)).
RM: Even if the math used to explain the invariance of the power law were wrong (which it's not) it is still easy to see that the power law is a side effect of control and, to the extent that it is taken to tell us something about how the observed movement was produced, it is a behavioral illusion.Â
RM: So it is rather ironic that people who are supposed to be experts on PCT and the behavioral illusion have succumbed to that illusion so thoroughly that they are trying to use math to show that an obvious example of a behavioral illusion -- the power law -- is not an illusion at all.
BestÂ
Rick

=====================================================

[From Bill Powers (931029.0750 MDT)]
Greg Williams (931028) --
Very nice job in picking up the quote on motor schemas. Don't

forget to include William James -- the constancy of ends and the
variability of means. All this shows that the problem has been
known for a very long time (100 years), but that the solution has
eluded a continuing search.

The critical misdirection is contained in
>Thus whatever is learned and stored in long-term memory cannot

be a specific set of muscle commands but must represent a more
generic or general set of specifications of how to reach the
desired goal.

Behind the idea of "specifications of how to reach the goal" is

still a picture of direct causality: the specification is for
_how_ to reach the goal, instead of _what goal to reach._ You can
see this same idea in Atkeson and Hollerbach:

"A strategy for gaining insight into planning and control

processes of the motor system is to look for kinematic
invariances in trajectories of movement. The significance of
straight-line movements is that they imply movement planning at
the hand or object level." (p. 2318)

In the discussion:
"Taken together, shape invariances for path and tangential

velocity profiles indicates that subjects execute only one form
of trajectory between any two targets when not instructed to do
otherwise. The only changes in the trajectory are simple scaling
operations to accomodate different speeds. ... Different subjects
use the same tangential velocity profile shape." (pp. 2325-6).

And making the problem even clearer:
"A number of issues remain with regard to these dynamic scaling

results. How are the initial torques for the first movement
generated? If the motor controller has the ability to fashion the
correct torques for one movement, why does it not use this same
ability for all subsequent movements rather than utilize the
dynamic scaling properties? Among the possibilities we are
considering, the first is a generalized motor tape where only one
movement between points need be known if the dynamic components
in Equation 6 are stored separately. ... A second possibility is
a modification of tabular approaches (Rabert, 1978) where the
dimensionality and parameter adjustment problem could be reduced
by separate tables for the four components in equation 6." (p.
2327).

The only possibilities being considered are those that involve

open-loop generation of the torques that will produce a pre-
planned trajectory. "Equation 6" is an equation expressing joint
torque as a nonlinear function of scaling factors applied to
torques previously produced for a known movement. As noted above,
the problem is how the torques for the known movement are
generated in the first place. This problem is not solved. What
the authors hope for is that by finding invariants such as
velocity profiles, they will be able to deduce a motor program
that will produce constant results in object space even when
variations in torque are required.

They simply haven't got far enough into the problem to see that

this quest is hopeless. If they did manage to come up with a
motor program that could realistically create several
trajectories of hand movement between different pairs of points,
they would then have to ask how this can work with different
loads. If they solved that problem, they would have to explain
how the trajectories are produced when the loads are vary
unexpectedly and the muscles progressively fatigue. And then they
would have to explain how the right torques can be produced under
varying loads (such as the varying friction between pencil and
paper) and with fatiguing muscles, when the task is to write the
subject's name. And then they would have to explain how a subject
can accomplish the same movements under the same uncertain
conditions for the tip of a pointer held in the hand. In truth,
they are extrapolating a long way ahead, and predicting success,
when they have not even found the simplest motor program of all:
that for moving quickly between two points under undisturbed
conditions. Their projected work simply expresses faith that
somehow the require movements in object space can be generated by
a clever enough motor programming device -- without ever taking
feedback into account, the feedback both kinesthetic and visual
that is _known_ to exist and that is _known_ to be essential for
skilled performance.

There is an explanation for the observed invariances of velocity

profiles that Atkeson and Hollerbach never consider: these
invariances might simply be the natural outcome of physical
processes of control. There might be no need at all for the motor
program to precompute them. The velocity profiles are
individually scaled both as to amplitude and duration in order to
generate the congruence that the authors found. If you do this
for the joint angular velocity profiles in Little Man V2, you
will find similar invariances, even though there is nothing
computing them. It just happens that when a control system is
given a step-change of reference-signal, the trajectory of the
controlled variable naturally scales up or down so that the
velocity rises and falls along the same generic curve. This is
purely a consequence of the mathematical relationships of control
and the passive dynamical properties of the arm; nothing is
acting to make sure that the trajectory follows any particular
path. The trajectory is a side-effect, not a planned movement.
Evidence of trajectory planning would appear only if the actual
trajectory departed from the one that can be explained as a step-
change in the reference signal of a control system from one fixed
value to another. For example, one can easily move a finger from
one point to another along a semi-circle or an S-shaped curve.
_That_ requires a "program" of velocity or position reference
signals. But it still doesn't require precomputing torques.
------------------------------------------------
The key idea to look for in all these sources is how the authors
propose to account for the forces that create movements. It's
clear in Atkeson and Hollerbach that the torques are going to be
computed so as to have the required object-space consequences and
that proprioceptive and visual feedback are not considered. All
approaches that propose to use inverse kinematic or inverse
dynamical computations are also attempting to solve the problem
open-loop. In all such approaches, the key idea that is missing
is comparison of the observed consequences with the desired
consequence _in real time_ as the means of producing the required
output signals.
------------------------------------------------
Atkeson, C. G., and Hollerbach, J.M.; Kinematic Features of
Unrestrained Vertical Arm Movements. The Journal oif Neuroscience
_5_, No. 9, pp. 2318-2330. Sept. 1985.
--------------------------------------------------------------
Best,

Bill P.
[From Bill Powers (950527.0950 MDT)]
Just got back from seeing our daughter Barbara off in the start of the

Iron Horse bike race, Durango to Silverton. The length is 45 miles, the
total climb over two main passes is 5500 feet (the highest pass, Molas,
is about 11,000 feet). Last year (her first, at age 35) she did it in
4:20; this year she hopes for under 4:00. The pro winning time last year
was 2:10. She should be about halfway right now, starting the four-mile
climb to Coal Bank Pass (2500 foot climb to over 10,000 ft). Go Bara!
-----------------------------------------------------------------------
Rick Marken, Bruce Abbott (continuing) --

When you push on a control system, it pushes back.

------------------------------
RE: trajectories vs. system organization

In a great deal of modern behavioral research, trajectories of movement

are examined in the hope of finding invariants that will reveal secrets
of behavior. This approach ties in with system models that compute
inverse kinematics and dynamics and use motor programs to produce
actions open-loop. These models assume that the path followed by a limb
or the whole body is specified in advance in terms of end-positions and
derivatives during the transition, so the path that is followed reflects
the computations that are going on inside the system.

It is this orientation that explains papers like
Atkeson, C. G. and Hollerback, J.M.(1985); Kinematic features of

unrestrained vertical arm movements. The Journal of Neuroscience _5_,
#9, 2318-2330.

In the described experiments, subjects move a hand in the vertical plane

at various prescribed speeds from a starting point to variously located
targets, and the positions are recorded as videos of the positions of
illuminated targets fastened to various parts of the arm and hand.

The authors constructed a tangential-velocity vs time profile of the

wrist movement for various speeds, directions, and distances of
movement. They normalized the profiles to a fixed magnitude, then to a
fixed duration, and found that the curves then had very nearly the same
shape. Using a "similarity" calculation, they quantified the measures of
similarity.

They were then able to compare these normalized tangential velocity

profiles across various directions and amounts of movement and show that
the treated profiles were very close to the same. They conclude:

    Taken together, shape invariance for path and tangential velocity

    profile indicates that subjects execute only one form of trajectory
    between any two targets when not instructed to do otherwise. The
    only changes in trajectory are simple scaling operations to
    accomodate different speeds. Furthermore, subjects use the same
    tangential velocity profile shape to make radically different
    movements, even when the shapes of the paths are not the same in
    extrinsic coordinates. Different subjects use the same tangential
    velocity profile shape.

    ... this would be consistent with a simplifying strategy for joint

    torque formation by separation of gravity torques from dynamic
    torques and a uniform scaling of the tangential velocity profile
    ... (p. 2325)

    ... if the motor controller has the ability to fashion correct

    torques for one movement, why does it not use this same ability for
    all subsequent movements rather than utilize the dynamic scaling
    properties? Among the possibilities we are considering, the first
    is a generalized motor tape where only one movement between points
    must be known if the dynanmic components in equation 6 are stored
    separately....A second possibility is a modification of tabular
    approaches [ref] where the dimensionality and parameter adjustment
    problem could be reduced by separate tables for the four components
    in equation 6. (p. 2326)

This paper was sent to me by Greg Williams as a source of data about

actual hand movements, for comparison with the hand movements generated
by Little Man v. 2, the version using actual arm dynamics for the
external part of the model. The model's hand movements were, as Greg
will attest, quite close to those shown in this paper, being slightly
curved lines connecting the end-points. Forward and reverse movements
followed somewhat different paths, and by adjustment of model parameters
this difference, too, could be reproduced.

What is interesting is that the fit between the Little Man and the real

data was found without considering tangential velocity profiles or doing
any scaling or normalization. In other words, the invariances noted by
the authors were simply side-effects of the operation of the control
systems of the arm interacting with the dynamics of the physical arm. In
the Little Man there is no trajectory planning, no storage of movement
parameters, no table-lookup facility, no computation of invariant
velocity profiles. The observed behavior is simply a reflection of the
organization of the control system and the physical plant.

The path which Atkeson, Hollerbach (and many others at MIT and

elsewhere) are treading is a blind alley, because no matter how
carefully the observations are made and the invariances are calculated,
there will be no hint of the control-system organization, the SIMPLE
control-system organization, that (I claim) is actually creating the
observed trajectories. No doubt a sufficiently complex trajectory-
control model, with just the right tables of coefficients and velocity
profiles, would ultimately be able to match the behavior. But this line
of investigation, with its underlying assumptions, will never lead to
the far simpler and anatomically correct PCT model.

In terms of the current discussion on the net, the observations made by

the authors were interesting as checks on the model, but were actually
irrelevant to what the control systems were doing. The control systems
(the first two levels of the Little Man model) controlled only three
kinds of variables that underlay the perceptual signals: angular
positions, angular velocities, and angular accelerations. They received
no information about wrist position in laboratory space. They contained
no provision for computing tangential velocities, or for computing
positions of points on the physical arm in space, or for computing
space-time invariants. The behavior of the control systems, in other
words, took place in a proprioceptive perceptual space that no outside
observer could see. In order to translate from this perceptual space
into variables that were observable, the computer program generated the
resulting arm positions and plotted them in a form suitable for visual
inspection. So a side-effect of the actual control process was presented
for comparison with a corresponding side-effect of the real control
process, as visible to an outside observer.

The approach of Atkeson and Hollerbach appears in many guises. We have

already talked about the apparent scaling and normalization of
trajectories seen when two hands move rapidly and simultaneously to
targets at different distances. In operant conditioning experiments, we
have seen how the control of reinforcement by behavior is obscured by
the fact that variations in behavior tend to stabilize reinforcement
rates, thus making reinforcement rate appear to be the independent
variable.

We have also seen a few -- a very few, so far -- studies in which the

PCT orientation was used, Srinivasan's being the most recent. What is
the difference? I think the difference is in whether the emphasis is on
seeing the behavior from the behaving systems's point of view, as best
we can imagine it, and seeing it strictly from the human observer's
point of view.

>From the human observer's point of view, it seems that we must account

for the detailed movements and physical interactions that are seen to
occur. This leads to trying to find invariances or striking mathematical
regularities of some sort in the observed behaviors. It leads to
imagining an internal system that is producing explicitly what we are
observing; if we observe a trajectory, there must be some generator that
is specifically calculating that trajectory.

But from the behaving system's point of view, we can consider only the

information that is available to the behaving system; we must look for
our explanations there. The trajectories of movement that result from
the system's operation are basically side-effects; they are not planned
and they are constant only in a constant environment. Furthermore, they
are unknown to the behaving system and play no part in the production of
behavior. We can deduce from the model of the behaving system what the
observable side-effects would be in a given environment, and so can
compare those side-effects with our external observations of the
behavior. But our explanation of the behavior is not based on those
side-effects.

Most important, when we simply describe behavior as a sequence of

physical happenings and relationships, we have no way of knowing whether
we are describing controlled variables or side-effects. When we see a
fly landing on a ceiling, it is perfectly possible that NOT A SINGLE
ASPECT OF WHAT WE SEE is perceived and controlled by the fly. When we
see the fly extending its legs just prior to landing, the fly may have
no perception of the configuration of its legs; to the fly, all that is
controlled may be two or three joint-angle signals, not even identified
by the fly as representing joint angle. When we see the wings stop
flapping, to the fly all that may be controlled is a sensation of
vibration. When we see the fly's body making a steep angle with the
surface, the fly may simply be experiencing a visual signal indicating,
as Rick guessed, a gradient of illumination or texture. Not one of the
variables we are observing may ever appear in the ultimate model of the
fly's internal organization, just as in the Little Man the actual arm
configuration and hand position never appear in the model of the first
two (kinesthetic) levels of control. Once we have the right model, we
can always compute how its operation will appear to an observer who is
focusing on various side-effects of the actions. But the model itself
says nothing about those appearances, and makes no use of them.
-----------------------------------------------------------------------
Best to all,

Bill P.
--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[rsmarken]

[Rick Marken 2018-08-05_19:01:11]

[Martin Taylor 2018.08.04.11.45]Â

MT: For the time being I have personally given up on trying to explain the issues to Rick...

MT: Meanwhile, the rest of us can continue to work on the science involved with the control of perception, and not worry about whether the science is in accord with what "PCT (Rick) says".

RM: Sounds like a great idea. I'm looking forward to seeing your perceptual control explanation of the power law.Â

BestÂ
Rick

Martin

[From Adam Matic]
My bet is you still don't know the difference between common uses of *trajectory* and *path*, Rick. Other that that, you keep saying mutually conflicting statements, or you keep flipping between two extreme positions on some issues. Your statements reveal confusion, but your are still very confrontational and trying to sound authoritative. That is just silly. You're on fast track to being a crank scientist. However, given your modus operandi of giving strong statements, then going back on them after a few days (weeks, months?) when you realize your mistake, you just might come back from the dark side. That is probably Martin's hope too.
RM: The math is irrelevant to seeing that the power law is a side effect of control. The power law is precisely analogous to the tangential velocity profiles discovered by Atkeson and Hollerbach and discussed by Bill Powers in the posts from the CSGNet archives that are copied below. Like the power law, tangential velocity profiles are an "invariant" side effect of making controlled movements of a limb. These side effects become a behavioral illusion when they are taken to tell us something about how the observed movement was produced.Â
AM: Yes, the power law is analogous to bell-shaped velocity profiles. They are not a controlled result (as Bill says), just like power law trajectories are not controlled results (this is my hypothesis). You, on the other hand, claim that power-law trajectories *are a controlled result*. You also published a model of ellipse tracing which has a 2/3 power law trajectory as the reference trajectory. That is where you are conflicting with yourself - if the trajectory is a side effect then it is not controlled. If it is controlled, then it is telling us that the reference trajectory is also a 2/3 power law trajectory.
AM: The behavioral illusion is one of the issues you have flipped up and down several times, I'll number it. After publishing two papers saying 1) the power law is behavioral illusion; sometime in May on csgnet you had no problems saying that 2) the "power law is not a behavioral illusion as in Powers(1978)", then 2) "it is maybe the illusion of control", then 3) maybe we should just stop telling people about this illusion business; and again back to 1) it is a behavioral illusion.

According to the 1978 paper, a behavioral illusion happens when someone mistakes a correlation between a stimulus and a response for a causal relationship. No one is taking the instantaneous curvatures and velocities of the response to be causally related, those are both properties of the response. There are hypotheses on causal role of the brain in the power law (the reference trajectory is a power law trajectory) and there is the unrelated causal role of the shape of the stimulus and the exponent of the power law (you get different exponents if you draw different shapes).Â
The math you and Shaffer present on the meaning of the power law exponent is based on some serious misunderstanding. The exponent of curvature - angular velocity power law just tells you how much an object traveling along a path is slowing down in corners. If the exponent is 1, the speed is going to be constant all over. If the exponent is lower, such as 2/3, it means the object is going a bit slower in the corners than in the straight parts. If is still lower, it means the object is slowing down a lot in the corners, and speeding up a lot in the straight parts. It is complete nonsense to speak about a "true exponent" (see, no need for math to show your bogus logic).

RM: So it is rather ironic that people who are supposed to be experts on PCT and the behavioral illusion have succumbed to that illusion so thoroughly that they are trying to use math to show that an obvious example of a behavioral illusion -- the power law -- is not an illusion at all.
AM:

It is ironic that you keep saying "PCT says" and then present your own bogus hypotheses. There is nothing in PCT that says the power law is a behavioral illusion or that the research on the power law is based on a mistake of confusing correlation for a causal relationship. You are saying that. And you are kinda shit at math, as you have been before, but you seem to have been quicker to admit it and did not make grand claims based on your "calculations". Maybe it was Bill who corrected you when you strayed over to crank territory, now you are just pushing forward.Â

Best,

Adam

[Rick Marken 2018-08-03_22:07:26]

[Martin Taylor 2018.07.30.01.08]

PY: In what way do you get from these equations:

to this equation:

PY: I see that R = V^3/what looks like some chain rule expression.

Can you please explain how you see V^3 in the denominator?

···

On Sat, Aug 4, 2018 at 9:15 AM Richard Marken <<mailto:csgnet@lists.illinois.edu>csgnet@lists.illinois.edu> wrote:

On 2018/07/29 7:38 PM, PHILIP JERAIR YERANOSIAN (<mailto:pyeranos@ucla.edu>pyeranos@ucla.edu via csgnet Mailing List) wrote:
MT: You are going the wrong way around, because any expression for R must be expressed in spatial variables alone. The point you want is to demonstrate that this is the case for the expression that starts with velocities and accelerations, which are expressions in space and time. The usual way to get there is to start from the expression in s, x, and y as did Viviani and Stucchi (whom Gribble and Ostry used as their source -- Marken and Shaffer used Gribble and Ostry to justify their equation) and work from there to the equation labelled (3). But I'll work it backwards for you.

RM: The math is irrelevant to seeing that the power law is a side effect of control. The power law is precisely analogous to the tangential velocity profiles discovered by Atkeson and Hollerbach and discussed by Bill Powers in the posts from the CSGNet archives that are copied below. Like the power law, tangential velocity profiles are an "invariant" side effect of making controlled movements of a limb. These side effects become a behavioral illusion when they are taken to tell us something about how the observed movement was produced.Â
RM: The mathematics presented by Dennis Shaffer and I shows why the invariance of power law is found for many controlled (and uncontrolled) movements. Powers' Little Man model of movement production shows why invariant velocity profiles are found for controlled movements. (note in [Bill Powers (931029.0750 MDT)] where Bill says: "The velocity profiles are individually scaled both as to amplitude and duration in order to generate the congruence that the authors found. If you do this for the joint angular velocity profiles in Little Man V2, you will find similar invariances, even though there is nothing computing them" (emphases mine--RM)).
RM: Even if the math used to explain the invariance of the power law were wrong (which it's not) it is still easy to see that the power law is a side effect of control and, to the extent that it is taken to tell us something about how the observed movement was produced, it is a behavioral illusion.Â
RM: So it is rather ironic that people who are supposed to be experts on PCT and the behavioral illusion have succumbed to that illusion so thoroughly that they are trying to use math to show that an obvious example of a behavioral illusion -- the power law -- is not an illusion at all.
BestÂ
Rick

=====================================================

[From Bill Powers (931029.0750 MDT)]
Greg Williams (931028) --
Very nice job in picking up the quote on motor schemas. Don't

forget to include William James -- the constancy of ends and the
variability of means. All this shows that the problem has been
known for a very long time (100 years), but that the solution has
eluded a continuing search.

The critical misdirection is contained in
>Thus whatever is learned and stored in long-term memory cannot

be a specific set of muscle commands but must represent a more
generic or general set of specifications of how to reach the
desired goal.

Behind the idea of "specifications of how to reach the goal" is

still a picture of direct causality: the specification is for
_how_ to reach the goal, instead of _what goal to reach._ You can
see this same idea in Atkeson and Hollerbach:

"A strategy for gaining insight into planning and control

processes of the motor system is to look for kinematic
invariances in trajectories of movement. The significance of
straight-line movements is that they imply movement planning at
the hand or object level." (p. 2318)

In the discussion:
"Taken together, shape invariances for path and tangential

velocity profiles indicates that subjects execute only one form
of trajectory between any two targets when not instructed to do
otherwise. The only changes in the trajectory are simple scaling
operations to accomodate different speeds. ... Different subjects
use the same tangential velocity profile shape." (pp. 2325-6).

And making the problem even clearer:
"A number of issues remain with regard to these dynamic scaling

results. How are the initial torques for the first movement
generated? If the motor controller has the ability to fashion the
correct torques for one movement, why does it not use this same
ability for all subsequent movements rather than utilize the
dynamic scaling properties? Among the possibilities we are
considering, the first is a generalized motor tape where only one
movement between points need be known if the dynamic components
in Equation 6 are stored separately. ... A second possibility is
a modification of tabular approaches (Rabert, 1978) where the
dimensionality and parameter adjustment problem could be reduced
by separate tables for the four components in equation 6." (p.
2327).

The only possibilities being considered are those that involve

open-loop generation of the torques that will produce a pre-
planned trajectory. "Equation 6" is an equation expressing joint
torque as a nonlinear function of scaling factors applied to
torques previously produced for a known movement. As noted above,
the problem is how the torques for the known movement are
generated in the first place. This problem is not solved. What
the authors hope for is that by finding invariants such as
velocity profiles, they will be able to deduce a motor program
that will produce constant results in object space even when
variations in torque are required.

They simply haven't got far enough into the problem to see that

this quest is hopeless. If they did manage to come up with a
motor program that could realistically create several
trajectories of hand movement between different pairs of points,
they would then have to ask how this can work with different
loads. If they solved that problem, they would have to explain
how the trajectories are produced when the loads are vary
unexpectedly and the muscles progressively fatigue. And then they
would have to explain how the right torques can be produced under
varying loads (such as the varying friction between pencil and
paper) and with fatiguing muscles, when the task is to write the
subject's name. And then they would have to explain how a subject
can accomplish the same movements under the same uncertain
conditions for the tip of a pointer held in the hand. In truth,
they are extrapolating a long way ahead, and predicting success,
when they have not even found the simplest motor program of all:
that for moving quickly between two points under undisturbed
conditions. Their projected work simply expresses faith that
somehow the require movements in object space can be generated by
a clever enough motor programming device -- without ever taking
feedback into account, the feedback both kinesthetic and visual
that is _known_ to exist and that is _known_ to be essential for
skilled performance.

There is an explanation for the observed invariances of velocity

profiles that Atkeson and Hollerbach never consider: these
invariances might simply be the natural outcome of physical
processes of control. There might be no need at all for the motor
program to precompute them. The velocity profiles are
individually scaled both as to amplitude and duration in order to
generate the congruence that the authors found. If you do this
for the joint angular velocity profiles in Little Man V2, you
will find similar invariances, even though there is nothing
computing them. It just happens that when a control system is
given a step-change of reference-signal, the trajectory of the
controlled variable naturally scales up or down so that the
velocity rises and falls along the same generic curve. This is
purely a consequence of the mathematical relationships of control
and the passive dynamical properties of the arm; nothing is
acting to make sure that the trajectory follows any particular
path. The trajectory is a side-effect, not a planned movement.
Evidence of trajectory planning would appear only if the actual
trajectory departed from the one that can be explained as a step-
change in the reference signal of a control system from one fixed
value to another. For example, one can easily move a finger from
one point to another along a semi-circle or an S-shaped curve.
_That_ requires a "program" of velocity or position reference
signals. But it still doesn't require precomputing torques.
------------------------------------------------
The key idea to look for in all these sources is how the authors
propose to account for the forces that create movements. It's
clear in Atkeson and Hollerbach that the torques are going to be
computed so as to have the required object-space consequences and
that proprioceptive and visual feedback are not considered. All
approaches that propose to use inverse kinematic or inverse
dynamical computations are also attempting to solve the problem
open-loop. In all such approaches, the key idea that is missing
is comparison of the observed consequences with the desired
consequence _in real time_ as the means of producing the required
output signals.
------------------------------------------------
Atkeson, C. G., and Hollerbach, J.M.; Kinematic Features of
Unrestrained Vertical Arm Movements. The Journal oif Neuroscience
_5_, No. 9, pp. 2318-2330. Sept. 1985.
--------------------------------------------------------------
Best,

Bill P.
[From Bill Powers (950527.0950 MDT)]
Just got back from seeing our daughter Barbara off in the start of the

Iron Horse bike race, Durango to Silverton. The length is 45 miles, the
total climb over two main passes is 5500 feet (the highest pass, Molas,
is about 11,000 feet). Last year (her first, at age 35) she did it in
4:20; this year she hopes for under 4:00. The pro winning time last year
was 2:10. She should be about halfway right now, starting the four-mile
climb to Coal Bank Pass (2500 foot climb to over 10,000 ft). Go Bara!
-----------------------------------------------------------------------
Rick Marken, Bruce Abbott (continuing) --

When you push on a control system, it pushes back.

------------------------------
RE: trajectories vs. system organization

In a great deal of modern behavioral research, trajectories of movement

are examined in the hope of finding invariants that will reveal secrets
of behavior. This approach ties in with system models that compute
inverse kinematics and dynamics and use motor programs to produce
actions open-loop. These models assume that the path followed by a limb
or the whole body is specified in advance in terms of end-positions and
derivatives during the transition, so the path that is followed reflects
the computations that are going on inside the system.

It is this orientation that explains papers like
Atkeson, C. G. and Hollerback, J.M.(1985); Kinematic features of

unrestrained vertical arm movements. The Journal of Neuroscience _5_,
#9, 2318-2330.

In the described experiments, subjects move a hand in the vertical plane

at various prescribed speeds from a starting point to variously located
targets, and the positions are recorded as videos of the positions of
illuminated targets fastened to various parts of the arm and hand.

The authors constructed a tangential-velocity vs time profile of the

wrist movement for various speeds, directions, and distances of
movement. They normalized the profiles to a fixed magnitude, then to a
fixed duration, and found that the curves then had very nearly the same
shape. Using a "similarity" calculation, they quantified the measures of
similarity.

They were then able to compare these normalized tangential velocity

profiles across various directions and amounts of movement and show that
the treated profiles were very close to the same. They conclude:

    Taken together, shape invariance for path and tangential velocity

    profile indicates that subjects execute only one form of trajectory
    between any two targets when not instructed to do otherwise. The
    only changes in trajectory are simple scaling operations to
    accomodate different speeds. Furthermore, subjects use the same
    tangential velocity profile shape to make radically different
    movements, even when the shapes of the paths are not the same in
    extrinsic coordinates. Different subjects use the same tangential
    velocity profile shape.

    ... this would be consistent with a simplifying strategy for joint

    torque formation by separation of gravity torques from dynamic
    torques and a uniform scaling of the tangential velocity profile
    ... (p. 2325)

    ... if the motor controller has the ability to fashion correct

    torques for one movement, why does it not use this same ability for
    all subsequent movements rather than utilize the dynamic scaling
    properties? Among the possibilities we are considering, the first
    is a generalized motor tape where only one movement between points
    must be known if the dynanmic components in equation 6 are stored
    separately....A second possibility is a modification of tabular
    approaches [ref] where the dimensionality and parameter adjustment
    problem could be reduced by separate tables for the four components
    in equation 6. (p. 2326)

This paper was sent to me by Greg Williams as a source of data about

actual hand movements, for comparison with the hand movements generated
by Little Man v. 2, the version using actual arm dynamics for the
external part of the model. The model's hand movements were, as Greg
will attest, quite close to those shown in this paper, being slightly
curved lines connecting the end-points. Forward and reverse movements
followed somewhat different paths, and by adjustment of model parameters
this difference, too, could be reproduced.

What is interesting is that the fit between the Little Man and the real

data was found without considering tangential velocity profiles or doing
any scaling or normalization. In other words, the invariances noted by
the authors were simply side-effects of the operation of the control
systems of the arm interacting with the dynamics of the physical arm. In
the Little Man there is no trajectory planning, no storage of movement
parameters, no table-lookup facility, no computation of invariant
velocity profiles. The observed behavior is simply a reflection of the
organization of the control system and the physical plant.

The path which Atkeson, Hollerbach (and many others at MIT and

elsewhere) are treading is a blind alley, because no matter how
carefully the observations are made and the invariances are calculated,
there will be no hint of the control-system organization, the SIMPLE
control-system organization, that (I claim) is actually creating the
observed trajectories. No doubt a sufficiently complex trajectory-
control model, with just the right tables of coefficients and velocity
profiles, would ultimately be able to match the behavior. But this line
of investigation, with its underlying assumptions, will never lead to
the far simpler and anatomically correct PCT model.

In terms of the current discussion on the net, the observations made by

the authors were interesting as checks on the model, but were actually
irrelevant to what the control systems were doing. The control systems
(the first two levels of the Little Man model) controlled only three
kinds of variables that underlay the perceptual signals: angular
positions, angular velocities, and angular accelerations. They received
no information about wrist position in laboratory space. They contained
no provision for computing tangential velocities, or for computing
positions of points on the physical arm in space, or for computing
space-time invariants. The behavior of the control systems, in other
words, took place in a proprioceptive perceptual space that no outside
observer could see. In order to translate from this perceptual space
into variables that were observable, the computer program generated the
resulting arm positions and plotted them in a form suitable for visual
inspection. So a side-effect of the actual control process was presented
for comparison with a corresponding side-effect of the real control
process, as visible to an outside observer.

The approach of Atkeson and Hollerbach appears in many guises. We have

already talked about the apparent scaling and normalization of
trajectories seen when two hands move rapidly and simultaneously to
targets at different distances. In operant conditioning experiments, we
have seen how the control of reinforcement by behavior is obscured by
the fact that variations in behavior tend to stabilize reinforcement
rates, thus making reinforcement rate appear to be the independent
variable.

We have also seen a few -- a very few, so far -- studies in which the

PCT orientation was used, Srinivasan's being the most recent. What is
the difference? I think the difference is in whether the emphasis is on
seeing the behavior from the behaving systems's point of view, as best
we can imagine it, and seeing it strictly from the human observer's
point of view.

>From the human observer's point of view, it seems that we must account

for the detailed movements and physical interactions that are seen to
occur. This leads to trying to find invariances or striking mathematical
regularities of some sort in the observed behaviors. It leads to
imagining an internal system that is producing explicitly what we are
observing; if we observe a trajectory, there must be some generator that
is specifically calculating that trajectory.

But from the behaving system's point of view, we can consider only the

information that is available to the behaving system; we must look for
our explanations there. The trajectories of movement that result from
the system's operation are basically side-effects; they are not planned
and they are constant only in a constant environment. Furthermore, they
are unknown to the behaving system and play no part in the production of
behavior. We can deduce from the model of the behaving system what the
observable side-effects would be in a given environment, and so can
compare those side-effects with our external observations of the
behavior. But our explanation of the behavior is not based on those
side-effects.

Most important, when we simply describe behavior as a sequence of

physical happenings and relationships, we have no way of knowing whether
we are describing controlled variables or side-effects. When we see a
fly landing on a ceiling, it is perfectly possible that NOT A SINGLE
ASPECT OF WHAT WE SEE is perceived and controlled by the fly. When we
see the fly extending its legs just prior to landing, the fly may have
no perception of the configuration of its legs; to the fly, all that is
controlled may be two or three joint-angle signals, not even identified
by the fly as representing joint angle. When we see the wings stop
flapping, to the fly all that may be controlled is a sensation of
vibration. When we see the fly's body making a steep angle with the
surface, the fly may simply be experiencing a visual signal indicating,
as Rick guessed, a gradient of illumination or texture. Not one of the
variables we are observing may ever appear in the ultimate model of the
fly's internal organization, just as in the Little Man the actual arm
configuration and hand position never appear in the model of the first
two (kinesthetic) levels of control. Once we have the right model, we
can always compute how its operation will appear to an observer who is
focusing on various side-effects of the actions. But the model itself
says nothing about those appearances, and makes no use of them.
-----------------------------------------------------------------------
Best to all,

Bill P.
--
Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.�
                --Antoine de Saint-Exupery

--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery