# Bogus mathematics, (was Re: L'état de PCT, c' est moi (was ...))

Following on from my last email, can anyone tell me that centripetal force isnâ€™t relevant to the power law?

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

···

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?

``````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.

``````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.

``````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?

Best

Rick

``````Martin
``````

Richard S. Marken

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
–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.
You do this by showing (correctly, I assume) that D^1/3 is
equivalent to V
(1/(R^1/3))

``````          so that
V
``````

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

``````            RM:
``````

But as I’ve said, that’s true of any equation. The fact
that
V
= R^1/3D^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.

``````                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.

``````                                RM:
``````

Just as it has proved useful to
analyze the total variance in
experiments (
MS.total) 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.

``````                                  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) = 1/3
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.

``````                                  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
these movements are produced, when
it doesn’t.

``````                                  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.

``````                                      RM:
``````

There are many other incorrect
to our paper, Martin. But I
think this is enough for now
(based on our alleged
mathematical mistake) seemed
to be central to your
argument.

Best

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
mathematics you have shown to be bogus. Your previous
response did not address this question.

``````            Martin
``````

Richard S. Marken

``````                                  "Perfection
``````

is achieved not when you have
nothing more to add, but when you
have
nothing left to take away.â€?

–Antoine de Saint-Exupery

Dr Warren Mansell

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

Tel: +44 (0) 161 275 8589

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

Oh, maybe I can carry on!

I donâ€™t think there is a rule that a disturbance has to be completely independent from the organism is there? 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?

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

All the best

Warren

···

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

Hi Rick, I donâ€™t see how the power law itself is incompatible with PCT,
just how people interpret it.

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.

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 â€˜skiddingâ€™ 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.

But I should probably stop now because I donâ€™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.

Best

Rick

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

Hi Warren

Richard S. Marken

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

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

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?

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.

My feeling is that the power law doesnâ€™t explain the physiological and perceptual process but merely some relationship between force constraints, but I could be completely wrong, and Iâ€™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

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

Richard S. Marken

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

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

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

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 doesnâ€™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).

WM: To answer Martinâ€™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.

WM: However, unlike Rick and Martin, Iâ€™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.

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) = 1/3log(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.

RM: So the “proof” 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.

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 â€˜skidâ€™ 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 best

Warren

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

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

Richard S. Marken

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

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 isnâ€™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.

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 – 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.

Best

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).

Richard S. Marken

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

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

Dr Warren Mansell

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

Tel: +44 (0) 161 275 8589

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?

``````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.

``````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.

``````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?

Best

Rick

``````Martin
``````

Richard S. Marken

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
–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.
You do this by showing (correctly, I assume) that D^1/3 is
equivalent to V
(1/(R^1/3))

``````          so that
V
``````

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

``````            RM:
``````

But as I’ve said, that’s true of any equation. The fact
that
V
= R^1/3D^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.

``````                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.

``````                                RM:
``````

Just as it has proved useful to
analyze the total variance in
experiments (
MS.total) 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.

``````                                  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) = 1/3
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.

``````                                  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
these movements are produced, when
it doesn’t.

``````                                  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.

``````                                      RM:
``````

There are many other incorrect
to our paper, Martin. But I
think this is enough for now
(based on our alleged
mathematical mistake) seemed
to be central to your
argument.

Best

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
mathematics you have shown to be bogus. Your previous
response did not address this question.

``````            Martin
``````

Richard S. Marken

``````                                  "Perfection
``````

is achieved not when you have
nothing more to add, but when you
have
nothing left to take away.â€?

–Antoine de Saint-Exupery

Hi Rick, sorry but itâ€™s not easy to write a reply to your last email for a few different reasons, so Iâ€™ll come back to it another time.

Warren

···

[Rick Marken 2018-07-25_09:29:22]

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

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.

WM: I donâ€™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.

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.

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.

Best

Rick

All the best

Warren

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

Hi Warren

Richard S. Marken

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

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

Hi Rick, I donâ€™t see how the power law itself is incompatible with PCT,
just how people interpret it.

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.

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 â€˜skiddingâ€™ 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.

But I should probably stop now because I donâ€™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.

Best

Rick

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

Hi Warren

Richard S. Marken

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

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

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?

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.

My feeling is that the power law doesnâ€™t explain the physiological and perceptual process but merely some relationship between force constraints, but I could be completely wrong, and Iâ€™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

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

Richard S. Marken

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

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

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

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 doesnâ€™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).

WM: To answer Martinâ€™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.

WM: However, unlike Rick and Martin, Iâ€™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.

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) = 1/3log(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.

RM: So the “proof” 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.

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 â€˜skidâ€™ 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 best

Warren

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

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

Richard S. Marken

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

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 isnâ€™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.

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 – 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.

Best

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).

Richard S. Marken

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

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

Dr Warren Mansell

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

Tel: +44 (0) 161 275 8589

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?

``````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.

``````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.

``````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?

Best

Rick

``````Martin
``````

Richard S. Marken

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
–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.
You do this by showing (correctly, I assume) that D^1/3 is
equivalent to V
(1/(R^1/3))

``````          so that
V
``````

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

``````            RM:
``````

But as I’ve said, that’s true of any equation. The fact
that
V
= R^1/3D^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.

``````                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.

``````                                RM:
``````

Just as it has proved useful to
analyze the total variance in
experiments (
MS.total) 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.

``````                                  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) = 1/3
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.

``````                                  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
these movements are produced, when
it doesn’t.

``````                                  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.

``````                                      RM:
``````

There are many other incorrect
to our paper, Martin. But I
think this is enough for now
(based on our alleged
mathematical mistake) seemed
to be central to your
argument.

Best

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
mathematics you have shown to be bogus. Your previous
response did not address this question.

``````            Martin
``````

Richard S. Marken

``````                                  "Perfection
``````

is achieved not when you have
nothing more to add, but when you
have
nothing left to take away.â€?

–Antoine de Saint-Exupery