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

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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}= V^{3}.

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 V^{3}

cancels out from numerator and denominator of the fraction that is

the expression for R. Instead, they leave V^{3}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^1/3*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))

```
so that
V
```

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

```
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^1/3*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.

```
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.

```
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

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.

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

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. 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

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Manchester M13 9PL

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