[From Rick Marken (2017.11.13.2250)]

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Richard Kennaway (2017.10.13 1448 GMT)–

RK: 1.Â Several people in the discussion are in error about the meaning of a mathematical equality, or a law of physics expressed as one.Â No such equation expresses any sort of causal relationship.

RK: For any smoothly traversed curve in a plane, we have V = D^(1/3) R^(1/3), where V is the speed, R the radius of curvature, and D the magnitude of the cross product of acceleration and velocity.Â This is simply a mathematical fact.Â

RM: Thanks. I think that should end Taylor and Abbott’s “mathematical” criticisms. But, of course, it won’t.Â Â

RK: 2.Â As mentioned above, V = D^(1/3) R^(1/3).Â This mathematically implies that if any two of these variables have a power-law relationship with each other, they each have power-law relationships with the third.Â However, no power-law relationships need be present at all.

RM: The paper looks at this in terms of data analysis. Power law researchers use log-log regression analysis to test for the existence of a power law relationship. That is, they regress measures of log (R) on log (V), where R and V are calculated from the x,y values of the trajectory per the equations in Gribble and Ostry. The equation above implies thatÂ log (V) = 1/3 log(D) + 1/3 log (R). So if only log (R) is used in the analysis, the coefficient found by the regression will be close to 1/3 but with some deviation that will be proportional to the covariance between the omitted variable (log (D), affine velocity) and the included variable (log (R)).Â

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RK: 3.Â If D is constant, V is proportional to R^(1/3).Â

RM: Yes, and if D is a constant, log(D) is a constant and the covariance between log(D) and log(R) is 0.0.

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RK: This is the classic power law. Conversely, the classic power law implies that D is constant.Â However, actual power laws in the literature have widely varying exponents.Â The 1/3 power, despite frequent citation, does not seem to play any special role.

Â RM: Right, and OVB analysis shows why the exponents vary.

RK: 4.Â Physically, a particle can be made to move along any path with any course of velocity over time, independently of the curvature of the path.Â For such an arbitrary course of velocity, the power law in general does not hold.Â

RM: What does it mean for the power law not to hold? I think what has to hold when the movement of the particle is measured in terms of V and R is thatÂ log (V) = 1/3 log(D) + 1/3 log (R). Even if a particle is made to “move along any path with any course of velocity over time” this equation holds. I’ve tried it with data where I moved quickly and slowly through curves and it holds; the deviation of the power coefficient from 1/3 is exactly predicted by OVB analysis.

RK: When the power law does hold, this is a non-trivial fact about the motion.Â

RM: What does it mean for the power law to hold or not hold?Â Are you saying that there can be another relationship between log(V) andÂ log(R) other than log (V) = 1/3 log(D) + 1/3 log (R)?Â

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RK: It is not guaranteed mathematically.Â

RM: But log (V) = 1/3 log(D) + 1/3 log (R) is guaranteed mathematically, is it not?Â

RK: When it does hold, the exponent can take any value.Â

RM: Again, what does it mean for the power law not to hold. The only time I think this could be true is when there are segments of the trajectory where the first or second derivative of x or y is 0. And even this is not a problem if you use non-linear regression.Â

RK: If in some class of situations the exponent is always found to be close to 1/3 (equivalently, that D is approximately constant), this is a further non-trivial fact.

RM: A fact is a fact. Our paper doesn’t say that power law analysis of movement trajectories yields trivial facts. Zago et al imply that this is what we are saying but we are not. What we are showing is that the power law is an illusion in the sense that it seems to say something about how the movement was generated, but it doesn’t.Â The fact that different trajectories are fit by a power law to different degrees is not a trivial fact – indeed, it’s probably of considerable interest to mathematicians and physicists; rather, its a *misleading fact*, an illusion that leads people who are studying “motor control” down a blind alley.Â

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RK: 5.Â If experimental data are obtained for V and R (from which D can be calculated), then any subset of those data selected for having a single value of D must mathematically obey the power law.Â

Â RM: Again, what does it mean to obey the power law? I’ve found that all the experimental data I have analyzed fits log (V) = 1/3 log(D) + 1/3 log (R) exactly.Â And when only log (R) is included in the regression, the exponent of log (R) deviates from 1/3 precisely by an amount proportional to the covariance between log(D) and log(R).

RK: This is true regardless of the source of the data and the mechanism that produces the data.Â

RM: And that is the point of our paper.Â

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RK: The power law is only non-trivial if no selection on D is made.

RM: The power law is never trivial. It just is. And we don’t “select” D; D is always there to be calculated. It’s the variation in the affine velocity of the trajectory. For some trajectories (like a perfect ellipse) affine velocity is constant so the regression of log(R) on log(V) gives a power exponent ofÂ exactly 1/3 and an R2 of 1.0. The point is that this is all a property of the nature of the trajectory itself; it has, as you note, nothing to do with how the trajectory was produced.

RK: If the power law is then observed, this is a nontrivial fact.Â

RM: Again, I don’t know what you mean when you say that the power law is not observed. But whatever you observe about the trajectory is just a fact, neither trivial nor non-trivial.Â

RK: So I disagree with Rick’s characterisation of the power law as omitted variable bias.

RM: We didn’t “characterize” the power law as OVB. We showed that when you regress log(R) on log(V) for movement trajectory data you always find a power coefficient of some value, though the R2 is sometimes quite low; R2 is generally higher when you regress log (C) on log (A).Â

RK: 6.Â An empirical regularity can rule out models that do not exhibit the same regularity, but cannot distinguish models that do.Â Many completely different models might give rise to the same observations.

RM: The empirical regularity of the power law was discovered with no understanding of the fact that behavior is control. Behavior was seen as a generated output so regularities in this output were taken to be informative about the processes that generated it. But we now know (thanks to Bill Powers) that behavior is a process of control. So we know that curved movements are consistent results produced by necessarily variable means – necessarily variable due to the need to compensate for variable disturbances. So when we are dealing with control, studying characteristics of the consistently produced results of control tells us nothing aboutÂ how these results were produced. But these regular characteristics of controlled results (invariances), such as the power law and tangential velocity profiles, are seductive facts that can lead us down blind alleys.Â

RM: I think that’s enough for now. I’ll just end by saying (again) that the fact that the power lawÂ tells us nothing about how intentionally produced movement trajectories are produced can be deduced simply from understanding that intentional (purposeful) behavior is a control process: variable means used to produce consistent results. Our mathematical/statistical analysis is presented only to show why a relatively consistent power law relationship between curvature and velocity is seen when analyzing curved trajectories.Â

Best regards

Rick

8.Â A particle moving with constant linear velocity along a path of varying curvature satisfies a power law, although of a degenerate sort: both exponents are zero.Â V = K D^0 R^0 where K is constant.Â This is not the sort of power law generally found in these experiments.

9.Â A pendulum swinging in a single plane (an example used by one of Rick Marken’s interlocutors), or a weight bobbing up and down on a spring, do not satisfy the power law.Â R is constant in both situations, while V varies.Â No tuning of the alpha parameter can produce plausibly straightish lines on a log-log plot.

10.Â Something immediately struck me as fishy about the power law, because it predicts that a straight line is traversed at infinite velocity.Â So I looked up Gribble and Ostry and found that the measure of radius that they use is not the actual radius R, but R* = R/(1 + alpha R), where alpha is a constant parameter.Â This is close to R when R is much smaller than alpha, and approaches 1/alpha as R tends to infinity.

This strikes me as something of a fudge.Â There are many ways in which we can impose a ceiling on R*; why this one?Â One might equally well choose, for example

R* = (2/(pi

alpha))R/2)atan(pialphaor

R* = (1/alpha)

tanh(alphaR)Both of these have the property of tending to R for small R and to 1/alpha for large R.

I could suppose that the experiments are all in the range of small R, but then there would be no need to introduce alpha, and G&O mention the power-law being observed for lemniscates (figure 8s).Â In these, if the curvature varies smoothly along the path, it must go to zero at least twice.Â So in some experiments, at least, alpha must play an essential role in bounding the predicted velocity.Â I would like to see how closely the power law fits the data when R is comparable to 1/alpha or greater, for each of these three definitions of R*.

It has been quipped that any data looks like a straight line when plotted on log-log paper. This is an exaggeration, but with this extra parameter alpha to fit, I wonder.Â On the other hand, G&O do find remarkably high correlation coefficients, on a par with those found in PCT experiments.Â So it does look, prima facie, as if their data non-trivially fit the power law.

G&O do not fit alpha, but use values from Viviani & Stucchi 1992, who do fit alpha to their data. I don’t know why G&O didn’t fit alpha to their own data – it is not a global fundamental constant, since they use different values in different experiments.

G&O allow themselves a further amount of freedom in fitting their model.Â The model is V(t) = K(t) (R*)^beta, where R* is as above.Â Notice that K is allowed to vary with time.Â If K is allowed to vary arbitrarily with time, then a suitable choice of K will make this hold regardless of V and R: just take K(t) = (R*)^beta / V(t).Â Only if K(t) cannot be arbitrarily chosen does the model have the ability to not fit a set of data (an essential property of any proposed model).Â G&O take K to be constant over each “segment” of the motion, calling it a scaling factor, presumably to account for the fact that the same movement can be done with different overall speeds.Â They refer to Viviano and Cenzato 1985 for this. But then for at least one example, they use multiple values of K over a single trajectory (the subject scribbles at random on paper).Â I am uncomfortable with the number of degrees of freedom this allows in fitting the model.

11.Â If a model is fitted to data, and fits very well, that does not imply that the model describes the mechanism that produced the data.

I am reminded of Zipf’s Law, a power law that Zipf originally found in linguistics (the frequency of a word is inversely proportional to its frequency rank). The same power law has been found in diverse situations (e.g. city population inversely proportional to rank order).Â The last time I looked at that, no-one knew why, in the sense of having a model of an underlying process that both produces Zipf’s Law statistics and can be shown to be present everywhere that Zipf’s Law is observed.

Rick Marken’s paper uses the example of people chasing toy helicopters, something that even our best robots cannot yet do.Â Other papers use the trajectories of crawling insects, or the tip of a pencil making scribbles.Â All of these physical mechanisms are very complex, and for a simple law to emerge, there must be some simple commonality among all these systems.Â I don’t think anyone knows what that is though.

It would be interesting to simulate different control systems performing a pursuit task, and see if the same power laws are observed, especially with control systems organised on very different lines.

–

Richard Kennaway

School of Computing Sciences, University of East Anglia, Norwich, UK

Cell and Developmental Biology, John Innes Centre, Norwich, UK

–

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 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery