[From Bruce Nevin (2011.11.06.12:20 ET)]

Rick Marken (2017.11.06.0910) –

I’ve had the same experience with papers about linguistics and perceptual control.Â Consider following the strategy that I adopted. ‘Publish’ the uncut paper on Researchgate with a footnote commenting and citing the Springer version.

You could take the opportunity to modify the uncut paper. You might try to justify having the same term covertly on both sides of an equation.Â Or you might consider how to reframe the mathematical discussion.

A demonstration that PCT accounts for the variety of power law observations seems to be what Alex is really after. My intuitive hunch is that a mathematical proof of that is not feasible. Experimental demonstration and mathematical proof are different beasts. The following two views are orthogonal, if not antithetical:

- A control system varies its motor movements in whatever way is sufficient to maintain equivalence of perception to reference.
- The observed trajectories of motor movements as plotted by an observer fit certain curves that are generated by a mathematical ‘power law’.

A mathematical proof seemingly would have to focus on the patterning of the output behavior (the curves of the trajectories), which is precisely what PCTÂ does not seek to characterizeÂ because it is a byproduct of control (“whatever output is sufficient”).Â

What the M&S paper doesÂ (in the final sections and the omitted sections) is demonstrate that a control model produces the observed trajectories for one kind of pursuit task (helicopters, and by inference baseball and [American]Â football). I do not have any grasp of the variety of power law observations–I see reference to different exponents. If these variant patterns show up in the observed behavior and in the model behavior, that strengthens the argument. But it is not a mathematical proof. AsÂ Adam Matic (2017.11.6) says, the generalization cannot be asserted as proof (“overgeneralization”).

Adam Matic (2017.11.6) –

Most studies on the production of the power law were done on tracing different curves, or drawing scribbles or letters,. The power law can be found in trajectories of very simple mass on spring systems, or coupled sinusoidal trajectories, and this doesn’t say anything definite about how this movement is produced in tracing experiments.

Right. I wonder if a more fruitful approach might be: Given the characteristics of control systems, and a presumption of least output to maintain control (an efficiency assumption, if you will), what constraints on output behavior are imposed byÂ inertial forcesÂ and other physiological andÂ environmental factors?Â

## ···

On Mon, Nov 6, 2017 at 12:11 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2017.11.06.0910)]

Bruce Nevin (2017.11.05.19:56 ET)

BN: I’ve been trying unsuccessfully to track this discussion, without being able to focus on it. I may be wildly off base, but this is my take after looking through the two papers this evening.

RM: I think this is a pretty good take on it. Our paper is heavy on the “statistical artifact” and light on the “PCT explanation” for a reason. The original paper had a more detailed discussion of the “PCT explanation” but one of the reviewers didn’t like that part but loved the “statistical artifact” part. So I had to trim the “PCT explanation” considerably. But PCT sneaks in at theÂ beginning of the article in this paragraph:

Although a third variable is a plausible explanation of

the correlation between curvature and velocity, it does not

explain why that correlation is consistently found to follow

a power law, per Eq. 1. A third variable explanation

requires that the cause of movement—thee muscle forces—

consistently affects curvature and velocity in ssuch a way

that velocity is a power function of curvature. However,

this explanation ignores the fact that different muscle forces

are required to produce the same movement trajectory on

different occasions due to variations in the circumstances

that exist each time the movement is produced (Marken

1988). For example, the forces required to move a finger in

an elliptical trajectory are different each time the movement

is produced due to slight changes in oneâ€™s orientation relative

to gravity. Therefore, muscle forces will not be consistently

related to the curvature and velocity of the movement;

the same power relationship between curvature and velocity

will be associated with somewhat different muscle forces

each time the same movement trajectory is produced.

RM: This is the PCT explanation of why we though the explanation of the power law might be found in the mathematics of how curvature and velocity are measured; and, as you note, a large part of our paper is dedicated to showing that this is, indeed, the case. But this PCT-based observation alone – that, due to varying disturbances, different muscle forces are being used to produce the same curved movement on different occasions; that variable means must be used to produce consistent resultsÂ – is enough to lead one to suspect that the power law that is found for the resulting movement can’t possibly be telling us anything about how that movement was produced. That’s why I have been rather surprised (and disappointed) that all these presumed PCT experts here on CSGNet are making out like I’m the great enemy of PCT for pointing out one of the most basic facts about behavior that we get from PCT;Â consistent results are produced by variable means. It’s called “control” and curved movements (indeed, all movements) are controlled (consistently produced) results of (necessarily variable) muscle forces.

Best

Rick

Rick Marken (2017.11.05.1220) –

What we showed is that the curved paths taken by both pursuers and a PCT model of those pursuers (a model that accounts for on average 93% of the variance in the curved paths taken by pursuers on 41 differentÂ trials ) exhibits a power law relationship between speed and curvature. This is evidence that the power law is a side-effect of the outputs that produced the curved paths as the means of controlling for intercepting

Sounds to me like this is the central finding. But it’s not so clear that this is the main point of Marken & Shaffer (2017)Â (posted by Alex on March 19, Subject: Power Law Publication). What you say there is “The present paper shows that the power law is actually a statistical artifact that results from mistaking a correlational for a causal relationship between variables… a mathematical consequence of the way that these variables are calculated.”

Zago, Matic, Flash, Gomez-Marin, and Lacquaniti (2017), which Alex posted at the start of this thread on 10/28, does not talk about control theory and mentions control systems only once (on p. 12 of the unpagenated reprint: an “experimental finding … implies that the control systems are [cap]able of establishing non-trivial co-regulations of path geometry and kinematics”). I may be mistaken, but it seems to me that the few and sporadic other uses of the word control in the article refer to motor control systems in a conventional way that does not invoke negative-feedback control.

We know that that negative-feedback control systems that use movements to control their input do not calculate the path geometry or kinematics of those movements, though the movements in the cases considered here can be described with path geometry and kinematics. Indeed, that is the final point of the Marken & Shaffer paper. The problem appears to be that the brief mention of this experimental finding at the end of the paper is dwarfed and obscured by the protracted critique that precedes it and which has every appearance of being presented as the main point of the paper.

Zago, Matic, et al. do not refer to the control-system model discussed in the short final sections of the Marken & Shaffer paper (called there a COV model), nor do they acknowledge the assertion thatÂ

The movements produced by the COV model accounted for an average of 93% of the variance in the movements of the actual pursuers over all trials … without any attempt to produce trajectories that followed a power law. Nevertheless, the model trajectories, like those of the actual pursuers, followed a power law with an exponent equivalent to that found in other studies of similarly curved movement trajectories… [T]he observed power law is a mathematical “side effect” of the model’s purposeful behavior. Specifically, it is a mathematical property of the trajectories that result from the model acting (varying ox and oy) to achieve its purpose of keeping the controlled perceptual variables…at the specified reference values.

Zago, Matic, et al say that Marken & Shaffer claim “that this power law is simply a statistical artifact, being a mathematical consequence of the way speed and curvature are calculated”. This is almost a direct quote of the passage cited above, here again: “a statistical artifact that results from mistaking a correlational for a causal relationship between variables… a mathematical consequence of the way that these variables are calculated.” Substitute “speed and curvature” for “variables”.

Zago, Matic, et al critique the assertion that “Since neither of these variables is manipulated under controlled conditions, any observed relationship between them cannot be considered to be causal.” However, the final claim at the end of Marken & Shaffer is that the power law is not a consequence of calculating speed and curvature, but rather a consequence ofÂ control. Isn’t this the real basis for the argument that correlation is not causation?Â

It appears to me thatÂ Â the rejoinder by Zago, Matic, et al overlooked the demonstration that is the real point of the paper, and that they did so because the critique of statistical methods of power law analysis takes up the central and largest sections of the Marken & Shaffer paper and seems to be its main argument. It also follows, I think, that however the mathematical quarrel between you and Martin is resolved, it will have no bearing on that substantial point: control systems produce ‘power law’ effects without doing power law calculations.

Thus, Zago, Matic, et al say “D cannot be considered an independent predictor of A (or V), because D itself depends on A (or V),” etc., echoing Martin’s objection to predicting V from V. But however the power law isÂ calculated, it is descriptive, whereas a control model is generative, and errors or misconstruals in that calculation are beside that main point.

They rather acknowledge this in concludingÂ

The issue that remains to be solved concerns the physiological origins of the power law. But this is a different topic to be covered in a forthcoming article.

It is a different topic which was covered in Marken & Shaffer (2017) only in the appendix-like concluding sections. I wonder, will their forthcoming discussion of “the physiological origins of the power law” recognize that control systems behave according to the power law without an elaborate physiological account? Will that future paper refer to the final sections of Marken & Shaffer (2017)?

/Bruce

–

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

On Sun, Nov 5, 2017 at 6:15 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2017.11.05.1515)]

Bruce Nevin (2017.11.05.1755 ET)

Rick Marken (2017.11.05.1220) –

What we showed is that the curved paths taken by both pursuers and a PCT model of those pursuers (a model that accounts for on average 93% of the variance in the curved paths taken by pursuers on 41 differentÂ trials ) exhibits a power law relationship between speed and curvature. This is evidence that the power law is a side-effect of the outputs that produced the curved paths as the means of controlling for intercepting

BN: Does this hold also for the paths taken by people catching baseballs?

RM:Yes, and it also holds for people catching footballs thrown to themselves (based on the data from Shaffer, D. M., Marken, R. S., Dolgov,

I. and Maynor, A. B. (2015) Catching objects thrown to oneself: Testing the

generality of a control strategy for object interception, *Perception,*44, 400-409).

BestÂ

Rick

–

Richard S. MarkenÂ

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

have nothing left to take away.â€?

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