# The speed�?curvature power law of movements: a reappraisal

[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:

1. A control system varies its motor movements in whatever way is sufficient to maintain equivalence of perception to reference.
2. 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”).

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

BN: 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 by inertial forces and other physiological and environmental factors?Â

Yep, from what I understand, the optimal control approach is more or less exactly that - trying to find what quantity is minimized in the production of curved trajectories. It’s been very fruitful in the sense that observed trajectories fit very nicely with predicted ones in many cases. Huh & Sejnowski (2015)Â https://www.ncbi.nlm.nih.gov/pubmed/26150514 show that human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related variables, but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.

···

On Mon, Nov 6, 2017 at 7:28 PM, Bruce Nevin bnhpct@gmail.com wrote:

[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:

1. A control system varies its motor movements in whatever way is sufficient to maintain equivalence of perception to reference.
2. 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”).

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

/Bruce

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—the muscle forcees—
consistently affects curvature and velocity in such a way

that velocity is a power function of curvature. However,
this explanation ignores the fact that different muscle forces
are required to produce the same movement trajectory on
different occasions due to variations in the circumstances
that exist each time the movement is produced (Marken
1988). For example, the forces required to move a finger in
an elliptical trajectory are different each time the movement
is produced due to slight changes in 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

[From Bruce Nevin (2017.11.0.20:06 ET)]

Martin Taylor 2017.11.06.15.16 –

I said (2011.11.06.12:20 ET)

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 by inertial forces and other physiological and environmental factors?

By “a presumption of least output to maintain control” I did not mean “trying to find what quantity is minimized in the production of curved trajectories” (Adam Matic 2017.11.6 at 2:30 PM). I was merely reflecting that it is often the case that a given goal can be achieved by any of a variety of outputs, so that for purposes of identifying environmental constraints this simplifying assumption would be useful. The phrase is a distraction. Leave it out, as follows:

Given the characteristics of control systems, what constraints on output behavior are imposed by inertial forces and other physiological and environmental factors?

I was responding to 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.

The trajectory of a “very simple mass-on-spring system” clearly must be determined exclusively by ‘environmental factors’ (inertial forces, etc.) with no negative-feedback control system involved.

IIRC, the movement trajectories of the pointing finger in the Little Man demo accord closely with the measured movement trajectories of Bill Powers’ pointing finger, and this is achieved by the interaction of hierarchical control loops with the (modeled) physical characteristics of the arm (physiological characteristics, if you will) in the environment, parameters of the control loops being tuned to reflect or ‘capture’ his personal behavior. Do those movements of the Little Man model display the power law? That would be an instance of what I am talking about when I ask about the interaction of control systems with physical constraints affecting the trajectory of control outputs.

But when you say “the optimal control approach is … trying to find what quantity is minimized in the production of curved trajectories”, I no longer feel that I can say “given the characteristics of control systems.” I am no longer confident that an understanding of what control systems are and how they work is a given in this thread. Martin’s (2017.11.06.15.16) comments seem to me to reflect the same concern: do we mean the same thing when we say “control system” and talk about the “control [system] approach” to this question of observed movement trajectories? Or are you talking about ‘motor control’ in the conventional sense without consideration of negative-feedback loops closed through the environment? I come to that speculation because that conventional usage seems to be the norm in Zago, Matic, Flash, Gomez-Marin, and Lacquaniti (2017), insofar as the word ‘control’ occurs there at all.

If the subject is tracing a curve that is perceptibly given by a template, the first assumption would be that the controlled perception is a relationship between the ‘current’ portion of the curve and the corresponding portion of the template–if the template is an overlay (or underlay with translucent medium) a simple distance relationship between the current point and the corresponding point; if the template is nearby, a relation of congruence between configuration perceptions (at a higher level of the hierarchy). I can easily see that such a relationship can be perceived, but how to disturb it without overwhelming control is not immediately obvious to me–lack of imagination, I’m sure, and lack of experience designing such experiments.

···

On Mon, Nov 6, 2017 at 3:39 PM, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2017.11.06.15.16]

BN: 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 by inertial forces and other physiological and environmental factors?

… There were attempts with minimization of torque and similar effort-related variables, but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.

The PCT question then would be whether jerk is perceived and is controlled. The difference between moving in a viscous environment and in a free one is that in a viscous environment f=ma does not hold, and increases in force do not translate directly into increases in acceleration. Jerk in a high-viscosity environment is always very low, and yet the power law is observed for a given task in free and (with a different power) in viscous environments. My guess is that jerk is not a controlled perception, but it would need some testing to discover whether the guess is right or wrong.

At least for me, one of the takeaways from long interaction with Powers and reading his writings is that maximization or minimization is rarely the reference value of a controlled perception. It is usually a side-effect of something, possibly just that the reference value is higher or lower than the system can attain, but often that some other variable is controlled and the environment produces a side-effect that maximizes or minimizes the observed variable.

Finding what perception(s) is (are) controlled should be the basis for figuring out how the power-law side-effect is produced by environmental constraints. Maybe it is jerk, maybe it is lateral deviation from a reference path, maybe lateral deviation of the reference curve from straight ahead for a fixed lead time around the curve, maybe something else entirely. One could hypothesize endlessly, always fruitlessly until the TCV is used. Is the hypothesized variable actually perceived, and is it directly influenced? When you disturb it, is the disturbance resisted?

Martin

[From Rick Marken (2017.11.06.1800)]

···

Bruce Nevin (2011.11.06.12:20 ET)

BN: A demonstration that PCT accounts for the variety of power law observations seems to be what Alex is really after.

RM: It’s the omitted variable bias (OVB) analysis that accounts for this; PCT simply allowed us to see that the power law was likely a side effect of the the production of curved movements that has nothing to do withÂ how these movements are produced. It allowed us to see that because PCT makes us aware of the fact that intentionally produced results, such as curved movement trajectories, are produced by variable means. The time varying position of the finger tracing out a curved trajectory is clearly a controlled variable; the means of control is varying muscle forces. Those varying muscle tensions are likely to be uncorrelated with the resulting variations in the intended position of the finger. So looking at mathematical properties of variations in the position of the finger – the controlled variable – for evidence of how those variations were produced is like looking at mathematical properties of variations in the cursor in a tracking task for evidence of how those variations were produced.

Â

BN: 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:

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

BN: 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

RM: I think what is necessary is a demonstration that the power law that fits intentionally produced curved movements can be quite different (in terms of exponent) than then the power law that fits the output variations that produced these movements. I’ve suggested that Adam do some experiments of this sort.Â This kind of research could be the basis for ending this conflict since I’m quite sure that only a PCT type control model could account for the results of such an experiment.Â

Â

BN: 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).

RM: No, it demonstrates (via OVB analysis) that the power law relationship between velocity and curvature that is found for curved trajectories, whether produced by living systems or toy helicopters, is a mathematical consequence of the way these variables are measured and the precise value of the power exponent that is found depends on characteristics of the movement trajectory itself and has nothing to to with how that movement trajectory was produced.Â

Â

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.

BN: 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

RM: As I said, I’ve suggested some research somewhat along these lines that Adam could do. The research I suggested is another way of testing whether there is any explanation of why certain power relationships between curvature and velocity are observed when people produce curved movements other than that these relatoinships are a mathematical artifact.Â

BestÂ

Rick

"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

Adam is working on it, and has done part of it already. But it is not that easy, nor one can solve the problem by rephrasing Powers 1973: one really has to get wet with the details.

···

On Tue, Nov 7, 2017 at 9:03 AM, Eetu Pikkarainen eetu.pikkarainen@oulu.fi wrote:

[Eetu Pikkarainen Â 2017-11-07 9:47]

Â

Â

AM: Yep, from what I understand, the optimal control approach is more or less exactly that - trying to find what quantity is minimized in the production of curved trajectories.
It’s been very fruitful in the sense that observed trajectories fit very nicely with predicted ones in many cases. Huh & Sejnowski (2015)Â https://www.ncbi.nlm.nih.gov/pubmed/26150514 show that
human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related variables,
but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.

Â

This fits with my earlier guess that the test subject controls for (her perception of) the steady (or smooth) speed (even if no instruction about speed is given) and (her perception of) staying on the line (which is also
instructed). I guess that these could be tested with an experiment where the subject is drawing onto the tablet where the target line is not already visible but drawn by the program. So the task would be similar like a tracking task except (at least) the cursor
would also draw its own trajectory. The subject is not instructed to â€œcatchâ€? the cursor but just to follow it as near or well as possible so that her pen or finger would draw similar curved line as the cursor draws. The disturbances would be produced just
by altering the speed and curves or the cursor.

Is here any sense? Done already?

Â

Â

Eetu

Â

Â Please, regard all my statements as questions,

Â no matter how they are formulated.

Â

Â

Â

[From Rick Marken (2017.11.07.0650)]

···

AM: Huh & Sejnowski (2015)Â https://www.ncbi.nlm.nih.gov/pubmed/26150514 show that human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related variables, but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.

RM: Could you post a functional diagram of jerk minimization system? That is, a diagram of how the observed movement trajectory is produced by the jerk minimization system (a system presumably implemented in the muscles and nervous system). It’s hard to tell how this work from just looking at the equations.

Thanks.

Best

Rick

Â

On Mon, Nov 6, 2017 at 7:28 PM, Bruce Nevin bnhpct@gmail.com wrote:

[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:

1. A control system varies its motor movements in whatever way is sufficient to maintain equivalence of perception to reference.
2. 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”).

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

/Bruce

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 movementâ€”the muscle forces—
consistently affects curvature and velocitty in such a way
that velocity is a power function of curvature. However,
this explanation ignores the fact that different muscle forces
are required to produce the same movement trajectory on
different occasions due to variations in the circumstances
that exist each time the movement is produced (Marken
1988). For example, the forces required to move a finger in
an elliptical trajectory are different each time the movement
is produced due to slight changes in 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

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

Â It’s hard to tell how this work from just looking at the equations.

EXACTLY! That is why one needs to learn math!Â

But, Rick, before you move on to the hard maths of Huh and Sejnowski (a paper, by the way, which shows EMPIRICAL data for human tracing where clear power laws hold withÂ exponentsÂ DIFFERENT than 2/3  and OVT or RCT cannoot dispute that!), you should get pass the mathematical and statistical flaws of your recent paper. It is like pretending to read general relativity when one still denies and cannot understand Newton’s laws.Â

···

On Tue, Nov 7, 2017 at 3:53 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2017.11.07.0650)]

AM: Huh & Sejnowski (2015)Â https://www.ncbi.nlm.nih.gov/pubmed/26150514 show that human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related variables, but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.

RM: Could you post a functional diagram of jerk minimization system? That is, a diagram of how the observed movement trajectory is produced by the jerk minimization system (a system presumably implemented in the muscles and nervous system). It’s hard to tell how this work from just looking at the equations.

Thanks.

Best

Rick

Â

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 Mon, Nov 6, 2017 at 7:28 PM, Bruce Nevin bnhpct@gmail.com wrote:

[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:

1. A control system varies its motor movements in whatever way is sufficient to maintain equivalence of perception to reference.
2. 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”).

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

/Bruce

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the muscle forcess
consistently affects curvature and velocity in such a way
tthat 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

[From Bruce Nevin (2017.11.07.14:27 ET)]

Alex, are you saying that it is not possible for you to draw a functional diagram of the negative-feedback control loops that constitute the jerk-minimization system which the mathematical expressions describe?

···

On Tue, Nov 7, 2017 at 10:05 AM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Â It’s hard to tell how this work from just looking at the equations.

EXACTLY! That is why one needs to learn math!Â

But, Rick, before you move on to the hard maths of Huh and Sejnowski (a paper, by the way, which shows EMPIRICAL data for human tracing where clear power laws hold withÂ exponentsÂ DIFFERENT than 2/3  and OVT or RCT cannot disspute that!), you should get pass the mathematical and statistical flaws of your recent paper. It is like pretending to read general relativity when one still denies and cannot understand Newton’s laws.Â

On Tue, Nov 7, 2017 at 3:53 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2017.11.07.0650)]

AM: Huh & Sejnowski (2015)Â https://www.ncbi.nlm.nih.gov/pubmed/26150514 show that human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related variables, but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.

RM: Could you post a functional diagram of jerk minimization system? That is, a diagram of how the observed movement trajectory is produced by the jerk minimization system (a system presumably implemented in the muscles and nervous system). It’s hard to tell how this work from just looking at the equations.

Thanks.

Best

Rick

Â

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 Mon, Nov 6, 2017 at 7:28 PM, Bruce Nevin bnhpct@gmail.com wrote:

[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:

1. A control system varies its motor movements in whatever way is sufficient to maintain equivalence of perception to reference.
2. 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”).

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

/Bruce

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the muscle forces
consistently affects curcurvature and velocity in such a way
that velocity is a power function of curvature. However,
this explanation ignores the fact that different muscle forces
are required to produce the same movement trajectory on
different occasions due to variations in the circumstances
that exist each time the movement is produced (Marken
1988). For example, the forces required to move a finger in
an elliptical trajectory are different each time the movement
is produced due to slight changes in 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

i am saying not all can be compressed in a simple (or complex) pct diagram, let alone complex maths. one does not substitute the other. nevertheless, one should try.

···

On Tue, Nov 7, 2017 at 10:05 AM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Â It’s hard to tell how this work from just looking at the equations.

EXACTLY! That is why one needs to learn math!Â

But, Rick, before you move on to the hard maths of Huh and Sejnowski (a paper, by the way, which shows EMPIRICAL data for human tracing where clear power laws hold withÂ exponentsÂ DIFFERENT than 2/3  and OVT or RCT cannot dispute that!), you should get pass the mathematical and statistical flaws of your recent paper. It is like pretending to read general relativity when one still denies and cannot understand Newton’s laws.Â

On Tue, Nov 7, 2017 at 3:53 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2017.11.07.0650)]

AM: Huh & Sejnowski (2015)Â https://www.ncbi.nlm.nih.gov/pubmed/26150514 show that human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related variables, but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.

RM: Could you post a functional diagram of jerk minimization system? That is, a diagram of how the observed movement trajectory is produced by the jerk minimization system (a system presumably implemented in the muscles and nervous system). It’s hard to tell how this work from just looking at the equations.

Thanks.

Best

Rick

Â

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 Mon, Nov 6, 2017 at 7:28 PM, Bruce Nevin bnhpct@gmail.com wrote:

[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:

1. A control system varies its motor movements in whatever way is sufficient to maintain equivalence of perception to reference.
2. 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”).

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

/Bruce

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the muscle forces
consistently affects curvature an and velocity in such a way
that velocity is a power function of curvature. However,
this explanation ignores the fact that different muscle forces
are required to produce the same movement trajectory on
different occasions due to variations in the circumstances
that exist each time the movement is produced (Marken
1988). For example, the forces required to move a finger in
an elliptical trajectory are different each time the movement
is produced due to slight changes in 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

Alex Gomez-Marin
behavior-of-organisms.org

[From Bruce Nevin (2017.11.07.18:50 ET)]

OK, rereading this, I think I’m not as confused as I was.

Bruce Nevin (2011.11.06.12:20 ET) –

Adam Matic (2017.11.6) [two successive posts with the same timestamp]–

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

from what I understand, the optimal control approach is more or less exactly that - trying to find what quantity is minimized in the production of curved trajectories. It’s been very fruitful in the sense that observed trajectories fit very nicely with predicted ones in many cases. Huh & Sejnowski (2015)Â https://www.ncbi.nlm.nih.gov/pubmed/26150514Â show that human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related variables, but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.

As you say,Â Huh & Sejnowski (2017) is an instance of what is called optimal control, "the process of determining control and state trajectories for aÂ dynamic systemÂ over a period of time to minimise a performance index."Â As context and motivation for their analysis, theyÂ assume that motor control is accomplished by planning movements and then issuing commands to muscles to execute the movements according to plan. This is evidenced in many places, e.g.

"These findings have implications for motor planning and predict that movements only depend on one radian of angle coordinate in the past and only need to be planned one radian ahead."Â

"The symmetric shape of the impulse response function (Eq.Â 12) implies that it requires information about the future and the past path shape to plan a curved movement. Moreover, the extent of planning is Â â‰ˆÂ 1 radian, which is the approximate width of the impulse response function. In contrast, the one-third power law has a flat frequency response (Fig. 8A), and thus a delta function impulse response in the angle domain (Fig. 8B), which predicts that the movement speed should be instantaneously determined by the local curvature without any planning.)"Â

“The angle coordinate representation may be relevant for interpreting neurophysiological data as well: Power laws have been reported in neural activity recorded in premotor and motor cortices (2122), and many neurons in the motor cortex show preference for the direction of movement (23). This is consistent with motor planning based on a form of the angle coordinate representation for generating motor kinematics and dynamics in an allocentric reference frame (24). Scale invariance may be a general principle used by the motor system for organizing complex movements, and analyzing neurophysiological data in the angle coordinate representation may reveal new aspects of neural control of movements.”

As we know, no one has ever demonstrated how this is possible (planning movements and then issuing commands to muscles to execute the movements according to plan) in an environment in which, in the interval between plan and execution, unpredictable disturbances may add to or subtract from the effects of motor efforts. And as we know, negative-feedback control systems do not have this difficulty because they automatically generate more or less precisely countervailing motor forces almost simultaneously with the disturbance forces.Â

The “minimum-jerk model” thatÂ Huh and SejnowskiÂ present is an ‘optimal control’ model; it is not a control-theory model. So it is pointless to ask how to disturb “minimum jerk” or how it might be perceived or controlled by a control system.

Huh and Sejnowski “investigate here conditions when a known regularity fails to adequately describe motor behavior, which adds new insights into how movements are generated by the nervous system.” The power lawÂ is “the inverse relationship between the speed and curvature of 2D hand movements observed when subjects are instructed to freely draw curved shapes such as ellipses.” It is “One of the best-studied movement regularities”, but it has exceptions. For example, drawing an S curve or lemniscate

when the curvature changes sign: The power law predicts the speed should diverge to infinity as the curvature approaches zero, which is physically implausible, whereas actual movements exhibit smooth, finite speed profiles at such inflection points. Deviations have also been observed for movements without inflection points … in complex movements…

To fit such exceptions into the observational generalization (for that is all that the power law is, an observational generalization with exceptions) the prior proposal has been that “complex movements could be generated by concatenating smaller and simpler movement segments, each of which separately obeys the one-third power law…” They acknowledge that all of this is “poorly understood.” However, theirÂ

minimum-jerk … optimal control modelÂ Â is known to reproduce accurately the one-third power law for simple, ellipse drawing movements…, as well as the apparent fragmentation observed for more complex movement trajectories without invoking the segmented control hypothesis… This suggests that there may be additional regularities underlying fragmentation, apart from the one-third power law relationship. Thus, a closer examination of the minimum-jerk model could lead to a more comprehensive understanding of the regularities in curved hand movements.

Indeed, a more comprehensive understanding is in order. The classic path of a scientific revolution is to seek out awkward anomalies in the established paradigm and demonstrate a simpler and more comprehensive resolution.

The task is to demonstrate thatÂ PCTÂ computer simulations generate outputs that conform to the power law. that they also generate the exceptions to the power law (under the exception conditions, e.g. an inflection in curvature), to explain how and why they do so, and why they must always do so when constructed and tuned to model the natural behavior of the modeled organism.Â

I gather that is the task you are working on.

···

On Tue, Nov 7, 2017 at 3:31 PM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

i am saying not all can be compressed in a simple (or complex) pct diagram, let alone complex maths. one does not substitute the other. nevertheless, one should try.

On Tue, 7 Nov 2017 at 20:28, Bruce Nevin bnhpct@gmail.com wrote:

[From Bruce Nevin (2017.11.07.14:27 ET)]

Alex, are you saying that it is not possible for you to draw a functional diagram of the negative-feedback control loops that constitute the jerk-minimization system which the mathematical expressions describe?

Alex Gomez-Marin
behavior-of-organisms.org

On Tue, Nov 7, 2017 at 10:05 AM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Â It’s hard to tell how this work from just looking at the equations.

EXACTLY! That is why one needs to learn math!Â

But, Rick, before you move on to the hard maths of Huh and Sejnowski (a paper, by the way, which shows EMPIRICAL data for human tracing where clear power laws hold withÂ exponentsÂ DIFFERENT than 2/3 — and OVT or RCT cannot dispute that!), you should get pass the mathematical and statistical flaws of your recent paper. It is like pretending to read general relativity when one still denies and cannot understand Newton’s laws.Â

On Tue, Nov 7, 2017 at 3:53 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2017.11.07.0650)]

AM: Huh & Sejnowski (2015)Â https://www.ncbi.nlm.nih.gov/pubmed/26150514 show that human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related variables, but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.

RM: Could you post a functional diagram of jerk minimization system? That is, a diagram of how the observed movement trajectory is produced by the jerk minimization system (a system presumably implemented in the muscles and nervous system). It’s hard to tell how this work from just looking at the equations.

Thanks.

Best

Rick

Â

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 Mon, Nov 6, 2017 at 7:28 PM, Bruce Nevin bnhpct@gmail.com wrote:

[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:

1. A control system varies its motor movements in whatever way is sufficient to maintain equivalence of perception to reference.
2. 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”).

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

/Bruce

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—tthe muscle forces—
consistently affects curvature and velocity inn such a way
that velocity is a power function of curvature. However,
this explanation ignores the fact that different muscle forces
are required to produce the same movement trajectory on
different occasions due to variations in the circumstances
that exist each time the movement is produced (Marken
1988). For example, the forces required to move a finger in
an elliptical trajectory are different each time the movement
is produced due to slight changes in 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

[From Bruce Nevin (2017.11.07.18:57 ET)]

Martin Taylor 2017.11.07.16.03 –

As I belatedly realized (Bruce Nevin 2017.11.07.18:50 ET),

The “minimum-jerk model” that Huh and Sejnowski present is an ‘optimal control’ model; it is not a control-theory model. So it is pointless to ask how to disturb “minimum jerk” or how it might be perceived or controlled by a control system.

And it is likewise pointless to ask for a functional diagram. Wrong tree, and nothing to bark at.

···

On Tue, Nov 7, 2017 at 5:30 PM, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2017.11.07.16.03]

``````  On 2017/11/7 3:31 PM, Alex Gomez-Marin
``````

wrote:

``````      i am saying not all can be compressed in a
``````

simple (or complex) pct diagram, let alone complex maths. one
does not substitute the other. nevertheless, one should try.

On Tue, 7 Nov 2017 at 20:28, Bruce Nevin <bnhpct@gmail.com >
wrote:

[From Bruce Nevin (2017.11.07.14:27 ET)]

``````            Alex, are you saying that it is not possible for you
``````

to draw a functional diagram of the negative-feedback
control loops that constitute the jerk-minimization
system which the mathematical expressions describe?

``````What would be required of such a functional diagram? What hypothesis
``````

would it test? Would it test whether the “minimum jerk” observation
was a direct result of jerk being controlled (a possibility) or a
side-effect of controlling something else, which then leads to the
power-law side-effect? One can’t draw a functional diagram without
having at least a hypothesis about what perception(s) is/are being
controlled. It can’t be “jerk” with a reference value of zero,
because that is most closely achieved by an actor that takes an
infinite time to trace the curve, whereas the actors in the various
experiments are much quicker, tracing curves that have quite
appreciable jerk.

``````Jerk is a strange quantity, the third derivative of position,
``````

velocity being the first and acceleration the second. The first two
have direct relations to applied force under different environmental
conditions. Velocity is proportional to force in a viscous medium,
whereas acceleration is proportional to force in a non-viscous
medium (including frictional effects along with viscosity). Jerk is
rate of change of acceleration, which in a non-viscous medium is
proportional to rate of change of force, or in a viscous medium to
the rate of change of the rate of change of force. It’s a quantity
unlikely to be controlled directly (though not impossible), so its
appearance in observed data is likely to be a side-effect of
something else.

``````Should we assume that under conditions where the power law is
``````

observed, the organisms are trying to move as fast as possible,
either to some target (fly larvae seeking food, Marken’s helicopter
chaser)? Probably not, unless “as fast as possible” refers to either
some environmental limit on speed as a function of curvature or to
some conflict with another controlled variable that occurs only
after some threshold velocity is achieved. In PCT, the words “as
possible” refer to an unachievable reference value for the
controlled perception in question.

``````When one is driving a car in the normal way to get somewhere rather
``````

than for sightseeing, one does not go “as fast as possible” around
curves or even on a straight (even in, places with no speed limit).
But different people will take a curve of a specific radius of
curvature at different speeds, a race driver in a race much faster
than a senior who drives to the shopping mall with great
deliberation. Neither is constrained by the power of the engine, so
one must presume they control for something other than going “as
fast as possible” that either sets a reference value for velocity or
sets up a conflict.

``````What might that "something other" be, and is the limiting factor
``````

conflict or reference setting (which would mean that the driver
controls for something else, such as perceived safety)? Clearly the
fly larva doesn’t control for perceived safety. Nor, I assume does a
person tracing a drawn curve or someone actually scribbling
randomly. But if “minimum jerk” describes their performance well,
for what might they be controlling, and is it the same in all the
conditions where the power-law relationship is found? If there’s a
reasonable hypothesis about that, then one could devise a functional
diagram. and having that diagram (or maybe even without it) one
might be able to use the TCV to test whether the hypothesis is
tenable.

``````Sorry to sound so negative, but I think it is not a trivial problem.

Martin
``````

[From Bruce Nevin (2017.11.08.21:01 ET)]

Erling Jorgensen (2017.11.08 1005 EST) –

Yes, Erling, as you describe it, I do much the same. It is certainly possible and even commonplace to control a perception of smooth and gradual acceleration and deceleration. In your example of driving a car, it serves the higher purpose of sparing your passengers the kind of experience that you remember having as a passenger in your father-in-law’s car. In my case it also subserves the higher purpose of improving efficiency as measured by gas mileage or by the driving-efficiency displays on my Prius (off Island) or my Leaf (on Island). I also recall a perception of myself being in better control of the car when, for example, I could stop at a light without the front end bouncing up off the suspension after forward motion stopped. Conversely, I think we all can recognize the appearance of pushing to the edge of controllability as means of demonstrating how masterfully one is in control.

“In curved hand movements around ellipses, the speed tends to scale inversely with the curvature”. As I understand this, the tighter the curve, the slower you go. Sounds like a driver, doesn’t it? A line that is straight is easiest and fastest to draw (more or less straight, given disturbances). It involves nothing like the inertial forces of a moving car, but is there something to the idea of slowing down so as not to ‘go off the road’, so to speak?

“These findings … predict that movements only depend on one radian of angle coordinate in the past and only need to be planned one radian ahead.”

When I was first learning to drive a car, following verbal instructions I tried to control the visual relation between the edge of the road and the right side margin of the car (hood and fenders) , and between the center line and the left side margin of the car. Visual ‘landmarks’ disappeared out of sight behind me at an alarming rate! Nervous oversteering at slow speed resulted. With more experience, I came to control the visual relationships to points much farther ahead down the road. visual 'landmarks at that distance are stable in the visual field for a considerably longer time. Going around a curve, my line of sight is curtailed. The tighter the curve, the shorter my line of sight, and the closer to me are the visual ‘landmarks’ at the edges of the traffic lane.

Of course the distance encompassed by one radian of arc (the radius of the arc) is shorter the tighter the curve.

So we know that the ease of maintaining control of driving a car is inversely proportional to the tightness of the curve; is the ease of maintaining control of drawing a curve inversely proportional to the tightness of the curve?

···

On Wed, Nov 8, 2017 at 10:42 AM, Erling Jorgensen EJorgensen@riverbendcmhc.org wrote:

[From Erling Jorgensen (2017.11.08 1005 EST)]

Bruce Nevin (2017.11.07.18:57 ET)

As I belatedly realized (Bruce Nevin 2017.11.07.18:50 ET),

The “minimum-jerk model” that Huh and Sejnowski present is an ‘optimal control’ model; it is not a control-theory model. So it is pointless to ask how to disturb “minimum jerk” or how it might be perceived or controlled by a control system.

And it is likewise pointless to ask for a functional diagram. Wrong tree, and nothing to bark at.

Hi Bruce,

EJ: I agree that the ‘optimum control’ model, as I understand it, does not produce “minimum jerk” by controlling input, but by calculating and generating output. However, I believe it is possible to control for minimum jerk, or smoothness of acceleration, in a PCT manner.

EJ: My father-in-law was notorious for driving with a very erratic use of the accelerator, even at running speed on straight roads with minimal disturbances. The accelerator was constantly surging or easing off, and it made for a noticeably uncomfortable ride as a passenger. I think that sensitized me to what a passenger might be feeling if I was being too aggressive with acceleration or deceleration (even at steady rates.) I’ve tried to notice, and minimize, that last bit of jerk just before coming to a full stop, by slightly easing up on the brake with a slight mini-coast to a stop for the final second. I’ve tried to be smoother or steadier with changes in accelerating.

EJ: I think I am perceptually noticing a couple of things, in order to control for smoothness. One is the pressure of the accelerator on the bottom of my right foot, and the degree of perceived effort. Along with that is the rate of change in the angle of my right ankle, trying to minimize sudden changes. A more pronounced indicator of jerkiness in the ride is whether my upper torso is making small uncontrolled movements forward or back. As the driver, these get buffered somewhat by the muscle tone of my arms gripping the steering wheel, and even by the pressure back against my right foot by the springy accelerator pedal itself. Because I have those dynamic buffers online by virtue of driving, I try to realize that the passengers may not be bracing themselves in a similar way, and so that is a reminder to try to make the journey even smoother than it feels to me.

EJ: I haven’t yet thought through how this may be applied to drawing ellipses or scribbles or other results of curved-path movements. Nor have I thought yet about potential disturbances to smoothness in those activities. But it seems the third derivative of position may have some applicability, at least for some situations of PCT control.

All the best,

Erling

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