Discussion of the speed-curvature Power Law

Research Motor control
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RM: I know. And you have the same problem in your responding to my criticism.

I don’t think I have seen you criticizing my arguments by showing what is wrong with them, by showing contradictions in or faulty premises and inferences. You just keep repeating your views and stating that because I disagree I am wrong.

RM: That’s because we, like all living things, are controllers, in fact, not just in theory.

No, we certainly are controllers but that does not mean that we must control for being always and absolutely right. We can also control for finding truth and it is possible only if we do not control for being already right. You are controlling for knowing the final truth about the power law phenomenon but I am not. I admit that I have only very vague hypotheses about it. I am sure only about one thing: that there are essential errors in your argumentation.

RM: …Affine velocity is not an “extra” predictor. It is one of the variables in the mathematical relationship between the measures of curvature and velocity that are used in determining whether or not there is a -1/3 power relationship between these measures.

No, the fact that both velocity and curvature can be measured from the same time based coordinate data of x and y positions and that there are similarly looking parts in both equations does not create a mathematical relationship between them. This should be clear because first the velocity and curvature can be measured also differently: Time is not needed to measure the curvature. Secondly exactly the same curvature can be moved through with different velocities, also with a stable velocity (=no correlation between curvature and velocity). So the velocity does not affect the measure of curvature (and in another way around) – even if they were measured from the same position data. So there is no mathematical relationship between velocity and curvature, even though there of course is mathematical relationship between the three variables. This has been told you at least ten times but you have never shown what is wrong in this.

RM: My argument is that the measurement method – calculating velocity and curvature from the same movement trajectory-- and the analysis method – simple linear regression – have “created” the power law phenomenon.

What you mean by “created” (in quotes)? Are we in the beginning of the dispute: that power law phenomenon is a bogus phenomenon, an illusion, a mirage, created by the researchers but not existing in reality? Do you meant that when a hand, a larva, a person or a helicopter moves along a curved trajectory, there is not a correlation between the velocity and curvature, and the velocity is not (sometimes or often) decreasing when the curvature is increasing, and there is not (sometimes or often) a power relationship between the velocity and curvature?

RM: OK, so you agree now that my equation: … is, indeed, exactly the same as the one derived by Moaz et al.

No one has ever denied that. The problem is not there. The problem is that from those equations does not follow what you claim and what Maoz et al possibly thought.

EP: The problem is that we can derive (or devise) an infinite amount of valid mathematical equations but it is totally a different question to what tasks they can be applied.

RM: I don’t understand what this means. Could you give me an example of some small subset of equations from that " infinite amount of valid equations".

I have already done that earlier but probably you have not read it - at least very carefully. All the equations of the form C^x * V^y = Z are mathematically as valid. And by using Z as an extra predictor in your “OVB” regression analysis you will get as many different power coefficients – and fake mathematical relationships – between C and V as you wish to. What is more important – let’s say it again – is that C and V need not even be the curvature and the velocity of any trajectory. As we have shown by the spreadsheets they can be any random variables which have no internecine correlation, and still your “OVB” analysis creates a fake mathematical relationship between them. What could be stronger evidence of a faulty method and mistaken mathematical reasoning?

EP: Maoz et al (2006) did apply that equation by deriving further an equation (6) [β = −1/3 + ξ/3] where beta β is the linear regression coefficient of log(v) versus log(κ) and ξ (xi) is the linear regression coefficient of log(α) versus log(κ). From this they saw that: “Hence, if log(α) and log(κ) are statistically uncorrelated, the linear regression coefficient between them, which we termed ξ, would be 0, and thus from (6) the linear regression coefficient of log(v) versus log(κ), which we named β, would be exactly −1/3. Therefore, any trajectory that produces log(α) and log(κ) that are statistically uncorrelated would precisely conform to the power law in (1)” which they earlier termed as “the two-thirds power law”. In a footnote they say: “α being constant is naturally a special case of uncorrelated log(α) and log(κ).”
Now, what is wrong in above mathematical reasoning of Maoz et al?

RM: That’s what I’d like to know!!

Perhaps I did not say it clearly enough. The problem in the mathematical reasoning of Maoz et al seems to be that they did not see that in a trajectory where curvature varies the only possibility for uncorrelated log(α) and log(κ) is that alpha is stable. The more alpha varies the more there is correlation because alpha is mathematically determined as a multiplication product of curvature and velocity (or them to the powers).

RM: So you have clearly not shown that there is anything wrong with Moaz et al’s equation (4) (and M&S equivalent equation).

Oh no, how many times it must be said that there nothing wrong in that equation (per se), only in the inferences made from it.

EP: What about (6) and inferences from it? The core here is the question when does a trajectory produce log(α) and log(κ) that are statistically uncorrelated (which is the same that when does a trajectory obey the power law)?

RM: No, that is not the core question at all. The core question is “Does equation 6 correctly (and exactly) predict the value of the power coefficient of curvature (Beta) when affine velocity is omitted as a predictor variable”? And the answer, which is easily demonstrated computationally, is that it does.

I think it should not be any wonder that it does. I will simplify. Say that Z = X * Y. Now if there is a certain statistical relationship, let’s call it w, between X and Y then there certainly must be some similar kind of statistical relationship also between X and Z, say v, so that the relationship between w and v can be expressed by some mathematical equation. You can test this computationally, if you want. It would be nice to see that!

EP: Now we can go to the problem the “OVB analysis”.

RM: I think i’ll stop here. Based on what you’ve already said it is clear to me (but not to you) that you do not understand regression or OVB analysis, (or to be nicer about it, you don’t understand these analyses the way I do). So we’re really not going to get anywhere with this.

Dismissal again! Couldn’t you try to teach me, not by repeating yourself because repetition is not a good method to teach, but by showing what is wrong in my understanding? What is wrong in my example of a proper way to apply OVB to find extra variables which affect school success and salary level? What is wrong in my view that that a method which gives the same exact power coefficient to all possible pairs of random variables is quite uncredible?

Happy New Year!
Eetu

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In practice, power law researchers use the same time derivatives of the X, Y and sometimes Z coordinates of the movement to calculate velocity and curvature. That’s why the algebraic solution of the relationship between between curature and velocity found by Maoz et al and M&S and others is perfectly legitimate.

It’s in quotes because it was the term you used.

OK, so we didn’t make a math error; we did the same analysis as Maoz et al, we just came to the wrong conclusion. And now you say Maoz et al came to the wrong conclusion as well. But Maoz et al came to the opposite conclusion we did. We concluded that the power law is a mathematical/statistical artifact and they concluded that it wasn’t. So what’s the correct conclusion, Eetu?

That’s not the regression equation I use. MIne is V = C^x*Z^y.

This makes no sense to me Eetu. I hthnk you will have to demonstrate this in a spreadsheet. I thought I had already demonstrated, in my Area calculation spreadsheet, that this is not the case.

This is such a hugh misunderstanding of what I showed in the Area analysis spreadsheet that I think we should just stop this here. I apparently wasted a ton of time developing that spreadsheet, all for naught.

Have a nice day.

Best, Rick

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RM: In practice, power law researchers us the same time derivatives of the X, Y and sometimes Z coordinates of the movement to calculate velocity and curvature. That’s why the algebraic solution of the relationship between between curature and velocity found by Maoz et al and M&S and others is perfectly legitimate.

No one has ever denied the legitimacy of that equation. However, it does not create or reveal any mathematical relationship between curvature and velocity which would coerce curvature to vary in a manner which depends on the variation of velocity, or in another way around.

RM: OK, so we didn’t make a math error, we just came to the wrong conclusion. And now you say MAoz et al came to the same wrong conclusion.

Well, depends on your concept of mathematics. No error in creating an equation but error in mathematical thinking about what that equation means and what follows from it. Maoz et al did NOT came to same conclusion. Their conclusion is much more benign and thus their error much smaller, if even significant because they connected their thinking only to effects of noise.

RM: That’s not the regression equation I use. MIne is V = C^x*Z^y.

Yes, sorry, so it is, but it does not change the situation. What I meant was that all equations of that form are mathematically as valid – for example the equation V = C * Z^-1 which is the same we used in spreadsheets about quadrangles (H = A / W).

EP: And by using Z as an extra predictor in your “OVB” regression analysis you will get as many different power coefficients – and fake mathematical relationships – between C and V as you wish to.

RM: This makes no sense to me Eetu. I htink you will have to demonstrate this in a spreadsheet. I thought I had already demonstrated, in my Area calculation spreadsheet, that this is not the case.

In the spreadsheets there were two independent random variables, let’s call them A and B. We can imagine them as sides of a rectangular quadrangle, H and W, and calculate the respective area: Ar = H * W. We can as well imagine them as instantaneous velocity and curvature (or logs of them) of curve in a trajectory, V and C, and calculate the alpha (Maoz et al) or D (M&S): D = V^3 * C. Then we can make an “OVB” regression analysis with the quadrangle equation H = W^-1 * Ar and find that the “true” regression coefficient between H and W is -1. But if we make that same analysis with the imagined trajectory data equation V = C^1/3 * D, will find that the “true” regression coefficient is 1/3 (and if we imagined that A and B are log(V) and log(C) then there exists 1/3 power relationship between V and C). So we can get different relationships between the same A and B just by changing the exponents in the equation. And what is more important: all these results are fake: In reality there is none such relationships between A and B because they are plain independent noncorrelated random variables. Thus the “OVB” method is bocus and fake trick to create results ex nihilo.

RM: This is such a hugh misunderstanding of what I showed in the spreadsheet that I think we should just stop this here. I apparently wasted a ton of time developing that spreadsheet, all for naught.

How so? Didn’t you show that from the same data, the two random variables, it was possible to calculate two different “true” regression coefficients, just by imagining that they represented either sides of quadrangles or velocity and curvature of a curve and by using the respective equations? What is wrong in this understanding?

RM: Please feel free to believe that you have won the argument: the power law is an really important phenomenon, a side effect of control that is very relevant and researchers should spend the rest of their careers studying.

I am afraid I have to believe that I cannot make you see your two mistakes. At least I have tried. All this has nothing to do about the relevance of a power law phenomenon. Probably it does not have that relevance which some research may have thought, but still it can have some relevance – I do not know what, but as Peirce said: do not block the way of inquiry. If you are interested only on listing controlled variables, then all side effects are irrelevant to you.

All that has been at the stake in this thread from my part has been the sanity of the argumentation is this forum. So I think this should have been under the category Dysfunctional discourse after all.

It would have been interesting to discuss also about motor control, side effects, and speed-curvature Power Law but maybe in the other time.

Thanks for this, I have studied and learned a lot because of this debate.
Eetu

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I wonder if this is relevant to varying velocity while moving on a path with curves of varying radius.

A demonstration by Viviani and Stucchi (1992) showed that when an object follows a curved path at constant velocity it is perceived to speed up by an amount that depends on the curvature of the path.

Viviani, P. and N. Stucchi. 1992. “Motor-Perceptual Interactions.” In Tutorials in Motor Behavior II. Edited by G. E. Stelmach & J. Requin, 229–248. Amsterdam: North-Holland.

Or is this ancient history in this discussion?

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I think it is relevant but perhaps in another way round: An object is perceived to move at a constant speed if it slows down by a certain amount that depends on the curvature of the path. Our perceptual functions have evolved to do so (perhaps, I think!) because any moving agent in “normal” physical conditions and using a constant amount of energy to its movement slows down in curves by an amount that depends on the steepness of those curvatures. In a curvature a certain part of the agent’s energy goes to the turning of its direction and is thus away from its production of speed. That is how I interpret the situation at the moment.

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Visual perception of (1) lateral velocity is very different from visual perception of (2) approach velocity of an object (expansion relative to surround in the visual field) or (3) forward velocity of the agent (more complex, peripheral relative to foveal). Movement around a curve combines 1 and 3. I think some of the organisms studied were not controlling visual input, but if so the same figure/ground principles may apply.

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It is ancient history but there is some good stuff in that history. In a post to CSGNet in April, 1993, on the subject of “IV-DV”, Bill Powers wrote the following about the Viviani and Stucchi paper:

In JEP-Human Perception and Performance, there was a good control-theory experiment:

Viviani, P. and Stucchi, N. Behavioral movements look uniform: evidence of perceptual-motor interactions (JEP-HPP 18 #3, 603- 623 (August 1992).

Here the authors presented subjects with spots of light moving in ellipses and “scribbles” on a PC screen, and had them press the “>” or “<” key to make the motion look uniform (as many trials as needed). The key altered an exponent in a theoretical expression used to relate tangential velocity to radius of curvature in the model. The correlation of the formula with an exponent of 2/3 (used as a generative model) with the subjects’ adjustments of the exponent was 0.896, slope = 0.336, intercept 0.090.

This is a nice psychophysical experiment. Based on what we now know about the power law, the results show that the subjects are perceiving constant velocity through different degrees of curvature when they adjust the speed (tangential velocity) through the curves so that affine velocity is constant. (We know this is the case because the regression of log Curvature on log Velocity will correspond exactly to a 2/3 power law only when affine velocity – the omitted variable in the analysis – is constant). To the extent that subjects are able to do this – to vary tangential velocity so as to keep affine velocity constant – the observed relationship between curvature and tangential velocity will be a power function with an exponent of -2/3. Given the fairly high correlation between the power law model and subjects’ behavior (0.896) it looks like subjects were able to do this pretty well.

I think the relevance of this result to the power law is: Something close to the 2/3 power law relationship between velocity and curvature will be found for any arbitrary curved movement made by man, beast or toy helicopter (Maoz et al, 2008); Marken & Shaffer, 2017). However, movements that consistently fit a power function almost exactly (R^2>.9) are made by agents who are controlling for moving at constant velocity, which means they are moving so as to keep affine velocity constant.

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Hi Rick, that is interesting.

You wrote: “However, movements that consistently fit a power function almost exactly (R^2>.9) are likely made by agents who are controlling for moving at constant velocity, which means they are moving so as to keep affine velocity constant.”

Earlier I think that you insisted that the power law relationship between the velocity and curvature is an insignificant side effect of control (and/or a statistical artifact caused by the researchers), but now you seem to be saying that it is a controlled variable. Do I interpret you right?

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Hi Eetu

I’ll answer the second part of your post first because my answer to the first part will explain why you misunderstood me (spoiler alert: it was my fault;-).

No, I still think that the power law is a side effect of control (it’s not insignificant because it has, and still is, directing research in the wrong direction). But I can see why you would think that I am now saying that the power law is a controlled variable. As you note, in my earlier post describing the results of research by Viviani and Stucchi (1992) I said:

“However, movements that consistently fit a power function almost exactly (R^2>.9) are likely made by agents who are controlling for moving at constant velocity, which means they are moving so as to keep affine velocity constant.”

It was misleading for me to say that the movements in that study were made by agents. In fact, the curved movements in that study were made by the computer; the agents’ movements were keypresses that increased or decreased the value of an exponent in an equation that determined the velocity of a spot on the screen that was moving along an ellipse or a “scribble” path. The equation was a power relationship between V and C: V(t) = k*C(t)^Beta. Variations in C were predetermined to produce the elliptical or scribble paths. Beta was initially set to a random value between -.25 & .67. The agents pressed the “<” or “>” key to decrease or increase the value of Beta.

So, as Powers noted in his comments, this is a control task where the agents are controlling the movements of a spot on the screen. The controlled variable is the perceived speed of movement of the spot on the screen; the outputs used to control that variable are the keypresses; the equation V(t) = k*C(t)^Beta is the feedback function relating output (keypresses) to controlled variable (speed of spot movement) via the effect of the output on Beta.

The results of this research are purported to show that constant velocity through curves is perceived when the velocity through curves is a 1/3 power function of the curvature. But looking at the data I see a huge amount of variation in power exponent across trials and subjects. So what the results suggest to me is that agents do a fairly poor job of controlling the velocity of movement through curves. But to the extent that an agent sets Beta close to 1/3, we know (from the omitted variable bias of the regression analysis) that they are doing this in order to perceive constant affine velocity.

I should add that this is relevant to the power law of movement inasmuch as it shows that, to the extent that a person is intending to make curved movements of constant velocity, the movements will follow a power law to the extent that the person is able to keep their perception of affine velocity constant.