Discussion of the speed-curvature Power Law

My attempted paraphrase got a passing grade from both Martin and Eetu, with some additional aspects contributed by Eetu.

Can you lay out a summary of their position in your own words, for purposes of mutual verification?

It is customary for a critical article in a journal to begin by summarizing the views that it will criticize. I’m wondering now how much churning of controversy might be avoided if the authors first sent their summary to the other side for verification that it is a fair and accurate representation. Equivocation can still elude detection, but the straw man form of equivocation would be harder to erect. (That’s a generalization, not a pointing finger!)

Here is my summary of what I think is their position. I’m basing it on Martin’s comment in Exp. Brain Res. I believe he is saying that our mistake was to consider the term in parentheses in equation 1 below to be the same as the term in parentheses in equation 2. The difference that we supposedly missed is that the derivatives of x and y in equation 1 are with respect to time while those in equation 2 are with respect to space.

If this were the case it would invalidate our simple algebraic derivation of equation 3, above, because we get the numerator of equation 3 – V^3 – by noting that the numerator in 2 is equivalent to equation 1 raised to the 3rd power.

Our derivation of the relationship between V and R, shown below, depends on the correctness of the derivation of equation 3.

If our derivation of equation 3 is incorrect then equation 4 would also be incorrect, not only because the numerator – V^3 – was incorrectly derived, but also because the first and second derivatives of x and y that make up the denominator, D, shown below, would be incommensurate with the derivatives that were used to calculate V on the left side of the equation.


So if we made the mistake Martin claims we made – failing to note that the derivatives of x and y in equations 1 and 2 are taken with respect to different dimensions, time and space, respectively – then our derivation of equation 7 relating V to R and D, would be incorrect.

My reply to this is that, to my knowledge, it is not true that the derivatives used in the calculation of V (equation 1) are taken with respect to time while those used in the calculation of R (equation 2) are taken with respect to space. The piece of evidence that supports that conclusion is the fact that the derivatives in equations 1 and 2 are all notated the same way, suggesting that they are all taken with respect to the same dimension and that dimension is time.

Equations 1 and 2 are used by power law researchers to calculate the velocity, V, and curvature, R, variables that are used in their regression analysis to test for conformity of their data to a power law. If these calculations really involved calculating derivatives with respect to different dimensions I can’t help but think that that very important fact would be noted in their papers.

Another reason for thinking that the derivatives of x and y are not different in equations 1 and 2 is because other power law researchers – specifically Pollick & Sapiro G (1997) Constant affine velocity predicts the 1/3 power law of planar motion perception generation. Vision Res, 37:347–353 and Maoz U, Portugaly E, Flash T, Weiss Y (2006) Noise and the 2/3 power law. Adv Neural Inf Proc Syst 18:851–858 – have derived the same relationship between V and R as did Marken & Shaffer (2017), the one shown as equation 7.

If this doesn’t answer the claim that my math is wrong then feel free to claim victory.

Best, Rick

A quick fact check. Again, I claim no mathematical expertise, but in [Martin Taylor 2016.], it appears to me that Martin’s objection is that (1) is with respect to a particular trajectory while (2) is an abstraction over an infinite set of possible trajectories. Specifically, he says that the V in one equation (velocity during a particular trajectory) is not necessarily the same as the V in the other (velocities during possible trajectories), and velocity refers to both time and space in both instances. Yes, (1) refers to V and R during a particular time interval while (2) is timeless, but time vs. space is not the issue that he stated.

The conversion of [Martin Taylor 2016.] to a Discourse post is difficult to read and omits some data, so I have created a PDF from my csgnet/gmail copy. The ‘Bruce’ that Martin mentions is Bruce Abbott.

That was in 2016, eight years ago. Is there someplace else that Martin abandons this objection and replaces it with an objection that the derivatives of one equation are in respect to time and the derivatives of the other are in respect to space?

First I was going to describe another current political discussion maneuver: mis-re-interpretation of what others have said, but perhaps it is enough to say just that my interpretation about what Martin said is quite different. For to support my interpretation I attach the Martin’s article. Nowhere there reads that “the derivatives of x and y in equation 1 are with respect to time while those in equation 2 are with respect to space**.”** (The numbered equations in Rick’s message I am replying to.) Instead there reads that Marken & Schaffer “noted the visual similarity between the expression under the square root sign in (4) [inside the bracket in eq. 1 mentioned above] and the expression inside the bracket of (5) [eq. 2 mentioned above] and treated them as being the same thing. In (4), however, ẋ (dx/dt) and ẏ (dy/dt) are values observed in an experiment and are used to compute the corresponding velocity, whereas in (5) they are arbitrary parameters, corresponding to any velocity whatever (including the observed velocity).”

I do not even understand what would mean the derivatives of x and y were with respect to space. Of course (or at least naturally) they all are time derivatives: the first derivatives (one dot above) mean x and y velocities and second derivatives (two dots) mean x and y accelerations. If the curve of the trajectory is not drawn or otherwise determined beforehand, then it must be calculated from the movement of the studied moving object. The object moves with some velocity which is calculated in every check point from the first derivatives of its x and y transitions. The curve is calculated from the changes of x and y velocities by the x and y accelerations. So the derivatives in both equations are in this relation similar. This is not your error – if I interpret Martin’s article right.

Instead, your error is to think that by replacing the contents of the brackets in eq. 2 with V you had unveiled a mathematical dependence relation or a function between R and V, curvature and velocity. In the first part (“Mathematical background”) of his article Martin proves with dimensional analysis and by testing different velocity values the case – which I think should be self-evident to everyone – that the curve and velocity are mathematically independent and the eq. 2 will give the right and same curvature independently from what is the velocity of the studied object.

So I would say (once again) that your error is not that you replace the contents of the brackets in eq. 2 with V (the measured velocity). You can of course do it. You can replace it with any possible velocity. But the error is to think that it somehow shows or creates a mathematical relationship between R and V. This is an astonishing error, but still bigger error waits in the OVB.

Taylor2018_Article_CommentsOnMarkenAndShafferTheP.pdf (841 KB)

Yes, and his justification for saying that is at the end of that post [[Martin Taylor 2016.]] where he says:

MT: Actually one does not have to use mathematics to see the falsity of Rick’s point 4*. As has been pointed out many times, curvature has no relation to time, whereas velocity does. At the risk of inducing boredom by repetition, curvature has the dimension (1/length), while velocity has a dimension (length/time). Although they may be related in experimental observations, they are not, and could not be, mathematically related.

  • My point 4. “So power law researchers are using linear regression to determine whether there is a linear relationship between variables that are mathematically linearly related, per equation 2.”

Martin made this point (that curvature is related to space while velocity is related to time) in his comment on Marken & Shaffer (2017) in Exp. Brain Res. That paper was the basis for my description of what I believed Martin thought was our mistake.

Actually it was six, not eight, years ago and the objections to my analysis of the power law have been stated in many different ways during that time and I’ve tried to answer them substantively but, apparently, my answers have never satisfied my opponents. And vice versa.

So after six years of debate about the power law, some of which has taken place on CSGNet, some as published papers in journals (Exp. Brain Res.), and many now in IAPCT Discourse, we are clearly no closer to agreement than we were when this all started.

However, I think the fact that this discussion has been so persistent and so heated suggests that there is something very basic at stack here. I was getting at what it is in my earlier post where I quoted Bill on PCT as a paradigm shift. I think the power law is an excellent example of what Bill was talking about – a problem that no longer needs to be solved due to the paradigm shift that is PCT. I think the power law debate shows that some people think of PCT as a paradigm shift, and some don’t.

I think you may have a point there. I’ll re-read Martin’s Mathematical Analysis section more carefully and see where I might have made my mistake.

While I am trying to better understand what you think is my math mistake in the derivation of the equation relating velocity to curvature of movement, I would appreciate it if you could tell me whether Maoz U, Portugaly E, Flash T, Weiss Y (2006) made the same mistake. After all, they derived the same equation we did (equation (4) below) for the relationship between velocity and curvature from the same equations for computing velocity, V, (what they call v(t)) and curvature, R, (what they call kappa(t), which is curvature measured as 1/R) as follows:

If they did make the same mistake just say “yes” and no further explanation is needed. But if they didn’t could you please explain why. I think that would go a long way toward helping me understand my own mistake.


I have now read and re-read the Mathematical Background section of your comment in Exp Brain Res and I still can’t understand how you got to your conclusion:

Any way the equation is examined, R and V are mathematically completely independent of each other, even if experiments suggest that in many situations they are not factually independent. The research question is why mathematical independence does not imply measured independence in those experimental and observational situations.

I understand this conclusion and, if it were correct, it would certainly mean that our mathematical/statistical explanation of the approximately 1/3 power relationship that is typically found between R on V is wrong.

I think your proof of the mathematical independence of V and R has something to do with the dimensional analysis of equation 3 (which is the crucial step in our derivation of the mathematical relationship between V and R). So if you could go over that I’d appreciate it. I’d also appreciate it if you could explain why this analysis does or doesn’t apply to the Moaz et al derivation of the mathematical relationship between R and V. They used the equivalent of your equation 3 to do it, as you can see here:


The right hand side of their equation 3 is the equivalent of your equation 3 except that V^3 is in the denominator because they are measuring curvature as 1/R rather than R. But that shouldn’t make a difference, should it? According to your comment they are still inappropriately deriving a mathematical dependence between V and R (actually, 1/R) in equation 4. So they must have made the same mistake we did, right?

Anyway, looking forward to your clarifications.

Best, Rick

The only point I have ever tried to make is that your original equation 4 is based on experimental data, but equation 5 applies to any curve and any velocity profile at all. That your data fit equation 5 is no surprise.

It may be relevant that Moaz et al. preface their derivation of their (3) and (4) from the Frenet-Serret formulas with the phrase “for any regular planar curve parameterized with t …”. If the distinction is between the specific and the general, both their formulae are universal generalizations.

My equation 4 is based on algebra, not experimental data. And equation 5 certainly does apply to any curve (with no straight segments). The fact that the the data from all curved movements is fit by equation 5 is not supposed to be surprising; it is a devastating proof that, for any curved movement, when log(R) is regressed on log(V), the data will be fit well by something close to a 1/3 power function when the correlation between the included predictor, log (R), and the omitted covariate, log (D), is small. And Moaz et al showed, via simulation, that the correlation between log(R) and log(D) for most randomly generated movements is, indeed, small.

I read your Exp. Brain Res paper criticizing Marken & Shaffer (2017) several times now and I find it completely unconvincing. Your conclusion that R and V are mathematically independent does not follow from the dimensional analysis in equation 3a that presumably shows this to be the case. And both Maoz et al and I have demonstrated via simulation that it is empirically false; R and V are demonstrably mathematically dependent on each other per equation 4 in Maoz et al (2006) and equation 5 in Marken & Shaffer (2017).

I predict with high confidence that you will reject my evaluation of your criticism of Marken & Shaffer (2017). But I am done with this. You have been able to successfully convince most everyone involved in this discussion, including yourself, that I have made a grave mathematical error in my analysis of the power law. You did it with some pretty snazzy mathematical razzle dazzle, but I am not fooled.

I have never claimed you made a mathematical mistake. I have claimed, and still do, that you interpreted the mathematics wrongly, by making the claims that you consistently have done.

Though it evidently appears to you that I have taken sides, Rick, my conscious purpose is that the two positions engage via the same terms with the same meanings for those terms. That’s a pretty basic requisite for discourse not to be ‘dysfunctional’.

I want to make sure that you both are talking about eq. 4 and eq. 5 in Marken & Shaffer (2017) (hereafter referred to as MS). For reference, a PDF of MS is here.

I have not found where anyone has uploaded a PDF of Gribble and Ostry. A PDF of GO is here.

On my inexpert reading, the equations in GO are based upon experimental data. Here is an excerpt (with a column break healed in the image):

Though MS does not explicitly say so, the juxtaposition of the text with two equations I think is clearly intended to say that the restatement in different notation is equivalent.


Parenthetically, may I note that the pairwise equivalence of these is not obvious to the neophyte, but I take it as given and accepted .

MS then says that that MS eq. 4 is algebraically equivalent to MS eq. 3:


If my reading of GO is correct, that GO equations 1 and 2 are based on experimental data, does algebra bequeath that dependency on experimental data from GO onto MS eq. 4? But maybe my reading of GO is wrong?

Viviani & Stucchi (1992) is on the APA site here (I don’t have institutional access). They provided some of the experimental data used by Gribble & Ostry. Interestingly, the abstract mentions an illusion, that movement is perceived to be of uniform velocity despite 200% change.

Maybe not. But that was one of the main complaints from the people who were debating me here. They seem to think that you had discovered what they called a mathematical mistake. But maybe they mean a mathematical interpretation error. I either case, it’s easy to show that I made neither.

And I claim (based on evidence) that your claim (based on what looks to me like highly questionable deductions from math) is consistently wrong. Please state clearly one of the wrong claims that you think I make and I will show you why the claim is not wrong or (more likely) that I didn’t make the claim (you very often attribute claims to me that I am not making).

Best, Rick

Your pointer is to Gribble and Ostry (G&O), not Marken and Shaffer (M&S here). But we are talking about more than eq. 4 and 5 in M&S. We are talking about equations 2 and 3 in M&S, which are equivalent to equations 8 and 9 in G&O, and equations 4 and 5 in Taylor (T here). These are the equations used to compute V and R values from movement data.

We are also talking about equations 4 and 5 in M&S. Equation 4 in M&S is equivalent to equation 3 in T and equation 3 in Moaz et al (MEA here). Equation 5 in M&S is equivalent to equation 6 in T and equation 4 in MEA.

This may be hard to keep track of but it seems to me that Martin and I are on the same page about which equations we are talking about. As Martin said in an earlier port, he is not claiming that my math was wrong (so you can all stop with the math error stuff now); I get the same mathematical results as he does. It’s just my interpretation of the math that he questions, for what I think are very poor reasons.

None of these equations are based on experimental data. These equations are used to compute variable aspects of the data (V and R) and to determine the relationship between these variables (the regression equations, such as V = kR^beta in G&O)

Your reading (actually, your understanding) is wrong. Equation 1 is a regression equation that is the basis for the analysis of the relationship between any pair of variables that are suspected to be related by a power relationship. The value of the exponent, beta, and the intercept constant, k, are the only components of equation 1 that are based on data. Equation 2 is just a way of normalizing the R data. The equation is not based on data; however, the value of alpha is a normalizing factor that is derived from the data. I’ve never seen the equivalent of equation 2 used in other studies of the power law.

Best, Rick

My only objection is that you claim that the general equation 5 in M&S fits only the velocity profile found in an experiment, whereas in fact it would equally well fit the velocity profile of a snail travelling at its constant top speed all the way around the curve. Or of a car driven along the curved road up to a viewpoint where the car stops so the driver can appreciate the view before continuing around the curve.

Thanks for the corrections.

I had both links ready, in fact copied to a temp text file to keep them straight, but still put the same link in the refs to both papers.

Well, then you have nothing to object to because I have never claimed that “equation 5 in M&S fits only the velocity profile found in an experiment”. Perhaps I’ve never said it explicitly but equation 5:
fits the velocity profiles of ALL curved movements such as the ones you mention as well as those of the movements of the planets in their orbits, the path of a fielder running to catch a fly ball, and randomly generated movements, such as those produced by Maoz et al (2016).

That’s an extremely important fact about equation 5 and quite counter intuitive, I think. I’m glad you pointed it out. I was blown away by the fact that it accounts for the velocity profiles of planetary orbits.

Best, Rick


RM: Perhaps I’ve never said it explicitly but equation 5:
fits the velocity profiles of ALL curved movements such as the ones you mention as well as those of the movements of the planets in their orbits, the path of a fielder running to catch a fly ball, and randomly generated movements, such as those produced by Maoz et al (2016).
That’s an extremely important fact about equation 5 and quite counter intuitive, I think. I glad you pointed it out. I was blown away by the fact that it accounts the velocity profiles of planetary orbits.

Perhaps that fact is counter-intuitive to you, but I think it has been self-evident to all others. What is more important is that it does not only fit to every movement as a trajectory but to every single point of every possible trajectory. Let’s think about an arbitrary point p. The studied object can pass p with different velocities even if the curvature remains the same and conversely the object can pass p with the same velocity even if the curvature in p were different.

So (5) does NOT show that there is a mathematical relationship between V and R. Quite the contrary it shows that the there is NO mathematical relationship between V and R and they can vary fully independently.

(5) is equivalent for example to D = V3 x 1/R which becomes a bit clearer if change the radius R to curvature C (C = 1/R) and thus we get:

(6) D = V3 x C

Now, let’s again compare this to an analogical but perhaps intuitively easier case of the size of a rectangular quadrangle. Area is Width times Height, or:

(7) A = W x H

Note that the forms of (6) and (7) are just similar so that there are two measurements (both based on different values of x and y variables) and from these two is calculated a third combined variable which describes something about the current combination of those two measures. As well as (5) or (6) fits to every single point of any trajectory as well (7) fits to every single rectangular quadrangle. And as well as (5) or (6) tells nothing about the relationship between the velocities and curvatures of these points as well (7) tells nothing about the relationship between the heights and width of these rectangular quadrangles.

Adam has stated principally this same argument to you in http://discourse.iapct.org/t/behavioral-illusions-the-basis-of-a-scientific-revolution/15614/118 and you did not answer it. Would you answer it now?

That’s great. But you still don’t seem to understand the implications of this fact – that equation (5) applies to all curved trajectories – for the PCT explanation of the power law as a behavioral illusion and, therefore, (as Powers said in the conclusion his 1978 Psych Review paper) as a problem that no longer needs to be solved. So I think you may be intuiting it the wrong way.

Actually, equation (5) shows that V and R do not vary independently. Variations in V are dependent on the interaction between R and D, the interaction being in the form of the product of R^3 and D^3. The interaction means that there is never an independent (mathematical) effect of R on V. Even if you created a curve (sequence of X, Y values) from just a sequence of R values and looked for the relationship between R and V for that curve you would not be seeing the independent effect of R on V because you would also have been implicitly creating a sequence of D values that also contribute to the observed values of V.

I think I just did. But, as I said earlier, if you don’t find the above a sufficient answer then just feel free to declare victory and go have a beer. I am perfectly content to lose because this discussion has already given me an idea for an essay on why PCT has not been more broadly accepted in the life sciences, even among those who accept it;-)

Best, Rick