# Discussion of the speed-curvature Power Law

The ‘power law’ is an attempted mathematical representation of the observation that motor control of movement slows velocity during a curve in the movement proportionately to how ‘sharp’ the curve is (as measured e.g. by shorter radius). Its relevance to a PCT model of such motor control has been subject of sometimes contentious discussions.

Those discussions are scattered in many places. Prior posts about the power law may be found among these search results.

It is suggested that future posts about it be entered into Discourse as replies to this topic.

Thanks for doing this Bruce. I’ll reply to Eetu’s post here.

I don’t believe that I have ever said that the power law is not an existing (real) phenomenon. It is, indeed, a real phenomenon. I just don’t think it is really a “law” since the observed exponent of the power relationship can vary so much around 1/3. And the observed fit to a power relationship can also vary considerably.

I do think the mathematical relationship between speed and curvature explains why we do (and often don’t) find a power relationship between these variables. But after re-reading what Powers said in his 1978 Psych Review paper about PCT being a paradigm shift that makes many old problems obsolete, I think it’s better to just stop looking for an explanation of the power “law” and start studying movement as a control phenomenon.

The mathematical explanation is really a statistical explanation based on an understanding of how multiple regression analysis works and it does explain why sometimes power relationships between speed and curvature and sometimes none. It’s all explained by omitted variable bias (OVB) analysis that we described in our first paper (and that Moaz et al described in theirs). When I have time tonight I will explain why this is the case and also show – by “showing my work” – why my math describing the power between speed, curvature and affine velocity isn’t wrong.

By the way, I am not at all interested in shutting you out of this conversation of this forum; I don’t think you (or any of my other opponents in this argument) are heretics. I just think you are wrong. Actually, I appreciate your efforts to try to convince me that you are right. It helps me improve my arguments. And, who knows, maybe you will eventually convince me that you are right.

Best, Rick

Here is my reply to the persistent charge that there is a mathematical mistake in our first paper (Marken & Shaffer, 2017) where we used mathematics (algebra) to show why the power relationship between velocity (V) and curvature (R) is found to have a power coefficient that is typically (but not always) close to 1/3.

First, here is Martin Taylor’s explanation of why our math is wrong:

Here are the two equations, (4) and (5), to which he refers:  This is what his equations for V and R look like in my copy of his paper. I presume these equations are the same as the one’s we used in our paper, which are the one’s used by power law researchers and are shown here:  Our mistake, according to Martin, was to take the first derivative of X and Y (X.dot and Y.dot) in his equation (4) and our equation (2) to be the same as those in the numerator of his (5) and our (3). In fact, they are exactly the same. In both cases they are computed from the X and Y values of the movement data, so that

Xdot = X.t - X.t-tau

and

Ydot = Y.t - Y.t-tau.

The second derivatives in the denominator of R are computed from the same data, so:

X2dot = Xdot.t - Xdot.t-tau

and

Ydot.t - Ydot.t-tau.

This is the way power law researchers compute the values of V and R for the regression analysis that is used to find whether the data are fit by a 1/3 power law.

The derivation of the mathematical relationship between V and R simply involves solving for V as a function of R. This is shown in our paper in two simple steps that follow from equations (2) and (3) So I think there can be no question that equation (5) describes the correct mathematical relationship between V and R. Tomorrow I will try to explain the implications of equation (5) for the regression analysis that is used to test for a 1/3 power relationship between V and R.

Best, Rick

I don’t suppose you will ever respond to the core point made in the quote you offer from my original critique, that the values in your equation 4 refer to experimentally observed data, whereas those in equation 5 apply to all and any velocity whatever (including the observed velocity). Despite that, you still assert that ONLY the observed velocity can fit equation 5.

The argument is the same as that if that dog is called Fido, then all dogs must be Fido.

I think there can be no question that equation (5) describes the correct mathematical relationship between V and R. So now I’ll try to explain how power law researchers go about testing whether a curved movement conforms to the power law. This testing is done using linear regression analysis. This is a statistical method that determines how well a straight line fits the relationship between two variables. In power law research, the two variables are V and R, the instantaneous velocity (V) and radial curvature (R) at each point in a curved movement. (nowadays, C, which is just 1/R, is the preferred measure of curvature but I’ll use R for now since that’s the variable used in equation 5 above).

Power law researchers want to test for a power relationship between V and R. That is, they want to test to see how well movement data is fit by the following equation:

Linear regression can be used to test whether this power relationship fits the observed relationship between V and R by using the log of these variables in the analysis . So in power law research, linear regression is used to see how well the data is fit by the following linear equation:

The regression analysis will find the intercept (log⁡( k)) and slope (β) values that fit the observed relationship between log(V) and log (R) for the movement data. If the solution finds β close to 1/3 with a good fit to the data (an R^2> .75) then it is concluded that the movement conforms to the power law.

Here is an example of the data that goes into the analysis:

The x and y values are 8 of the first 10 samples of 1000 samples of a curved movement that lasted about 80 seconds. The first two samples are not shown because it is impossible to calculate the second derivatives of x and y for those samples. So in this case the regression analysis if done on only 98 of the 1000 pairs of x, y values that define the curved movement.

The columns labeled X.dot, Y.dot, X2.dot and Y2.dot are the first and second derivatives of the x and y values of the curve. These values are then used to compute the instantaneous values of V, R and D per equations 2, 3 and 4 in Marken & Shaffer (2017). The final three columns are, of course, the log values of V, R and D.

A power law researcher would then run a regression of log(R) on log(V). The result for this data, which actual human cursor movement data, is β = .37, log(k) = - .14 and R^2 = .85. This movement would be considered power law conforming since .37 is close to .33 and the R^2 is pretty high.

Power law researchers would consider this finding to be the result of the motor processes that generated the movement and/or the physics of the situation that constrained the relationship between the velocity and curvature of the movement to conform to a power law.

But what is shown by equation 5 above is that the 1/3 power relationship between V and R is constrained, not by physics, but, rather, by the mathematical relationship that exists between V and R due to the way V and R are computed. The mathematically constraining variable is D, which turns out to be a measure of affine velocity. This constraint is purely mathematical and results from the degree to which the variables R and D are correlated in any particular movement.

Note that in equation 5, which describes the mathematical relationship between V and R, the power coefficient of R is 1/3. This means that if D were not also a variable in the equation, a regression of log (R) on log (V) would always find β equal to exactly 1/3 and an R^2 or 1.0. But D is in the equation and mathematically it is a covariate that contributes to the variations V along with those associated with variations in R. So when log (R) is regressed on log (V) the observed estimate of the power coefficient of R is biased away from 1/3 to the extent that the contribution of variations in D to variations in V are correlated with variations in R.

The extent to which the correlation between D and R in any particular movement biases the observed coefficient of R away from its true mathematical value, 1/3, can be determined mathematically, by omitted variable bias (OVB) analysis, which is described in Marken & Shaffer (2017).

What this means is that the degree to which on finds a power law (1/3 power exponent) relationship between R and V when doing a linear regression analysis depends completely on mathematical/statistical properties of the movement itself and not on how the movement was produced and/or the physics of the situation in which the movement was produced.

I thought that was exactly what I was responding to.

Here are my equations 4 and 5. Are you saying that V (and possibly D and R) in equation 5 is somehow different than V (and possibly D and R) in equation 4? There are no “values” in either of the equations, by the way, except for the exponents, which are operators that have constant values (3 or 1/3).

The variables R, V and D in both equations (the denominator on the right in equation 4 is equivalent to D) refer to measures of variable aspects of any curved movement (such as variations in the instantaneous velocity of the movement, V, over time).

By the way, X.dot, X2.dot, Y.dot and Y2.dot are also variables that are measures of variable aspects of a curved movement. When you analyze a particular curved movement you compute the instantaneous values of X.dot, X2.dot, Y.dot and Y2.dot from data and use equations 2, 3 and 4 in Marken & Shaffer (2017) to compute the values of V, R and D for that movement.

Hope that helps. I’ll give a more detailed explanation of how these equations are used to compute the variables involved in the power law and analyze individual examples of curved movement in my next installment on statistical tests of the power law.

Best, Rick

No, it does not help at all, since it doesn’t even attempt to touch on the issue at the core of the problem.

Well, I tried. I would really appreciate it if you tell me exactly how I failed to touch on the core of the problem.

This is a typical example of the discourse dysfunctions discussed in another thread (http://discourse.iapct.org/t/discourse-dysfunction/15960)

It seems(!) like Rick is so sure about being right that he cannot understand what Martin means by his criticism and so he (Rick) just repeats his original claims.

I try to help.

In the eq. 4 the dotted xs and ys are measurement values by which the relationship between V and R are calculated in the trajectory which is under study. Those measurement values are specific to just that trajectory and they determine the specific relationship between V and R in just this trajectory. I repeat: relationship between these two variables: V and R in this specific trajectory, and not in some other trajectory. (So it is possible to determine the relationship between V and R if it is done in one specific trajectory and if those x and y (dots) values are available!)

Instead eq. 5 is a general truth about the relationship between these THREE variables: V, R, and D in any possible trajectory, but it says nothing about the relationship between V and R. (I thought you had already understood this in your earlier reply to my criticism.) From this equation you can, however, infer that if D remains stable then also the relationship between V and R remains stable and obeys 1/3 power law, but this is a full tautology.

This situation is analogical to a much simpler case of determining the relationship between the height and width of a changing rectangular. Analogically with the eq. 4 you can calculate the relationship from then x and y components of the heights and widths of differently sized rectangulars. Instead, with the equation: width x height = area you cannot determine that relationship because it fits to all possible rectangulars independently of their form. But if you know that the area of a changing rectangular remains stable, then you know that there must be a linear reversed relation between width and height.

(I hope but quite weakly that this is of any help.)

I’ll attempt a paraphrase of what Eetu, Martin, et al. are saying. If it’s not actually a paraphrase, they’ll tell us, and you can ignore this, Rick, without responding to it.

Richard Fineman famously advised us that “The first principle is that you must not fool yourself and you are the easiest person to fool.” I think these folks are saying that you are fooling yourself. The form of the self-deception is equivocation, taking two different things to be the same.

Everyone agrees that the V and R variables in (4) are instantiated with values of “the movement data” (Rick), “the measurement values” (Eetu) of particular movement paths. For each path, the relationship between V and R is given in the data, and a particular value of D can be calculated.

According to Martin and Eetu and others the variables in (5) are not instantiated with any particular values because (5) is a generalization over all possible values of all three variables, V, R, and D. No ratio or proportion between V and R can be calculated from (5) unless D is held constant. (This is true generally of an equation with three variables, no?)

Holding D constant is kinda like instantiating D with data, except that values of D are not observational data. D is not observed as such, it is calculated from observed values of V and R. In (4), particular values of D are dependent upon and derived from particular observed values of V and R. In the different specific instantiations of V and R in (4), substituting specific measurements of velocity and curvature, D can have a range of values, and the proportional relation of V and R is variable.

When D is made a constant rather than being dependent upon the variable values of V and R, the proportional relation of V and R is constant. Making D a constant with some arbitrary value in (5) kinda looks like presuming the conclusion (petitio principii, ‘begging the question’).

In (4), “the movement data” are data for a particular trajectory and D is derived from V and R; in (5), “the movement data” are all possible values of all three variables V, R, D and no constant proportion between V and R (or any pair of them) can be derived unless one of the three values is given a constant value. “All possible values” is a generalization.

A finite set of observational data can be organized in a table Tn. Holding any one variable to a constant value excerpts a subset of that tabulation and in that subset the ratio of the other two values is constant. The values tabulated in Tn are not all possible values. The variables V, R, D in (5) ‘refer to’ imagined values in an imagined tabulation T of all possible values for them. In imagined table T the only way to get a constant ratio of any two values is by excerpting a subset of T in which the third value has a fixed value.

“Refer to” has two meanings. In (4), the variables R and V ‘refer to’ the data for each trajectory, one at a time, and D ‘refers to’ these data indirectly, by solving equation (4) for those particular values. In (5), the variables R, V, and D ‘refer to’ ranges of observed values for R and V and calculated values for D. These may be imagined to be in a tabulation of all possible values, though in practice only a subset of all possible values are the observed data and the generalization of (5) enables unobserved values for two variables to be extrapolated from any stipulated value of the third.

Probably best, Rick, for you not to respond to this until we find out whether or not they agree that I have paraphrased what they are saying. If I’m way off base, there’s no need for you to waste your time on it.

I think I finally understand what Martin et al are getting at and I can see why they think I am being evasive. So I will start working on an explanation that, I hope, they will find to be responsive. I’ll try to post something by this evening. If not, it will have to wait until Friday.

Best, Rick

Now that I’ve re-read your and Eetu’s posts I think this is good advice. I’m not going to reply to your post until I hear whether Martin and Eetu agree with it.

But while we’re waiting, could you please tell me if you agree with Eetu that my post to which he replied was a good example of dysfunctional discussion?

Best, Rick

I am not the judge. It may appear that I am because I am talking about how to make such a judgement. This is important, because the quality of discussion is a function of collective control that any of us can participate in and, in my opinion, we are each responsible for this.

I think Eetu is irked that <Eetu-perception>your responses have been non sequiturs </Eetu-perception>. Eetu, please correct me if this guess is wrong about your CV and the disturbance.

I believe (subject to correction by Eetu and Martin) that the following illustrates why your replies appear to them to be non sequiturs.

• Martin and Eetu have stated what they perceive as the core problem in your position.
• In your replies you appear to them to restate your position without acknowledging or referring to the core problem.
• Consequently, they perceive your replies as non sequiturs.

You evidently believe conversely that you have ‘answered their objections’.

This has the hallmarks of people talking past each other, i.e. thinking that they are talking about the same thing when they are talking about different things (equivocation à deux?). No dispute can be resolved in which the parties are talking about different things.

To be sure of talking about the same thing, go back a step and ask a different question: what is the core of the problem as they see it? The way to ask that question is to study their statements of the core issue and then paraphrase them, in your own words, and then to ask them if your restatement is a good paraphrase with the same meaning. Note that if you let any whisp of refutation intrude it does not have the same meaning. You must see things from their point of view. (That should sound familiar.) Repeat as necessary until they agree that you understand what they perceive to be the problem.

They have given you their paraphrases of your understanding of the relation between equations (4) and (5), i.e. that ( - ) in (4) and ( - ) in (5) are interchangeable. You have not objected to these paraphrases.

There follow some quotations and links to help you have a clear focus on their formulation of the core issue as they see it, so you can paraphrase it, without at this time adding any “but that’s wrong because”. If you have a good refutation later, it will be more effective for being expressed in their terms. To limit confusion, use the notation and equation numbers that they’re using.

Here’s Martin’s restatement in post #4, with a reference to “the quote you offer from [his] original critique” in post #3.

In your image (screenshot?) of that quotation in post #3, the equations were distorted. Here’s a clear screenshot of the entire relevant passage as I see it in my PDF. I have taken the liberty of healing a page break in this screenshot, and the search term “critical mistake” was highlighted by my PDF reader.

My paraphrase of this, to which no one so far has objected, is that (4) refers to actual experimental measurements (the values of and are “determined by the velocity observed in an experiment”, emphasis added), a finite set, whereas (5) refers to the infinite set of all possible velocities. In Martin’s words, “In (4) ẋ and ẏ are determined by the velocity observed in an experiment, whereas in (5) they are arbitrary parameters, valid for any velocity whatever (including the observed velocity), which is not the same at all.”

In post #9 Eetu has rooted this in the observation that (4) concerns experimental values of two variables V and R for a specific trajectory but (5) concerns the general relationship of V, R, and D for any possible trajectory, that it says nothing about the relationship of V and R unless a stable value is stipulated for D, in which case by tautology the relationship of V and R conforms to the 1/3 power law.

I don’t understand the need to wait until Eetu and I say whether we agree with Bruce, but if it matters to you, Rick, I say I do.

The reasons for waiting were not Rick’s but mine, my acute awareness of my limitations—particularly limitations of my mathematical competence and limitations of my competence to impute motivations.

[I tried to reply from email but discourse cut the message from beginning. It is now corrected here.]

First I want to say that we are now discussing only the first (“preparatory”) part of Rick’s argument. I think it is no use to waste too much energy to this stage because the substantial grave error happens in second part. But the error begins already here.

I’ll attempt a paraphrase of what Eetu, Martin, et al. are saying. If it’s not actually a paraphrase, they’ll tell us, and you can ignore this, Rick, without responding to it.

Thank’s Bruce. for me it seems that you are at least nearly paraphrasing what I tried to say, but there are perhaps some unnecessary complications. Let’s see.

Richard Fineman famously advised us that “The first principle is that you must not fool yourself and you are the easiest person to fool.” I think these folks are saying that you are fooling yourself. The form of the self-deception is equivocation, taking two different things to be the same.

This is at least part of the diagnosis.

Everyone agrees that the V and R variables in (4) are instantiated with values of “the movement data” (Rick), “the measurement values” (Eetu) of particular movement paths. For each path, the relationship between V and R is given in the data, and a particular value of D can be calculated.

Yes.

According to Martin and Eetu and others the variables in (5) are not instantiated with any particular values because (5) is a generalization over all possible values of all three variables, V, R, and D. No ratio or proportion between V and R can be calculated from (5) unless D is held constant. (This is true generally of an equation with three variables, no?)

Yes, this is generally true of all this kind of three variable equations. We can know or infer something about the behavior of one or two variables if we know something about the behavior of the other variables. From this kind of strictly defined relationship between three variables we can infer nothing about the relationship between two of them if we do not know how the third behaves.

Holding D constant is kinda like instantiating D with data, except that values of D are not observational data. D is not observed as such, it is calculated from observed values of V and R. In (4), particular values of D are dependent upon and derived from particular observed values of V and R. In the different specific instantiations of V and R in (4), substituting specific measurements of velocity and curvature, D can have a range of values, and the proportional relation of V and R is variable.

D need not be necessarily held constant because it remains constant quite spontaneously IF the trajectory happens to obey the 1/3 power law (about the relationship between V and R). If the trajectory does not obey (or manifest) just that variant of power relationships between V (velocity) and R (curvature) then D is not constant but varies somehow.

When D is made a constant rather than being dependent upon the variable values of V and R, the proportional relation of V and R is constant. Making D a constant with some arbitrary value in (5) kinda looks like presuming the conclusion (petitio principii, ‘begging the question’).

I cannot say whether or that Rick thinks that D is always constant, because he admits that not all possible trajectories obey 1/3 power law. Rather he seems to think that there is some kind of “mathematical pressure” for all trajectories to come near that power law. But there is no (mathematical) mechanism which could create this kind of pressure. D has no privilege over other possible third elements in those kinds of three-part equations between V, R, and some third variable. Neither there is any metaphysical power or controller which would try to stabilize that D or any other third variable. D stabilizes if there happens to prevail that special power law relationship between V and R. And the question remains, why they behave so sometimes but not always? rsmarken:
In both cases [i.e. in both equation 4 and equation 5] they are computed from the X and Y values of the movement data

In (4), “the movement data” are data for a particular trajectory and D is derived from V and R; in (5), “the movement data” are all possible values of all three variables V, R, D and no constant proportion between V and R (or any pair of them) can be derived unless one of the three values is given a constant value. “All possible values” is a generalization.

Yes.

A finite set of observational data can be organized in a table Tn. Holding any one variable to a constant value excerpts a subset of that tabulation and in that subset the ratio of the other two values is constant. The values tabulated in Tn are not all possible values. The variables V, R, D in (5) ‘refer to’ imagined values in an imagined tabulation T of all possible values for them. In imagined table T the only way to get a constant ratio of any two values is by excerpting a subset of T in which the third value has a fixed value.

For me this idea of observational table here feels like a complication, but it can be helpful for someone else. rsmarken:
The variables R, V and D in both equations (the denominator on the right in equation 4 is equivalent to D) refer to measures of variable aspects of any curved movement (such as variations in the instantaneous velocity of the movement, V, over time).

“Refer to” has two meanings. In (4), the variables R and V ‘refer to’ the data for each trajectory, one at a time, and D ‘refers to’ these data indirectly, by solving equation (4) for those particular values. In (5), the variables R, V, and D ‘refer to’ ranges of observed values for R and V and calculated values for D. These may be imagined to be in a tabulation of all possible values, though in practice only a subset of all possible values are the observed data and the generalization of (5) enables unobserved values for two variables to be extrapolated from any stipulated value of the third.

Yes, they refer in different ways in different stages – there happens quite of a metamorphosis. First the instantaneous values of R and V are determined in (2) and (3) based on the dotted (time derivative) values of x and y:  But then Rick thinks that contents of the brackets in in both equations are equivalent and replaces the bracketed part of (3) with D and gets the equations (4) and (5). However, as Martin said, these bracketed parts are not equivalent. It is perhaps not very easy to see their difference just by looking at equations. But think that (2) is a measurement of that current velocity and (3) is the measurement of that current curvature. Then note that the curvature is not dependent of the velocity of the object that is moving through it. The object can move through just the same curve with different velocities: you can drive the same bend of the road sometimes slower and sometimes faster. So the bracketed part – which is the velocity – is needed in (3) but it can be any velocity, not just that measured velocity of (2). Why seeing the difference between the bracketed parts of these equations is especially difficult to Rick, is that he thinks already beforehand that there must be some mathematical dependence between velocity and curvature – he needs it to show that power law research is pointless tinkering.

So the eq. (5) has lost its connection to the measurements of the original trajectory and it is valid for any velocity and any curvature – and as such tells nothing about the relationship between velocity and curvature, except if we happen to know that D remains constant.

Eetu

Thanks Bruce, good analysis and good advices!

Rick discusses like politicians. If a politician is asked a question which she cannot understand or cannot answer or she knows that the answer would be disadvantageous to her then she does not even try to answer that question but answers some other unasked question which she can answer with a story she thinks is advantageous to her career.

Again, Discourse cut my message, but I corrected it to: http://discourse.iapct.org/t/discussion-of-the-speed-curvature-power-law/15990/16?u=eetup

Of course we all attribute motivations to others, a propensity that probably goes quite far back in evolutionary time. It can be important in collective control (though not the accidental or coincidental collective control that results from confluence or conflict of effects of control outputs in a shared environment).

I think we attribute motives by associating observed behavior with memory of (apparently) controlled consequences of such actions in the past. If they are memories of our own actions (of course we do perceive our own behavioral outputs), then we also remember what CVs we were controlling. Either way, we can imagine what we would control if we were doing what that individual is doing. So there is something of confession in every attribution (the quip “the hand with the pointing finger has three more pointing back”.)

PCT admonishes us to verify these imagined perceptions. Do we know how to do that?

Please note that I did not claim that Rick would have similar motivation as that ideal typical politician. I meant rather that for me it causes a similar frustration when he does not answer the criticism but says that he does not agree because … and then states just the same claim again which was criticized.