Discussion of the speed-curvature Power Law

No you did not answer, not well, not badly, not at all. You just repeat your credo again and again. Perhaps I must say this more clearly: Would you please answer these questions and with arguments, not just that you believe so:

a) Think (or make) a program which draws rectangular quadrangles and between every drawing it randomly changes the length of either the width or the height of the next quadrangle. (Hopefully this is understandable English!) Now, do you think that there is a mathematical dependence relationship between the width and height of this randomly changing quadrangle?

b) If you think that there is a mathematical dependence between the width and the height, do you think so because the area of these differently sized and formed quadrangles can be calculated from their current width and height with the eq. (7) of my previous message: A = W x H ?

c) If you think that there isn’t a mathematical dependence between the width and the height, then why there should be such relationship between the velocity and the curvature of a trajectory just because the D (a cube root of so called affine velocity) can be calculated with the eq. (6) from the velocity and the curvature?

After that I will return to comment your reply.

Minor point. A behavioural illusion is perceived by the controller, whereas the curvature-velocity relation in any specific case has to be determined by a third-party analyst. So the power-law relation is not a behavioural illusion. It is an analytical fact, a side-effect of whatever the moving controller is controlling.

(My guess is the lateral acceleration that is equivalent to a lateral force that might exceed the traction of the moving entity. This force can be sensed by the moving entity, which makes it a candidate for being a controlled perception. But that’s just a guess.)


did you perhaps mean to say that a behavioral illusion is a perception of an observer? (not controller)

I’m betting he meant controller and that he will explain why;-)

Best, Rick

I guess “behavioural illusion” means different things to different people. I must say that I never thought about it before — or at least, not recently. As I had thought about it, Eetu is right, but it was a mistake on my part that led to an unnecessary side-track in the main discussion.

What I wanted to get across is that the power-law relationship frequently observed by analysts in different situations is not a controlled variable, but is a side-effect of whatever perceptual variables are controlled.

Well done. You proved me wrong!

Right, it will be a side effect of control of any perceptual variable(s) as long as curved movements are required to control that variable. It will also be see in any curved movement whether that movement is part of a control loop or not. A mathematical characteristic of the relationship between V and R will always result in close to a 1/3 power relationship when R is regressed on V and D is left out of the analysis.

The power law is Boojam, not a Snark!! I just don’t want you guys to “softly and suddenly vanish away”.

Best, Rick

I have been answering with arguments. There may be no way for me to help you to understand. But it’s fun trying so here we go.

I actually made a little spreadsheet just for you (well, actually, for everyone who wants to understand what’s going on) that does exactly what you ask. Download it and open in Excel.

The first three columns do what you ask, producing 100 rectangles of randomly varying height (H) and width (W). Each row in the columns is a different, randomly generated rectangle The first column is the area (A) and the 2nd and 3rd are H and W , respectively of the rectangle. I’ve made it possible to raise H to any exponent by entering a value into cell I1. The current value is 1 so that A = H * W.

You don’t need the spreadsheet to tell you that; no, there is no mathematical dependence between H and W. However there is a statistical (correlational) relationship between H and W as shown in cell I3, labeled r(H,W). I also show the correlation between the logs of H and W below that. Pressing key F9 generates 100 new rectangles with random values for H and W. This changes the correlation between H and W but, because they are statistically as well as mathematically independent the correlations are typically very small(<.2)

I don’t think there is a mathematical dependences between H and W.

This is why I made the spreadsheet. The equation for area,

A = H *W

is exactly analogous to the equation for velocity,

V = R^1/3*D^1/3

where the implicit exponents of 1 for H and W are analogous to the explicit exponents of 1/3 for R and D.

You ask “why there should be such relationship between the velocity and the curvature of a trajectory just because the D (a cube root of so called affine velocity) can be calculated with the eq. (6) from the velocity and the curvature?” The answer is that the relationship between velocity (V) and curvature (R) is not found because D can be calculated from the velocity and the curvature. It is found because V is mathematically related to R, just as a relationship between H and A will be found because A is mathematically related to H, not because W can be calculated from A and H.

The spreadsheet shows what you would find if you did a regression of just H on A, using the regression equation A = k+ BetaH, which is exactly analogous to what power law researchers are doing when they regress R on V. But since power law researchers are expecting a power relationship between R and V they regress log (R) on log (V) using the regression equation

log (V) = log(k) + Beta*log (R)

so the analogous regression for H on A is to regress log (H) on log (A) using the regression equation

log (A) = log(k) + BetaH*log (H).

The results of this regression are shown in cells H6 - J6 labeled BetaH, k and r2, respectively. BetaH is the estimate of the exponent of H in A = H * W, which is 1. k is the intercept constant and r2 is a measure of the goodness of fit of the regression equation.

As you can see by repeatedly pressing F9, the estimate of BetaH for different sets of 100 randomly generated rectangles varies around the true value, 1.0, occasionally being exactly 1.0 but usually deviating from 1.0 by as much as .15. The r2 for the fit is generally pretty poor, as one would expect, rarely getting above .3 but occasionally getting up around .68.

The results of the regression of log (H) on log (A) is exactly what is going on in power law research when researchers regress log (R) on log (V). They often get an estimate of Beta, the exponent of R in M&R equation (5), that is close to the true, mathematical value,1/3, but sometimes they get values that deviate from 1/3 by quite a bit.

Of course, we get the results we do from the regression of log(H) on log (A) because we have left out the other variable that we know determines the value of A, the width of the rectangle, W. When we include W in the regression we get an exact estimate of the true exponent of H, 1.0, as shown in H8. The regression equation used was

log (A) = log(k) + BetaHlog(H) + BetaW log (W)

and this regression always gets the exact, true value for both BetaH and BetaW, and the r2 is always 1.0.

You can run an even closer analogy to the situation with the power law by changing the setting the exponent of H in cell I1 to something other than 1.0 (that is greater than 0 and less than 1). If, for example, you set it to .33 then the equation for A becomes

A = (H^.33) * W

and you will see that the estimate of the exponent of H when log (H) is regressed on log (A) (in cell H6) is close to .33 rather than to 1, and the estimate of the exponent of H when regression includes both log (H) and log(W) (in cell H8) will give it’s exact true value, .33 in this case.

The degree to which the estimate of the coefficient of H will deviate from its true value (whatever was set in cell I1) can be calculated if one knows the value of the variable omitted from the analysis, which is the width variable, W. The deviation of the exponent observed in the regression of log (H) on log (A) from its true value is exactly equal to the covariance of log (H) with log (W) divided by the variance of log (H). That value is shown under the delta in cell K6. Notice that the value of delta shown in K6 shows exactly how much the value of BetaH in cell H6 deviates from the true value in I1.

I hope I have answered all your questions. But if not or if you have any trouble understanding what is going on in the spreadsheet and how what is being calculated in the spreadsheet is analogous to what is calculated by power law researchers feel free to ask.

Best, Rick

Thank you Rick, it was nice and very kind that you made that spreadsheet. I hope it will help the continuation of the discussion. But I am sorry I cannot continue the discussion immediately, because of the conference of Finnish Educational Research Association here in Oulu in Thursday and Friday. I have there a presentation where I talk about basics of PCT and I now have to concentrate on finalizing it.

Thanks Eetu. No rush. I hope you have a successful and fun time at the conference.