# The speed‐curvature power law of movements

[From Richard Kennaway (2017.10.13 1448 GMT)]

Thanks, Warren, for prodding me to take a look at this.

I've read Rick's paper, and some of the background papers, but not yet Zago et al's response or most of the rather vituperous discussion on CSGNET. These are just a few preliminary notes, but if I wait until I know everything I want to say on the subject I'll never post anything.

1. Several people in the discussion are in error about the meaning of a mathematical equality, or a law of physics expressed as one. No such equation expresses any sort of causal relationship.

For any smoothly traversed curve in a plane, we have V = D^(1/3) R^(1/3), where V is the speed, R the radius of curvature, and D the magnitude of the cross product of acceleration and velocity. This is simply a mathematical fact. We can equally well arrange the equation as D R = V^3, R = V^3/D, etc. All of these forms mean the same thing. None of them imply any causal relationship among the variables, any more than the expression 2+2 causes 4 or 4 causes 2+2.

One of Newton's laws is F = M A, force equals mass times acceleration. This can with exactly the same meaning be written in any mathematically equivalent way, such as M A = F, M = F/A or A = F/M, or F - M A = 0. This is not, contra [Dag Forssell 2017.11.12 19:45 PST], something that engineers do but really shouldn't. All of these equations mean the same thing. If one is true, all are true. Certainly you can apply a force to a mass to cause an acceleration, and cannot apply a force to an acceleration to produce a mass, but those causal facts have nothing to do with any way of writing F = M A. Consider also the more drastic rearrangements of Newton's laws that are the Lagrangian and Hamiltonian formulations. All of these describe the same physical reality.

In physics, none of the equations mean that the left hand side is caused by the right hand side. If you seek causality in physics, look for it in the properties of its differential equations, that allow the future to be calculated from the present. These properties have nothing to do with which terms happen to be on the left hand side of an equals sign.

[Digression: in econometrics there is something called an SEM (Structural Equational Model). This is a set of equations whose left hand sides are single variables, and which are intended to be read with causal meaning: the things represented by variables appearing on the right hand side are asserted to cause the thing represented by the left hand side. However, some SEM modellers are confused about this and do not clearly distinguish between the causal and non-causal reading of an equation. They might be helped by using different symbols for the different concepts, instead of the equals sign for both. I mention this just in case anyone in this discussion is familiar with SEMs. In the present discussion, none of the equations should be read causally.]

2. As mentioned above, V = D^(1/3) R^(1/3). This mathematically implies that if any two of these variables have a power-law relationship with each other, they each have power-law relationships with the third. However, no power-law relationships need be present at all.

3. If D is constant, V is proportional to R^(1/3). This is the classic power law. Conversely, the classic power law implies that D is constant. However, actual power laws in the literature have widely varying exponents. The 1/3 power, despite frequent citation, does not seem to play any special role.

4. Physically, a particle can be made to move along any path with any course of velocity over time, independently of the curvature of the path. For such an arbitrary course of velocity, the power law in general does not hold. When the power law does hold, this is a non-trivial fact about the motion. It is not guaranteed mathematically. When it does hold, the exponent can take any value. If in some class of situations the exponent is always found to be close to 1/3 (equivalently, that D is approximately constant), this is a further non-trivial fact.

5. If experimental data are obtained for V and R (from which D can be calculated), then any subset of those data selected for having a single value of D must mathematically obey the power law. This is true regardless of the source of the data and the mechanism that produces the data. The power law is only non-trivial if no selection on D is made. If the power law is then observed, this is a nontrivial fact. So I disagree with Rick's characterisation of the power law as omitted variable bias.

6. An empirical regularity can rule out models that do not exhibit the same regularity, but cannot distinguish models that do. Many completely different models might give rise to the same observations.

7. Consider simple harmonic motion in two dimensions. The general path is an ellipse. Let its semi-axes be P and Q. Taking these to be the coordinate axes, the path is x = P cos(wt), y = Q sin(wt) where w is the (abstract) angular velocity (not to be confused with the angular velocity of an object moving like this) and t is the time.

I calculated that for this example, the value of D is constant, in fact equal to A B w^3. Constant D implies the power law. Thus two-dimensional harmonic motion always satisfies the power law, regardless of the mechanism.

Suppose you swing your arm, using only the shoulder joint, so that the hand describes a small ellipse at a moderate velocity, with none of the muscles working very hard, relative to their tension when the arm is held still. With all the relevant variables being small, we may expect a linear analysis of the mechanism of this system to be reasonably accurate. This linearity implies that the hand must be performing simple harmonic motion in two dimensions. But as just seen, two-dimensional SHM necessarily obeys the power law, regardless of mechanism. Therefore an experiment studying this type of movement would offer no insight into the mechanism.

8. A particle moving with constant linear velocity along a path of varying curvature satisfies a power law, although of a degenerate sort: both exponents are zero. V = K D^0 R^0 where K is constant. This is not the sort of power law generally found in these experiments.

9. A pendulum swinging in a single plane (an example used by one of Rick Marken's interlocutors), or a weight bobbing up and down on a spring, do not satisfy the power law. R is constant in both situations, while V varies. No tuning of the alpha parameter can produce plausibly straightish lines on a log-log plot.

10. Something immediately struck me as fishy about the power law, because it predicts that a straight line is traversed at infinite velocity. So I looked up Gribble and Ostry and found that the measure of radius that they use is not the actual radius R, but R* = R/(1 + alpha R), where alpha is a constant parameter. This is close to R when R is much smaller than alpha, and approaches 1/alpha as R tends to infinity.

This strikes me as something of a fudge. There are many ways in which we can impose a ceiling on R*; why this one? One might equally well choose, for example

R* = (2/(pi*alpha))*atan(pi*alpha*R/2)

or

R* = (1/alpha)*tanh(alpha*R)

Both of these have the property of tending to R for small R and to 1/alpha for large R.

I could suppose that the experiments are all in the range of small R, but then there would be no need to introduce alpha, and G&O mention the power-law being observed for lemniscates (figure 8s). In these, if the curvature varies smoothly along the path, it must go to zero at least twice. So in some experiments, at least, alpha must play an essential role in bounding the predicted velocity. I would like to see how closely the power law fits the data when R is comparable to 1/alpha or greater, for each of these three definitions of R*.

It has been quipped that any data looks like a straight line when plotted on log-log paper. This is an exaggeration, but with this extra parameter alpha to fit, I wonder. On the other hand, G&O do find remarkably high correlation coefficients, on a par with those found in PCT experiments. So it does look, prima facie, as if their data non-trivially fit the power law.

G&O do not fit alpha, but use values from Viviani & Stucchi 1992, who do fit alpha to their data. I don't know why G&O didn't fit alpha to their own data -- it is not a global fundamental constant, since they use different values in different experiments.

G&O allow themselves a further amount of freedom in fitting their model. The model is V(t) = K(t) (R*)^beta, where R* is as above. Notice that K is allowed to vary with time. If K is allowed to vary arbitrarily with time, then a suitable choice of K will make this hold regardless of V and R: just take K(t) = (R*)^beta / V(t). Only if K(t) cannot be arbitrarily chosen does the model have the ability to not fit a set of data (an essential property of any proposed model). G&O take K to be constant over each "segment" of the motion, calling it a scaling factor, presumably to account for the fact that the same movement can be done with different overall speeds. They refer to Viviano and Cenzato 1985 for this. But then for at least one example, they use multiple values of K over a single trajectory (the subject scribbles at random on paper). I am uncomfortable with the number of degrees of freedom this allows in fitting the model.

11. If a model is fitted to data, and fits very well, that does not imply that the model describes the mechanism that produced the data.

I am reminded of Zipf's Law, a power law that Zipf originally found in linguistics (the frequency of a word is inversely proportional to its frequency rank). The same power law has been found in diverse situations (e.g. city population inversely proportional to rank order). The last time I looked at that, no-one knew why, in the sense of having a model of an underlying process that both produces Zipf's Law statistics and can be shown to be present everywhere that Zipf's Law is observed.

Rick Marken's paper uses the example of people chasing toy helicopters, something that even our best robots cannot yet do. Other papers use the trajectories of crawling insects, or the tip of a pencil making scribbles. All of these physical mechanisms are very complex, and for a simple law to emerge, there must be some simple commonality among all these systems. I don't think anyone knows what that is though.

It would be interesting to simulate different control systems performing a pursuit task, and see if the same power laws are observed, especially with control systems organised on very different lines.

···

--
Richard Kennaway
School of Computing Sciences, University of East Anglia, Norwich, UK
Cell and Developmental Biology, John Innes Centre, Norwich, UK

[From Bruce Abbott (2017.11.13.1235 EST)]

Richard Kennaway (2017.10.13 1448 GMT)

RK: Thanks, Warren, for prodding me to take a look at this.

RK: I’ve read Rick’s paper, and some of the background papers, but not yet Zago et al’s response or most of the rather vituperous discussion on CSGNET. These are just a few preliminary notes, but if I wait until I know everything I want to say on
the subject I’ll never post anything.

BA: Thanks for this, Richard. If you haven’t already, I recommend that you read Huh and Sejnowsky (2015), because if offers a different approach to the analysis of curvature that yields insight into why the power-law exponent takes the values it does, depending on the shapes that are being drawn.

RK: 1. Several people in the discussion are in error about the meaning of a mathematical equality, or a law of physics expressed as one. No such equation expresses any sort of causal relationship.

BA: Although the equal sign per se does not imply causality, causality may enforce the relationship describe
d by the equal sign, yes? The racecar driver example I used earlier assumes that the driver is attempting to control several variables while moving around the racecourse: maximize the average speed, maintain adhesion of the tires to the road surface, and several others. Taking just the two mentioned, they could achieve both goals by going as fast as possible on the straightaways but slowing for the curves so as not to generate lateral forces that exceed the adhesion of the tires. These two factors produce a relation between V and R. Although R is not a direct cause of V, the changes in V are causally related to the changes in R, although this relationship is mediated by the control-system mechanism. Or have I gone off the rails here?

RK: For any smoothly traversed curve in a plane, we have V = D^(1/3) R^(1/3), where V is the speed, R the radius of curvature, an
d D the magnitude of the cross product of acceleration and velocity. This is simply a mathematical fact. We can equally well arrange the equation as D R = V^3, R = V^3/D, etc. All of these forms mean the same thing. None of them imply any causal relationship among the variables, any more than the expression 2+2 causes 4 or 4 causes 2+2.

RK: One of Newton’s laws is F = M A, force equals mass times acceleration. This can with exactly the same meaning be written in any mathematically equivalent way, such as M A = F, M = F/A or A = F/M, or F - M A = 0. This is not, contra [Dag Forssell 2017.11.12 19:45 PST], something that engineers do but really shouldn’t. All of these equations mean the same thing. If one is true, all are true. Certainly you can apply a force to a mass to cause an acceleration, and cannot apply a force to an acceleration to produ
ce a mass, but those causal facts have nothing to do with any way of writing F = M A. Consider also the more drastic rearrangements of Newton’s laws that are the Lagrangian and Hamiltonian formulations. All of these describe the same physical reality.

RK: In physics, none of the equations mean that the left hand side is caused by the right hand side. If you seek causality in physics, look for it in the properties of its differential equations, that allow the future to be calculated from the present. These properties have nothing to do with which terms happen to be on the left hand side of an equals sign.

BA: Bill Powers emphasized the same point to me many years ago, using the same example of F = M A. In the formula for the radius of curvature, R, one can solve for the tangent
ial velocity V that went into the computation of R, given that you know the value of the cross products. But this does not imply that this V was the cause of R, nor that R was the cause of V.

RK: [Digression: in econometrics there is something called an SEM (Structural Equational Model). This is a set of equations whose left hand sides are single variables, and which are intended to be read with causal meaning: the things represented by variables appearing on the right hand side are asserted to cause the thing represented by the left hand side. However, some SEM modellers are confused about this and do not clearly distinguish between the causal and non-causal reading of an equation. They might be helped by using different symbols for the different concepts, instead of the equals sign for both. I mention this just in case anyone in this discussion is familiar with SEMs. In the present discussion, none of the equations should be read causally.]

RK: 2. As mentioned above, V = D^(1/3) R^(1/3). This mathematically implies that if any two of these variables have a power-law relationship with each other, they each have power-law relationships with the third. However, no power-law relationships need be present at all.

RK: 3. If D is constant, V is proportional to R^(1/3). This is the classic power law. Conversely, the classic power law implies that D is constant. However, actual power laws in the literature have widely varying exponents. The 1/3 power, despite frequent citation, does not seem to play any special role.

RK: 4. Physically, a particle can be made to move along any path with any course of velocity over time, independently of the curvature of the path. For such an arbitrary course of velocity, the power law in general does not hold. When the power law does hold, this is a non-trivial fact about the motion. It is not guaranteed mathematically. When it does hold, the exponent can take any value. If in some class of situations the exponent is always found to be close to 1/3 (equivalently, that D is approximately constant), this is a further non-trivial fact.

BA: Points 2-4 are precisely what Alex, Martin, and I have been asserting.

RK: 5. If experimental data are obtained for V and R (from which D can be calculated), then any subset of those data selected for having a single value of D must mathematically obey the power law. This is true regardless of the source of the data and the mechanism that produces the data. The power law is only non-trivial if no selection on D is made. If the power law is then observed, this is a nontrivial fact. So I disagree with Rick’s characterisation of the power law as omitted variable bias.

RK: 6. An empirical regularity can rule out models that do not exhibit the same regularity, but cannot distinguish models that do. Many completely different models might give rise to the same observations.

BA: This is consistent with the oft stated dictum that different theories may make the same predictions under many circumstances. To distinguish them, one must find conditions in which the predictions made are different, and then do the appropriate empirical tests to determine which prediction is correct. If all the predictions are the same, then at some level the theories are the same even if they appear to be different. Sometimes the predictions differ but at present the theories cannot be distinguished because the differences are within the limits of experimental error.

RK: 7. Consider simple harmonic motion in two dimensions. The general path is an ellipse. Let its semi-axes be P and Q. Taking these to be the coordinate axes, the path is x = P cos(wt), y = Q sin(wt) where w is the (abstract) angular velocity (not to be confused with the angular velocity of an object moving like this) and t is the time.

RK: I calculated that for this example, the value of D is constant, in fact equal to A B w^3. Constant D implies the power law. Thus two-dimensional harmonic motion always satisfies the power law, regardless of the mechanism.

BA: I recall reading this somewhere in the literature, so apparently it is not unknown to researchers in this area.

RK: Suppose you swing your arm, using only the shoulder joint, so that the hand describes a small ellipse at a moderate velocity, with none of the muscles working very hard, relative to their tension when the arm is held still. With all the relevant variabl
es being small, we may expect a linear analysis of the mechanism of this system to be reasonably accurate. This linearity implies that the hand must be performing simple harmonic motion in two dimensions. But as just seen, two-dimensional SHM necessarily obeys the power law, regardless of mechanism. Therefore an experiment studying this type of movement would offer no insight into the mechanism.

RK: 8. A particle moving with constant linear velocity along a path of varying curvature satisfies a power law, although of a degenerate sort: both exponents are zero. V = K D^0 R^0 where K is constant. This is not the sort of power law generally found in these experiments.

RK: 9. A pendulum swinging in
a single plane (an example used by one of Rick Marken’s interlocutors), or a weight bobbing up and down on a spring, do not satisfy the power law. R is constant in both situations, while V varies. No tuning of the alpha parameter can produce plausibly straightish lines on a log-log plot.

RK: 10. Something immediately struck me as fishy about the power law, because it predicts that a straight line is traversed at infinite velocity. So I looked up Gribble and Ostry and found that the measure of radius that they use is not the actual radius R, but R* = R/(1 + alpha R), where alpha is a constant parameter. This is close to R when R is much smaller than alpha, and approaches 1/alpha as R tends to infinity.

BA: Wouldn’t using Curvature C = 1/R instead of R solve this problem? The curvature C of a straight line is zero, which avoids the infinity encountered at the inflection points when using R.

RK: This strikes me as something of a fudge. There are many ways in which we can impose a ceiling on R*; why this one? One might equally well choose, for example

R* = (2/(pialpha))atan(pialphaR/2)

or

R* = (1/alpha)tanh(alphaR)

Both of these have the property of tending to R for small R and to 1/alpha for large R.

I could suppose that the experiments are all in the range of small R, but then there would be no need to introduce alpha, and G&O mention the power-law being observed for lemniscates (figure 8s). In these, if the curvature varies smoothly along the path, it must go to zero at least twice. So in some experiments, at least, alpha must play an essential role in bounding the predicted velocity. I would like to see how closely the power law fits the data when R is comparable to 1/alpha or greater, for each of these three definitions of R*.

It has been quipped that any data looks like a straight line when plotted on log-log paper. This is an exaggeration, but with this extra parameter alpha to fit, I wonder. On the other hand, G&O do find remarkably high correlation coefficients, on a par with those found in PCT experiments. So it does look, prima facie, as if their data non-trivially fit the power law.

G&O do not fit alpha, but use values from Viviani & Stucchi 1992, who do fit alpha to their data. I don’t know why G&O didn’t fit alpha to their own data – it is not a global fundamental constant, since they use different values in different experiments.

G&O allow themselves a further amount of freedom in fitting their model. The model is V(t) = K(t) (R*)^beta, where R* is as above. Notice that K is allowed to vary with time. If K is allowed to vary arbitrarily with time, then a suitable choice of K will make this hold regardless of V and R: just take K(t) = (R*)^beta / V(t). Only if K(t) cannot be arbitrarily chosen does the model have the ability to not fit a set of data (an essential property of an
y proposed model). G&O take K to be constant over each “segment” of the motion, calling it a scaling factor, presumably to account for the fact that the same movement can be done with different overall speeds. They refer to Viviano and Cenzato 1985 for this. But then for at least one example, they use multiple values of K over a single trajectory (the subject scribbles at random on paper). I am uncomfortable with the number of degrees of freedom this allows in fitting the model.

RK: 11. If a model is fitted to data, and fits very well, that does not imply that the model describes the mechanism that produced the data.

BA: But any candidate mechanism must reproduce the relationships f
ound in the data. Failure to do so is grounds for rejecting the candidate, as you noted earlier.

RK: I am reminded of Zipf’s Law, a power law that Zipf originally found in linguistics (the frequency of a word is inversely proportional to its frequency rank). The same power law has been found in diverse situations (e.g. city population inversely proportional to rank order). The last time I looked at that, no-one knew why, in the sense of having a model of an underlying process that both produces Zipf’s Law statistics and can be shown to be present everywhere that Zipf’s Law is observed.

BA: This is the problem that Alex Gomez-Marin brought to the table when he first joined CSGnet: Given the power-law relation observed for movements, what mechan
ism could we identify that would reproduce that relationship? Alex was looking for an application of PCT to this problem, a control-system mechanism or mechanisms that would agree with the observations.

RK: Rick Marken’s paper uses the example of people chasing toy helicopters, something that even our best robots cannot yet do. Other papers use the trajectories of crawling insects, or the tip of a pencil making scribbles. All of these physical mechanisms are very complex, and for a simple law to emerge, there must be some simple commonality among all these systems. I don’t think anyone knows what that is though.

RK:It would be interesting to simulate different control systems performing a pursuit task, and see if the same power laws are observed, especially with control systems organ
ised on very different lines.

BA: Yes, indeed. Meanwhile, I look forward to your more in-depth discussion, once you’ve had a chance to digest more of the relevant literature. I appreciate your efforts.

Bruce

[Richard Kennaway (2017.10.14 1309 GMT)]

[From Rick Marken (2017.11.13.2250)]
RM: The equation above implies that log (V) = 1/3 log(D) + 1/3 log ®. So if only log ® is used in the analysis, the coefficient found by the regression will be close to 1/3

RK: No, it need not be anything like 1/3.

RM: but with some deviation that will be proportional to the covariance between the omitted variable (log (D), affine velocity) and the included variable (log ®).

Any law can be “saved” by calling departures from it “deviations”. But if the deviations are as large as the law, the law is wrong, not approximately right.

···

RK:

There are three separate issues mixed together in this discussion.

The first is that curve-fitting tells you nothing about mechanism, except for ruling out mechanisms that cannot reproduce the data. In general that will leave many possibilities. I think everyone is agreed on that, within and outside CSGNET.

The second is under what circumstances trajectories exhibit a power law relating V and R. By mathematical necessity, they do satisfy log (V) = 1/3 log(D) + 1/3 log ®. (Verifying this experimentally only tells you that your equipment is working and you made
the right calculations. It tells you nothing about the system you are looking at.) This is not a power law relating V and R, because D depends on V. There is no reason to expect any system a priori to behave as log(V) = c + 1/3 log®, or even as log(V)
= c + k log® for constants c and k. If you are going to call log (V) = 1/3 log(D) + 1/3 log ® a power law between V and R, you can call any relationship whatever between variables X and Y a power law, by defining Z = X/Y^q. Then X = Z Y^q.

So, under what circumstances will a power law be observed? Broadly speaking, in two: either the data really exhibit such a power law, or they are tortured to fit one. The claim in all the power law papers is that the data really exhibit such a power law. This
is something the data, before you have collected them, might do, or not do: it is an empirical matter, not a mathematical one.

In the first case, you simply observe V and R for some system, make a log-log plot, and see the data clustered closely about a single straight line that is neither horizontal nor vertical. Some systems behave like that, e.g. two-dimensional simple harmonic
motion. Some systems do not, e.g. an orbiting planet, whose plot of log(V) against log® is an inverted U shape. (Notice that the pericentron and apocentron have the same radius of curvature, but the minimum and maximum orbital velocity respectively.) I’ve
attached plots of log(V) against log®, log(D), and log(D*R) for an example orbit. The last is necessarily a straight line of gradient 3, but the first two are nothing like straight lines. Orbiting planets do not obey a power law of motion.

One way to torture the data is to control for a fixed value of D. This will mathematically guarantee log(V) = c + 1/3 log®, and is therefore a worthless method of investigation. I have not observed anyone doing this, and therefore the fact that it is possible
to do this is not an argument against the validity of these papers.

Another method of torture would be to chop the trajectory up into intervals so short that a straight line can be fitted to the logV-logR plot. Everything is linear to first order! Some segmentation is indeed made by some of the papers on the power law of
movement, but I cannot say at the moment whether it is enough to cast doubt on the reality of the empirical power laws.

The third issue is whether any of the existing work is informed by PCT (my impression is not, but I’ve only read a small part of these papers), and what an investigation on the basis of PCT would do differently. This has nothing to do with the
existence or otherwise of the empirical power laws. If the power laws are empirically real, they require explanation; if not, not.

Richard Kennaway

School of Computing Sciences, University of East Anglia, Norwich, UK

Cell and Developmental Biology, John Innes Centre, Norwich, UK

[From Richard Kennaway 2017.11.14 15:33 GMT]

···

On 13 Nov 2017, at 17:35, Bruce Abbott <bbabbott@frontier.com> wrote:

BA: Thanks for this, Richard. If you haven't already, I recommend that you read Huh and Sejnowsky (2015), because if offers a different approach to the analysis of curvature that yields insight into why the power-law exponent takes the values it does, depending on the shapes that are being drawn.

I've quickly read through it, and the flexibility they allow in their curve-fitting rather reduces my confidence that they have found any real regularity beyond velocity tending to increase with increasing radius of curvature. E.g. I find their figure 3 less than convincing. Their minimum jerk model does rather better than a power law, but of course does not imply that the real mechanism is optimising for minimum jerk. Then they get into a mixture of power laws depending on frequency, which looks perilously close to the tautology that log V = 1/3 log D + 1/3 log R. Even for a simple continuous path consisting of two ellipses (figure 1B) they fit a pair of straight lines rather than just one, and they don't fit the parts of the trajectory connecting the ellipses.

--
Richard Kennaway
School of Computing Sciences, University of East Anglia, Norwich, UK
Cell and Developmental Biology, John Innes Centre, Norwich, UK