The speed�?curvature power law of movements

[From Rick Marken (2017.11.13.2250)]

···

:

Richard Kennaway (2017.10.13 1448 GMT)–

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

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, and D the magnitude of the cross product of acceleration and velocity.Â This is simply a mathematical fact.Â

RM: Thanks. I think that should end Taylor and Abbott’s “mathematical” criticisms. But, of course, it won’t.Â Â

RK: 2.Â As mentioned above, V = D^(1/3) R^(1/3).Â 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.Â However, no power-law relationships need be present at all.

RM: The paper looks at this in terms of data analysis. Power law researchers use log-log regression analysis to test for the existence of a power law relationship. That is, they regress measures of log (R) on log (V), where R and V are calculated from the x,y values of the trajectory per the equations in Gribble and Ostry. The equation above implies thatÂ log (V) = 1/3 log(D) + 1/3 log (R). So if only log (R) is used in the analysis, the coefficient found by the regression will be close to 1/3 but with some deviation that will be proportional to the covariance between the omitted variable (log (D), affine velocity) and the included variable (log (R)).Â

Â

RK: 3.Â If D is constant, V is proportional to R^(1/3).Â

RM: Yes, and if D is a constant, log(D) is a constant and the covariance between log(D) and log(R) is 0.0.

Â

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

Â RM: Right, and OVB analysis shows why the exponents vary.

RK: 4.Â 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.Â For such an arbitrary course of velocity, the power law in general does not hold.Â

RM: What does it mean for the power law not to hold? I think what has to hold when the movement of the particle is measured in terms of V and R is thatÂ log (V) = 1/3 log(D) + 1/3 log (R). Even if a particle is made to “move along any path with any course of velocity over time” this equation holds. I’ve tried it with data where I moved quickly and slowly through curves and it holds; the deviation of the power coefficient from 1/3 is exactly predicted by OVB analysis.

RK: When the power law does hold, this is a non-trivial fact about the motion.Â

RM: What does it mean for the power law to hold or not hold?Â Are you saying that there can be another relationship between log(V) andÂ log(R) other than log (V) = 1/3 log(D) + 1/3 log (R)?Â

Â

RK: It is not guaranteed mathematically.Â

RM: But log (V) = 1/3 log(D) + 1/3 log (R) is guaranteed mathematically, is it not?Â

RK: When it does hold, the exponent can take any value.Â

RM: Again, what does it mean for the power law not to hold. The only time I think this could be true is when there are segments of the trajectory where the first or second derivative of x or y is 0. And even this is not a problem if you use non-linear regression.Â

RK: 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.

RM: A fact is a fact. Our paper doesn’t say that power law analysis of movement trajectories yields trivial facts. Zago et al imply that this is what we are saying but we are not. What we are showing is that the power law is an illusion in the sense that it seems to say something about how the movement was generated, but it doesn’t.Â The fact that different trajectories are fit by a power law to different degrees is not a trivial fact – indeed, it’s probably of considerable interest to mathematicians and physicists; rather, its a misleading fact, an illusion that leads people who are studying “motor control” down a blind alley.Â

Â

RK: 5.Â 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.Â

Â RM: Again, what does it mean to obey the power law? I’ve found that all the experimental data I have analyzed fits log (V) = 1/3 log(D) + 1/3 log (R) exactly.Â And when only log (R) is included in the regression, the exponent of log (R) deviates from 1/3 precisely by an amount proportional to the covariance between log(D) and log(R).

RK: This is true regardless of the source of the data and the mechanism that produces the data.Â

RM: And that is the point of our paper.Â

Â

RK: The power law is only non-trivial if no selection on D is made.

RM: The power law is never trivial. It just is. And we don’t “select” D; D is always there to be calculated. It’s the variation in the affine velocity of the trajectory. For some trajectories (like a perfect ellipse) affine velocity is constant so the regression of log(R) on log(V) gives a power exponent ofÂ exactly 1/3 and an R2 of 1.0. The point is that this is all a property of the nature of the trajectory itself; it has, as you note, nothing to do with how the trajectory was produced.

RK: If the power law is then observed, this is a nontrivial fact.Â

RM: Again, I don’t know what you mean when you say that the power law is not observed. But whatever you observe about the trajectory is just a fact, neither trivial nor non-trivial.Â

RK: So I disagree with Rick’s characterisation of the power law as omitted variable bias.

RM: We didn’t “characterize” the power law as OVB. We showed that when you regress log(R) on log(V) for movement trajectory data you always find a power coefficient of some value, though the R2 is sometimes quite low; R2 is generally higher when you regress log (C) on log (A).Â

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

RM: The empirical regularity of the power law was discovered with no understanding of the fact that behavior is control. Behavior was seen as a generated output so regularities in this output were taken to be informative about the processes that generated it. But we now know (thanks to Bill Powers) that behavior is a process of control. So we know that curved movements are consistent results produced by necessarily variable means – necessarily variable due to the need to compensate for variable disturbances. So when we are dealing with control, studying characteristics of the consistently produced results of control tells us nothing aboutÂ how these results were produced. But these regular characteristics of controlled results (invariances), such as the power law and tangential velocity profiles, are seductive facts that can lead us down blind alleys.Â

RM: I think that’s enough for now. I’ll just end by saying (again) that the fact that the power lawÂ tells us nothing about how intentionally produced movement trajectories are produced can be deduced simply from understanding that intentional (purposeful) behavior is a control process: variable means used to produce consistent results. Our mathematical/statistical analysis is presented only to show why a relatively consistent power law relationship between curvature and velocity is seen when analyzing curved trajectories.Â

Best regards

Rick

8.Â 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.Â V = K D^0 R^0 where K is constant.Â This is not the sort of power law generally found in these experiments.

9.Â 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.Â R is constant in both situations, while V varies.Â No tuning of the alpha parameter can produce plausibly straightish lines on a log-log plot.

10.Â Something immediately struck me as fishy about the power law, because it predicts that a straight line is traversed at infinite velocity.Â 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.Â 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.Â There are many ways in which we can impose a ceiling on R*; why this one?Â 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).Â In these, if the curvature varies smoothly along the path, it must go to zero at least twice.Â So in some experiments, at least, alpha must play an essential role in bounding the predicted velocity.Â 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.Â On the other hand, G&O do find remarkably high correlation coefficients, on a par with those found in PCT experiments.Â 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.Â The model is V(t) = K(t) (R*)^beta, where R* is as above.Â Notice that K is allowed to vary with time.Â 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).Â 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).Â 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.Â 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).Â I am uncomfortable with the number of degrees of freedom this allows in fitting the model.

11.Â 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).Â 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.Â Other papers use the trajectories of crawling insects, or the tip of a pencil making scribbles.Â 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.Â 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

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

[From Rick Marken (2017.11.17.1240)]

···

Richard Kennaway (2017.11.14 15:33 GMT)–

BA: Thanks for this, Richard.Â 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: 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.Â 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.

RM: Power law researchers use linear regression analysis to determine whether or not a particular movement trajectory follows a power law. The variables used in this regression are either R and V, as measured per the equations in Gribble and Ostry, or C and A, where C is 1/R and A = V/R. The equation that you call a “tautology” – V = 1/3 log D + 1/3 log R.-- is just the mathematical relationship that exists between R and V calculated per the Gribble/Ostry equations. The corresponding equation relating measures of velocity and curvature measured as A and C is log A = 1/3 log D + 2/3 log C.Â

RM: These equations say that if you do a multiple regression analysis with log V as the criterion variable and log D and log R as theÂ predictors the b weights for log D and log R will be 1/3 and the R^2 will be 1.0 and if you do a multiple regression analysis with log A as the criterion variable and log D and log C as theÂ predictors the b weights for log D and log C will be 1/3 and 2/3 respectively and the R^2 will be 1.0Â And this will be true for ANY curved movement trajectory where velocity is measured as V and curvature is measured as RÂ or velocity is measured as A and curvature is measured as C.Â

RM: And, indeed, this is precisely true. I have done multiple regression analyses on many different mathematically defined trajectories (including pendulum trajectories, circular trajectories and random squiggle trajectories) as well as for the 41 different actual and modeled pursuit trajectories of the pursuers of toy helicopters, and the toy helicopter trajectories themselves; I’ve done it for the trajectories of the mouse doing two dimensional tracking or making arbitrary two dimensional movements; I’ve done it for filtered versions of these trajectories (using varying filter bandwidths) and I’ve done it for the raw trajectory data. In all cases, the multiple regression coefficients of both D and R are exactly 1/3 and the coefficients of D and C are exactly 1/3 and 2/3. And the multiple R^2 value in all cases is always exactly 1.0.Â

RM: So your claim, Richard, that the coefficients of R and D in the equationÂ V = 1/3 log D + 1/3 log R can take on any value is wrong, unless you were talking about what could happen if only log D or log R were the predictors in a linear regression analysis. The equation with both log D and log R as predictors describes the mathematical relationship between V, R and D for ALL curvedÂ trajectories. I think this shows pretty convincingly that power law researchers have been studying a phenomenon (the power law) that is a statistical artifact of how V and R (or A and C) are calculated. It also shows that the complex explanations of the observed variations in the coefficient of the “power law” in the paper by Huh and Sejnowsky (2015) can be easily explained by the OVB hypothesis. Since, like all power law researchers, Huh and Sejnowsky use linear regression to determine the power coefficient relating velocity measured as V to curvature measured as R for various movement trajectories, the deviations of the power coefficient from 1/3 (or from -1/3 in the case of Huh and Sejnowsky, since their predictor was actually 1/R ) for all their different trajectories can be perfectly predicted from equations 11 and 12 in Marken and Shaffer(2017):

where I is the included variable (log R or log C), O is the omitted variable (log D) in the linear regression and beta.true is the true coefficient that relates R to V (1/3) or A to C (2/3) or, as in HuhÂ and Sejnowsky, 1/R to V (-1/3).Â

RM; I think this shows why all the “physical” arguments against the analysis in Marken & Shaffer are irrelevant. The power relationship that is found for curved movements has nothing to do with the physics of curved movements; the regression used to find the power law doesn’t care whether the variables included are derivatives with respect to time or space; it doesn’t care about the temporal relationship between the values of the variables that are entered into the regression. The power law is an artifact of the way the velocity and curvature variables are computed and how the power law is found (using linear regression that includes just the curvature measure as a predictor and the velocity measure as a criterion variable, omitting the D variable).Â

RM: But the fact that power law is an example of a behavioral illusion was obvious without any need for mathematics. The movements studied by power law researchers are presumably produced intentionally so they are unquestionably a controlled consequence of neural outputs that are not correlated with the observed movement due to the fact that they are also busy compensating for disturbances to that movement. This alone should tell anyone who understands PCT that the power law relationship between curvature and velocity is an example of a behavioral illusion. The analysis in Marken & Shaffer just explains why a power law with a coefficient close to 1/3 or 2/3 is consistently observed, making the illusion particularly compelling.Â

RM: Since I’m sure this will convince no one on this list that the power law is a statistical artifact that creates the illusion that it says says something important about how curved movement are generated I will confined my further comments on this to my rebuttals in refereed journals.

BestÂ

Rick

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

This is difficult to read, Rick. Velocity and curvature were not invented by Gribble and Ostry or Lacquanitiy or whoever, this is centuries old math. You can’t just use OVB on two independent variables. And then get error-less predictions and claim success. I hope you get to your senses soon.

···

On Fri, Nov 17, 2017 at 9:40 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2017.11.17.1240)]

Richard Kennaway (2017.11.14 15:33 GMT)–

BA: Thanks for this, Richard.Â 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: 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.Â 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.

RM: Power law researchers use linear regression analysis to determine whether or not a particular movement trajectory follows a power law. The variables used in this regression are either R and V, as measured per the equations in Gribble and Ostry, or C and A, where C is 1/R and A = V/R. The equation that you call a “tautology” – V = 1/3 log D + 1/3 log R.-- is just the mathematical relationship that exists between R and V calculated per the Gribble/Ostry equations. The corresponding equation relating measures of velocity and curvature measured as A and C is log A = 1/3 log D + 2/3 log C.Â

RM: These equations say that if you do a multiple regression analysis with log V as the criterion variable and log D and log R as theÂ predictors the b weights for log D and log R will be 1/3 and the R^2 will be 1.0 and if you do a multiple regression analysis with log A as the criterion variable and log D and log C as theÂ predictors the b weights for log D and log C will be 1/3 and 2/3 respectively and the R^2 will be 1.0Â And this will be true for ANY curved movement trajectory where velocity is measured as V and curvature is measured as RÂ or velocity is measured as A and curvature is measured as C.Â

RM: And, indeed, this is precisely true. I have done multiple regression analyses on many different mathematically defined trajectories (including pendulum trajectories, circular trajectories and random squiggle trajectories) as well as for the 41 different actual and modeled pursuit trajectories of the pursuers of toy helicopters, and the toy helicopter trajectories themselves; I’ve done it for the trajectories of the mouse doing two dimensional tracking or making arbitrary two dimensional movements; I’ve done it for filtered versions of these trajectories (using varying filter bandwidths) and I’ve done it for the raw trajectory data. In all cases, the multiple regression coefficients of both D and R are exactly 1/3 and the coefficients of D and C are exactly 1/3 and 2/3. And the multiple R^2 value in all cases is always exactly 1.0.Â

RM: So your claim, Richard, that the coefficients of R and D in the equationÂ V = 1/3 log D + 1/3 log R can take on any value is wrong, unless you were talking about what could happen if only log D or log R were the predictors in a linear regression analysis. The equation with both log D and log R as predictors describes the mathematical relationship between V, R and D for ALL curvedÂ trajectories. I think this shows pretty convincingly that power law researchers have been studying a phenomenon (the power law) that is a statistical artifact of how V and R (or A and C) are calculated. It also shows that the complex explanations of the observed variations in the coefficient of the “power law” in the paper by Huh and Sejnowsky (2015) can be easily explained by the OVB hypothesis. Since, like all power law researchers, Huh and Sejnowsky use linear regression to determine the power coefficient relating velocity measured as V to curvature measured as R for various movement trajectories, the deviations of the power coefficient from 1/3 (or from -1/3 in the case of Huh and Sejnowsky, since their predictor was actually 1/R ) for all their different trajectories can be perfectly predicted from equations 11 and 12 in Marken and Shaffer(2017):

where I is the included variable (log R or log C), O is the omitted variable (log D) in the linear regression and beta.true is the true coefficient that relates R to V (1/3) or A to C (2/3) or, as in HuhÂ and Sejnowsky, 1/R to V (-1/3).Â

RM; I think this shows why all the “physical” arguments against the analysis in Marken & Shaffer are irrelevant. The power relationship that is found for curved movements has nothing to do with the physics of curved movements; the regression used to find the power law doesn’t care whether the variables included are derivatives with respect to time or space; it doesn’t care about the temporal relationship between the values of the variables that are entered into the regression. The power law is an artifact of the way the velocity and curvature variables are computed and how the power law is found (using linear regression that includes just the curvature measure as a predictor and the velocity measure as a criterion variable, omitting the D variable).Â

RM: But the fact that power law is an example of a behavioral illusion was obvious without any need for mathematics. The movements studied by power law researchers are presumably produced intentionally so they are unquestionably a controlled consequence of neural outputs that are not correlated with the observed movement due to the fact that they are also busy compensating for disturbances to that movement. This alone should tell anyone who understands PCT that the power law relationship between curvature and velocity is an example of a behavioral illusion. The analysis in Marken & Shaffer just explains why a power law with a coefficient close to 1/3 or 2/3 is consistently observed, making the illusion particularly compelling.Â

RM: Since I’m sure this will convince no one on this list that the power law is a statistical artifact that creates the illusion that it says says something important about how curved movement are generated I will confined my further comments on this to my rebuttals in refereed journals.

BestÂ

Rick

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

[From Rick Marken (2017.11.18.1100)]

···

On Fri, Nov 17, 2017 at 1:20 PM, Adam Matic adam.matic@gmail.com wrote:

AM: This is difficult to read, Rick. Velocity and curvature were not invented by Gribble and Ostry or Lacquanitiy or whoever, this is centuries old math.

RM: I was not criticizing the Gribble/Ostry equations (or the Viviani/Stuchhi equations from which they are derived). Indeed, I rely on their equations being correct. I think you may be confusing me with Martin Taylor, of all things.Â

AM: You can’t just use OVB on two independent variables.

RM: You keep saying that but you never explain why you think I can’t do this? I actually can “just use” OVB on two predictor variables (they are not independent variables in any sense of that word) because I have done it.Â

AM: And then get error-less predictions and claim success.

RM: Why not? Please read our paper carefully.Â We get error-less predictions because the equation relating R to VÂ

V = R^1/3*D^1/3Â Â Â Â

or, in a form appropriate to a linear analysisÂ Â

log V = 1/3 log R + 1/3 log D

provides the true value of the coefficient (beta.true) for log R (1/3) and the true value of the coefficient (beta.omit) for omitted variable, log D (1/3) – the variable that is omitted when the regression is done using only log R to predict log V. Then the value of the power coefficient that will be observed when log R is regressed on log V (beta’.obs), as in done in power law research, can be calculated from the following

where Cov (I,O)/Var(I) can be calculated from the data.Â The prediction of beta’.obs is perfect because the the formula relating V and R derived from the Gribble/Ostry equations is log V = 1/3 log R + 1/3 log D.Â

AM: I hope you get to your senses soon.

RM: And the same to you, everyone in your lab, all power law researchers and everyone on CSGNet who disagrees with me;-)

Best

Rick

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

On Fri, Nov 17, 2017 at 9:40 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2017.11.17.1240)]

Richard Kennaway (2017.11.14 15:33 GMT)–

BA: Thanks for this, Richard.Â 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: 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.Â 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.

RM: Power law researchers use linear regression analysis to determine whether or not a particular movement trajectory follows a power law. The variables used in this regression are either R and V, as measured per the equations in Gribble and Ostry, or C and A, where C is 1/R and A = V/R. The equation that you call a “tautology” – V = 1/3 log D + 1/3 log R.-- is just the mathematical relationship that exists between R and V calculated per the Gribble/Ostry equations. The corresponding equation relating measures of velocity and curvature measured as A and C is log A = 1/3 log D + 2/3 log C.Â

RM: These equations say that if you do a multiple regression analysis with log V as the criterion variable and log D and log R as theÂ predictors the b weights for log D and log R will be 1/3 and the R^2 will be 1.0 and if you do a multiple regression analysis with log A as the criterion variable and log D and log C as theÂ predictors the b weights for log D and log C will be 1/3 and 2/3 respectively and the R^2 will be 1.0Â And this will be true for ANY curved movement trajectory where velocity is measured as V and curvature is measured as RÂ or velocity is measured as A and curvature is measured as C.Â

RM: And, indeed, this is precisely true. I have done multiple regression analyses on many different mathematically defined trajectories (including pendulum trajectories, circular trajectories and random squiggle trajectories) as well as for the 41 different actual and modeled pursuit trajectories of the pursuers of toy helicopters, and the toy helicopter trajectories themselves; I’ve done it for the trajectories of the mouse doing two dimensional tracking or making arbitrary two dimensional movements; I’ve done it for filtered versions of these trajectories (using varying filter bandwidths) and I’ve done it for the raw trajectory data. In all cases, the multiple regression coefficients of both D and R are exactly 1/3 and the coefficients of D and C are exactly 1/3 and 2/3. And the multiple R^2 value in all cases is always exactly 1.0.Â

RM: So your claim, Richard, that the coefficients of R and D in the equationÂ V = 1/3 log D + 1/3 log R can take on any value is wrong, unless you were talking about what could happen if only log D or log R were the predictors in a linear regression analysis. The equation with both log D and log R as predictors describes the mathematical relationship between V, R and D for ALL curvedÂ trajectories. I think this shows pretty convincingly that power law researchers have been studying a phenomenon (the power law) that is a statistical artifact of how V and R (or A and C) are calculated. It also shows that the complex explanations of the observed variations in the coefficient of the “power law” in the paper by Huh and Sejnowsky (2015) can be easily explained by the OVB hypothesis. Since, like all power law researchers, Huh and Sejnowsky use linear regression to determine the power coefficient relating velocity measured as V to curvature measured as R for various movement trajectories, the deviations of the power coefficient from 1/3 (or from -1/3 in the case of Huh and Sejnowsky, since their predictor was actually 1/R ) for all their different trajectories can be perfectly predicted from equations 11 and 12 in Marken and Shaffer(2017):

where I is the included variable (log R or log C), O is the omitted variable (log D) in the linear regression and beta.true is the true coefficient that relates R to V (1/3) or A to C (2/3) or, as in HuhÂ and Sejnowsky, 1/R to V (-1/3).Â

RM; I think this shows why all the “physical” arguments against the analysis in Marken & Shaffer are irrelevant. The power relationship that is found for curved movements has nothing to do with the physics of curved movements; the regression used to find the power law doesn’t care whether the variables included are derivatives with respect to time or space; it doesn’t care about the temporal relationship between the values of the variables that are entered into the regression. The power law is an artifact of the way the velocity and curvature variables are computed and how the power law is found (using linear regression that includes just the curvature measure as a predictor and the velocity measure as a criterion variable, omitting the D variable).Â

RM: But the fact that power law is an example of a behavioral illusion was obvious without any need for mathematics. The movements studied by power law researchers are presumably produced intentionally so they are unquestionably a controlled consequence of neural outputs that are not correlated with the observed movement due to the fact that they are also busy compensating for disturbances to that movement. This alone should tell anyone who understands PCT that the power law relationship between curvature and velocity is an example of a behavioral illusion. The analysis in Marken & Shaffer just explains why a power law with a coefficient close to 1/3 or 2/3 is consistently observed, making the illusion particularly compelling.Â

RM: Since I’m sure this will convince no one on this list that the power law is a statistical artifact that creates the illusion that it says says something important about how curved movement are generated I will confined my further comments on this to my rebuttals in refereed journals.

BestÂ

Rick

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

[From Rick Marken (2017.11.18.1120)]

···

Â

[From Erling Jorgensen (2017.11.17 1626 EST)]Â

Richard Kennaway (2017.11.14 15:33 GMT)Â

Rick Marken (2017.11.17.1240)Â

RK:Â …[Huh and Sejnowsky (2015)] 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. …

RM:Â I have done multiple regression analyses on many different mathematically defined trajectories (including pendulum trajectories, circular trajectories and random squiggle trajectories) as well as for the 41 different actual and modeled pursuit trajectories of the pursuers of toy helicopters, and the toy helicopter trajectories themselves; I’ve done it for the trajectories of the mouse doing two dimensional tracking or making arbitrary two dimensional movements; I’ve done it for filtered versions of these trajectories (using varying filter bandwidths) and I’ve done it for the raw trajectory data. In all cases, the multiple regression coefficients of both D and R are exactly 1/3 and the coefficients of D and C are exactly 1/3 and 2/3. And the multiple R^2 value in all cases is always exactly 1.0.Â

EJ:Â The fact that all these different situations come out to exactly those predicted coefficients is evidence to me that it probably is dealing with a tautology.Â

RM: This reminds me of an early experience I had when trying to explain PCT to a colleague of mine. Of course, I showed him the results of the basic tracking experiment, how outputs are uncorrelated with inputs and how well the model accounts for variation in the output despite their lack of correlation with the input (so an input-output model wouldn’t work). The correlation between model and output was typically greater than .99, which led my colleague to conclude that the results were trivial. That was back in 1978 or so. Plus Ã§a change,Â plusÂ c’est la mÃªme chose.Â

Â

RM:Â … the regression used to find the power law doesn’t care whether the variables included are derivatives with respect to time or space; it doesn’t care about the temporal relationship between the values of the variables that are entered into the regression.Â

EJ:Â If spatial variables are beingÂ confounded with time derivatives, this suggests to me that regression analysis is the wrong tool for exploring these distinctions, whether its linear or multiple regression.Â

RM: Why does it suggest that to you?Â

BestÂ

Rick

Â

All the best,

Erling

Disclaimer: This message is intended only for the use of the individual or entity to which it is addressed, and may contain information that is privileged, confidential and exempt from disclosure under applicable law. If the reader of this message is not the intended recipient, or the employer or agent responsible for delivering the message to the intended recipient, you are hereby notified that any dissemination, distribution or copying of this communication is strictly prohibited. If you have received this communication in error, please notify the sender immediately by telephone and delete the material from your computer. Thank you for your cooperation.

Â

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

[Martin Taylor 2017.11.18.14.51]

[From Rick Marken (2017.11.18.1100)]

``````This is quite true. There's nothing wrong with the statistical
``````

analysis, only with the interpretation of it. Since the formula for
D is V3 S, where “S” is a function of spatial variables
(as you have now agreed), the Gribble-Ostry equation in log form is

``````       log V = 1/3 log R + log V + 1/3 log S

But S is actually 1/R, as one can see by plugging V=1 into the
``````

Gribble-Ostry formula for R. So

``````       log V = 1/3 log R + log V - 1/3 log R

or

log V = log V.

It's not really surprising that the result comes out exactly when
``````

you don’t expand D and then pretend it is independent of V, so that
you can make the equation look as though it is log V = 1/3 log R +
constant.

``````Martin
``````

···

On Fri, Nov 17, 2017 at 1:20 PM, Adam
wrote:

``````            AM: This is difficult to read, Rick.
``````

Velocity and curvature were not invented by Gribble and
Ostry or Lacquanitiy or whoever, this is centuries old
math.

``````          RM: I was not criticizing the Gribble/Ostry equations
``````

(or the Viviani/Stuchhi equations from which they are
derived). Indeed, I rely on their equations being correct.
I think you may be confusing me with Martin Taylor, of all
things.

``````            AM: You can't just use OVB on two
``````

independent variables.

``````          RM: You keep saying that but you never explain why you
``````

think I can’t do this? I actually can “just use” OVB on
two predictor variables (they are not independent
variables in any sense of that word) because I have done
it.

``````            AM: And then get error-less predictions and
``````

claim success.

``````          RM: Why not? Please read our paper carefully.  We get
``````

error-less predictions because the equation relating R to
V

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

or, in a form appropriate to a linear analysis

log V = 1/3 log R + 1/3 log D

``````          provides the true value of the coefficient (beta.true)
``````

for log R (1/3) and the true value of the coefficient
(beta.omit) for omitted variable, log D (1/3) – the
variable that is omitted when the regression is done using
only log R to predict log V. Then the value of the power
coefficient that will be observed when log R is regressed
on log V (beta’.obs), as in done in power law research,
can be calculated from the following

``````          where Cov (I,O)/Var(I) can be calculated from the
``````

data. The prediction of beta’.obs is perfect because the
the formula relating V and R derived from the
Gribble/Ostry equations is log V = 1/3 log R + 1/3 log D.

[From Rick Marken (2017.11.19.1050)]

Martin Taylor (2017.11.18.14.51)-Â

MT: But S is actually 1/R, as one can see by plugging V=1 into the Gribble-Ostry formula for R. So

Â Â Â Â Â Â log V = 1/3 log R + log V - 1/3 log R
or
Â Â Â Â Â Â log V = log V.

RM: Well, that's kind of going around Hogan's barn to show it. But that is sure true.
Â

MT: It's not really surprising that the result comes out exactly when you don't expand D and then pretend it is independent of V, so that you can make the equation look as though it is log V = 1/3 log R + constant.

RM: No pretending involved.Â The regression comes out exactly because the relationship between the calculated values of V, R and D is exactlyÂ
log V = 1/3 log R + 1/3 log DÂ Â (1)
RM: The log D term is a variable, not a constant. Though D can certainly be constant for some trajectories.Â So we are not saying that V = 1/3 log R + constant. We are saying what is said in equation 1.Â Nevertheless, equation 1 certainly implies that log V = log V.Â Since 1/3 log R + 1/3 log D = log V, we can substitute log V for 1/3 log R + 1/3 log D in equation 1 and getÂ log V = log V.Â
RM: Equatoin 1 shows that with enough df the regression will always find that the best fitting coefficients for predicting log V are 1/3 for both log R and log D with an intercept of 0.0. This prediction will be perfect so R^2 will be 1.0. And this will be true for ALL curved trajectories.Â
RM: If, however, the regression analysis is done using only log R to predict log V, the regression equation is:
log V = a + b*log R
RM: The results of the regression, in terms of the value of b that is found, depends on the covariance between the variables log R and log D for the particular trajectory being analyzed. If this covariance is 0 then b will be found to be exactly 1/3, "confirming" an apparent 1/3 power law relationship between V and R. Trajectories for which the covariance between log R and log D differs from 0 will result in b values that differ from 1/3 (and R^2 values that might be quite low depending on the proportion of variance in V accounted for by the omitted variable log D). The point being that whether or not you find a power law relationship between curvature (R or C) and velocity (V or A) for a particular trajectory depends on the nature of the trajectory itself and not on how that trajectory was generated.Â
Have a nice PCT day.Â
Best
Rick

···

--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery