Behavioral Illusions: The Basis of a Scientific Revolution

Fundamentals
81

Hi Eetu

EP: An object or point can move with a stable speed absolutely independently of its direction of movement AND of the changes of its direction of the movement.

RM: True.

EP: A quick change of the direction of the movement is a turn or angle, a slower change of direction is a curvature. This is a basic mathematical and logical truth which should not be muddled up by such complicated subtleties as affine velocity etc.

RM: The muddling results from the fact that the power law relationship between curvature and velocity is determined by regression analysis on variables that are mathematically related. Power law researchers understand this and that the results of this regression depend on the variance of an “omitted variable”, affine velocity. And they knew this before I did (see references below) They just come to a different conclusion than I do about the implications of that fact.

RM: The different conclusions of the power law researchers come from not understanding that the movements produced by living organisms are controlled results of muscle forces. They still think of movement as results that are produced open loop and that the observed power law relationship that characterizes these movements results from environmental constraints on movement generated open-loop. Why they are interested in PCT is, therefore, beyond me.

EP: As Adam wrote: “the ‘physical’ meaning of the speed-curvature power law - when the curvature is high (‘corners’ of the ellipse) the speed will tend to be lower than when the curvature is low (‘flat parts’ of the ellipse).

RM: Which is true as long as the R^2 for the regression derived power law relationship between the measures of velocity and curvature is close to 1.0. In that case, speed (measured as tangential, V, or angular, A, velocity) is lower when curvature (measured as radial, R, or degree of, C, curvature) is high. That is really happening physically in that case, in the sense that V, A, R and C are functions of physical variables. But there is no physical law that is making this happen. Nor is there any behavioral law making this happen.

EP: And this does not happen always,

RM: RIght! The power law relationship only happens when the affine velocity in the movement under study is uncorrelated with the measure of curvature used in the regression analysis.

EP: so it is not any mathematical dependence but a real physical

RM: The mathematical dependence between measures of speed (V,A) and curvature (R, C) does reflect a real physical dependence since the variables V,A,R and C are measures of movement based on actual physical movements (the time changes in the X and Y coordinates of the moved entity). But those measures – V,A, R and C – are unquestionably mathematically dependent:

image
image

RM: which means that any observed relationship between V and R or A and C depends on how a third variable, D (affine velocity) varies.

EP: and explanation requiring phenomenon.

RM: And I have provided one: The power law is a statistical side-effect of regression analysis with the variable D omitted. The results of the regression depend on what movement the organism is producing on purpose (i.e. controlling for), not on how that movement is produced.

RM: The only explanation of the power law that I have seen from the power law people is that the power law results from the open-loop production of trajectories that follow a power law. This figure from M & S, 2018 shows that this explanation is wrong:
image

Best

Rick

References

1.Pollick F, Sapiro G (1997) Constant affine velocity predicts the 1/3 power law of planar motion perception generation. Vision Res , 37:347–353
2. Maoz U, Portugaly E, Flash T, Weiss Y (2006) Noise and the 2/3 power
law. Adv Neural Inf Proc Syst 18:851–858

82

Rick,

First there seems to be a bad contradiction in your thinking. First you agree that speed is mathematically and logically independent of curvature but then you claim that “the power law relationship between curvature and velocity is determined
by regression analysis on variables that are mathematically related”.

The whole idea of that ‘power law’ is that in some (quite common) conditions the speed of the observed object changes in relation to the gradient of the curve of the movement path. Because speed and curve are mathematically independent
then that empirical dependence must be caused and explained by something else. Whatever this cause and explanation could be it is NOT the mathematical interdependence between speed and curvature because – as you agreed – that interdependence does not exist.
No regression analysis can produce that interdependence ex nihilo – except if the method is erroneous.

Second, I can’t understand how you have read your references (1, 2). Neither of them says that there is a mathematical interdependence between speed and curvature. They both seek and suggest other reasonable explanations for the power law
phenomenon. Reference (2) suggests that the phenomenon could be caused by noise in research methods OR in the movement producing mechanism. (1) suggests that the movement (and/or perception) producing mechanism uses affine geometry (see
https://en.m.wikipedia.org/wiki/Affine_geometry) instead of Euclidean.

Affine velocity is kind of an affine transformation (see
https://en.m.wikipedia.org/wiki/Affine_transformation ) from the usual Euclidean velocity and as such there is of course the mathematical dependence between them. If you know one you can always calculate other. That’s why it may not be very useful to add
affine velocity to the predictors of Euclidean velocity in regression analysis?

83

Hi Eetu

EP: First there seems to be a bad contradiction in your thinking.

RM: Wouldn’t surprise me. I contain multitudes;-)

EP: First you agree that speed is mathematically and logically independent of curvature but then you claim that “the power law relationship between curvature and velocity is determined by regression analysis on variables that are mathematically related”.

RM: Actually, I think what I agreed to was this:

EP: An object or point can move with a stable speed absolutely independently of its direction of movement AND of the changes of its direction of the movement.

RM: I’m not sure that’s true. But true or not, what you are talking about here is physical, not mathematical independence. I think it is true because I can keep the speed of my car at X mph while driving through curves of varying degrees of curvature. But I am sure that, if you measured my instantaneous speed and curvature as V and R while I was doing it, you would find the relationship between those variables to be exactly:

image

EP: The whole idea of that ‘power law’ is that in some (quite common) conditions the speed of the observed object changes in relation to the gradient of the curve of the movement path. Because speed and curve are mathematically independent then that empirical dependence must be caused and explained by something else.

RM: For the sake of argument, let’s agree that what you say is true; that the empirical dependence between speed and curvature must be caused and explained by “something else”. This dependence has been known since Alfred Binet discovered it in 1893 and yet there is still no explanation for it. From my point of view, that (the lack of explanation) is what needs an explanation, not the power law.

EP: Whatever this cause and explanation could be it is NOT the mathematical interdependence between speed and curvature because – as you agreed – that interdependence does not exist. No regression analysis can produce that interdependence ex nihilo – except if the method is erroneous.

RM: The mathematical interdependence between speed and curvature (see equation above) IS the explanation of the power law. Unless someone can come up with a better explanation I will go with that. And since no one has come up with an explanation of the power law in over 100 years – years in which the likes of Einstein and Feynman walked the earth – I think it’s safe to say that I will not be expecting any alternative explanations of the power law (other than “it suggests… blah blah”) in the near future.

EP: Second, I can’t understand how you have read your references (1, 2). Neither of them says that there is a mathematical interdependence between speed and curvature.

RM: Here is the mathematical interdependence between V and R (actually 1/R) in Maoz et al (2006) Noise and the 2/3 power law. Adv Neural Inf Proc Syst 18:851–858

image
v(t) is V in M&S (2017), apha(t) is affine velocity (D in M & S, 2017) and kappa(t) is C = 1/R (hence the negative exponent). It was on this basis that Maoz realized that if affine velocity was constant you would find a perfect -1/3 power law relationship between V and C (equivalent to a 1/3 relationship between V and R). Since they didn’t understand this in terms of multiple regression they didn’t realize that you would also find a perfect -1/3 power law by regressing C on R if the correlation between a non-constant alpha(t) and C was zero.

EP: They both seek and suggest other reasonable explanations for the power law phenomenon.

RM: Their reasonable suggested explanation is basically mathematical, just like mine. They say that the power law will be found if affine velocity is constant; that’s a mathematical fact. If they think this happens only when affine velocity is controlled then that would be a non-mathematical explanation that would have to be tested. But we really don’t need to test it because we know that you also get a perfect power law if alpha(t) and C are uncorrelated, which can happen when affine velocity is not constant (controlled) at all.

EP: Reference (2) suggests that the phenomenon could be caused by noise in research methods OR in the movement producing mechanism.

RM: I knoow. That’s the conclusion of all power law research: “It could be…blah blah”. “The results suggest …blah blah”. It turns out that my little demo in our reply paper – of a power law conforming movement being the result of non-power law conforming movement – rules out both the “noise” and “movement mechanism” explanations, vague though they be.

RM: Maoz et al actually had the true explanation right in front of them. They found that most randomly produced trajectories – produced by a computer algorithm, not people – follow the power law. This implies that the correlation between alpha(t) and C for most trajectories is close to 0. So the question of why the power law is found really boils down to a mathematical question – why is it that most trajectories follow the power law? Now that I think of it, this is a job for our resident PCT mathematician, Richard Kennaway. I doubt that he’s listening in but I’ll send him a message asking if he’s interested in finding the mathematical explanation of the power law.

ET: (1) suggests that the movement (and/or perception) producing mechanism uses affine geometry

RM: This could be tested if they would have provided a description of the “movement producing mechanism”.

Best

Rick

84

Fine, Liability to contradictions and errors is human; it is also understandable to think that one has solved a scientific problem which has bedeviled for over a century. If I had made such a mathematical invention – as non-mathematician
– I would have first described it to mathematician. So it is an excellent idea to ask Richard Kennaway! So I will stop that correspondence for a while and wait for Richard’s verdict.

Best

85

Hi Eetu

First, I must correct myself. I said Maoz et al (2006) Noise and the 2/3 power law. Adv Neural Inf Proc Syst 18:851–858 didn’t view the power law calculation in terms of multiple regression, as I did, and simply assumed that the power coefficient would be 1/3 (or 2/3) if affince velocity was constant. I’m wrong. In fact, Maoz et al did exactly the same analysis I did – what is called “omitted variable analysis” in multiple regression analysis, and came to the same conclusion I did: the degree to which regression of curvature on velocity results in the 1/3 (or 2/3) power law depends on the covariance between curvature and affine velocity. They just came to a different conclusion than I did about what that implies about the the behavior that produced the power law,

EP: So it is an excellent idea to ask Richard Kennaway! So I will stop that correspondence for a while and wait for Richard’s verdict.

RM: OK, will do.

Best

Rick

86

I didn’t notice the thread, mainly because I find Discourse rather awkward for reading simply the most recent messages. All I can see is the most recently posted-to threads, and I wouldn’t have noticed I was tagged here if Richard Marken hadn’t mentioned it in email.

On the emergence of power laws from random data (and data in general):

  1. Whether random trajectories follow the power law depends on the random process that generates the data. Suppose that you generate a random waveform for speed, and an independent random waveform for direction, both expressed as functions not of time, but of distance travelled so far. Then since curvature is the rate of change of direction with respect to distance, there will be no relationship between speed and curvature.

However, when one generates random waveforms, the natural thing to do is to make them functions of time. Random trajectories of speed and direction as functions of time will show a relationship between speed and curvature. For a given rate of change of direction, the curvature will be inversely proportional to the velocity.

If you generate random trajectories for X and Y instead of speed and direction, varying smoothly and independently of each other, then again high velocity will tend to go with low curvature. High velocity implies that both X and Y have high velocities. But because of the smoothness, their velocities will not change fast, and so the path will have low curvature. Low velocity implies that the rates of change of both X and Y are small, and they can easily change direction, giving high curvature.

  1. Simple harmonic motion in two dimensions obeys the 2/3 power law exactly. So a pendulum that can swing in two dimensions obeys it. Anything similar to simple harmonic motion is going to obey the power law approximately. Hence (as for the phenomenon of “gait planarity”) pretty much any smooth oscillation of X and Y of the same period will tend to produce the power law.

  2. I haven’t tried analysing the movements of Archy the robot to see if its foot movements obey the power law. That would be interesting.

  3. The people who study this give themselves an extraordinary number of degrees of freedom. They allow the power law to have any exponent. They add a parameter to avoid the problem of predicting infinite velocity at inflection points. They segment the trajectory into pieces and fit power-law curves to each section separately. They have multiple different forms of the power law and mix them as they please. You could pretty much fit any experimental results that way, especially given the tendency for such a relationship to exist even for random data.

– Richard

87

@RichardK, The most recently posted-to threads are those with the most recent posts. When you go to the Category view (by clicking the logo in the upper left corner), the page has three columns: Category, Topics, and Latest. In the central Topics column for each of the listed categories there is a number (the number of topics) and “n unread” which links you directly to those topics in that category which you have not yet read.

If you want to find posts in which your name has been mentioned, you can search on your name and then sort by “latest” (the default is “relevance”, whatever the system makes of that). There may be a way to have the system notify you if your name is mentioned, but I’m not clear how it works. It may be that they have to use the @username tag as I have done at the top of this post. Did it generate a notification to you?

88

Hi Eetu

EP: Fine, Liability to contradictions and errors is human; it is also understandable to think that one has solved a scientific problem which has bedeviled for over a century. If I had made such a mathematical invention – as non-mathematician– I would have first described it to mathematician. So it is an excellent idea to ask Richard Kennaway! So I will stop that correspondence for a while and wait for Richard’s verdict.

RM: As you can see, Richard’s verdict is in. I look forward to seeing what you think of it.

Best regards

Rick

89

Thank you Richard, this message was very chastening for me.

(I have found it easiest to follow IAPCT Discourse similarly as CSGnet by setting “Enable mailing list mode" in Profile -> Prefeerences -> Emails.)

I was wrong earlier claiming that curvature is the same as change of direction and thus fully independent of speed. Now I see better that curvature is a result of (caused by) both speed and change of direction. (I still believe that direction can change is – at least in principle - independently of the speed.)

This means that if I drive a car with a certain speed and keep the steering wheel turned by a certain angle (=change of direction) that creates a curve with a certain radius or sharpness. If I accelerate the speed and keep the wheel in same position then it will produce a sharper curve. On the other hand If I want to drive a certain curve with a higher speed then I must turn the wheel more. It seems possible that I could create an elliptical trajectory by keeping the wheel stable in certain angel and then just changing the speed suitably.

Of course this does not mean that this mathematical interdependence of speed, direction and curvature would force me to drive an elliptic (or any) curvature always with a speed which is in 2/3 power law (or any other) relation to the radius of the curvature. I can change or not change my direction as I wish and compensate it with changing or not changing the angle of the steering wheel.

So I don’t believe that this interdependence would finally solve the research problems of that so called power law research and if I understood right you are not claiming this either unlike Rick. So I would have liked to hear your opinion about Rick’s claims that the equation A = D1∕3C2∕3 forces every curvilinear movement to obey 2/3 power law and that this could be empirically proved by adding D as a predictor (with C) to a multiple regression analysis to predict A.

90

Hi,

A quick interim report that I started to doubt the correctness of my musings in the last message. If the observed object is moving stepwise like for example a kangaroo jumping or a horse galloping then it might be true but not if the object is moving on wheels like a car. (Unfortunately I have not had a chance to make good test with a car. I am waiting for snow so that the trails of my care were more visible.)

If the object moves jumping it can change its direction only when it touches the ground. Now if the speed of the object accelerates by increasing the length of the jumps and if every jump changes the direction as much then the curvature of the created path might depend on the speed.

But if the object moves on wheels and thus can change its direction in every moment of time then the speed should not similarly affect the curvature.

Anyway the speed does not determine the changes of the direction and thus neither the curvature.

91

I was wrong earlier claiming that curvature is the same as change of direction and thus fully independent of speed.

Curvature is rate of change of direction with distance. It is different from rate of change of direction with time.

(I still believe that direction can change is – at least in principle - independently of the speed.)

Yes, in principle direction and speed can vary independently of each other. But most processes tend to produce a correlation of high speed with low curvature, for various reasons, some of which I described in my previous message, and none of which have to do with a power law relationship being either planned or a controlled variable.

If I accelerate the speed and keep the [steering] wheel in same position then it will produce a sharper curve.

No, you will travel the same curve, but faster.

So I would have liked to hear your opinion about Rick’s claims that the equation A = D1∕3C2∕3 forces every curvilinear movement to obey 2/3 power law and that this could be empirically proved by adding D as a predictor (with C) to a multiple regression analysis to predict A.

For a fixed D, mathematically you have a power law relating A and C. But this is just a mathematical fact that says nothing about the mechanism, nor about whether D does or does not vary. If D is not known to be fixed, there need be no power law between A and C. The “classical” power law of A being proportional to C2/3 is mathematically equivalent to the claim that D remains constant. The various relaxations of the power law, such as different exponents, or fixed offsets, are inconsistent with constant D: they require that D varies.

The relation A = D1∕3C2∕3 is not something that anything tries to maintain or control. It is a mathematical fact that is true of any smooth trajectory whatsoever, howsoever produced. Nothing is trying to establish it any more than the planets are trying to stay in their orbits and might wander if they suffered a lapse of attention.

92

Hi Richard

RM: Thanks for this.

RK: …in principle direction and speed can vary independently of each other. But most processes tend to produce a correlation of high speed with low curvature, for various reasons, some of which I described in my previous message, and none of which have to do with a power law relationship being either planned or a controlled variable.

RM: I’ll just add that one of these processes is a wheel spinning with constant angular velocity. Assume the wheel has radius r. The tangential velocity, V, of a point on the wheel at radius r is greater than that of a point closer to the hub. The direction of movement of a point at radius r also differs from that of a point closer to the hub; the point at radius r is moving though a flatter curve than point near the hub. So the tangential velocity of a point on the wheel is proportional to r – the greater r, the greater V. Indeed, V = r*w where w is angular velocity. Since curvature, C, is inversely related to r, (C = 1/r) we can say that the tangential velocity of any point on the wheel decreases as curvature increases (r decreases): V = (1/C)*w. So points on a wheel are moving more slowly as curvature through which they move increases; speed decreases with an increase in curvature. I think this is is why we see a power law relationship between curvature and speed in most curved movements where speed appears to slow down through curves. Maybe you, Richard, could show why this might be the case.

EP: So I would have liked to hear your opinion about Rick’s claims that the equation A = D1∕3C2∕3 forces every curvilinear movement to obey 2/3 power law and that this could be empirically proved by adding D as a predictor (with C) to a multiple regression analysis to predict A.

RK: For a fixed D, mathematically you have a power law relating A and C. But this is just a mathematical fact that says nothing about the mechanism, nor about whether D does or does not vary.

RM: This is precisely the point we made in M & S 2017. So, Eetu, now that it comes from a real mathematician, can you finally accept the fact that the relationship between speed and curvature, which you called a “mathematical invention” developed by a non-mathematician (me;-), is a mathematical fact that tells you nothing about the mechanism that produced it?

RK: If D is not known to be fixed, there need be no power law between A and C. The “classical” power law of A being proportional to C2/3 is mathematically equivalent to the claim that D remains constant.

RM: Since regression analysis is used to determine the power law you will get a perfect 1/3 or 2/3 power relationship between speed and curvature if the correlation (or covariance) between the curvature predictor (R or C) and the omitted D variable is zero. This is true even if the variance of D is greater than zero (that is, if D is not constant). This is shown in the spreadsheet below:

image
image800×392 19.3 KB

RM: This sheet analyzes the fit of the power law to squiggly movements (like the one in the top left) which are created by 2 low pass periodic waveforms, one of which determines the X and the other the Y position of the movement trajectory over time. For this particular squiggle the regression of curvature on speed results in a perfect power law relationship between these variables; a power coefficient of .33 (1/3) for the relationship between R and V and a power coefficient of .67 (2/3) for the relationship between C and A. The reason for the perfect fit is that the covariance between the measures of curvature and affine velocity (D) is 0 (see upper right, Cov (r,D)). But note that the D variable is not constant; the variance of D ((var(D)) is .27. This probably accounts for the fact that the R^2 is not 1.0; the simple regression didn’t pick up the variance in speed attributable to D. For movements where D is, indeed, constant, a simple regression of curvature on speed will result in an R^2 of 1.0 since there is no variance in speed attributable to D.

RM: This next figure shows what happens when the covariance between the predictor variable (curvature) and D is greater than 0:

image
image800×392 19.5 KB

Now you don’t talk so loud, now you don’t seem so proud … oops, wrong song. Now you don’t see a perfect power law relationship between curvature and speed using regression and leaving out the variable D. Here the covariance is .25 and the power coefficient is.16 (rather than .33) for the relationship between R and V and .84 (rather than .67) for the relationship between C and A. And the R^2 values are quite low due to the rather large variance of D (.5).

RM: So there you have it; whether or not you see something close to a power law relationship between speed and curvature – actually,the closeness of the relationship to the “power law” – depends on the variability of D (affine velocity) in that movement and the covariance between D and the measure of curvature used as a predictor in the regression. In other words, it depends only on the nature of the trajectory produced and tells you nothing at all about how it was produced.

Best regards

Rick

93

@RichardK

For a fixed D, mathematically you have a power law relating A and C. But this is just a mathematical fact that says nothing about the mechanism, nor about whether D does or does not vary. If D is not known to be fixed, there need be no power law between A and C. The “classical” power law of A being proportional to C2/3 is mathematically equivalent to the claim that D remains constant. The various relaxations of the power law, such as different exponents, or fixed offsets, are inconsistent with constant D: they require that D varies.

The relation A = D1∕3C2∕3 is not something that anything tries to maintain or control. It is a mathematical fact that is true of any smooth trajectory whatsoever, howsoever produced. Nothing is trying to establish it any more than the planets are trying to stay in their orbits and might wander if they suffered a lapse of attention.

You did not answer the question. The question is not weather the relation of speed and curvature is a controlled variable.

The question is, do you need to use D to correct the relationship between A(or V) and C, because V and C are mathematically related?

No comment on the other stuff.

94

Suppose each of these is a perceptual variable. “Rate of change of direction with distance” is perceived inertially as lateral acceleration, for example (centrifugal force). The ‘power law’ describes how these two variables co-vary, such that as the first increases, the second decreases, and vice versa. Is there a plausible function over them (such as e.g. their sum) which could be a perceptual variable that is held constant?

95

BN: Is there a plausible function over them (such as e.g. their sum) which could be a perceptual variable that is held constant?

If you look at curvature and speed as instantaneous variables, both aspects of behavior (as they are in the speed-curvature power law), then no, there is no plausible controlled variable as function of those two.

If you look at curvature as a property of the path to trace, like a road you need to drive on, then there could be a function of current speed and the path in front of you that is controlled and maintained constant.

96

Hi Adam

RM: I’ll just quickly reply to this myself.

RK: The relation A = D1∕3C2∕3 is not something that anything tries to maintain or control. It is a mathematical fact that is true of any smooth trajectory whatsoever, howsoever produced.

AM: The question is, do you need to use D to correct the relationship between A(or V) and C, because V and C are mathematically related?

RM: The variable D doesn’t “correct” the relationship between A (or V) and C; it is simply part of the mathematical fact that is the relationship between A (or V) and C (or R) and D. You are using regression analysis to determine whether there is a power law relationship between speed (A or V) and curvature (C or R) of movement . You do this using linear regression of the log of C (or R) on the log of A (or V).

RM: If you leave log (D) out of the regression, then the relationship between speed and curvature that you find using simple regression will depend on the correlation of the criterion variable, speed (A or V) and the predictor variable, curvature (C or R) with D. So you are not seeing the true, mathematical relationship between speed and curvature – which is a perfect 1/3 power relationship between R and V or a perfect 2/3 power relationship between C and A – if you leave the variable D out of the regression analysis.

RM: By the way, I think it is very likely that the perfect mathematical power relationship between speed (A or V) and curvature (C or R) has speed decreasing as curvature increases can be explained by the basic physical fact that the speed (tangential velocity) of a point on a wheel moving at constant angular velocity decreases with its radial distance (r) from the center of the wheel: V = rA

RM: Curvature increases and tangential velocity decreases as r decreases. So the smaller the distance of a point from the center of a wheel moving with constant angular velocity the greater the curvature and the lower the velocity; point on the rim of the wheel will be moving at a high speed through a lower curvature than a point near the center of that wheel. So a point that moves from the rim to the center of a wheel moving at constant velocity will appear to be slowing down as curvature increases when, in fact, it is always moving with the same speed (angular velocity).

Best

Rick

97

RM: So you are not seeing the true, mathematical relationshi
p between speed and curvature – which is a perfect 1/3 power relationship between R and V or a perfect 2/3 power relationship between C and A – if you leave the variable D out of the regression analysis.

So, you are using D to correct the apparent relationship between C and A to get the true relationship?

RM So a point that moves from the rim to the center of a wheel moving at constant velocity will appear to be slowing down as curvature increases when, in fact, it is always moving with the same speed (angular velocity).

No. The tangential speed is really different for points near the center, and higher for points near the edge. Angular velocity is the same.

In the speed-curvature power law, the angular velocity measured is not the the same as angular velocity in a spinning wheel. The angle is not measured from the center of the ellipse (or other shape). It is measured angle of direction of motion, and then the angular velocity is the rate of change of this direction of motion.

And again, the speed is really lower in the parts of higher curvature, this is empirical data without any statistics, completely beside the issue of the power law. I did not even calculate the curvature, nor the angular velocity, just speed.

download

98

This only happens at relatively high speeds of movement. The color bar on the side shows the range of speeds, and the x and y axis show the size of the shape in milimeters.

When the target is moving fast - people cannot follow the speed at each moment, and decide to follow or control something else. The average speed still stays the same as target’s average speed, for example.

This also discards D (affine velocity) as a controlled variable at high speeds, as it is not constant in a constant moving target, and it is ‘more-less’ constant in the subject’s trajectory.

When the target is moving slow, then people don’t have a problem with tracking its speed, as well as position. You don’t get a power law.

99

Hi Adam:

RM: So you are not seeing the true, mathematical relationship between speed and curvature – which is a perfect 1/3 power relationship between R and V or a perfect 2/3 power relationship between C and A – if you leave the variable D out of the regression analysis.

AM: So, you are using D to correct the apparent relationship between C and A to get the true relationship?

RM: That’s kind of an odd way to say it but it’s OK I suppose.

RM: The true relationship between, say, A and C is:

log (A) = 2/3 * log (C ) + 1/3 * log (D)

RM: If you just do a regression of log (C ) on log (A) the regression equation is:

log (A) = k + B*log (C )

RM: where k and B are the regression coefficients found by the regression analysis. So the value of B that you get from this analysis will not be equal to exactly 2/3 (its true mathematical value) unless the covariance between the variable omitted from the analysis, log (D), and log (C ) is precisely 0. The closer that covariance is to 0, the closer the value of B found in the “omitted variable” regression will be to 2/3. This means that the fit of the data to a power law depends completely on the covariance between log(C ) and log(D). I believe there is evidence that the covariance between log(C ) and log(D) depends on the nature of the trajectory that was produced, not on how it was produced.

Best

Rick

100

AM: So, you are using D to correct the apparent relationship between C and A to get the true relationship?

RM: That’s kind of an odd way to say it but it’s OK I suppose.

Good. So, that is my question to Richard ( @RichardK ). Do you need to use D to correct the apparent relationship between C and A, to get the real ?

Why not take the empirical relationship as the true relationship? You measure C, you measure V, and you just put them on a log graph, and you get what you get. That is the true relationship.