[Rick Marken 2018-03-18_17:15:07]
[Martin Taylor 2018.03.17.13,45]
RM: Thus, we have the power "law", V= R1/3, an observed relationship between the curvature (R) and velocity (V) of curved movements that presumably tells us something about the mechanisms that produced these movements. Â
MT: I think we all agree that when we find what variables are controlled when fly larvae approach food, and when people and other animals make movements that approximate power-law relationships between curvature and velocity, we will understand why the power is often near 1/3 (or 2/3, depending on which relation you look at), and under what conditions it has some other value.
RM: Great. But you won't discover those variables without testing to see what they are.Â
RM: However, both Pollack and Sapiro (1997) and Moaz et al (2006) have shown that there is a mathematical relationship between V and R of the form:Â V=Â R1/3Â *Â alpha1/3, where alpha =Â VR-1/3, the affine velocity.
MT: ... What remains unclear to me is why this might be interesting when we are looking for reasons why the person, animal, or larva might under some conditions appear to control for constant "affine velocity" and under other conditions not control for velocity, affine or otherwise.
RM: You can't look for the conditions under which organisms control for constant affine velocity until you find at least one condition under which they do. This involves testing for the controlled variable.
RM: So it's true that when alpha is constant, V will be proportional to R1/3. But this just proves the point we made in our paper: the empirical power law is a mathematical artifact.
MT: I do wish you would respond to at least one of the criticisms I made in my published comment, or for that matter, in over a year of CSGnet interaction, that show your second sentence to be a non-sequitur.
RM: I did respond to your criticisms in my rebuttal to your paper. If you think they are not correct, feel free to respond to them and explain why they are not.
RM: The "law" is found to hold to the extent that the affine velocity throughout a curved movement trajectory is constant. If the trajectory is one where affine velocity is constant throughout the movement then V will be found to be proportional to R1/3.
MT: As you said, "affine velocity" is the D in your paper. You cite Pollack and Shapiro, but not my simpler way of doing the maths they do to show that D is easily decomposable into V time a pure function of geometric variables, and is therefore of no value in an expression that purports to explain or describe the value of V.
RM: But that's not what they showed. They showed the V is a function of both C and D, as we did. And that when D is constant, the relationship between V and C will follow a 1/3 power relationship. They proved it mathematically and tested it using regression of C on V and we and Maoz et al proved it statistically and tested it using OVB analysis.Â
MT: It is not clear why you cite Pollack and Shapiro approvingly, but fail to approve my criticism, instead refusing to mention it in your rebuttal, instead substituting an entirely different criticism that you pulled out of thin air. If you believe Pollack and Shapiro, you have to believe me, too. You can't have one without the other.
RM: Actually, I can. I cite Pollack and Shapiro because they and Maoz et al discovered the same thing we discovered in our paper; that when you regress C on V omitting the variable D from the analysis you will find that the relationship between C and V is fit by a power law with a coefficient that approximates 1/3 to an extent determined by the correlation between C and the omitted variable D. I don't believe this is what you showed in your criticism of our paper.
RM: If the trajectory is one where affine velocity varies along with curvature  then V will be found to be proportional to R raised to an exponent other than 1/3.
MT: Suppose "affine velocity" varies as radius of curvature minus 6 cm, what power will be found when the radius of curvature varies from 10 cm to 2 cm in different parts of, say, an ellipse?
 RM: I have no idea. To do it exactly you would have to determine, either mathematically or computationally, how these variations affect the correlation between D (affine velocity) and C. I could work it out with my spreadsheet models if you are really interested.
RM: So the extent to which the observed relationship between V and R conforms to the power law depends on what movement trajectory was produced, not on how it was produced.Â
MT: There's no "So" there. It's a simple fact. If the curve tracer chooses to stop a while or reverse direction during the track, the observed relation between V and R will not conform well to the power law, whatever exponent you choose.
RM: Reversals and stops result in different trajectories that will follow the 1/3 or 2/3 power law to the extent that affine velocity is not correlated with curvature throughout the trajectory. The "hows" I'm referring to are the neural and muscular mechanisms that result in the trajectory. That is what the power law is supposed to tell us about. Our paper shows that it's those "hows" that the power law can't tell us about.
RM: Affine velocity may be one of the perceptual variables people control when they produce curved movements.Â
MT: Maybe it is, but what evidence can you adduce to suggest that it is other than a simple possibility?
RM: I explained why affine velocity was a possible controlled variable if a person is trying to control for making a curved movement with constant speed. But the test for the controlled variable always starts with a hypothesis about the controlled variable that is a "simple possibility". Then you iteratively work toward getting a better definition of the controlled variable(s). That's what PCT based research is about.Â
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MT: After all, other experimenters have found that minimum jerk (rate of change of acceleration) is usually close to fitting the data, which makes perception of jerk an equally plausible possibility for a variable people control, with a reference value of zero. Personally, i find the concept of "zero" to be less complex than that of "affine velocity", so the Ockham's Razor vote would go to that, if they were the only two variables in question.
RM: Those are output-generation models; minimum jerk is used as a parameter of the output waveform that generates the trajectory open loop. Such models would fail if the trajectory were produced against known disturbances. The model of trajectory production must be a control model (as Figure 1 in our rebuttal paper shows) and if there is a way to make minimum jerk a perceptual variable, it could be a candidate for a controlled variable.Â
MT: And how does it explain the wide range of exponents actually observed in experiments?
 RM: It is explained by the wide range of correlations between affine velocity and curvature for different movement trajectories. That's what our OVB analysis shows;
MT: Again, it would be really nice if you would address the published criticism of that statement, instead of refuting a criticism never made.
RM: I believe we did refute it in the section entitled "Tangled up in statistics" (at least you ya gotta love the section titles).Â
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MT: Even if your "refutation" had addressed the actual criticism, you would still have to answer why and under what conditions the correlations change so consistently.
RM: I don't know if we "had to" do that. But I agree that it would be interesting to do. I think the paper on movement in water versus air is what you're talking about. Maybe I will do what you suggest with that data.Â
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MT: To the CSGnet readership more generally, I apologise for this message, since I said I was going to avoid putting my hands on the curvature-velocity tar-baby again. But now I suppose I have got stuck again. I will try to wash my hands of it, but with what success, I do not know.
RM: Well I think it's useful. But do what makes you feel best. I hope you can come to the IAPCT meeting so we can discuss it there. Should make good theater;-)
Best
Rick
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--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery
