[From Bruce Abbott (2018.08.16.0930 EDT)]
[Rick Marken 2018-08-15_18:27:22]
[From Bruce Abbott (2018.08.15.1030 EDT)]
BA: Pollick and Shapiro (1995) performed an analysis of the velocity-based formula for curvature to determine the condition under which one would expect the regression of V on C to follow the 2/3 power law across observations. They found that the 2/3 exponent will hold only if the affine velocity is constant across observations. If affine velocity is not constant, then the exponent will differ from 2/3.
BA: Their analysis was essentially identical to yours (both add affine velocity to the regression equation as a separate factor) but their conclusion was not. They did not conclude that the curvature formula enforced, as you put it, “a true (mathematical) power coefficient� of 2/3. They didn’t, because it doesn’t. Instead, they concluded that the power coefficient of 2/3 will appear across observations only if the affine velocity is constant across observations. The added factor (which you call the omitted variable) is just a measure of by how much the affine velocity varies from constancy across observations. The 2/3 exponent is not a mathematical consequence of how curvature is computed, but rather, a mathematical consequence of a constant affine velocity.
RM: Maoz et al. (2006) also found what we had found and came to the same mistaken conclusion about what it meant as did Pollick and Shapiro (1995). We addressed this in our rebuttal:
Z/M say that the message of the Maoz et al. (2006) study is the opposite of ours, their message being that empirical speed–curvature power laws are not mathemattical/statistical artifacts but, rather “…real and require aa critical investigation of the properties of D to account for compliance or deviation of empirical β values relative to the prototypical 2/3 value found in elliptic drawingsâ€? (p. 12). But this is the message of Moaz et al. only if one assumes that the “trueâ€? value of the power coefficient (− 1/3 for Moaz et al.) is the one that results from the physiological processes that produce the movement and that the variance in the affine velocity variable, α (or, equivalently, D), that results in compliance or deviation from that value is the result of “noiseâ€? factors. This amounts to assuming that one’s theory of the cause of the power law is correct and any deviation of data from the theory is the fault of the data. In fact, when the results of the Moaz et al. study are interpreted correctly we see the message of their and other similar studies (e.g., Pollick, and Sapiro 1997) as being perfectly consistent with ours, which is that **the power law coefficient that is found using a regression analysis that omits the cross-product (or affine velocity) variable depends on characteristics of the trajectory itself and says nothing about the mechanisms that produced those trajectories. ** (Emphasis mine-- RM)
The fact that you just restated this misconception is proof that you either did not read, did not understand, or have chosen to ignore, the analysis I presented that should be the subject of your reply. That analysis is a refutation of the position you take above in the second bolded section.
RM: But ignoring the mathematics for now, I’m still interested in knowing what you think is the correct application of PCT to the power law. My application of PCT to the power law has been pretty thoroughly trashed so you must have some idea of what the correct application of PCT to the power law is. Or does all this trashing of my analysis mean that we have now come to the conclusion that PCT does not apply to the power law?
Ignoring the mathematics? Let’s not try to duck the issue by changing the subject. At this point there are only two appropriate responses to the analysis I laid out: (1) admit that you have made a serious error of mathematical logic, or (2), provide a mathematical proof that invalidates my refutation.
To be clear, the serious error to which I refer is the following: That the formula for curvature implies that for a time series of V,C pairs, “the true (mathematical) power coefficient (beta) relating these variables is 2/3.�
To get you started along this path, the first thing I need from you is evidence that you have read the argument against your conclusion and have understood it well enough to restate it correctly in your own words.
In your reply, insert restatement here:
Bruce