[From Rick Marken (2016.07.30.1030)]

RM: Well, you can imagine my surprise this morning when I opened my mail to find a post from Bruce Abbott entitled “The Power Law: Why Rick’s Derivation is Wrong”. And this just as I was preparing a reply to an earlier post from Martin Taylor who was trying to show the same thing. Since I’m kind of busy at the moment I won’t have time to give a detailed rebuttal to these posts. So I’ll just post my spreadsheet demonstration of the correctness my analysis of the power law, which shows that the law tell you nothing about how curved movements are produced. The spreadsheet is attached.

RM: The spreadsheet opens to a Main tab that shows a set of buttons. Each button, when pressed, produces a different temporal pattern of two dimensional movement. To the right of the buttons are two graphs; the top graph shows the movements in spatial coordinates (X,Y space) and bottom graph shows the pattern of movements over time. To the right of those graphs are tables that contain the results of regression analyses of the movement pattern displayed in those graphs. One regression is equivalent to those used to test for the power law relationship between curvature and velocity. It is a regression of log® on log(V) (top table) or of log© on log(A) (third table down). The other regressions include log (D) as a predictor variable in the regression. Each regression table reports the results of the regression in terms of (1) the value of the power law coefficient, b, which is the coefficient of log ® or log© (2) the value of the a coefficient, which is the coefficient of log(D) when that variable is included in the regression, and (3) the R^{2} of the regression.

PowerLawRegression07.30.xlsm (649 KB)

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RM: Finally, the graphs to the right of the regression table are plots of log (V) vs log ® and log (A) vs log©, which are like the graphs found in papers on the power law for movement patterns.

RM: The button labelled “Human” is the movement pattern I produced. The button labelled “Perfect Ellipse” is a mathematically produced elliptical pattern. The buttons labelled “Controlled Ellipse FB= 1” and “Controlled Ellipse FB= .5” are elliptical movement produced by a control system model, one operating in an non-resistive environment where the feedback function (FB) was = 1.0 and the other operating in a resistive environment where the feedback function was = 0.5. The button labelled “Concentric Ellipse w Noise” produces concentric elliptical movements to which some random noise has been added. And finally the button labelled “Random Pattern” produces a different random pattern of movement each time it’s pressed.

RM: There is much to learn from this demo but for now the most informative lesson might come from comparing the results of the regressions for human movement to those for the random movement patterns. For this particular human movement the b value for the regression of log® on log(V) is .34 and the b value for the regression of log© on log(A) is .67, consistent with the findings of power law research. Now look at the same regression coefficients for different random patterns that are produced by repeatedly pressing the “Random Pattern” button. The b value for the regression of log® on log(V) varies for different patterns but ranges from around .27 to .34 and the b value for the regression of log© on log(A) varies for different patterns but ranges from around .64 to .75. So you will often find that the b value for the regression analysis of a random movement pattern produced mathematically will match the b value for the analyses of the random movement pattern produced by the human (.34 and .67)

RM: The fact that the mathematically generated random patterns of movement produce b values that are in the range of those found for studies of movement produced by living systems shows that these coefficients have nothing to do with *how* the movements were produced (whether, for instance, they were produced by a living organism, a control model or a waveform generator. The power coefficient has only to do with the nature of the pattern of movement itself (as per my “wrong” analysis). Also, whatever the pattern of movement, the b value found for the regression of log® on log (V) will be in a band around .33 and the b value found for the regression of log© on log (A) will be in a band around .67.

RM: Also note that the R^{2} of the regression of log® on log (V) is always lower than the R^{2} of the regression of log© on log (A), which has been typically found in studies of the power law. This may be the reason why recent studies of the power law have preferred the use of C and A to R and V as measures of curvature and velocity, respectively.

RM: Finally, notice that the b value for the regression of log® on log(V) is always .33 and the b value for the regression of log© on log(A) is always .67 when log(D) is included in the regression analysis, as predicted by my “wrong” analysis. And in both cases the coefficient of log D is .33, as predicted above and the R^{2} value equals 1.0.

RM: Finally, you can go to the spreadsheet tab labelled “Regression” to see how the variables involved in the regressions – in particular R, V, C and A – are calculated.

RM: Let me know if have any problems using the spreadsheet, unless your problem is that it shows that my analysis is correct. I can’t help you there;-)

Best regards

Rick

–

Richard S. Marken

“The childhood of the human race is far from over. We

have a long way to go before most people will understand that what they do for

others is just as important to their well-being as what they do for

themselves.” – William T. Powers