# The Power Law: Why Rick's Derivation is Right

[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 R2 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 R2 of the regression of log® on log (V) is always lower than the R2 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 R2 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

[From Bruce Abbott (2016.07.30.1700 EDT)]

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.

BA: How does the spreadsheet generate the â€œperfect ellipseâ€??Â In particular, I want to know why the velocity around the ellipse varies with position.Â Iâ€™d like to see a version in which the tangential velocity is constant and can be set by the user to different values.

Bruce

[From Rick Marken (2016.07.30.1450)]

Â Bruce Abbott (2016.07.30.1700 EDT)–

BA: How does the spreadsheet generate the âperfect ellipseâ??Â In particular, I want to know why the velocity around the ellipse varies with position.Â Iâd like to see a version in which the tangential velocity is constant and can be set by the user to different values.

RM: The ellipse was created in the same way as on an oscilloscope; by having motion in the X and Y dimensions be sine and cosine waves of equal frequency and unequal amplitude. Note that the result of the regression of log(R) on log (V) gives the same result as Gribble and Ostry get when they analyze a pure ellipse like this; beta coefficient of .33 and R^2 = 1.0.Â

RM: If you would like to see a versionÂ of a movement pattern that has a tangential velocity that is constant and can be set by the user to different values please feel free to create it yourself and send it to me and I’ll test it in the spreadsheet. I don’t know why you would want to do that but go ahead.

RM: The spreadsheet shows that virtually any movement pattern will result in a beta coefficient for log(R) predicting log (V) of approximately .33 and a beta coefficient for log(C) predicting log (A) of approximately .67. It also shows that every movement pattern can be perfectly accounted for (R^2 = 1.0) using log(R) and log (D) as predictors of log (V) or log (C) and log (D) as predictors of log (A); and the coefficient of log (R) will always be .33 and the coefficient for log (C) will always be .67, as predicted by my “wrong” equations:Â

log (V) = .33log(D) + .33log(R)

log (A) = .33log(D) + .67log(C)

RM: But it shouldn’t have been necessary for me to develop the spreadsheet demonstration to convince you and Martin that studies of the power law are making a grave mistake. The problem with these studies is easily detected by anyone who understands the nature of control as seen in the behavior of living organisms. The problem should certainly have been obvious to you two, who have been ostensible student of PCT for many years. And yet, instead of seeing the problem and helping me clarify it to people who don’t understand PCT, you guys come up with reason after reason why the PCT explanation of the results of the power law research can’t possibly be correct. What gives? Â Â

BestÂ

Rick

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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

[From Bruce Abbott (2016.07.31.1130 EDT)]

[From Rick Marken (2016.07.30.1450)]

> Bruce Abbott (2016.07.30.1700 EDT)--

> BA: How does the spreadsheet generate the “perfect ellipse�?? In particular, I want to know why the velocity around the ellipse varies with position. I’d like to see a version in which the tangential velocity is constant and can be set by the user to different values.

RM: The ellipse was created in the same way as on an oscilloscope; by having motion in the X and Y dimensions be sine and cosine waves of equal frequency and unequal amplitude. Note that the result of the regression of log(R) on log (V) gives the same result as Gribble and Ostry get when they analyze a pure ellipse like this; beta coefficient of .33 and R^2 = 1.0.

RM: If you would like to see a version of a movement pattern that has a tangential velocity that is constant and can be set by the user to different values please feel free to create it yourself and send it to me and I'll test it in the spreadsheet. I don't know why you would want to do that but go ahead

If you don't know why, then you aren't thinking.

By the way, your spreadsheet has errors. X2dot and Y2dot each need to be divided by dt. How did you not notice that the value of R was ridiculous?

Bruce

[From Rick Marken (2016.07.31.1750)]

···

Bruce Abbott (2016.07.31.1130 EDT)

RM: If you would like to see a version of a movement pattern that has a tangential velocity that is constant and can be set by the user to different values please feel free to create it yourself and send it to me and I’ll test it in the spreadsheet. I don’t know why you would want to do that but go ahead

BA: If you don’t know why, then you aren’t thinking.

RM: So you’re not going to provide it? It would sure be nice. I actually asked because I have no idea how to generate the movement pattern you want. SO it wasn’t so much that I wasn’t thinking; it was just that I was not capable of thinking at that level.

BA: By the way, your spreadsheet has errors. X2dot and Y2dot each need to be divided by dt. How did you not notice that the value of R was ridiculous?

RM: Good catch on seeing that I didn’t divide X2dot and Y2dot dt. I changed it but it changed nothing in the analysis since you are just dividing a variable by a constant.

RM: And what value of R (I presume you mean R^2) is ridiculous? All the values of R^2 (and, thus of R) looked perfectly reasonable to me.

Best

Rick

Bruce

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

[From Bruce Abbott (2016.08.01.1030 EDT)]

Rick Marken (2016.07.31.1750) –

Bruce Abbott (2016.07.31.1130 EDT)

RM: If you would like to see a version of a movement pattern that has a tangential velocity that is constant and can be set by the user to different values please feel free to create it yourself and send it to me and I’ll test it in the spreadsheet. I don’t know why you would want to do that but go ahead

BA: If you don’t know why, then you aren’t thinking.

RM: So you’re not going to provide it? It would sure be nice. I actually asked because I have no idea how to generate the movement pattern you want. SO it wasn’t so much that I wasn’t thinking; it was just that I was not capable of thinking at that level.

BA: When I said you weren’t thinking, I meant that you weren’t thinking about why I would want to see movement at a constant tangential rate along the ellipse. If you read and understood the explanation I gave for what the formula for computing R gives us (or Martin’s further analysis of it), you would understand the significance of running your analysis of motion around an ellipse with a constant tangential velocity point. But you say you don’t know why I would want to do that, so it is evident to me that you are not even trying to understand the explanations that Alex, Martin, and I have offered you in hopes that you will think about them carefully and finally understand why the V you compute by solving for V in the formula for R is not velocity that an object can have while traveling around the ellipse.

BA: As for computing X and Y at a constant tangential velocity around the ellipse, I don’t know how to do that either, although I have a good idea (I think) of how to go about it by solving numerically. But right now I’m swamped with other work (I shouldn’t have been devoting as much time to this exchange as I have been!) so unless Alex is willing to provide the X,Y pairs under the constant velocity restriction that can be entered into your spreadsheet analysis (preferable for at least two different constant speeds), that solution is going to have to wait.

BA: What will the results tell us? If Alex, Martin, and I are right, then V (the tangential velocity V) will be constant at each speed but R will vary depending on the position of the point on the ellipse at each dt. Because V is constant, the power law will be violated.

BA: By the way, your spreadsheet has errors. X2dot and Y2dot each need to be divided by dt. How did you not notice that the value of R was ridiculous?

RM: Good catch on seeing that I didn’t divide X2dot and Y2dot dt. I changed it but it changed nothing in the analysis since you are just dividing a variable by a constant.

BA: It changed the value of R by a factor of 120 (based on your dt=1/120).

RM: And what value of R (I presume you mean R^2) is ridiculous? All the values of R^2 (and, thus of R) looked perfectly reasonable to me.

BA: No, I mean R. R is the computed radius of curvature of the path at a given point along the path. Consider your example of a perfect ellipse, which is ± 5 units wide by ± 15 units high. The spreadsheet begins the analysis with X = 0 and Y = 15, which is located at the top center of the ellipse. Time starts at zero and increments by 1/120 seconds per step. As we move just past the top of the ellipse with the first time-step, R (the radius of curvature) should be close to the radius of a circle of R = 5 units (the radius of the minor axis of the ellipse). Instead it is computed to be 798.64 units.

BA: If we divide this number by dt = 120, we get 6.656 units, which is reasonable.

BA: As an experiment I tried creating a new set of Y values for the perfect ellipse by multiplying the original values by 2/3. This changes the ellipse into a circle since the major and minor axis dimensions are now ± 5 units. A circle has a constant radius, so the computed radius of curvature R should be 5 units. The sine and cosine waves used to generate the circle should generate a constant tangential velocity as well. Both were confirmed to an approximation (after fixing the computations for X2dot and Y2dot). (There is some variation due to the errors produced by rendering time as discrete changes of size dt.)

BA: By the way, the graph in the first sheet showing the figure traced is not being rendered correctly (at least on my screen). For the ellipse it shows one with X as the major axis but the data are for an ellipse with Y as the major axis, and the circle is rendered as an ellipse.

BA: I have not taken the time to proof the spreadsheet completely, so there may be other computational errors. In addition, as Alex noted some time ago, velocities and accelerations computed from differences can result in wild fluctuations, especially if dt is relatively large. These are probably the source of cyclical variations in computed R that were observed in my circle test.

Bruce

[From Bruce Abbott (2016.08.01.1030 EDT)]

···

Rick Marken (2016.07.31.1750) –

``````              BA: I have not taken the time to
``````

proof the spreadsheet completely, so there may be
other computational errors. In addition, as Alex
noted some time ago, velocities and accelerations
computed from differences can result in wild
fluctuations, especially if dt is relatively large.