Looking at the Power Law through Laputasian Glasses

[Abbott]

[From Bruce Abbott (2016.08.07.0905 EDT)]

Rick Marken (2016.08.06.1200)

Bruce Abbott (2016.08.05.1055 EDT)

RM: The power law is just an artifact; an embarrassing result of looking at behavior through causal rather than control theory glasses. Isn’t it about time for you to change your prescription.

BA: Time for you to change yours. Attached is my own spreadsheet investigation of V versus R. …

BA: So here is an example that violates the power rule. Velocity along the curve (in units of length/time) is not a fixed power function of the circle’s radius, R, contrary to your assertion that V is a power function of R with an exponent of 1/3.

RM: I’m afraid that your nice spreadsheet does not provide an example that violates what you call the “power rule” (it’s called the “power law” in the literature). The problem is that your conclusion (that it does) is based on a misunderstanding of how the power law is determined. The power law is determined (for the 100th time) using linear regression on the log values of R and V (or C and A) to solve for the power coefficient b as in :

log (V) = a +b*log(R)

RM: The regression analysis solves for b, the power coefficient, and the R^2 of the regression tells how well the power law (the linear fit of the log-log values) fits the data. So I took your nice quarter circle (X,Y values), put it into my spreadsheet that performs the log-log regression and found the following regression equation:

log (V) = -.53+.5*log(R)

with an R^2 of .74, a pretty good fit of of the power law. So if your perfect quarter circle were the movement made by a human being a power law researcher would likely conclude that the power coefficient relating R to V is .5. You get the same result, by the way, for a complete circle – a power coefficient of .5 with R^2 = .74 – though the intercept value changes to .32.

BA: Ah, poor Rick! As a scientist of Laputa, you have attempted to extract sunshine from cucumbers, and now this – attempting to extract tangential velocity around a cirrcle, measured in units of distance/time, from a spreadsheet in which time is nowhere represented!

BA: In my spreadsheet, V is the tangential “velocity� measured in distance traveled along the circumference of a circle per degree of change in the central angle of the circle, and is measured in units of distance/degree. This “velocity� is a characteristic of any circle of a given radius. It is not the velocity (in distance covered per unit of time) with which a person might draw the circle.

BA: But alas, as an upstanding member of the Laputa scientific community, you will be unfazed by such criticism, such is your delusional state. You will continue to insist that the only proper way to analyze these data is by performing a regression analysis on the two fixed values that characterize the shape we call a circle – the circlee’s radius (R) and the length of the arc subtended by a one-degree change in angle (V). You will remain blissfully unaware that your regression of log(R) onto log(V) should blow up (because of the absence of any variance in R and V), but doesn’t because of the tiny variances introduced by taking discrete differences to compute dX, dY, ddX, and ddY when using the formulas to compute V and R from the motion of the point along the circumference.

BA: It’s all there to see in my spreadsheet (compare the actual radius and velocity (in distance/degree) to the values of R and D as estimated by the formulas for computing these values from changes in X and Y. But you didn’t bother to look, did you?

BA: But wait, you say, look how beta in the regression equation quickly approaches 1/3 as I transform the circle into a slightly elliptical form! Why does that happen? Why, it must be because people can’t draw perfect circles! Well, I imagine that makes sense to you and your scientific colleagues on Laputa, but elsewhere it just means that the distances traveled around the perimeter of an ellipse with each degree of change in the central angle (V in distance/degree) systematically varies, as does the radius of curvature (R). As you increase the ratio of the ellipse’s length to width, this variation in V and R increases, quickly swamping out the “noise� in the computed values of V and R. But again, you are finding a relationship that is a fixed characteristic of a given oval.

BA: the “Vâ€? in these computations has nothing to do with how quickly or slowly one draws a circle or ellipse. That’s a different “Vâ€? entirely, one measured in units of distance/time. I do wish you would have a sudden epiphany and understand all this, but your responses to all previous attempts by earthbound scientists to do so have been frustratingly ineffective. I guess it is a consequence of your seeing the world through Laputasian glasses. Please, I implore you – take them off!

Your humble servant,

Gulliver

[rsmarken]

[From Rick Marken (2016.08.07.1320)

PowerLawRegression08.07.xlsm (581 KB)

···

Bruce Abbott (2016.08.07.0905 EDT)–

RM: I’m afraid that your nice spreadsheet does not provide an example that violates what you call the “power rule” (it’s called the “power law” in the literature).

BA: But alas, as an upstanding member of the Laputa scientific community, you will be unfazed by such criticism, such is your delusional state.

RM: Actually, I was quite fazed. You were doing the same calculations as I was but I was getting different results in my spreadsheet for the waveform in your spreadsheet. But I finally figured out the problem; I was dividing my derivatives by what I thought was a constant time difference, dt – the one both Alex and you said I should include – and you weren’t. So I changed that in my spreadsheet and got the same results as you! The result being that, for your part circle wave form the b coefficient for the regression of log R on log V is 0.00.

RM: So you were right; the part circle movement you sent does not fit the power law, at least in terms of the relationship between V and R. The regression equation I get is:

log V = -2.8+ 0.0 * log(R) with an R^2 value of .17

RM: But interestingly, the part circle movement does fit a power law relationship between A and C. The regression equation I get is:

log A = -2.8 + 1.0 * log(C) with an R^2 of 1.0

RM: I bet you can figure out why that is. This finding also means that researchers investigating the power law in terms of the relationship between C and A would conclude that the power law does hold for perfect part circle movements. That would lead to a big fight between the R vs V and C vs A power law researchers.

RM: Also, interestingly, when log D is included in the regression I get the solutions predicted by my derivations for the relationship between both V and R and between A and C. That is, I find

log V = .33 * log(D) + .33 * log(R) with an R^2 of 1.0

and

log A = .33 * log(D) + .67 * log(C) with an R^2 of 1.0

RM: I’ve attached a revised version of the spreadsheet. It now includes a button that lets you analyze part circles of different degrees of arc. The results are the same (in terms of the b value for log (R)) for all part circles – including a whole circle, 360 degrees. I also fixed the ellipse calculator so now a perfect circle (an ellipse with a ratio of major to minor axis length of 1.0) gives the same result as for the “part circle” segment that is 360 degrees. Now any deviation of an ellipse from a perfect circle results in estimates of power coefficients for log (R) and log (C) that are exactly what is predicted by my equations, .33 and .67 respectively. So the relationship between R and V is not a power law only for a perfect circle.

BA: Your humble servant,

RM: You don’t seem that humble to me but I appreciate you fact checking my work. It lets me keep\ improving my spreadsheet as well as my understanding of the problems with the power law research.

RM: So here’s what we have learned so far from you perfect circle movement:

1. Perfectly circular movement does not follow a power law relationship between R and V. (b = 0, R^2 = .17)

2. Perfectly circular movement follows a power law relationship between C and A exactly (b = 1.0, R^2 = 1.0).

3. Perfectly circular movement exactly follows a power law relationship between R and V and between C and A when the variable D is included in the regression. And the power coefficients of R vs V and of C vs A are .33 and .67 respectively, which ar exactly those predicted by the equations relating V to R and A to C:

V = D^1/3 *R^1/3

and

A = D^1/3 *C^2/3

RM: All these results are consistent with the PCT view of movement produced by living organisms; an observed pattern of movement is either a controlled variable (as per my PCT model) or a side effect of controlling some other variable(s). The pattern of movement contains no information about how it was produced. Thus, any observed relationship between variable aspects of a pattern of movement – such as a power relationship between R and V or C and A-- must depend on characteristics of the pattern itself and have nothing to do with how the pattern was generated. The fact that the nature (in terms of the size of the power coefficient) and fit (in terms of R^2) of a power relationship differs depending on the pattern of movement being analyzed supports this conclusion. The equations above showing that a power relationship exists between R and V and between C and A for any movement pattern also supports this conclusion.

RM: The equations above also show why power law researchers would be expected to find variability in their estimates of the power coefficient and the R^2 measure of fit for different movement patterns (rather than always finding a power coefficient of exactly 1/3 for R vs V and 2/3 for C vs A). The relationship between R and V and C and A also depends on the variable D. This variable is not taken into account when researchers study the power law. They look only at the relationship between log (R) and log (V) or log(C) and log(A). Leaving the variable D out the the regression results in variations in the estimates of the power coefficients for R and C. When log(D) Is included in the regression, the power coefficients of R and C are .33 and .67, for all patterns, as predicted by the equations above (and as demonstrated in the attached spreadsheet).

RM: So thanks again for persisting with this, Bruce. I think the analysis is getting better and better. And, of course, if you find any other errors in my spreadsheet analysis let me know. I was going to chide you for working so hard against PCT but I now think you contribute more to PCT by being it’s vigorous (and very capable) foe, rather than its friend. Oh, and on that note, could you (and/or Martin) tell me what you think is wrong with my PCT model of intentionally produced movement patterns.

Thanks again and greetings from Laputa.

Best

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

[Abbott]

[From Bruce Abbott (2016.08.07.1935 EDT)]

Rick Marken (2016.08.07.1320) –

Bruce Abbott (2016.08.07.0905 EDT)–

RM: I’m afraid that your nice spreadsheet does not provide an example that violates what you call the “power rule” (it’s called the “power law” in the literature).

BA: But alas, as an upstanding member of the Laputa scientific community, you will be unfazed by such criticism, such is your delusional state.

RM: Actually, I was quite fazed. You were doing the same calculations as I was but I was getting different results in my spreadsheet for the waveform in your spreadsheet. But I finally figured out the problem; I was dividing my derivatives by what I thought was a constant time difference, dt – the one both Alex and you said I should include – and you weren’t. So I changed that in my spreadsheet and got the same results as you! The result being that, for your part circle wave form the b coefficient for the regression of log R on log V is 0.00.

RM: So you were right; the part circle movement you sent does not fit the power law, at least in terms of the relationship between V and R. The regression equation I get is:

log V = -2.8+ 0.0 * log(R) with an R^2 value of .17

RM: But interestingly, the part circle movement does fit a power law relationship between A and C. The regression equation I get is:

log A = -2.8 + 1.0 * log(C) with an R^2 of 1.0

RM: I bet you can figure out why that is. This finding also means that researchers investigating the power law in terms of the relationship between C and A would conclude that the power law does hold for perfect part circle movements. That would lead to a big fight between the R vs V and C vs A power law researchers.

RM: Also, interestingly, when log D is included in the regression I get the solutions predicted by my derivations for the relationship between both V and R and between A and C. That is, I find

log V = .33 * log(D) + .33 * log(R) with an R^2 of 1.0

and

log A = .33 * log(D) + .67 * log(C) with an R^2 of 1.0

RM: I’ve attached a revised version of the spreadsheet. It now includes a button that lets you analyze part circles of different degrees of arc. The results are the same (in terms of the b value for log (R)) for all part circles – including a whole circle, 360 degrees. I also fixed the ellipse calculator so now a perfect circle (an ellipse with a ratio of major to minor axis length of 1.0) gives the same result as for the “part circle” segment that is 360 degrees. Now any deviation of an ellipse from a perfect circle results in estimates of power coefficients for log (R) and log (C) that are exactly what is predicted by my equations, .33 and .67 respectively. So the relationship between R and V is not a power law only for a perfect circle.

BA: Your humble servant,

RM: You don’t seem that humble to me but I appreciate you fact checking my work. It lets me keep\ improving my spreadsheet as well as my understanding of the problems with the power law research.

RM: So here’s what we have learned so far from you perfect circle movement:

1. Perfectly circular movement does not follow a power law relationship between R and V. (b = 0, R^2 = .17)

2. Perfectly circular movement follows a power law relationship between C and A exactly (b = 1.0, R^2 = 1.0).

3. Perfectly circular movement exactly follows a power law relationship between R and V and between C and A when the variable D is included in the regression. And the power coefficients of R vs V and of C vs A are .33 and .67 respectively, which ar exactly those predicted by the equations relating V to R and A to C:

V = D1/3 *R1/3

and

A = D1/3 *C2/3

RM: All these results are consistent with the PCT view of movement produced by living organisms; an observed pattern of movement is either a controlled variable (as per my PCT model) or a side effect of controlling some other variable(s). The pattern of movement contains no information about how it was produced. Thus, any observed relationship between variable aspects of a pattern of movement – such as a power relationship between R and V or C and A-- must depend on characteristics of the pattern itself and have nothing to do with how the pattern was generated. The fact that the nature (in terms of the size of the power coefficient) and fit (in terms of R^2) of a power relationship differs depending on the pattern of movement being analyzed supports this conclusion. The equations above showing that a power relationship exists between R and V and between C and A for any movement pattern also supports this conclusion.

RM: The equations above also show why power law researchers would be expected to find variability in their estimates of the power coefficient and the R^2 measure of fit for different movement patterns (rather than always finding a power coefficient of exactly 1/3 for R vs V and 2/3 for C vs A). The relationship between R and V and C and A also depends on the variable D. This variable is not taken into account when researchers study the power law. They look only at the relationship between log (R) and log (V) or log(C) and log(A). Leaving the variable D out the the regression results in variations in the estimates of the power coefficients for R and C. When log(D) Is included in the regression, the power coefficients of R and C are .33 and .67, for all patterns, as predicted by the equations above (and as demonstrated in the attached spreadsheet).

RM: So thanks again for persisting with this, Bruce. I think the analysis is getting better and better. And, of course, if you find any other errors in my spreadsheet analysis let me know. I was going to chide you for working so hard against PCT but I now think you contribute more to PCT by being it’s vigorous (and very capable) foe, rather than its friend. Oh, and on that note, could you (and/or Martin) tell me what you think is wrong with my PCT model of intentionally produced movement patterns.

RM: Thanks again and greetings from Laputa.

BA: You’re welcome, I guess.

BA:  I’m puzzled why you wish to pursue this fantasy. Perhaps it is the rarefied air up there on Laputa that has affected your thinking, or some strange radiation emanating from the rocks that keep the island of Laputa floating so high in the sky. Perhaps the explanation is more mundane – maybe you simply have lost too many of thosee little gray cells over the years. It’s got to be something, because otherwise I am at a loss to explain why you would continue to insist, against all logic, that the velocity computed in my spreadsheet is the speed (distance/time) that an object must necessarily travel while navigating turns of various curvature.

BA: I imagine that your scientific colleagues on Laputa will enthusiastically recommend publishing your results in the Proceedings of the Laputasian Royal Society, but, sorry to say, you will probably fare less well if you submit them to one of the earthbound scientific journals. You may believe you have extracted velocity as distance/time from length of arc subtended by a change of central angle (distance/degree), and no doubt your Laputasian colleagues will agree that you have. So from your perspective congratulations are in order, I suppose. I wish you similar success in your continuing efforts to extract sunshine from cucumbers.

Regretfully yours,

Gulliver

[rsmarken]

[From Rick Marken (2016.08.07.2130)]

···

Bruce Abbott (2016.08.07.1935 EDT)

BA: I’m puzzled why you wish to pursue this fantasy.

RM: As I’ve said, it’s because this “fantasy” perfectly illustrates the nightmare Bill Powers described in his 1978 Psych Review paper.

BA: It’s got to be something, because otherwise I am at a loss to explain why you would continue to insist, against all logic, that the velocity computed in my spreadsheet is the speed (distance/time) that an object must necessarily travel while navigating turns of various curvature.

RM: I’ve never insisted this about the velocity computer in your spreadsheet (or mine or in the power law research). You and Martin are the one’s doing the insisting that this is what I’m insisting. All I’m insisting that is V (as measured in power law experiments and now in our spreadsheets) and R ( (also as measured in power law experiments and now in our spreadsheets) are related to each other as

V = D^1/3 *R^1/3

and, of course, that

A = D^1/3 *C^2/3

RM: I don’t care what you call what V (or R) measures. All I care about is how these variables are computed. And you compute the value of these variables the same way I do (and the same way power law researchers do).

BA: I imagine that your scientific colleagues on Laputa will enthusiastically recommend publishing your results in the Proceedings of the Laputasian Royal Society, but, sorry to say, you will probably fare less well if you submit them to one of the earthbound scientific journals.

RM: I’m quite sure it will be quite difficult. It’s much easier to get non-PCT stuff published. But I think it’s worth the try.

BA: You may believe you have extracted velocity as distance/time from length of arc subtended by a change of central angle (distance/degree),

RM: I don’t believe any of the things you (and Martin) think I believe or keep asserting I believe. But I understand why you guys are insisting it. You are controlling people, after all. And nightmares are certainly something you would want to control for avoiding. And you’re doing a great job. And I really do appreciate the resistance; it helps me tune up my argument for the paper I plan to have rejected by the non-Laputan journals. So keep up the good work.

Best

Rick

and no doubt your Laputasian colleagues will agree that you have. So from your perspective congratulations are in order, I suppose. I wish you similar success in your continuing efforts to extract sunshine from cucumbers.

Regretfully yours,

Gulliver

–
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

[MartinT]

[Martin Taylor 2016.08.08.09.38]

[From Rick Marken (2016.08.07.2130)]

That's very good to know. What that comment says is that your method

of communicating is very unclear. What I believe you believe is that
the “V” in V = D^1/3*R^1/3 has something to do
with the speed with which a controller moves an object around a
curved path. If you don’t believe this, then I have been misreading
your messages ever since the one in which you first calimed to have
discovered a new behavioural illusion [[From Rick Marken
(2016.07.16.1415)] and a bit before.

If all my efforts (and Bruce's more recently) to show you how this

cannot be true have been off the point and you don’t in fact believe
your V is related to actual movement speeds, then we have all wasted
an awful lot of CSGnet time that could have been spent looking for a
PCT answer to Alex’s question.

Martin

···

RM: I don’t believe any of the things you [Bruce] (and
Martin) think I believe or keep asserting I believe.

[rsmarken]

[From Rick Marken (2016.08.08.0930)]

···

Martin Taylor (2016.08.08.09.38)–

MT: That's very good to know. What that comment says is that your method

of communicating is very unclear.

RM: Or it means that your listening is unclear.

MT: What I believe you believe is that

the “V” in V = D^1/3*R^1/3 has something to do
with the speed with which a controller moves an object around a
curved path. If you don’t believe this, then I have been misreading
your messages ever since the one in which you first calimed to have
discovered a new behavioural illusion [[From Rick Marken
(2016.07.16.1415)] and a bit before.

RM: I know that I have discovered a new behavioral illusion and it has nothing to do with what I or you or anyone thinks V measures.

MT: If all my efforts (and Bruce's more recently) to show you how this

cannot be true have been off the point and you don’t in fact believe
your V is related to actual movement speeds, then we have all wasted
an awful lot of CSGnet time that could have been spent looking for a
PCT answer to Alex’s question.

RM: I don’t think we have wasted any time. If you think I haven’t already provided the PCT answer to Alex’s question then we clearly have a ways to go. As a start I will post my slightly revised PCT model of movement production. The slight revision involves placing the equations for V and A out in the environment. Hopefully this makes clear that the PCT explanation of the power law is that the observed power law relationship between V and R and A and C is simply a mathematical property of curved movements. The variables V, R, A and C are the variables that are actually measured in these studies; they are computed from the varying X, Y position of the movement using the computational formulas given in papers on the power law, the same formulas Bruce in our spreadsheet simulations. Therefore, what these variables “really” measure is not relevant to the analysis; the variables V, R, A and C are data and the model below explains why power law researchers find values that vary around .33 for the power coefficient of R and around .67 for the power coefficient of C when they do a log - log regression and leave out the variable D. What they are studying is properties of the environment (the movement pattern produced), not properties of the system that produced those variations in the environment (the movement pattern). This is certainly a version of Powers’ behavioral illusion.

          RM: I don't believe any of the things you [Bruce] (and

Martin) think I believe or keep asserting I believe.

–
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

RM: I hope this serves as a reasonably clear explanation of why all the mathematics about what V really measures are irrelevant.

Best

Rick

[MartinT]

[Martin Taylor 2016.08.09.13.25]

[From Rick Marken (2016.08.08.0930)]

Same thing. My listening to Chinese is also unclear.

That's two beliefs, neither of which I had considered. But I am

surprised by the second, because your “new behaviural illusion”
seems to depend on your belief " that the “V” in V = D^1/3*R^1/3
has something to do with the speed with which a controller moves an
object around a curved path."

Since you won't say whether you actually hold that belief, one

cannot say whether your belief in your “new behavioural illusion” is
misguided or illogical. It’s one or the other. I’d like to think
it’s misguided, because that befits a scientist better than that it
is illogical. But that’s just me.

I suppose they are irrelevant, if the "illogical" hypothesis about

your beliefs is correct. But they aren’t irrelevant to Alex’s
problem, and they aren’t irrelevant if the “misguided” hypothesis is
correct. Alex’s V is what people and other organisms do when they
move around curves. Your V is a formal parameter, sometimes called
an “intervening variable” in a formula for curvature. Your “D” is
defined as V³/R, so it’s not very surprising that V = D^1/3*R^1/3
= (V³/R)^1/3*R^1/3 = V. That’s hardly
justification for claiming you have found a “new behavioural
illusion”.

Martin

···

Martin Taylor (2016.08.08.09.38)–

            MT: That's very good to know. What that comment says is

that your method of communicating is very unclear.

RM: Or it means that your listening is unclear.

                      RM: I don't believe any of the things you

[Bruce] (and Martin) think I believe or keep
asserting I believe.

            MT: What I believe

you believe is that the “V” in V = D^1/3*R^1/3
has something to do with the speed with which a
controller moves an object around a curved path. If you
don’t believe this, then I have been misreading your
messages ever since the one in which you first calimed
to have discovered a new behavioural illusion [[From
Rick Marken (2016.07.16.1415)] and a bit before.

          RM: I know that I have discovered a new behavioral

illusion and it has nothing to do with what I or you or
anyone thinks V measures.

        RM: I hope this serves as a

reasonably clear explanation of why all the mathematics
about what V really measures are irrelevant.

[rsmarken]

[From Rick Marken (2016.08.09.1310)]

···

Martin Taylor (2016.08.09.13.25)–

MT: That's two beliefs, neither of which I had considered. But I am

surprised by the second, because your “new behaviural illusion”
seems to depend on your belief " that the “V” in V = D^1/3*R^1/3
has something to do with the speed with which a controller moves an
object around a curved path."

RM: No, it doesn’t depend on my belief about what V measures. The idea that it does is just your belief;-)

MT: Since you won't say whether you actually hold that belief, one

cannot say whether your belief in your “new behavioural illusion” is
misguided or illogical.

RM: So the only possibilities are that I am misguided or illogical? I actually can think of another possibility;-)

MT: It’s one or the other.

RM: No, the far more probable possibility is that I am correct and you are suffering from the behavioral illusion yourself because of your belief that the observed power law relationship between V and R (or between A and C) tells us something about how outputs produce controlled inputs. In fact, the power law is a statistical artifact; a side effect of the way the relationship between two measures of the same curved movement is analyzed.

Best

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

          RM: I know that I have discovered a new behavioral

illusion and it has nothing to do with what I or you or
anyone thinks V measures.

[Abbott]

[From Bruce Abbott (2016.08.09.1910 EDT)]

Rick Marken (2016.08.08.0930) –

Martin Taylor (2016.08.08.09.38)–

RM: I don’t believe any of the things you [Bruce] (and Martin) think I believe or keep asserting I believe.

MT: That’s very good to know. What that comment says is that your method of communicating is very unclear.

RM: Or it means that your listening is unclear.

MT: What I believe you believe is that the “V” in V = D1/3*R1/3 has something to do with the speed with which a controller moves an object around a curved path. If you don’t believe this, then I have been misreading your messages ever since the one in which you first calimed to have discovered a new behavioural illusion [[From Rick Marken (2016.07.16.1415)] and a bit before.

RM: I know that I have discovered a new behavioral illusion and it has nothing to do with what I or you or anyone thinks V measures.

MT: If all my efforts (and Bruce’s more recently) to show you how this cannot be true have been off the point and you don’t in fact believe your V is related to actual movement speeds, then we have all wasted an awful lot of CSGnet time that could have been spent looking for a PCT answer to Alex’s question.

RM: I don’t think we have wasted any time. If you think I haven’t already provided the PCT answer to Alex’s question then we clearly have a ways to go. As a start I will post my slightly revised PCT model of movement production. The slight revision involves placing the equations for V and A out in the environment. Hopefully this makes clear that the PCT explanation of the power law is that the observed power law relationship between V and R and A and C is simply a mathematical property of curved movements. The variables V, R, A and C are the variables that are actually measured in these studies; they are computed from the varying X, Y position of the movement using the computational formulas given in papers on the power law, the same formulas Bruce in our spreadsheet simulations. Therefore, what these variables “really” measure is not relevant to the analysis; the variables V, R, A and C are data and the model below explains why power law researchers find values that vary around .33 for the power coefficient of R and around .67 for the power coefficient of C when they do a log - log regression and leave out the variable D. What they are studying is properties of the environment (the movement pattern produced), not properties of the system that produced those variations in the environment (the movement pattern). This is certainly a version of Powers’ behavioral illusion.

RM: I hope this serves as a reasonably clear explanation of why all the mathematics about what V really measures are irrelevant.

BA: No, but it is a reasonably clear explanation of why you believe that all the mathematics about what V really measures is irrelevant. Martin showed how time is really irrelevant in the calculation of R (which actually reflects how much the path of the particle bends as the particle changes position). But if that confuses you, let’s just stick with V as distance/time, because that’s what appears in the equations we are dealing with.

BA: The V you observe in your data representing the tangential velocity of a point moving along a curve is the same V that enters into the equation for determining the radius of curvature. On the one hand it measures the speed of a particle as it traces curves of various sorts. On the other hand, divided by what you have labeled “D,� it yields the radius of curvature of the point’s motion at a given instant in time. Getting this estimate of curvature requires that you compute not only the velocity of the point along both the x and y dimensions, but also how the point is accelerating along those dimensions. V in the numerator provides the tangential velocity of the point, or how fast the point is moving along the curve The velocities and accelerations in the denominator provide the speed with which the point is both moving and accelerating along the x and y directions, which reflect the tightness of the curve. For a curve of a given radius, the effect of the denominator is such as to give the same radius of curvature regardless of the value of V. The velocities and accelerations cancel out in the equation, leaving only the distance R.

BA: So where are we? For a given curve, the computed radius of curvature R will be independent of the V at any given instant (except for approximation errors in the computations). Consequently, the point theoretically can move along that curve at any velocity whatsoever. The tangential velocity of the point at a given instant does not depend on R. The point can speed up or slow down and the variables in the denominator will take these changes into account when computing the radius of the curve along which the point is moving at that moment. Richard Kennaway (2016.08.08 14:14 BST) demonstrated this independence in a thoroughly mathematical way involving several examples.

BA: At this point we have Alex, Martin, Richard K., and me agreeing on the independence of V from R, and you asserting that V is a function of R that is “simply a mathematical property of curved movement.� Might it be time to rethink your position?

BA: This conclusion agrees with common experience. For example, you can drive your car at 35 mph around a long, lazy curve or a somewhat tighter one. You can even speed up or slow down while navigating the curve. The curvature of the road will not force you to travel at a velocity dictated by your rearrangement of the formula for the radius of curvature.

Bruce

[rsmarken]

[From Rick Marken (2016.08.09.1700)]

···

Bruce Abbott (2016.08.09.1910 EDT)

BA: The V you observe in your data representing the tangential velocity of a point moving along a curve is the same V that enters into the equation for determining the radius of curvature. …

Â

BA: At this point we have Alex, Martin, Richard K., and me agreeing on the independence of V from R, and you asserting that V is a function of R that is “simply a mathematical property of curved movement.â€? Might it be time to rethink your position?

RM: Actually, you can throw in Warren Mansell and, as far as I can tell, everyone else on CSGNet and possibly everyone off it. One reason I’m not ready to rethink my position is because no one – not even Richard K. – has addressed it.Â

RM: My position is that when a researcher does a regression analysis on log(R) vs. log(V) or of log (C) vs. log (A) (as is done in power law research) she is bound to find a power function relationship between these variables with a power coefficient around .33 (for log (R) on log (V)) or around .67 (log (C) on log (A), with an R^2 that can range from about .5 to 1.0. The variation in the value of the power coefficient and R^2 is a result of not taking the variable that I called D into account. This is true whether the curved movement is produced by a person, a fly larva of an equation. This statistical artifact results from the fact that:Â

V = D^1/3Â *R^{1/3 Â Â Â}

andÂ

A = D^1/3Â *C^{2/3 Â Â Â}

RM: The independence of V from R or A from C have nothing to do with my position. I’m saying that the power law is a statistical artifact and says nothing about the relationship between control system output and the controlled results produced by those outputs. But this was obvious just from knowing PCT; you can’t measure the means used to produce a result by looking at just the result itself. And the power law is based on measures of nothing other than the result itself – the curved movement pattern.Â

BestÂ

Rick

Â

Â

BA: This conclusion agrees with common experience. For example, you can drive your car at 35 mph around a long, lazy curve or a somewhat tighter one. You can even speed up or slow down while navigating the curve. The curvature of the road will not force you to travel at a velocity dictated by your rearrangement of the formula for the radius of curvature.

Â

Bruce

Â

–
Richard S. MarkenÂ

“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

[MartinT]

[Martin Taylor 2016.08.09.17.34]

[From Rick Marken (2016.08.09.1310)]

If you don't intend your V to represent something to do with the

speed with which a controller moves something around a curved path,
on what basis do you claim the discovery of a new behavioural
illusion?

Let me get this straight. Alex posed a question about the relation

between the speed with which a variety of controllers move around
curves with varying radius of curvature. You notice that there is a
variable that can be called “V” for velocity in the formula for
radius of curvature. You then express this “V” in terms of the
radius of curvature R and a new variable “D” which you define as V³ /R.
You also notice that your V = D^1/3*R^1/3 , which
holds for all curves for which R is neither zero nor infinite. You
notice that if you include the “D” variable V³ /R, your V
varies as R^1/3 , and on this basis say that the
experiments that often find the speed with which controllers move
objects around curves also varies as R^1/3 is a
behavioural illusion. And now you say that it is irrelevant whether
your V has any relation to the speed measured in experiments?

Am I right so far?

What two measures? Angular velocity and linear velocity? The

relationship is quite clear; ds/dθ = R if theta is measured in
radians, and so (ds/dt)/(dθ/dt) = R. There’s no statistical effect
at all, just a straightforward geometric relationship. In case you
were wondering, ds/dt is linear velocity along the curve, and dθ/dt
is angular velocity.

You have an unusual way of using the word "fact". I suppose it's a

different “fact” when the power law exponent changes to 1/4 or no
power law relation is found? Is there a partial behavioural illusion
then? Does it go away entirely?

In my suggested experiment, to which motion does the "statistical

artifact" apply? Both perceptual and motor ellipses have a curved
movement, but at different phases (the same kind of variation you
used in your bimanual rotation demo, which I confess to having had
in the back of my mind when I considered the experiment). Do you
expect the same power law relationship to apply to both if it
applies to either? Apparently you do. At least you asked a question
that presupposed that it might [From Rick Marken (2016.08.09.1430)].

  RM: So if the power law holds for mouse

movement, you’ll find one result of your experiment; if it holds
for perception of the track, you’ll find a different result. So
what will you find if the power law holds for both?

But I suppose you could be forgiven, since I only said I was using

Bill’s Circle-Square demo without saying explicitly that I was using
his technique of mapping the mouse movement (circle) onto the cursor
movement (square), and in any case I probably shouldn’t have
expected you to know what that was. Here’s an amended diagram that
shows the algorithm visually. The line extending from the centre of
the ellipses and the two circles marking where that line crosses the
ellipses are not visible. They just indicate the relationship
between a mouse position and the cursor position corresponding to
that mouse position at a given moment if the cursor is on the target
trajectory. If it isn’t, the cursor is at least on that line.

![SqareCircleEllipsesMoment.jpg|334x149](upload://o83E6x8fyzVkDzTszDMkJaodMSr.jpeg)

I think this should show why they can't both conform to the same

power law. Their speeds are anti-correlated. Most of the time when
one moves faster along its curve the other moves slower.

  MT: I know that logic has no relevance for you

RM: Now, now, Martin. No need to be insulting.

You are quite right to chastise me. I apologize and plead extreme

frustration with your repeated explicit assertion that mathematical
logic is irrelevant to approaching Alex’s question over the last 5
weeks. I had hoped that by introducing this novel experiment we
could get around that problem, and was unpleasantly surprised to
find that it did not work.

Martin

···

Martin Taylor (2016.08.09.13.25)–

            MT: That's two beliefs, neither of which I had

considered. But I am surprised by the second, because
your “new behaviural illusion” seems to depend on your
belief " that the “V” in V = D^1/3*R^1/3
has something to do with the speed with which a
controller moves an object around a curved path."

          RM: No, it doesn't depend on my belief about what V

measures. The idea that it does is just your belief;-)

                        RM: I know that I have discovered a new

behavioral illusion and it has nothing to do
with what I or you or anyone thinks V
measures.

            MT: Since you won't

say whether you actually hold that belief, one cannot
say whether your belief in your “new behavioural
illusion” is misguided or illogical.

          RM: So the only possibilities are that I am misguided

or illogical? I actually can think of another
possibility;-)

            MT: It's one or the

other.

          RM: No, the far more probable possibility is that I am

correct and you are suffering from the behavioral illusion
yourself because of your belief that the observed power
law relationship between V and R (or between A and C)
tells us something about how outputs produce controlled
inputs. In fact, the power law is a statistical artifact;
a side effect of the way the relationship between two
measures of the same curved movement is analyzed.

[Abbott]

[From Bruce Abbott (2016.08.10.1700 EDT)]

Rick Marken (2016.08.09.1700)]

Bruce Abbott (2016.08.09.1910 EDT)

BA: The V you observe in your data representing the tangential velocity of a point moving along a curve is the same V that enters into the equation for determining the radius of curvature. …

BA: At this point we have Alex, Martin, Richard K., and me agreeing on the independence of V from R, and you asserting that V is a function of R that is “simply a mathematical property of curved movement.� Might it be time to rethink your position?

RM: Actually, you can throw in Warren Mansell and, as far as I can tell, everyone else on CSGNet and possibly everyone off it. One reason I’m not ready to rethink my position is because no one – not even Richard K. – has addressed it.

RM: My position is that when a researcher does a regression analysis on log(R) vs. log(V) or of log (C) vs. log (A) (as is done in power law research) she is bound to find a power function relationship between these variables with a power coefficient around .33 (for log (R) on log (V)) or around .67 (log (C) on log (A), with an R^2 that can range from about .5 to 1.0. The variation in the value of the power coefficient and R^2 is a result of not taking the variable that I called D into account. This is true whether the curved movement is produced by a person, a fly larva of an equation. This statistical artifact results from the fact that:

V = D1/3 *R1/3

and

A = D1/3 *C2/3

RM: The independence of V from R or A from C have nothing to do with my position. I’m saying that the power law is a statistical artifact and says nothing about the relationship between control system output and the controlled results produced by those outputs. But this was obvious just from knowing PCT; you can’t measure the means used to produce a result by looking at just the result itself. And the power law is based on measures of nothing other than the result itself – the curved movement pattern.

BA: I’ve noticed that in your replies to me you simply excise my explanation for why your position is flat out wrong and then simply repeat what you have already asserted several times over. It’s like a Creationist, after being presented with overwhelming, converging evidence for the evolution of species across billions of Earth years, replies “You are wrong, because the Bible says . . .�

BA: I can’t help feeling that you didn’t even read my explanation, or if you did, made no attempt to understand it. Convince me that you did read it, understood it, but have some compelling reason for rejecting it. Thus far I haven’t heard one, just an assertion that the math is irrelevant to your position.

I want to hear your reasoning as to why the following is wrong (or at least irrelevant to your claim that the power law as found in curved movement patterns is a “statistical artifact.� Below I’ve repeated my explanation of why this assertion is false, which you deleted from your reply to me:

BA: The V you observe in your data representing the tangential velocity of a point moving along a curve is the same V that enters into the equation for determining the radius of curvature. On the one hand it measures the speed of a particle as it traces curves of various sorts. On the other hand, divided by what you have labeled “D,� it yields the radius of curvature of the point’s motion at a given instant in time. Getting this estimate of curvature requires that you compute not only the velocity of the point along both the x and y dimensions, but also how the point is accelerating along those dimensions. V in the numerator provides the tangential velocity of the point, or how fast the point is moving along the curve The velocities and accelerations in the denominator provide the speed with which the point is both moving and accelerating along the x and y directions, which reflect the tightness of the curve. For a curve of a given radius, the effect of the denominator is such as to give the same radius of curvature regardless of the value of V. The velocities and accelerations cancel out in the equation, leaving only the distance R.

BA: On the one hand you have been asserting that V varies with changes in R (and D), and that this relationship is nothing more than a characteristic of all curved lines, telling us nothing about how such lines may have been produced. On the other you admit that V and R can vary independently, as seen in situations such as a car navigating a winding road at a constant velocity,  or accelerating around a curve of constant radius, but claim that this fact is irrelevant. I for one am having trouble understanding how you can simultaneously hold that V both depends on R and varies independently of R. How can this relationship be both a “statistical artifact� that holds constant for all curves, regardless of how quickly or slowly they are traversed, and an observed relationship between R and V that may or may not conform to the power law?

BA: Please explain. And while you are at it, please tell me what you think this magical “Dâ€? variable represents – the one that, when included in the regression, rraises R2 to 1.0. Couch your explanation in terms of how D is calculated, not in terms of what it does when added to your regression equation.

Bruce

[rsmarken]

[From Rick Marken (2016.08.10.1900)]

···

Martin Taylor (2016.08.09.17.34)–

MT: If you don't intend your V to represent something to do with the

speed with which a controller moves something around a curved path,
on what basis do you claim the discovery of a new behavioural
illusion?

RM: Because what I would intend V to represent has nothing to do with the behavioral illusion. V is a variable measured in power law experiments; it it data! The illusion is that an observed relationship (or lack thereof) between this variable and another variable, R, that is also measured in power law experiments, has something to do with how curved movements are produced. It doesn’t.

MT: Let me get this straight. Alex posed a question about the relation

between the speed with which a variety of controllers move around
curves with varying radius of curvature.

RM: No, Alex posed a question about how PCT would explain the power law. Specifically, he asked “Any ideas why or how “the control of perception” may give rise to this power law…”. My PCT model of the control of curved movement trajectories (below) answers that question:

RM: So far, I have heard no legitimate criticisms of this model. And I have been able to demonstrate that this model can account for the data collected in power law experiments.

MT: You notice that there is a

variable that can be called “V” for velocity in the formula for
radius of curvature.

RM: I didn’t just “notice” this variable. This is the variable measured in power law experiments; the other variable is R.

MT: You then express this "V" in terms of the

radius of curvature R and a new variable “D” which you define as V³/R.

RM: I did this because I noticed that the two variables that are measured as the independent and dependent variables in power law research – R and V, respeectively - are mathematically related. So I solved the computational formulas for V and R simultaneously and got the result that you mention next:

MT: You also notice that your V = D^1/3*R^1/3 , which
holds for all curves for which R is neither zero nor infinite.

RM: Yes, and it was a stunning result. There is a mathematical power relationship between V and R and the coefficient of this power relationship is 1/3, the average value of the power coefficient found in power law experiments by regression of log(R) on log (V)! I figured this was unlikely to be a coincidence but there was that other pesky term, D^1/3, in the equation. But I realized that, since that term is left out of the regressions used to determine the power coefficient, its absence from the regressions might be responsible for the variability of the power coefficient estimates around .33 and for the fact that sometimes these estimates vary around other values, like .25. So I built the regression analysis spreadsheet to test this and found that, indeed, the value of the power coefficient found using regression of just log(R) on log(V) depend on the type of movement pattern produced and, presumably, how the unaccounted for variance in log(D) for each different pattern contributes to the variance in log(V), independent of log(R).

MT: You

notice that if you include the “D” variable V³ /R, your V
varies as R^1/3,

RM: V varies as R^1/3 whether D is there or not.

MT: and on this basis say that the

experiments that often find the speed with which controllers move
objects around curves also varies as R^1/3 is a
behavioural illusion.

RM: Close. First of all, power law experimenters don’t find that "the speed which controllers move objects around curves also varies as R^1/3 ". All they observe is an average estimate of a power coefficient of ~1/3 for some movement patterns or of 1/4 for others (like those produced under water), movement patterns which may or may not have been produced by controllers. My spreadsheet regression analysis program shows that these estimates depend on the movement pattern observed and have nothing to do with how the movement pattern was produced. The “illusion” is that these calculated power law estimates say something about how controllers move objects around curves. In fact, all they tell you about is the mathematical relationship between V and R when D is left out of the analysis. When D is included in the analysis the power coefficient of R is always .33, for all patterns of movement, regardless of how those patterns were produced.

MT: And now you say that it is irrelevant whether

your V has any relation to the speed measured in experiments?

RM: It depends on what you mean by “irrelevant”. My analysis is relevant to the fact that V (called instantaneous tangential velocity by power law researchers and is presumably a measure of speed) is calculated in the way the power law researchers say they calculate it. It’s also relevant to the fact that this measure of speed, V, is not a reflection of the speed with which the controller produced this speed of movement, if a controller did produce it. That is, at any instant, the speed of the controller’s movement is not necessarily the same as the speed of the object being moved (remember that the state of a controlled variable, such as the rate of movement of an object by a controller, depends on both the controller’s output and environmental disturbances; see the PCT model diagram above; I will have a demo of this soon).

MT: Am I right so far?

RM: I guess I’d have to say “no”.

MT: What two measures? Angular velocity and linear velocity?

No, between log(R) and log(V) or between log(C) and log (A).

MT:  There's no statistical effect

at all…

RM: It’s a statistical ARTIFACT not a statistical effect. The observed power law is an artifactual result of trying to find the power relationship between R and V or between C and A by looking at just the regression coefficient relating log(R) to log(V) or log(C) to log (A), respectively. If you include log(D) in the regression analysis (so that it is now a multiple regression) you will always find the power coefficient relating R to V to be .33 and the power coefficient relating C to A to be .67.

MT: You have an unusual way of using the word "fact". I suppose it's a

different “fact” when the power law exponent changes to 1/4 or no
power law relation is found?

RM: No, you will find an average power law exponent of 1/4 (averaged over several patterns) when you regress just log(R) on log (V) for certain patterns of movement (such as ellipses made under water). But you will find the that the power coefficient for such movements to be .33 when log(D) is included in the regression.

MT: In my suggested experiment, to which motion does the "statistical

artifact" apply? Both perceptual and motor ellipses have a curved
movement, but at different phases (the same kind of variation you
used in your bimanual rotation demo, which I confess to having had
in the back of my mind when I considered the experiment). Do you
expect the same power law relationship to apply to both if it
applies to either? Apparently you do. At least you asked a question
that presupposed that it might [From Rick Marken (2016.08.09.1430)].

RM: The statistical artifact will apply to both the perceptual and motor ellipses. And if they are both ellipses, as you expert, then the same (or very similar) power law relationship (in terms of estimates of b) will be found for both. But I think it should be possible to develop a demo where the estimates are quite different.

RM: I think that’s enough for now.

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

          RM: ... In fact, the power law is a statistical artifact;

a side effect of the way the relationship between two
measures of the same curved movement is analyzed.

[rsmarken]

[From Rick Marken (2016.08.11.1225)]

···

Bruce Abbott (2016.08.10.1700 EDT)

BA: I’ve noticed that in your replies to me you simply excise my explanation for why your position is flat out wrong…

Â

BA: I can’t help feeling that you didn’t even read my explanation, or if you did, made no attempt to understand it…

Â

BA: I want to hear your reasoning as to why the following is wrong (or at least irrelevant to your claim that the power law as found in curved movement patterns is a “statistical artifact.� Below I’ve repeated my explanation of why this assertion is false, which you deleted from your reply to me:

RM: OK, it’s a deal!

Â

BA: The V you observe in your data representing the tangential velocity of a point moving along a curve is the same V that enters into the equation for determining the radius of curvature.Â

RM: The problem here is that you imply that V is something I came up with. I would agree with this statement if it had started with “The V that is observed by power law researchers…” instead of “The V you observe…” You should also eliminate the phase “in your data…” since V is not “in data” it IS data, the data that is used in power law research to determine the power law.Â

Â

BA: On the one hand it measures the speed of a particle as it traces curves of various sorts. On the other hand, divided by what you have labeled “D,� it yields the radius of curvature of the point’s motion at a given instant in time. Getting this estimate of curvature requires that you compute not only the velocity of the point along both the x and y dimensions, but also how the point is accelerating along those dimensions. V in the numerator provides the tangential velocity of the point, or how fast the point is moving along the curve The velocities and accelerations in the denominator provide the speed with which the point is both moving and accelerating along the x and y directions, which reflect the tightness of the curve. For a curve of a given radius, the effect of the denominator is such as to give the same radius of curvature regardless of the value of V. The velocities and accelerations cancel out in the equation, leaving only the distance R.

RM: All this is fine as long as it is clear that this is not just true of “my” V but of V as measured in all power law research. Â

Â

BA: On the one hand you have been asserting that V varies with changes in R (and D), and that this relationship is nothing more than a characteristic of all curved lines, telling us nothing about how such lines may have been produced. On the other you admit that V and R can vary independently, as seen in situations such as a car navigating a winding road at a constant velocity,  or accelerating around a curve of constant radius, but claim that this fact is irrelevant.Â

RM: I disagree with the “On the other hand…” part of this. I have never “admitted” any of that stuff that you say I admitted. What I have said about a car navigating a winding road is that the movement path of the car is a result of combining forces produced by the driver (o) with environmental disturbance forces (d). The resulting movement path, q.i, is a controlled variable and it is that variable from which the measures of V and R are obtained in power law research. So it’s not V and/or R that are varied when navigating a car along a winding road; V and R are measures of the controlled result of this navigating (controlling). Â

Â

BA: I for one am having trouble understanding how you can simultaneously hold that V both depends on R and varies independently of R.Â

RM: Â I don’t believe V varies independently of R. I have shown that V (as measured by power law researchers) is related to R (as measured by power law researchers) by the equation:

V = Â D^1/3Â *R^{1/3Â Â} Â Â Â

BA: How can this relationship be both a “statistical artifact� that holds constant for all curves, regardless of how quickly or slowly they are traversed, and an observed relationship between R and V that may or may not conform to the power law?

RM: This is a hard question to answer because the statistical artifact doesn’t “hold constant” for all curves. What holds constant for all curves is the fact thatÂ

V = Â D^1/3Â *R^{1/3Â Â} Â

RM: The statistical artifact is the value of the power coefficient of R that is found from regression analysis using only log(R) to predict log(V). The power coefficient that is found as the solution to the regression equation (and the fit of the power law to the data in terms of the size of R^2 for this regression) will not be constant but will vary depending on the pattern of movement analyzed. For elliptical movements the regression of log(R) on log(V) will yield a power coefficient that will be close to .33; for movements that deviate from perfectly elliptical, such as elliptical movements made underwater, the regression of log(R) on log(V) will yield a power coefficient that will be close to .25; other movement patterns result in other power coefficients. If a person could move their hand in a perfect circle the power coefficient for R on V would be 0; but for C on V it would be 1.0. Even the slightest deviation from circularity will bring these coefficients to their “expected” values: .33 and .67 respectively.Â

 BA: Please explain.Â

RM: That’s my explanation.Â

BA: And while you are at it, please tell me what you think this magical “Dâ€? variable represents – thhe one that, when included in the regression, raises R2 to 1.0. Couch your explanation in terms of how D is calculated, not in terms of what it does when added to your regression equation.

RM: As Richard Kennaway pointed out:

RK: where D is defined as |XdotYdotdot -
XdotdotYdot|
 we can recognise D as being the magnitude of the
cross product of V = (Xdot,Ydot) and the acceleration Vdot = (Xdotdot,Ydotdot).

RM: So that’s what D is; the cross product of V and Vdot. As you instruct, I will say nothing about what D does when added to the regression. Although I will point out that, as Richard K. also noted, D is a constant for some movement patterns (such the circle traced out by a pendulum swinging in three dimensions) and a variable for others (such as the elliptical motion of the planets). The implications of this for regression analysis should  be obvious.

Best regards

Rick

–
Richard S. MarkenÂ

“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

[rsmarken]

[From Rick Marken (2016.08.11.1235)]

Errata: I said:

···

I should have said:

If a person could move their hand in a perfect circle the power coefficient for R on V would be 0; but for C on A it would be 1.0

You can’t be too careful around opponents of PCT!

Best

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

If a person could move their hand in a perfect circle the power coefficient for R on V would be 0; but for C on V it would be 1.0.

[_Boris_Hartman]

[From Rick Marken (2016.08.11.1235)]

Errata: I said:

If a person could move their hand in a perfect circle the power coefficient for R on V would be 0; but for C on V it would be 1.0.

I should have said:

If a person could move their hand in a perfect circle the power coefficient for R on V would be 0; but for C on A it would be 1.0

You can’t be too careful around opponents of PCT!

HB : You probably meant opponents of selfrgulation J. You are not a PCT thinker Rick. For you »Behavor is Control«… So…. You are the opponent of of PCT. In PCT »Perception is Controlled«.

Best,

Boris

Best

Rick

···

From: Richard Marken [mailto:rsmarken@gmail.com]
Sent: Thursday, August 11, 2016 9:38 PM
To: csgnet@lists.illinois.edu
Subject: Re: Looking at the Power Law through Laputasian Glasses

–

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

[Abbott]

[From Bruce Abbott (2016.08.11.1820 EDT)]

Rick Marken (2016.08.11.1225) –
/o:p>

Bruce Abbott (2016.08.10.1700 EDT)

BA: I’ve noticed that in your replies to me you simply excise my explanation for why your position is flat out wrong…

BA: I can’t help feeling that you didn’t even read my explanation, or if you did, made no attempt to understand it…

BA: I want to hear your reasoning as to why the following is wrong (or at least irrelevant to your claim that the power law as found in curved movement patterns is a “statistical artifact�). Below I’ve repeated my explanation of why this assertion is false, which you deleted from your reply to me:

RM: OK, it’s a deal!

BA: The V you observe in your data representing the tangential velocity of a point moving along a curve is the same V that enters into the equation for determining the radius of curvature.

RM: The problem here is that you imply that V is something I came up with.

BA: Huh?

I would agree with this statement if it had started with “The V that is observed by power law researchers…” instead of “The V you observe…” You should also eliminate the phase “in your data…” since V is not “in data” it IS data, the data that is used in power law research to determine the power law.

BA: Are not the values of V you compute in your spreadsheet not in the data you present in your spreadsheet? I am referring specifically to the values you computed, not to values observed by power law researchers. Your values, presented in your spreadsheet as proof of your conjecture, are the values I refer to.

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BA: On the one hand it measures the speed of a particle as it traces curves of various sorts. On the other hand, divided by what you have labeled “D,� it yields the radius of curvature of the point’s motion at a given instant in time. Getting this estimate of curvature requires that you compute not only the velocity of the point along both the x and y dimensions, but also how the point is accelerating along those dimensions. V in the numerator provides the tangential velocity of the point, or how fast the point is moving along the curve The velocities and accelerations in the denominator provide the speed with which the point is both moving and accelerating along the x and y directions, which reflect the tightness of the curve. For a curve of a given radius, the effect of the denominator is such as to give the same radius of curvature regardless of the value of V. The velocities and accelerations cancel out in the equation, leaving only the distance R.

RM: All this is fine as long as it is clear that this is not just true of “my” V but of V as measured in all power law research.

BA: Your V is computed in the same way they compute theirs. Your results, however, are due to the way you computed your data points that describe an ellipse. Theirs is an empirical relationship in which V and R could turn out to have any relationship, not just a power one.

BA: On the one hand you have been asserting that V varies with changes in R (and D), and that this relationship is nothing more than a characteristic of all curved lines, telling us nothing about how such lines may have been produced. On the other you admit that V and R can vary independently, as seen in situations such as a car navigating a winding road at a constant velocity, or accelerating around a curve of constant radius, but claim that this fact is irrelevant.

RM: I disagree with the “On the other hand…” part of this. I have never “admitted” any of that stuff that you say I admitted. What I have said about a car navigating a winding road is that the movement path of the car is a result of combining forces produced by the driver (o) with environmental disturbance forces (d). The resulting movement path, q.i, is a controlled variable and it is that variable from which the measures of V and R are obtained in power law research. So it’s not V and/or R that are varied when navigating a car along a winding road; V and R are measures of the controlled result of this navigating (controlling).

BA:Â So to be clear, you are saying that V (the velocity of the car) and R (the radius of the turn) can be related in different ways depending on what the driver was controlling, but that the V and R measures computed by power law researchers are constrained in a particular way to be in a 1/3 power relation?

BA:  I agree that V and R are measures of the controlled result of the driver’s navigating.  I don’t see how this fact makes any difference to the fact that the driver could vary either speed or rate of turn (as computed from the data and the power researchers’ equations) in a variety of relations, none of which necessarily conforms to a power law. Velocity is speed and R is a measure of turning radius. You could measure these with a ruler and compass and get the same results.

BA: I for one am having trouble understanding how you can simultaneously hold that V both depends on R and varies independently of R.

RM: I don’t believe V varies independently of R. I have shown that V (as measured by power law researchers) is related to R (as measured by power law researchers) by the equation:

V = D1/3 *R1/3

BA: How can this relationship be both a “statistical artifact� that holds constant for all curves, regardless of how quickly or slowly they are traversed, and an observed relationship between R and V that may or may not conform to the power law?

RM: This is a hard question to answer because the statistical artifact doesn’t “hold constant” for all curves. What holds constant for all curves is the fact that

V = D1/3 *R1/3

RM: The statistical artifact is the value of the power coefficient of R that is found from regression analysis using only log(R) to predict log(V). The power coefficient that is found as the solution to the regression equation (and the fit of the power law to the data in terms of the size of R^2 for this regression) will not be constant but will vary depending on the pattern of movement analyzed. For elliptical movements the regression of log(R) on log(V) will yield a power coefficient that will be close to .33; for movements that deviate from perfectly elliptical, such as elliptical movements made underwater, the regression of log(R) on log(V) will yield a power coefficient that will be close to .25; other movement patterns result in other power coefficients. If a person could move their hand in a perfect circle the power coefficient for R on V would be 0; but for C on V it would be 1.0. Even the slightest deviation from circularity will bring these coefficients to their “expected” values: .33 and .67 respectively.

BA: You have been led astray by a coincidence. When you create an ellipse by incrementing in constant degrees per iteration you get a power law with a 1/3 exponent. You take this as proof that the equations used by power law researchers must produce a power law relationship between their observed V and calculated R (with D included in the regression).

BA: What you are missing is the understanding that in those calculations of R in which R = V³/D, V and D change together in such a way as to make R potentially independent of V. Consequently, when researchers compute the velocity of a fingertip and the radius of curvature based on the changing values of x and y as a curve is drawn, the equation for computing R in no way enforces any particular relation between R and V. However, when you draw an ellipse in your spreadsheet by incrementing an angle a constant amount per iteration, you thereby enforce the very relationship you believe to be dictated by those equations.

BA: And while you are at it, please tell me what you think this magical “Dâ€? variable represents – the one that, when included in the regression, raises R2 to 1.0. Couch your explanation in terms of how D is calculated, not in terms of what it does when added to your regression equation.

RM: As Richard Kennaway pointed out:

RK: where D is defined as |XdotYdotdot - XdotdotYdot|

we can recognise D as being the magnitude of the cross product of V = (Xdot,Ydot) and the acceleration Vdot = (Xdotdot,Ydotdot).

RM: So that’s what D is; the cross product of V and Vdot. As you instruct, I will say nothing about what D does when added to the regression. Although I will point out that, as Richard K. also noted, D is a constant for some movement patterns (such the circle traced out by a pendulum swinging in three dimensions) and a variable for others (such as the elliptical motion of the planets). The implications of this for regression analysis should be obvious.

BA: But it’s important to understand that the cross product of V and Vdot, when divided into V, gives the radius of curvature, a value that has nothing to do with velocity (it’s just a length). Because velocity (decomposed into x and y components) appears in D (along with acceleration, similarly decomposed), you have velocity (and its derivative) on both sides of your equation, V = D1/3 *R1/3. Which is why solving it for V as you do makes little sense, as if V is forever locked into a 1/3 power relation with R.

BA: Power law researchers use the equation for R to get curvature from V and D. The V is computed directly from the changes in the position of the observed point along its trajectory. D is computed from that same velocity and the accelerations. The V observed can in principle be any value no matter what the R; they are the same variables recorded in the data of the car rounding a curve,  or in a the CVs of a control system model designed to explain the relationship between the observed V and R. Their relationship is in no way an artifact of the way R is computed from the data.

Bruce

[Abbott]

[From Bruce Abbott (2016.08.11.1830 EDT) –

Rick Marken (2016.08.11.1235)]

RM: Errata: I said:

If a person could move their hand in a perfect circle the power coefficient for R on V would be 0; but for C on V it would be 1.0.

RM: I should have said:

RM: If a person could move their hand in a perfect circle the power coefficient for R on V would be 0; but for C on A it would be 1.0

PM. You can’t be too careful around opponents of PCT!

BA:Â Yes, and I am indeed a staunch opponent of Perniciously Confused Thinking (my own included).

Bruce

[rsmarken]

[From Rick Marken (2016.08.12.0830)]

···

Bruce Abbott (2016.08.11.1820 EDT)]

BA: Your V is computed in the same way they compute theirs. Your results, however, are due to the way you computed your data points that describe an ellipse. Theirs is an empirical relationship in which V and R could turn out to have any relationship, not just a power one. Â

RM: I’ve used all kinds of movement patterns, including human movements, and I get the same results that the power law researchers get. So before we continue this discussion I really need to see your model of movement control showing how you think the power law fits into it. Otherwise I have no idea why you think the results of my model are wrong. Again, here’s my model:Â

RM: It says that intentionally produced curved movements (variations in q.ix, q.iy) are a controlled variable  and the observed power relationship (or lack thereof) between measures of curvature (R or C) and measures of velocity (V or A) of the variations in q.ix, q.iy are a statistical artifact that results from using regression analysis to estimate the relationship between two variables that are related mathematically while failing to include the third variable that is a mathematical component of that relationship.Â

RM: What is your model of how the power law fits into movement control?Â

Â

BA:Â So to be clear, you are saying that V (the velocity of the car) and R (the radius of the turn) can be related in different ways depending on what the driver was controlling,

RM: Yes, different patterns of variation of q.ix, q.iy will result in different relationships between R and V (and between D and V and R and D). Therefore, a researcher will calculate  different values for the power coefficient (beta) relating log (R) to log (V) for different movement patterns when using linear regression that leaves the variable log(D) out of the analysis.

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BA: but that the V and R measures computed by power law researchers are constrained in a particular way to be in a 1/3 power relation?

RM: No. The power coefficient is not constrained to be any particular value. Indeed, if people could produce perfect circles the power coefficient would be found to be 0.0. My model predicts that the power coefficient found for human movement depends completely on the type of movement produced. And so far my tests confirm this – circular movements produce a coefficient near .33; back and forth movement produce a coefficient near .5. Other movements produce other values of the coefficient. Of course, when the variable D is taken into account in the regression, the coefficient of both R and D for all movement patterns is exactly .33 and the R^2 is 1.0.Â

RM: So, again, before we go on, please show me your model of how the power law fits into movement control.Â

BestÂ

Rick

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BA:  I agree that V and R are measures of the controlled result of the driver’s navigating. I don’t see how this fact makes any difference to the fact that the driver could vary either speed or rate of turn (as computed from the data and the power researchers’ equations) in a variety of relations, none of which necessarily conforms to a power law. Velocity is speed and R is a measure of turning radius. You could measure these with a ruler and compass and get the same results.

Â

BA: I for one am having trouble understanding how you can simultaneously hold that V both depends on R and varies independently of R.Â

Â

RM: Â I don’t believe V varies independently of R. I have shown that V (as measured by power law researchers) is related to R (as measured by power law researchers) by the equation:

Â

V = Â D1/3Â *R1/3Â Â Â

 Â

BA: How can this relationship be both a “statistical artifact� that holds constant for all curves, regardless of how quickly or slowly they are traversed, and an observed relationship between R and V that may or may not conform to the power law?

RM: This is a hard question to answer because the statistical artifact doesn’t “hold constant” for all curves. What holds constant for all curves is the fact thatÂ

Â

V = Â D1/3Â *R1/3Â Â Â

Â

RM: The statistical artifact is the value of the power coefficient of R that is found from regression analysis using only log(R) to predict log(V). The power coefficient that is found as the solution to the regression equation (and the fit of the power law to the data in terms of the size of R^2 for this regression) will not be constant but will vary depending on the pattern of movement analyzed. For elliptical movements the regression of log(R) on log(V) will yield a power coefficient that will be close to .33; for movements that deviate from perfectly elliptical, such as elliptical movements made underwater, the regression of log(R) on log(V) will yield a power coefficient that will be close to .25; other movement patterns result in other power coefficients. If a person could move their hand in a perfect circle the power coefficient for R on V would be 0; but for C on V it would be 1.0. Even the slightest deviation from circularity will bring these coefficients to their “expected” values: .33 and .67 respectively.Â

Â

BA: You have been led astray by a coincidence. When you create an ellipse by incrementing in constant degrees per iteration you get a power law with a 1/3 exponent. You take this as proof that the equations used by power law researchers must produce a power law relationship between their observed V and calculated R (with D included in the regression).

Â

BA: What you are missing is the understanding that in those calculations of R in which R = V³/D, V and D change together in such a way as to make R potentially independent of V. Consequently, when researchers compute the velocity of a fingertip and the radius of curvature based on the changing values of x and y as a curve is drawn, the equation for computing R in no way enforces any particular relation between R and V. However, when you draw an ellipse in your spreadsheet by incrementing an angle a constant amount per iteration, you thereby enforce the very relationship you believe to be dictated by those equations.

Â

BA: And while you are at it, please tell me what you think this magical “Dâ€? variable represents – the onee that, when included in the regression, raises R2 to 1.0. Couch your explanation in terms of how D is calculated, not in terms of what it does when added to your regression equation.

Â

RM: As Richard Kennaway pointed out:

Â

RK: where D is defined as |XdotYdotdot - XdotdotYdot|

 we can recognise D as being the magnitude of the cross product of V = (Xdot,Ydot) and the acceleration Vdot = (Xdotdot,Ydotdot).

Â

RM: So that’s what D is; the cross product of V and Vdot. As you instruct, I will say nothing about what D does when added to the regression. Although I will point out that, as Richard K. also noted, D is a constant for some movement patterns (such the circle traced out by a pendulum swinging in three dimensions) and a variable for others (such as the elliptical motion of the planets). The implications of this for regression analysis should  be obvious.

Â

BA: But it’s important to understand that the cross product of V and Vdot, when divided into V, gives the radius of curvature, a value that has nothing to do with velocity (it’s just a length). Because velocity (decomposed into x and y components) appears in D (along with acceleration, similarly decomposed), you have velocity (and its derivative) on both sides of your equation, V =  D1/3 *R1/3. Which is why solving it for V as you do makes little sense, as if V is forever locked into a 1/3 power relation with R.

Â

BA: Power law researchers use the equation for R to get curvature from V and D. The V is computed directly from the changes in the position of the observed point along its trajectory. D is computed from that same velocity and the accelerations. The V observed can in principle be any value no matter what the R; they are the same variables recorded in the data of the car rounding a curve,  or in a the CVs of a control system model designed to explain the relationship between the observed V and R. Their relationship is in no way an artifact of the way R is computed from the data.

Â

Bruce

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–
Richard S. MarkenÂ

“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

[_Erling_Jorgensen]

[From Erling Jorgensen (2016.0812 1135 EDT)]

Rick Marken (2016.08.12.0830)

Bruce Abbott (2016.08.11.1820 EDT)

Hi Rick,

Would you respond to the following part of Bruce’s post? This is the part where I keep getting hung up, in trying to follow these postings.

Because velocity (decomposed into x and y components) appears in D (along with acceleration, similarly decomposed), you have velocity (and its derivative) on both sides of your equation, V = D1/3 *R1/3. Which is why solving it for V as you do makes little sense…

Thanks.

Erling

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