# Testing the Curvature- Velocity Power Law (was Velocity versus "Velocity")

[From Rick Marken (2016.07.27.1220)]

···

Martin Taylor (2016.07.27.14.26)–

``````Typo. You mean a multiplication sign, as in earlier messages, don't
``````

you?

RM: Yes, sorry. Getting logged out;-) Correct form is:

or

V = D1/3.* R1/3

``````MT: Regardless of that, you simply
``````

can’t call it an equation that presents V as a function of R when
the expression for V includes terms in dV/dt .

MT: You just can’t.

RM: Actually, I can and I’ll show you how it works when I distribute the spreadsheet version (the last one I sent had an error in it).

RM: By the way, what do you think the power law shows that is relevant to 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

V = |dx/dt*d2y/dt2 - d2X/dt2*dy/dt|1/3. +
R1/3

V = |dx/dt*d2y/dt2 - d2X/dt2dy/dt|1/3. R1/3

[Martin Taylor 2017.07.27.17.17]

[From Rick Marken (2016.07.27.1220)]

``````If your spreadsheet is designed to solve the differential equation,
``````

then I will grant that you can. Otherwise, you can, in the same
sense that you can ignore a red light, but the result in either case
may be not quite what you would hope for.

``````Nothing. But PCT may (and I expect will) be able to solve the three
``````

decade-old problem of why the along-track speed of so many curved
tracks has the same power law relation as does the shape
relationship between curvature and rate of change of tangent angle
per unit along-track distance. That’s unlikely to be a coincidence,
but that it doesn’t happen under specific conditions should be a
clue as to what is being controlled that makes it it happen in so
many cases. Your studies of the power relation for shapes isn’t
going to tell us anything about it.

``````What we need is an application of the Test for the Controlled
``````

Variable. I proposed disturbing the perception of the curve ahead of
the current position of the moving body. Partly I suggested that
because of my strong feeling as a driver that if I can’t see a few
tens of yards ahead (such as in a fog or around a cliffside corner)
I go very slowly. Maybe there are other hypothetical perceived and
controlled variables that could be disturbed, but at the moment only
viscosity has been used as a disturbance, and so far as I know, it
hasn’t been proposed as a perceived and controlled variable. It does
affect the observed performance, though. What does the observer see
as stabilized against this kind of disturbance, and why? That’s the
kind of question I would be asking, rather than endlessly
rearranging the equations that describe curvature.

``````Martin
``````
···

Martin Taylor (2016.07.27.14.26)–

``````            Typo. You mean a multiplication sign, as in
``````

earlier messages, don’t you?

RM: Yes, sorry. Getting logged out;-) Correct form is:

or

V = D1/3.* R1/3

``````          </sup>
``````
``````            MT: Regardless of that, you simply
``````

can’t call it an equation that presents V as a function
of R when the expression for V includes terms in dV/dt .

``````            MT: You just can't.
``````
``````          RM: Actually, I can and I'll show you how it works when
``````

I distribute the spreadsheet version (the last one I sent

V = |dx/dt*d2y/dt2 -
d2X/dt2*dy/dt|1/3. +
R1/3

V = |dx/dt*d2y/dt2 -
d2X/dt2*dy/dt|1/3. *
R1/3

``````                </sup>

RM: By the way, what do you think the power law shows
``````

that is relevant to 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

Hi Rick, my block is more about my own mental simulation than being precious about earlier research. You know I am not. Our joint papers reveal that.

It’s just that when I do a fixed curve myself I know that I can change my velocity at will. So that doesn’t obey the law. Are you saying that ‘to the extent that movement obeys the power law then it is a property of curved motion at a given on-track velocity?’ Thus, deviations from the power law would indicate either some purposeful change in velocity for the same curvature (by a PCT system) or a shift in the physics of the environment that the organism cannot immediately counter. I might agree with that. ‘On-track velocity’ is the velocity they your car would register going round a curve, irrespective of its angular velocity. Caravan go different speeds round the same curve.

So please put our thoughts about the history of science aside and help me get my head round what you are saying (and what you are not saying).

Warren

···

On Wed, Jul 27, 2016 at 9:56 AM, Warren Mansell wmansell@gmail.com wrote:

W: Hi Rick, what variable is D though, in a PCT loop?

RM: D is not a variable in a PCT loop. D is the term in my derivation of the mathematical relationship between V
and R (or A and C) for any curved movement regardless of how its produced. Here’s the derivation again:

V = D1/3 *R1/3 and A = D1/3 *C2/3

RM: where A = V/R , C = 1/R and D = |X.dotY.2dot-X.2dotY.dot|

RM: So in a log-log regression, for any curve you will find that:

V = .33* log ( |X.dotY.2dot-X.2dotY.dot|) +.33*log (R)

accounts for all the variance in the curve: R^2 = 1.0.

RM: If you just use log (R) as the predictor, you will get coefficients other than .33 as the power coefficient for log (R) and that’s because the variance of |X.dotY.2dot-X.2dotY.dot| is being absorbed into the constant log (K).

WM: I am a bit worried you are experiencing the behavioral illusion.

RM: No, you just don’t understand my analysis. I think it’s because, like Martin and Bruce A., you just can’t believe that 40+ years of power law research that was was aimed at determining how organisms produce curved movement could not possibly have revealed anything about how organisms produce curved movements.

WM: Are you not assuming that the trace of a larva is the purpose of the larva?

RM: Absolutely not! I’m saying that any curved movement will show a power law relationship, whether the curved movement was intentionally produced or a side effect of doing something else (as with the larvae or agents in the crowd program) or produced by a robot, a software simulation or equations. The power law is just a mathematical property of curve
movement; it has nothing to do with how those movements are produced.

WM: Surely the larva cannot plot its movements in X and Y dimensions from an observer’s perspective like in your model? Surely the little larva is controlling for some perception much simpler from it own perspective? Like forward motion on the retina? Then it might control for more intense food smells and veer off in one direction from this path. But it doesn’t have a reference perception of it’s aerial X and Y coordinates does it? Do we necessarily? And other animals?

RM:No. I just think you have a very strong desire to not offend anyone who shows an interest in PCT. But one
reason I like the discovery about the power law is that it reveals, more clearly than anything I’ve ever seen, why scientific psychologists – even ones who are ostensibly fans of PCT – always get all upset about PCT at some point. It’s because PCT always ends up shows that something that some scientific psychologist considers to be holy writ is not even true.

Best regards

Rick

Warren

On 27 Jul 2016, at 17:30, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.27.0930)]

Dr Warren Mansell
School of Psychological Sciences
2nd Floor Zochonis Building
University of Manchester
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - Principles-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check www.pctweb.org for further information on Perceptual Control Theory

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

On Wed, Jul 27, 2016 at 3:55 AM, Warren Mansell wmansell@gmail.com wrote:

WM: …except you don’t always find it because of all those exceptions which we need to understand…

RM: If you mean you don’t always find a perfect 1/3 (.33) power law (for V versus R; let’s stick to that for now) then this observation is explained by my analysis which shows that the relationship between V and R for any movement pattern is:

log V = .33log (D) + .33 log (R).&nbsp
;

RM: where D is a variable.

RM: The power law is tested using just log(R) as a predictor of log (V) in the regression analysis. So the regression equation is:

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

RM: where K is a constant. This suggests to me that the regression analysis will only yield a b value that is exactly equal to .33 when the average variance of the .33*log(D) value for a movement pattern allows the regression to find a value for the constant log(K) that results in a b value of exactly .33. Based on the data below, it looks like that is the case:

<image.png>

RM: These a
re r
esults for different ellipses drawn by a simulation model. Notice that the variation in the b value estimates correspond to variation in the log (K) value estimates. Indeed, there is a pretty strong negative correlation (-.63) between the log (K) values and deviations of the b values from .33. This suggests that, indeed, leaving the log(D) term out of the regression is the reason for the variation in estimates of the b value in the power law research

RM: If there is anyone out there who understands my analysis of the power law – especially one who is a skilled mathematician who understands multiple regression analysis in linear algebraic detail (I’m looking at you Richard Kennaway) – it would be nice if you could show exactly how differences in the log (D) term in the equation that describes the relat
ionship between log (V) and log (R) for any figure will affect the value of log(K) and b that is found in a simple linear regression of log (R) on log (V) when log (D) is not included as a predictor variable.

Best regards

Rick

… it doesn’t it mean that V and A have opposing relationships with curvature but as V and A are themselves related, doesn’t this lead to a conflict situation?

Warren

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

On Tue, Jul 26, 2016 at 11:36 PM, Richard Marken &l
t;rsmarken@gmail.com> wrote:

[From Rick Marken (2016.07.26.1535)]

On Tue, Jul 26, 2016 at 3:23 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Mmm, one more thing…

RM: C = 1/R

RM: I guess if R is
a measure of radius then curvature is inversely related to radius. And maybe curvat
ure does increase with angular velocity and decrease with on-track velocity (whatever that is). And maybe they (whoever they are) do meet in the middle or have some sort of pay off against one another. But, again, what does this have to do with the fact that the power law is a mathematical property of movement in curved paths and has nothing to do with how those movements were created? As I said once before, you will find the 1/3 power law (between V and R or 2/3 between A and C) for any curve, whether it was produced by an equation, a robot, a fly larva or a homicide detective lighting his cigar.( I do feel a lot like one of Colombo’s suspects being toyed with;-)

Best

Rick

Dr Warren Mansell
School of Psychological Sciences
2nd Floor Zochonis Building
University of Manchester
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - Principles-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check www.pctweb.org for further information on Perceptual Control Theory

On Tue, Jul 26, 2016 at 10:58 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.26.1455)]

RM: If you were Colombo you would have prefaced your question with “Oh, one more thing…”

RM: Anyway, my evasive answer is: I have no idea what you’re talking about. Where did I say anything about radius being inversely proportional to curvature? Or about on-track and angular velocity? And what does whether or not they meet in the middle have to do with the fact that the power law tells us nothing about how people produce curved movement? Is it somet
hing you want to bring home to Mrs Columbo?

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

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:
WM: Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, th
is means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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 Rick Marken (2016.07.28.0805)]

···

Martin Taylor (2017.07.27.17.17)–

``````MT: Nothing. But PCT may (and I expect will) be able to solve the three
``````

decade-old problem of why the along-track speed of so many curved
tracks has the same power law relation as does the shape
relationship between curvature and rate of change of tangent angle
per unit along-track distance. …

RM: That’s seems like a bit more than “nothing”; nothing will come of nothing, you know;-) But I thought the question was how PCT would explain the consistently observed power law relationship between curvature and velocity that is observed in studies of movement control. Is that the same as what you say the question is: how PCT explains why the along-track speed of so many curved tracks has the same power law relation as does the shape relationship between curvature and rate of change of tangent angle per unit along-track distance…?

``````MT: What we need is an application of the Test for the Controlled
``````

Variable. I proposed disturbing the perception of the curve ahead of
the current position of the moving body. Partly I suggested that
because of my strong feeling as a driver that if I can’t see a few
tens of yards ahead (such as in a fog or around a cliffside corner)
I go very slowly.

RM: This has nothing to do with the power law. But it certainly is an interesting question. Drivers certainly do slow down around steep turns. They are doing it to control for some variable. So what variable do you think is being controlled. You have to come up with a hypothesized controlled variable in order to do the test. But I think we should take this up after we’ve agreed on the PCT explanation of the power law.

Best

Rick

``````Maybe there are other hypothetical perceived and
``````

controlled variables that could be disturbed, but at the moment only
viscosity has been used as a disturbance, and so far as I know, it
hasn’t been proposed as a perceived and controlled variable. It does
affect the observed performance, though. What does the observer see
as stabilized against this kind of disturbance, and why? That’s the
kind of question I would be asking, rather than endlessly
rearranging the equations that describe curvature.

``````Martin
``````

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: By the way, what do you think the power law shows
``````

that is relevant to 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

[Martin Taylor 2016.07.29.11.19]

[From Rick Marken (2016.07.28.0805)]

``````It's an interesting idea, that we can come up with a PCT explanation
``````

of some observation, and afterwards find out what variable is being
controlled. It seems to my idiosyncratic view of PCT that we start
from the observation that some variable seems to be being
controlled, do the TCV, which includes tests as to whether the
presumed controlling organism could perceive that variable and
influence it, and then do the theoretical work of modelling to see
whether we can produce a model that mimics the observed control
under all the different circumstances where we observe it.

``````(I hope you notice that I am agreeing with your dictum [From Rick Marken
(2016.07.06.1915)]: "RM:  Control is the fact to be explained;
``````

control of perception is the theory that explains that fact.")

``````Martin
``````
···

``````            MT: What we need is an application
``````

of the Test for the Controlled Variable. I proposed
disturbing the perception of the curve ahead of the
current position of the moving body. Partly I suggested
that because of my strong feeling as a driver that if I
can’t see a few tens of yards ahead (such as in a fog or
around a cliffside corner) I go very slowly.

``````          RM: This has nothing to do with the power law. But it
``````

certainly is an interesting question. Drivers certainly do
slow down around steep turns. They are doing it to control
for some variable. So what variable do you think is being
controlled. You have to come up with a hypothesized
controlled variable in order to do the test. But I think
we should take this up after we’ve agreed on the PCT
explanation of the power law.

[From Rick Marken (2016.07.28.0845)]

···

On Thu, Jul 28, 2016 at 12:02 AM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, my block is more about my own mental simulation than being precious about earlier research. You know I am not. Our joint papers reveal that.

It’s just that when I do a fixed curve myself I know that I can change my velocity at will. So that doesn’t obey the law.

RM: But it does. Or, at least, I’m sure it would fit the power law if you would send me the X,Y coordinates of the movement over time. Again, here is a movement pattern I made:

RM: The top graph is the spatial pattern and the lower graph is the temporal pattern of movement in X, Y coordinates. The time trace shows that the speed of movement was changing throughout. The power law coefficient resulting from regressing log (R) on log (V) was .34. The R^2 is only .64. When the log of the D variable is included in the regression the power coefficients for both log(R) and log (D) are exactly .3333 and the R^2 of the regression is 1.0, as predicted by my derivation of the relationship between R and V.

WM: Are you saying that ‘to the extent that movement obeys the power law then it is a property of curved motion at a given on-track velocity?’

RM: No. I’m saying that all curved motion seems to obey the power law.

WM: Thus, deviations from the power law would indicate either some purposeful change in velocity for the same curvature (by a PCT system) or a shift in the physics of the environment that the organism cannot immediately counter.

RM: No, I’m saying that the power law has nothing to do with how the curved movement is produced.

WM: I might agree with that. ‘On-track velocity’ is the velocity they your car would register going round a curve, irrespective of its angular velocity. Caravan go different speeds round the same curve.

RM: Right. But if you measured R and V for the pattern of movement traced out by the car you would find the power law relationship between V and R (or A and C). It’s just a mathematical property of curved paths. You slow down and vary your angle of attack through curves because you are controlling your path against force disturbances like gravity, friction, centripetal forces, etc. How we do that is a really interesting question and should be studied using the test, as Martin suggested. But the resulting path of the car will obey the power law because the power law is a property of curves paths, regardless of how they were created!! You will find the power law whether the path is a controlled variable (as it is with hand movements and car paths) or not (as in the case of fly larvae and the agents in the crowd demo).

WM: So please put our thoughts about the history of science aside and help me get my head round what you are saying (and what you are not saying).

RM: I hope the above helps. I know it’s kind of astounding to contemplate but, when you think of it, it’s really basic PCT. Movement paths are just something organisms do (intentionally or not) and PCT tells us that you can’t tell what organisms are doing (or how they are doing it) by just looking at what they are doing.

RM: Oh, and I’m still tuning up my spreadsheet demo. I think it could be used to help you understand what’s going on. And I think the best way to do this is for you to record some movement patterns that you think would violate the power law – get at least 1000 equally temporally spaced samples of X,Y points – and lets see whether it does, indeed, violate the power law.

Best regards

Rick

On 27 Jul 2016, at 18:43, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.27.1040]

On Wed, Jul 27, 2016 at 9:56 AM, Warren Mansell wmansell@gmail.com wrote:

W: Hi Rick, what variable is D though, in a PCT loop?

RM: D is not a variable in a PCT loop. D is the term in my derivation of the mathematical relationship between V
and R (or A and C) for any curved movement regardless of how its produced. Here’s the derivation again:

V = D1/3 *R1/3 and A = D1/3 *C2/3

RM: where A = V/R , C = 1/R and D = |X.dotY.2dot-X.2dotY.dot|

RM: So in a log-log regression, for any curve you will find that:

V = .33* log ( |X.dotY.2dot-X.2dotY.dot|) +.33*log (R)

accounts for all the variance in the curve: R^2 = 1.0.

RM: If you just use log (R) as the predictor, you will get coefficients other than .33 as the power coefficient for log (R) and that’s because the variance of |X.dotY.2dot-X.2dotY.dot| is being absorbed into the constant log (K).

WM: I am a bit worried you are experiencing the behavioral illusion.

RM: No, you just don’t understand my analysis. I think it’s because, like Martin and Bruce A., you just can’t believe that 40+ years of power law research that was was aimed at determining how organisms produce curved movement could not possibly have revealed anything about how organisms produce curved movements.

WM: Are you not assuming that the trace of a larva is the purpose of the larva?

RM: Absolutely not! I’m saying that any curved movement will show a power law relationship, whether the curved movement was intentionally produced or a side effect of doing something else (as with the larvae or agents in the crowd program) or produced by a robot, a software simulation or equations. The power law is just a mathematical property of curve
movement; it has nothing to do with how those movements are produced.

WM: Surely the larva cannot plot its movements in X and Y dimensions from an observer’s perspective like in your model? Surely the little larva is controlling for some perception much simpler from it own perspective? Like forward motion on the retina? Then it might control for more intense food smells and veer off in one direction from this path. But it doesn’t have a reference perception of it’s aerial X and Y coordinates does it? Do we necessarily? And other animals?

RM:No. I just think you have a very strong desire to not offend anyone who shows an interest in PCT. But one
reason I like the discovery about the power law is that it reveals, more clearly than anything I’ve ever seen, why scientific psychologists – even ones who are ostensibly fans of PCT – always get all upset about PCT at some point. It’s because PCT always ends up shows that something that some scientific psychologist considers to be holy writ is not even true.

Best regards

Rick

Warren

On 27 Jul 2016, at 17:30, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.27.0930)]

Dr Warren Mansell
School of Psychological Sciences
2nd Floor Zochonis Building
University of Manchester
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - Principles-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check www.pctweb.org for further information on Perceptual Control Theory

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

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

On Wed, Jul 27, 2016 at 3:55 AM, Warren Mansell wmansell@gmail.com wrote:

WM: …except you don’t always find it because of all those exceptions which we need to understand…

RM: If you mean you don’t always find a perfect 1/3 (.33) power law (for V versus R; let’s stick to that for now) then this observation is explained by my analysis which shows that the relationship between V and R for any movement pattern is:

log V = .33log (D) + .33 log (R).&nbsp
;

RM: where D is a variable.

RM: The power law is tested using just log(R) as a predictor of log (V) in the regression analysis. So the regression equation is:

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

RM: where K is a constant. This suggests to me that the regression analysis will only yield a b value that is exactly equal to .33 when the average variance of the .33*log(D) value for a movement pattern allows the regression to find a value for the constant log(K) that results in a b value of exactly .33. Based on the data below, it looks like that is the case:

<image.png>

RM: These a
re r
esults for different ellipses drawn by a simulation model. Notice that the variation in the b value estimates correspond to variation in the log (K) value estimates. Indeed, there is a pretty strong negative correlation (-.63) between the log (K) values and deviations of the b values from .33. This suggests that, indeed, leaving the log(D) term out of the regression is the reason for the variation in estimates of the b value in the power law research

RM: If there is anyone out there who understands my analysis of the power law – especially one who is a skilled mathematician who understands multiple regression analysis in linear algebraic detail (I’m looking at you Richard Kennaway) – it would be nice if you could show exactly how differences in the log (D) term in the equation that describes the relat
ionship between log (V) and log (R) for any figure will affect the value of log(K) and b that is found in a simple linear regression of log (R) on log (V) when log (D) is not included as a predictor variable.

Best regards

Rick

… it doesn’t it mean that V and A have opposing relationships with curvature but as V and A are themselves related, doesn’t this lead to a conflict situation?

Warren

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

On Tue, Jul 26, 2016 at 11:36 PM, Richard Marken &l
t;rsmarken@gmail.com> wrote:

[From Rick Marken (2016.07.26.1535)]

WM: Mmm, one more thing…

RM: C = 1/R

On Tue, Jul 26, 2016 at 3:23 PM, Warren Mansell wmansell@gmail.com wrote:
RM: I guess if R is
a measure of radius then curvature is inversely related to radius. And maybe curvat
ure does increase with angular velocity and decrease with on-track velocity (whatever that is). And maybe they (whoever they are) do meet in the middle or have some sort of pay off against one another. But, again, what does this have to do with the fact that the power law is a mathematical property of movement in curved paths and has nothing to do with how those movements were created? As I said once before, you will find the 1/3 power law (between V and R or 2/3 between A and C) for any curve, whether it was produced by an equation, a robot, a fly larva or a homicide detective lighting his cigar.( I do feel a lot like one of Colombo’s suspects being toyed with;-)

Best

Rick

Dr Warren Mansell
School of Psychological Sciences
2nd Floor Zochonis Building
University of Manchester
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - Principles-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check www.pctweb.org for further information on Perceptual Control Theory

On Tue, Jul 26, 2016 at 10:58 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.26.1455)]

RM: If you were Colombo you would have prefaced your question with “Oh, one more thing…”

RM: Anyway, my evasive answer is: I have no idea what you’re talking about. Where did I say anything about radius being inversely proportional to curvature? Or about on-track and angular velocity? And what does whether or not they meet in the middle have to do with the fact that the power law tells us nothing about how people produce curved movement? Is it somet
hing you want to bring home to Mrs Columbo?

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

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:
WM: Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, th
is means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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

Ok will do! Can you send me the software again to record movements as X and Y coordinates over time?

Warren

···

On Thu, Jul 28, 2016 at 12:02 AM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, my block is more about my own mental simulation than being precious about earlier research. You know I am not. Our joint papers reveal that.

It’s just that when I do a fixed curve myself I know that I can change my velocity at will. So that doesn’t obey the law.

RM: But it does. Or, at least, I’m sure it would fit the power law if you would send me the X,Y coordinates of the movement over time. Again, here is a movement pattern I made:

<image.png>

<image.png>

RM: The top graph is the spatial pattern and the lower graph is the temporal pattern of movement in X, Y coordinates. The time trace shows that the speed of movement was changing throughout. The power law coefficient resulting from regressing log (R) on log (V) was .34. The R^2 is only .64. When the log of the D variable is included in the regression the power coefficients for both log(R) and log (D) are exactly .3333 and the R^2 of the regression is 1.0, as predict
ed by my derivation of the relationship between R and V.

WM: Are you saying that ‘to the extent that movement obeys the power law then it is a property of curved motion at a given on-track velocity?’

RM: No. I’m saying that all curved motion seems to obey the power law.

WM: Thus, deviations from the power law would indicate either some purposeful change in velocity for the same curvature (by a PCT system) or a shift in the physics of the environment that the organism cannot immediately counter.

RM: No, I’m saying that the power law has nothing to do with how the curved movement is produced.

WM: I might agree with that. ‘On-track velocity’ is the velocity they your car would register going round a curve, irrespective of its angular velocity. Caravan go different speeds round the same curve.

RM: Right. But if you measured R and V for the pattern of movement traced out by the car you would find the power law relationship between V and R (or A and C). It’s just a mathematical property of curved paths. You slow down and vary your angle of attack through curves because you are controlling your path against force disturbances like gravity, friction, centripetal forces, etc. How we do that is a really inte
resting question and should be studied using the test, as Martin suggested. But the resulting path of the car will obey the power law because the power law is a property of curves paths, regardless of how they were created!! You will find the power law whether the path is a controlled variable (as it is with hand movements and car paths) or not (as in the case of fly larvae and the agents in the crowd demo).

WM: So please put our thoughts about the history of science aside and help me get my head round what you are saying (and what you are not saying).

RM: I hope the above helps. I know it’s kind of astounding to contemplate but, when you think of it, it’s really basic PCT. Movement paths are just something organisms do (intentionally or not) and PCT tells us that you can’t tell what organisms are doing (or how they are doing it) by just looking at what they are doing.

RM: Oh, and I’m still tuning up my spreadsheet demo. I think it could be used to help you understand what’s going on. And I think the best way to do this is for you to record some movement patterns that you think would violate the power law – get at least 1000 equally temporally spaced samples of X,Y points – and lets see whether it does, indeed, violate the power law.

Best regards

Rick

On 27 Jul 2016, at 18:43, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.27.1040]

On Wed, Jul 27, 2016 at 9:56 AM, Warren Mansell wmansell@gmail.com wrote:

W: Hi Rick, what variable is D though, in a PCT loop?

RM: D is not a variable in a PCT loop. D is the term in my derivation of the mathematical relationship between V
and R (or A and C) for any curved movement regardless of how its produced. Here’s the derivation again:

V = D1/3 *R1/3 and A = D1/3 *C2/3

RM: where A = V/R , C = 1/R and D = |X.dotY.2dot-X.2dotY.dot|

RM: So in a log-log regression, for any curve you will find that:

V = .33* log ( |X.dotY.2dot-X.2dotY.dot|) +.33*log (R)

accounts for all the variance in the curve: R^2 = 1.0.

RM: If you just use log (R) as the predictor, you will get coefficients other than .33 as the power coefficient for log (R) and that’s because the variance of |X.dotY.2dot-X.2dotY.dot| is being absorbed into the constant log (K).

WM: I am a bit worried you are experiencing the behavioral illusion.

RM: No, you just don’t understand my analysis. I think it’s because, like Martin and Bruce A., you just can’t believe that 40+ years of power law research that was was aimed at determining how organisms produce curved movement could not possibly have revealed anything about how organisms produce curved movements.

WM: Are you not assuming that the trace of a larva is the purpose of the larva?

RM: Absolutely not! I’m saying that any curved movement will show a power law relationship, whether the curved movement was intentionally produced or a side effect of doing something else (as with the larvae or agents in the crowd program) or produced by a robot, a software simulation or equations. The power law is just a mathematical property of curve
movement; it has nothing to do with how those movements are produced.

WM: Surely the larva cannot plot its movements in X and Y dimensions from an observer’s perspective like in your model? Surely the little larva is controlling for some perception much simpler from it own perspective? Like forward motion on the retina? Then it might control for more intense food smells and veer off in one direction from this path. But it doesn’t have a reference perception of it’s aerial X and Y coordinates does it? Do we necessarily? And other animals?

RM:No. I just think you have a very strong desire to not offend anyone who shows an interest in PCT. But one
reason I like the discovery about the power law is that it reveals, more clearly than anything I’ve ever seen, why scientific psychologists – even ones who are ostensibly fans of PCT – always get all upset about PCT at some point. It’s because PCT always ends up shows that something that some scientific psychologist considers to be holy writ is not even true.

Best regards

Rick

Warren

On 27 Jul 2016, at 17:30, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.27
.0930)]

Dr Warren Mansell
School of Psychological Sciences
2nd Floor Zochonis Building
University of Manchester
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - Principles-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check www.pctweb.org for further information on Perceptual Control Theory

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

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

On Wed, Jul 27, 2016 at 3:55 AM, Warren Mansell wmansell@gmail.com wrote:

WM: …except you don’t always find it because of all those exceptions which we need to understand…

RM: If you mean you don’t always find a perfect 1/3 (.33) power law (for V versus R; let’s stick to that for now) then this observation is explained by my analysis which shows that the relationship between V and R for any movement pattern is:

log V = .33log (D) + .33 log (R).&nbsp
;

RM: where D is a variable.

RM: The power law is tested using just log(R) as a predictor of log (V) in the regression analysis. So the regression equation is:

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

RM: where K is a constant. This suggests to me that the regression analysis will only yield a b value that is exactly equal to .33 when the average variance of the .33*log(D) value for a movement pattern allows the regression to find a value for the constant log(K) that results in a b value of exactly .33. Based on the data below, it looks like that is the case:

<image.png>

RM: These a
re r
esults for different ellipses drawn by a simulation model. Notice that the variation in the b value estimates correspond to variation in the log (K) value estimates. Indeed, there is a pretty strong negative correlation (-.63) between the log (K) values and deviations of the b values from .33. This suggests that, indeed, leaving the log(D) term out of the regression is the reason for the variation in estimates of the b value in the power law research

RM: If there is anyone out there who understands my analysis of the power law – especially one who is a skilled mathematician who understands multiple regression analysis in linear algebraic detail (I’m looking at you Richard Kennaway) – it would be nice if you could show exactly how differences in the log (D) term in the equation tha
t describes the relat
ionship between log (V) and log (R) for any figure will affect the value of log(K) and b that is found in a simple linear regression of log (R) on log (V) when log (D) is not included as a predictor variable.

Best regards

Rick

… it doesn’t it mean that V and A have opposing relationships with curvature but as V and A are themselves related, doesn’t this lead to a conflict situation?

Warren

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

On Tue, Jul 26, 2016 at 11:36 PM, Richard
Marken &l
t;rsmarken@gmail.com> wrote:

[From Rick Marken (2016.07.26.1535)]

WM: Mmm, one more thing…

RM: C = 1/R

On Tue, Jul 26, 2016 at 3:23 PM, Warren Mansell wmansell@gmail.com wrote:
RM: I guess if R is
a measure of radius then curvature is inversely related to radius. And maybe curvat
ure does increase with angular velocity and decrease with on-track velocity (whatever that is). And maybe they (whoever they are) do meet in the middle or have some sort of pay off against one another. But, again, what does this have to do with the fact that the power law is a mathematical property of movement in curved paths and has nothing to do with how those movements were created? As I said once before, you will find the 1/3 power law (between V and R or 2/3 between A and C) for any curve, whether it was produced by an equation, a robot, a fly larva or a homicide detective lighting his cigar.( I do feel a lot like one of Colombo’s suspects being toyed with;-)

Best

Rick

Dr Warren Mansell
School of Psychological Sciences
2nd Floor Zochonis Building
University of Manchester
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - Principles-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check www.pctweb.org for further information on Perceptual Control Theory

On Tue, Jul 26, 2016 at 10:58 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.26.1455)]

RM: If you were Colombo you would have prefaced your question with “Oh, one more thing…”

RM: Anyway, my evasive answer is: I have no idea what you’re talking about. Where did I say anything about radius being inversely proportional to curvature? Or about on-track and angular velocity? And what does whether or not they meet in the middle have to do with the fact that the power law tells us nothing about how people produce curved movement? Is it somet
hing you want to bring home to Mrs Columbo?

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

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:
WM: Hi Rick, I am Colombo
. So radius is inversely proportional to curvature? If so, th
is means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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 Rick Marken at the airport

I’m off to visit my 2 1/2 year old granddaughter and though she is very interested in robotics she will probably find the powerline stuff as uninteresting as does everyone else here on CSGNET. The program I used to collect the XY positions of my movement what the Nextel program that didn’t work very well. I will eventually write a program in JavaScript to collect the data but maybe you guys could do it over there with all those very bright programmers you have. It should be trivial my easy.

Best

Rick

···

On Thu, Jul 28, 2016 at 12:02 AM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, my block is more about my own mental simulation than being precious about earlier research. You know I am not. Our joint papers reveal that.

It’s just that when I do a fixed curve myself I know that I can change my velocity at will. So that doesn’t obey the law.

RM: But it does. Or, at least, I’m sure it would fit the power law if you would send me the X,Y coordinates of the movement over time. Again, here is a movement pattern I made:

<image.png>

<image.png>

RM: The top graph is the spatial pattern and the lower graph is the temporal pattern of movement in X, Y coordinates. The time trace shows that the speed of movement was changing throughout. The power law coefficient resulting from regressing log (R) on log (V) was .34. The R^2 is only .64. When the log of the D variable is included in the regression the power coefficients for both log(R) and log (D) are exactly .3333 and the R^2 of the regression is 1.0, as predict
ed by my derivation of the relationship between R and V.

WM: Are you saying that ‘to the extent that movement obeys the power law then it is a property of curved motion at a given on-track velocity?’

RM: No. I’m saying that all curved motion seems to obey the power law.

WM: Thus, deviations from the power law would indicate either some purposeful change in velocity for the same curvature (by a PCT system) or a shift in the physics of the environment that the organism cannot immediately counter.

RM: No, I’m saying that the power law has nothing to do with how the curved movement is produced.

WM: I might agree with that. ‘On-track velocity’ is the velocity they your car would register going round a curve, irrespective of its angular velocity. Caravan go different speeds round the same curve.

RM: Right. But if you measured R and V for the pattern of movement traced out by the car you would find the power law relationship between V and R (or A and C). It’s just a mathematical property of curved paths. You slow down and vary your angle of attack through curves because you are controlling your path against force disturbances like gravity, friction, centripetal forces, etc. How we do that is a really inte
resting question and should be studied using the test, as Martin suggested. But the resulting path of the car will obey the power law because the power law is a property of curves paths, regardless of how they were created!! You will find the power law whether the path is a controlled variable (as it is with hand movements and car paths) or not (as in the case of fly larvae and the agents in the crowd demo).

WM: So please put our thoughts about the history of science aside and help me get my head round what you are saying (and what you are not saying).

RM: I hope the above helps. I know it’s kind of astounding to contemplate but, when you think of it, it’s really basic PCT. Movement paths are just something organisms do (intentionally or not) and PCT tells us that you can’t tell what organisms are doing (or how they are doing it) by just looking at what they are doing.

RM: Oh, and I’m still tuning up my spreadsheet demo. I think it could be used to help you understand what’s going on. And I think the best way to do this is for you to record some movement patterns that you think would violate the power law – get at least 1000 equally temporally spaced samples of X,Y points – and lets see whether it does, indeed, violate the power law.

Best regards

Rick

On 27 Jul 2016, at 18:43, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.27.1040]

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

On Wed, Jul 27, 2016 at 9:56 AM, Warren Mansell wmansell@gmail.com wrote:

W: Hi Rick, what variable is D though, in a PCT loop?

RM: D is not a variable in a PCT loop. D is the term in my derivation of the mathematical relationship between V
and R (or A and C) for any curved movement regardless of how its produced. Here’s the derivation again:

V = D1/3 *R1/3 and A = D1/3 *C2/3

RM: where A = V/R , C = 1/R and D = |X.dotY.2dot-X.2dotY.dot|

RM: So in a log-log regression, for any curve you will find that:

V = .33* log ( |X.dotY.2dot-X.2dotY.dot|) +.33*log (R)

accounts for all the variance in the curve: R^2 = 1.0.

RM: If you just use log (R) as the predictor, you will get coefficients other than .33 as the power coefficient for log (R) and that’s because the variance of |X.dotY.2dot-X.2dotY.dot| is being absorbed into the constant log (K).

WM: I am a bit worried you are experiencing the behavioral illusion.

RM: No, you just don’t understand my analysis. I think it’s because, like Martin and Bruce A., you just can’t believe that 40+ years of power law research that was was aimed at determining how organisms produce curved movement could not possibly have revealed anything about how organisms produce curved movements.

WM: Are you not assuming that the trace of a larva is the purpose of the larva?

RM: Absolutely not! I’m saying that any curved movement will show a power law relationship, whether the curved movement was intentionally produced or a side effect of doing something else (as with the larvae or agents in the crowd program) or produced by a robot, a software simulation or equations. The power law is just a mathematical property of curve
movement; it has nothing to do with how those movements are produced.

WM: Surely the larva cannot plot its movements in X and Y dimensions from an observer’s perspective like in your model? Surely the little larva is controlling for some perception much simpler from it own perspective? Like forward motion on the retina? Then it might control for more intense food smells and veer off in one direction from this path. But it doesn’t have a reference perception of it’s aerial X and Y coordinates does it? Do we necessarily? And other animals?

RM:No. I just think you have a very strong desire to not offend anyone who shows an interest in PCT. But one
reason I like the discovery about the power law is that it reveals, more clearly than anything I’ve ever seen, why scientific psychologists – even ones who are ostensibly fans of PCT – always get all upset about PCT at some point. It’s because PCT always ends up shows that something that some scientific psychologist considers to be holy writ is not even true.

Best regards

Rick

Warren

On 27 Jul 2016, at 17:30, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.27
.0930)]

Dr Warren Mansell
School of Psychological Sciences
2nd Floor Zochonis Building
University of Manchester
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - Principles-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check www.pctweb.org for further information on Perceptual Control Theory

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

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

On Wed, Jul 27, 2016 at 3:55 AM, Warren Mansell wmansell@gmail.com wrote:

WM: …except you don’t always find it because of all those exceptions which we need to understand…

RM: If you mean you don’t always find a perfect 1/3 (.33) power law (for V versus R; let’s stick to that for now) then this observation is explained by my analysis which shows that the relationship between V and R for any movement pattern is:

log V = .33log (D) + .33 log (R).&nbsp
;

RM: where D is a variable.

RM: The power law is tested using just log(R) as a predictor of log (V) in the regression analysis. So the regression equation is:

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

RM: where K is a constant. This suggests to me that the regression analysis will only yield a b value that is exactly equal to .33 when the average variance of the .33*log(D) value for a movement pattern allows the regression to find a value for the constant log(K) that results in a b value of exactly .33. Based on the data below, it looks like that is the case:

<image.png>

RM: These a
re r
esults for different ellipses drawn by a simulation model. Notice that the variation in the b value estimates correspond to variation in the log (K) value estimates. Indeed, there is a pretty strong negative correlation (-.63) between the log (K) values and deviations of the b values from .33. This suggests that, indeed, leaving the log(D) term out of the regression is the reason for the variation in estimates of the b value in the power law research

RM: If there is anyone out there who understands my analysis of the power law – especially one who is a skilled mathematician who understands multiple regression analysis in linear algebraic detail (I’m looking at you Richard Kennaway) – it would be nice if you could show exactly how differences in the log (D) term in the equation tha
t describes the relat
ionship between log (V) and log (R) for any figure will affect the value of log(K) and b that is found in a simple linear regression of log (R) on log (V) when log (D) is not included as a predictor variable.

Best regards

Rick

… it doesn’t it mean that V and A have opposing relationships with curvature but as V and A are themselves related, doesn’t this lead to a conflict situation?

Warren

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

On Tue, Jul 26, 2016 at 11:36 PM, Richard
Marken &l
t;rsmarken@gmail.com> wrote:

[From Rick Marken (2016.07.26.1535)]

WM: Mmm, one more thing…

RM: C = 1/R

On Tue, Jul 26, 2016 at 3:23 PM, Warren Mansell wmansell@gmail.com wrote:
RM: I guess if R is
a measure of radius then curvature is inversely related to radius. And maybe curvat
ure does increase with angular velocity and decrease with on-track velocity (whatever that is). And maybe they (whoever they are) do meet in the middle or have some sort of pay off against one another. But, again, what does this have to do with the fact that the power law is a mathematical property of movement in curved paths and has nothing to do with how those movements were created? As I said once before, you will find the 1/3 power law (between V and R or 2/3 between A and C) for any curve, whether it was produced by an equation, a robot, a fly larva or a homicide detective lighting his cigar.( I do feel a lot like one of Colombo’s suspects being toyed with;-)

Best

Rick

Dr Warren Mansell
School of Psychological Sciences
2nd Floor Zochonis Building
University of Manchester
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Advanced notice of a new transdiagnostic therapy manual, authored by Carey, Mansell & Tai - Principles-Based Counselling and Psychotherapy: A Method of Levels Approach

Available Now

Check www.pctweb.org for further information on Perceptual Control Theory

On Tue, Jul 26, 2016 at 10:58 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.26.1455)]

RM: If you were Colombo you would have prefaced your question with “Oh, one more thing…”

RM: Anyway, my evasive answer is: I have no idea what you’re talking about. Where did I say anything about radius being inversely proportional to curvature? Or about on-track and angular velocity? And what does whether or not they meet in the middle have to do with the fact that the power law tells us nothing about how people produce curved movement? Is it somet
hing you want to bring home to Mrs Columbo?

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

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:
WM: Hi Rick, I am Colombo
. So radius is inversely proportional to curvature? If so, th
is means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

[From MK (2016.07.28.2100 CET)]

Rick Marken (2016.07.28.0845)--

You will find the power law whether the path is a controlled variable (as it is with hand movements and car paths) or not (as in the case of fly larvae and the agents in the crowd demo).

Didn't Alex write that he didn't observe the relationship in fish?

Perhaps he has discovered the krank-fish; the species of fish that never turns.

M

[Martin Taylor 2016.07.29.15.08]

``````
[From MK (2016.07.28.2100 CET)]
Rick Marken (2016.07.28.0845)--
``````
``````You will find the power law whether the path is a controlled variable (as it is with hand movements and car paths) or not (as in the case of fly larvae and the agents in the crowd demo).
``````
``````
Didn't Alex write that he didn't observe the relationship in fish?
Perhaps he has discovered the krank-fish; the species of fish that never turns.
M
``````
``````I very much doubt that what I write will help Rick, but in case it
``````

might help others to understand what is going on, I’m posting it
anyway. The bottom line is, as has often been pointed out, that Rick
is confusing a measure of curve shape with a measure of speed.
Here’s the derivation, slowly.

``````We start with the idea of a parameterized curve. What that means is
``````

that the curve is embedded in a space, in this case a plane in which
every point has an x-y coordinate pair. To describe the curve one
can pick a starting point on the curve and say that if one follows
the curve for a distance s, one will be at a point (x(s), y(s)). If
one is also keeping track of a clock, one could say that after one
has followed the curve for a time t, one will be at a point (x(t),
y(t)). Of course, to locate the point (x(t), y(t)) after an
arbitrary time t, you have to specify a velocity pattern or at least
the integral of velocity up to time t. The velocity pattern is
completely arbitrary except that it can’t go to zero or become
negative, but some patterns make calculations easier than do others.
The Wikipedia article on curvature uses a constant velocity of 1
distance unit per time unit, so that numerically t = s. It’s the
same curve, whichever way you choose to parameterize it.

``````How the "t" (time) parameter parameterizes the curve at any
``````

particular moment is given by v = ds/dt at that moment. The value of
v can be written in a variety of ways, one of which is v =
sqrt((dx/dt)2 + (dy/dt)2) because ds2
= dx2 + dy2, and another is v = sqrt((dx/ds)2
+ (dy/ds)2 )*ds/dt. The value of v at any point on the
curve is, of course, completely arbitrary, since you can choose to
speed up and slow down you parameterization anywhere along the
curve, so long as you don’t stop or back up. If you stopped or
backed up, the correspondence between s and t would no longer be
one-to-one.

``````It's the same curve and the same arbitrary velocity pattern
``````

whichever way you choose to write it. But the parameter has to
locate the curve in the embedding space at every point along the
curve. Every value of the parameter specifies a point on the curve
and every point on the curve specifies a unique value of the
parameter.

``````Let's suppose we have parameterized the curve by "z" other than "s"
``````

or “t”. For example, if the curve is an uphill road that has no
level or downhill stretches, z could be the altitude of the road at
any point along it. If we do that, we might coin a term “zelocity”
for v = ds/dz, the rate of change of s as a function of z, just as
“velocity” is a word for ds/dt. Using the “z” parameterization, v =
sqrt((dx/dz)2 + (dy/dz)2 ). It’s still the same
curve in the x-y plane, and the relation between z and (x. y) can be

``````Now curvature. Curvature is defined as 1/R where R is the radius of
``````

the “osculating circle”, the circle that just touches the curve and
has the same curvature as the curve at the point where they are
tangent to each other.

``````C=1/R.

I'm not even going to try to derive the next expression Rick uses.
``````

It’s in Wikipedia under “Curvature”.

``````C = (dx/dz*d<sup>2</sup>y/dz<sup>2</sup> -dy/dz*d<sup>2</sup>x/dz<sup>2</sup>    )
``````

/ ((dx/dz)2+(dy/dz2))3/2

``````</sup>    The numerator of this second expression is Rick's "D" factor,
``````

and the denominator is our arbitrary v, “zelocity”, cubed. Rick uses
the time parameterization of the curve, but that’s irrelevant. It’s
the same curve and has the same curvature however you parameterize
it.

``````So, we can write the second expression for the curvature more
``````

succinctly:

``````C = D/v<sup>3</sup>, or

R = v<sup>3</sup>/D, from which

v<sup>3</sup> = D*R or v = D<sup>1/3</sup>*R<sup>1/3</sup> for any
``````

curve, no matter how you parameterize it.

``````How you parameterize it does affect how you label things and how you
``````

label them affects how you think of them. With the z
parameterization it is “zelocity”. With a time parameterization, it
is “velocity”. The numerical value is the same either way, but if
you call it “velocity”, you can easily be misled into thinking it
says something about the speed with which the curve is drawn. It
doesn’t. It says something about the relation between the t
parameter and the distance along the curve, and that was set by the
arbitrary velocity pattern that defined the t parameterization, No
matter what the curve, the parameterization function v(s) that you
freely chose to link t to a position along the curve has that same
1/3 power relationship with the product of the local radius of
curvature and Rick’s D expression. It has nothing whatever to do
with the speed of something that might follow the curve in real
time, such as the pencil that draws it.

``````The point here is that everything about Rick's "behavioural
``````

illusion" has to do only with the shape of a curve described in
parametric form. As such it has been known for at least 30 years,
and probably much longer, and has nothing whatever to do with the
velocity of a thing that actually produces or follows the track in
real time.

``````Martin
``````

[From Rick Marken (2016.07.24.2150)]

(Attachment image0065.png is missing)

···

Fred Nickols (2016.07.24.1534 ET)

FN: It seems to me, Rick, that you might be onto something of extreme and profound significance and with considerable import for research and for PCT.

FN: I will continue following this thread. So, I imagine, will others.

RM: Thanks Fred. I do think it’s of considerable import for research on PCT. But I hope that it will also be useful to everyone interested in PCT. It’s really about the fact that understanding human behavior is a matter of understanding their controlling. What people do is control – we are controlling people (where have I heard that before;-) – and PCT explains the controlling that people do.

Best

Rick

Fred Nickols

From: Richard Marken [mailto:rsmarken@gmail.com]
Sent: Sunday, July 24, 2016 3:22 PM
To: csgnet@lists.illinois.edu
Cc: Richard Marken
Subject: Testing the Curvature- Velocity Power Law (was Velocity versus “Velocity”)

[From Rick Marken (2016.07.24.1220)]

Martin Taylor (16.07.20.21.19)

MT: If “V” is a true velocity, then the dots signify derivatives with respect to time. If “R” is the radius of curvature, it’s a length and the dots signify derivatives with respect to arc length along the curve. They aren’t the same thing, but you use your “kindergarten math” as though they are…

MT: I’m leaving the rest of my message in this one so you won’t have to look back a few hours in the archive if you ever decide you want to read the explanation of why I am quite certain your V = |dXd2Y-d2XdY| 1/3 *R1/3 formula not just wrong, but just so nonsensical as to be neither right nor wrong.

RM: The power law is tested using linear regression with log(R) as the predictor variable and log(V) as the criterion variable. The variables R (the measure of instantaneous curvature) and V (the measure of instantaneous velocity) are computed from the movements that produce the final pattern. For example, here is an example of an elliptical movement pattern:

RM: The top figure shows the two dimensional elliptical pattern created by the counter-clockwise movement of a point in the X and Y dimension simultaneously over time, as shown in the lower figure (the red line is movement in the X dimension and the black line is movement in the Y dimension).

RM: To do the regression of log(R) on log (V) the values of R and V are computed for each point in time during which the movement is made. I found the formulas for computing R and V in Gribble & Ostry (1996). J. Neurophysiology, 76(5), 2853-2860. The value of V (velocity) at each point in the movement pattern is computed as as follows:

and the value of R (curvature) is computed at each point as follows:

RM: These are formulas I used for computing V and R in my spreadsheet analysis of the power law. If R and V are related by a power law then there will be a linear relationship between log (R) and log (V) and the slope of the relationship will be a measure of the coefficient of the power function. Below is a graph of the relationship between log (R) and log (V) for an elliptical pattern of movement like that shown above:

RM: This graph is from Wann, Nimmo-Smith & Wing (1988).JEP: HPP, 14(4), 622. The two lines are for elliptical movements that were generated in two different ways: 1) using two sine waves (a Lissajous pattern) and 2) using a movement generation process called a “jerk” model. In both cases the relationship between log (R) and log (V) is precisely linear (R^2 = 1.0) and the coefficient of the power function is .33. I found the same relationship between log (R) and log (V) for ellipsoidal movement created by a control model, even for control models with different output and feedback functions.

RM: The fact that a power law (with a coefficient of .33) was found for elliptical movements that had been produced by many different processes suggested to me that the relationship between log (R) and log (V) may depend only on the nature of the movement pattern itself and not on how that movement pattern was produced. Since R and V are both measured from the movement pattern (the same values of the derivatives of X and Y movement are used in the computation of both R and V) I looked to see if there might be a mathematical relationship between V and R.

RM: Looking at the formulas for V and R I noticed that V2 = X.dot2+Y.dot2 , which is a term in the numerator in the formula for R. See for yourself in the equations for V and R above. So the equation for R can be re-wriitten as

R = (V2)3//2 |/ |X.dotY.2dot-X.2dotY.dot|

And from there it’s a couple steps to:

V = D1/3 *R1/3

where D = |X.dotY.2dot-X.2dotY.dot|

RM: So the math shows that there is a power relationship between R and V with a coefficient of .33 (1/3) which should hold for all two dimensional movement patterns as long as the variable D is taken into account. I tested this out for several movement patterns using multiple regression on the logs of variables R and D. The multiple regression equation was of the form:

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

where both a and b are predicted to be .33. The result for all movement patterns was that the regression picked up all the variation in log (V) (R^2 always = 1.0) and the a and b coefficients were .33.

RM: The research on the power law relationship between V and R has resulted in the finding of coefficients for a power law relationship between R and V other than .33 for patterns of movement that differ from an ellipse and for patterns produced by movements in contexts other than in air and by species other than humans. I thought this might be because these studies had included only log (R) as a predictor of log (V) in the regression analyses. And my initial spreadsheet analyses suggested that this was the case. But after rechecking and correcting some of my spreadsheet calculations I have found that regressing just log (R) on log (V) will result in an R^2 of 1.0 and a power coefficient of .33 for any pattern of movement as long as there is no point in the pattern where the first derivatives of X and Y are exactly equal. Why that should be, I don’t know. Perhaps someone who is more math savvy than I can figure it out.

RM: But here is an example of a movement pattern that results in a perfect power law relationship between V and R with a power coefficient of exactly 1/3.

RM: This movement pattern was created by a human (me) moving a mouse around the computer screen. Here’s the pattern of movement over time:

RM: A regression analysis of this pattern of movement with log (R) as the predictor and log (V) as the criterion results in am R^2 value of 1.0 and a b coefficient (the power law coefficient) of .33. I get the same results (R^2 = 1.0, b = .33) for every movement pattern I have analyzed , as long as there was no point in the pattern where the derivative of X, X.dot, equaled the derivative of Y, Y.dot.

RM: These results show that it is impossible to learn anything about how movements are produced by looking at the relationship between measures of the movements themselves. The observed relationship between variable aspect of the movement, such as V and R, reflects nothing more than a mathematical relationship which, in this case, can be written as

V = D1/3 *R1/3

RM: And this equation can be found by simply observing that the equation that defines V is part of the equation that defines R.

RM: So the big question is why did all the researchers in this area fail to see either that this mathematical relationship between V and R exists or, if they did know that it exists, why did they fail to see its implication, which is that any observed relationship between V and R is determined by math, not by anything about how the movement is generated? It’s certainly not because these researchers are not good at math; nor is it because they are not extremely intelligent. I think the only possible explanation – and the one that is very relevant to PCT – is that these researchers were (and still are) blinded by the wrong view of what behavior is: a view that sees behavior as a step – usually the last one – in a sequential causal process. They can’t help seeing a pattern of movement – in terms of the degree of curvature of that movement measured as the R (or C) – as caused by a pattern of forces that move a point though those curves at varying velocity (measured by V). I think this same view of behavior is the reason why my discovery of the mathematical relationship between V and R led to such dismay on CSGNet.

RM: The power law is important because it shows that you have to be able to look at behavior though control theory glasses – see that behavior is control – before you can correctly apply the theory of control – PCT – to behavior. The truth is that because I understood this fact about behavior I knew that the power law could tell us nothing about how organisms produce movement patterns before I discovered the mathematical relationship between R and V. When output is produced in a control loop you can’t “see” the output function that is producing the controlled result. This is one of the reasons why the conventional approach to research is guaranteed to produce results that are misleading (as per Powers 1978 Psych Review article). But it’s often very difficult to show this clearly to researchers. That’s why my discovery of the fact that there is a mathematical relationship between R and V was so exciting; it shows as clearly as I can imagine that looking at behavior – like drawing a squiggle pattern – as a step in a causal sequence is the wrong way to look at the behavior of a control system. And in this case, research based on this view of behavior is misleading in the worst possible way; it’s leading researchers to take a mathematical fact about curved movement as a fact of behavior to be explained.

RM: The power law research shows why PCT has had a hard time getting accepted (by mainstream psychologists) or understood (by many of its fans). PCT is a model of a different phenomenon than the one studied by mainstream psychologists. Mainstream psychologists, like the ones doing the power law research, are studying (and trying to explain) the phenomenon of output generation; control theorists (of the PCT persuasion) are studying (and trying to explain with PCT models) the phenomenon of input control. I think the power law can show why it’s hard for mainstream psychologists to get excited about a theory (PCT) that is not an explanation of the behavior they want to explain; indeed, PCT is not only an explanation of a phenomenon that is not the one that conventional psychologists want to explain; PCT also shows that the phenomenon that conventional psychologists want to explain is an illusion.

RM: I’m attaching the spreadsheet that I used to do the calculations to test for a 1/3 power law relationship between V and R (and A and C) for various movement patterns. It’s not very user friendly (though you can use the buttons to load in some of the movement patterns) and I don’t have time to explain it now but those of you who are interested and can read spreadsheet equations can check it out and at least see how I did the calculations of the variables (and see if all seems correct). It’s got macros in it so you’ll get a warning. But there are no viruses in it.

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

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