Review of the Power Law Discussion: The Perils of Refusing to Look at Behavior Through Control Theory Glasses

[rsmarken]

[From Rick Marken (2016.08.04.1540)]

Martin Taylor (2016.07.21.23.23)–

MT: This discussion is a waste of time, and moreover likely to prove a strong deterrent for any serious scientist who might think that CSGnet could be a serious place to discuss real PCT problems.

RM: I believe this discussion has been anything but a waste of time. It is indeed likely to prove a deterrent to scientists (if any are listening) who don’t understand the nature of control in living systems – scientists who refuse to look at behavior through control theory glasses. But it may attract those who have the courage to peek through control theory glasses and see that for over 100 years behavioral scientists have been studying mostly irrelevant side effects of control. It takes courage to look through control theory glasses because doing so reveals the “nightmare” that Powers was referring to in his 1978 Psychological Review paper when he said: “The nightmare of
any experimenter is to realize too late that his results were forced by his
experimental design and do not actually pertain to behavior. This nightmare has
a good chance of becoming a reality for a number of behavioral
scientists”. Looking at the research on the power law through control theory glasses reveals this nightmare with mathematical precision.

MT: The problem Alex initially posed is an interesting one. It would be nice if someone would try to address it.

RM: I did address it but you, Bruce, Warren and Alex certainly didn’t appreciate the way I addressed it. So, for the benefit of a couple of people who asked me in private emails to review my analysis – and also to let me see if I am the only one who thinks that I have correctly addressed, from a PCT perspective, the problem Alex posed-- I will give a synopsis of how I addressed the problem Alex posed.

RM: Here is the problem Alex initially posed (to CSGNet on July 6):

Alex: Any ideas why or how “the control of perception” may give rise to this power law constraining geometry and kinematics in humans, and now in fruit fly larvae?
http://biorxiv.org/content/early/2016/07/05/062166

RM: The pointer is to a recent paper by Alex and two other authors which describes the finding of “The velocity-curvature power law in Drosophila larval locomotion”. The first figure in that paper nicely illustrates what was found:

RM: The finding is that the relationship between angular velocity, A(t), and curvature, C(t), at each instant (t) during movements made over time (the blue squiggle) is the same power function (the equation in the upper right) for humans tracing a line (the hand with pencil) and Drosophila larvae following a path (the worm). The “same power function” means that a power function equation (in the upper right of the figure) fits the human and larval movement data nearly perfectly. The fit of a power function to the data is determined using linear regression with the log of the C(t) values as the predictor variable and the log of the A(t) values as the criterion. The resulting R^2 values are quite high, typically greater than .92 (the C(t) values account for about 92% of the variance in the A(t) values).The coefficient, b, that gives the best fit is around .67 for humans and .75 for the larvae. While the coefficient for the larvae is higher than that for the human, it is about the same as the coefficient found for humans making hand movements in water rather than in air.

RM: Research on the power law goes back at least to 1983 and a paper by Lacquaniti, Terzuolo & Viviani (Acta psychologica, 54, 115-130). Because they were finding that the b coefficient of the power law relating C(t) to A(t) for human movement was always close to .67, they proposed the “two-thirds power law” which say the “true” power function relating curvature to angular velocity in human movement is:

A(t) = kC(t)^2/3

RM: The power law is sometimes expressed as a relationship between tangential (rather than angular) velocity, V(t),and radius of curvature (Rather than curvature), R(t),in which case the power law follows the one-third power law:

V(t) = kR(t)^1/3

RM: The change in exponent results from the fact that V(t) and R(t) are proportional to A(t) and C(t) as follows:

V(t) = A(t)/R(t) and R(t) = 1/C(t)

RM: So what is the PCT explanation of this finding of a consistent 2/3 power law relationship between curvature, C(t), and angular velocity, A(t) or the equivalent consistent 1/3 power law relationship between radius of curvature, R(t), and tangential velocity, V(t)?The researchers in this area believe that the power law reflects a biological constraint on how curved movements are produced. That is, curvature, C(t) or R(t), is seen as an independent variable that constrains how quickly the movement around the curve, A(t) or V(t), can be produced. And the models that have been developed to produce movements that follow the power law have been caused-output models that are developed to cause outputs – instantaneous movement velocity, A(t) or V(t) – that obey the constraint that is imposed by the curvature through which the movement is taking place.

RM: The first step in my PCT analysis of the power law was to develop a PCT model of a person intentionally making squiggly drawing movements in X,Y space, like those in the figure above. The PCT model is shown below:

RM: I didn’t expect the movement pattern produced by this model to fit a power law because I was assuming that the power law had something to do with either the nature of the feedback connection of output to controlled input (k.f in the diagram) or the nature the output function itself (o.x and o.y). But in order to test this I had to obtain measures of A(t) and C(t) or V(t) and R(t) to see how well a power function fits the relationship between thee variables. I was able to find computational formulas for V(t) and R(t) in a paper by Gribble & Ostry (1996, Journal of Neurophysiology, 76, 2853-2860). The computational formulas are below:

RM: Since these are formulas for the instantaneous values of the variables the implication is that what is being computed is V(t) and R(t). Alex then gave me the formulas for deriving A(t) and C(t) from these measures of V(t) and R(t):

A(t) =V(t)/R(t) and C(t)=1/R(t)

RM: I then ran a log-log regression on the movement pattern generated by the model and was surprised to find that the power law fit all of them pretty well and the best fit b coefficient was always close to 1/3 for the regression of log R(t) on log V(t) and 2/3 for the regression of log C(t) on log A(t). This made me think that the observed power law may just be a property of curved movement itself and may have nothing to do with how the movement is generated. So I generated a few different curved movements from equations (rather than from models of behavior) and found that the power law held for these as well.

RM: The idea that the power law might just be a property of curved movement led me to wonder whether there was a mathematical power relationship between the variables used in studies of the power law: between R(t) and V(t) and between A(t) and C(t). After all, both variables are measures of the same movement pattern at the same instant. So I looked at the formulas for V and R (above) and noticed that V² occurs in the numerator of R. That is:

V = (X² + Y² )^1/2

R = [(X² + Y² )^3/2]/|X.dotY.2dot-X.2dotY.dot|

Since V² = (X² + Y² )

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

R = V³ |/ |X.dotY.2dot-X.2dotY.dot|

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

where D =

X.dotY.2dot-X.2dotY.dot|

RM: So the variable V is mathematically in a 1/3 power relationship to R, the very power relationship found in power law studies that is presumed to tell something about how the instantaneous tangential movement (V) around a curve (R) is generated.

RM: Substituting the appropriate formulas for A and C into equation (1) we get the following power relationship between A and C:

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

RM: So the variable A is mathematically in a 2/3 power relationship to C, again the very power relationship found in power law studies that is presumed to tell something about how the instantaneous angular movement (A) around a curve (C) is generated.

RM: The variable D in equations (1) and (2) accounts for the fact that the b coefficient found in power law studies varies around 1/3 (for V as a function of R) or 2/3 (for A as a function of C) for the different movement patterns. Since these studies use only log R or log C as predictors the b coefficient found by the regression will differ from 1/3 or 2/3 depending on the nature of the variance in D that exists in the pattern of movement. When log D is included in the regression, the b coefficient for R or C is always exactly 1/3 (.33) or 2/3 (.67), respectively. (The spreadsheet demonstrating this fact is attached).

RM: So my analysis of the power law, based on PCT, is that the power law is simply a property of curved movement and reveals nothing about how they are produced. It doesn’t matter whether the movement pattern is produced intentionally (is a controlled variable, as it is in my PCT model above and when people intentionally draw squiggles) or a side effect of controlling for something else (as in the case fly larva movement paths). It is based on PCT because I was led to this realization by producing a PCT model of curved movement production. That model showed that any curved movement produced by the model will follow a power law – regardless of how it was produced (in terms of the parameters of the output function).

RM: The implication of equations (1) and (2) is clear; the power law research, which has been going on for over 40 years, is the nightmare Powers warned of: the results were forced by the research design (measuring two related aspects of the same movement pattern) and do not actually pertain to behavior (how the movement pattern was produced). The nightmare exists because researchers took one variable (curvature) to be an environmental variable that is a cause of the other, a behavioral variable (velocity of movement). So the nightmare exists because of a failure to understand that behavior is a control (not a causal) process. Intentionally produced curved movement is a controlled variable.

RM: At any instant the state of a controlled variable, q.i, is a combined result of the effect of system output and environmental disturbances: q.i = q.o+d. So there is no way to learn about the nature of the output, q.o, that produced the observed controlled variable, q.i, by just looking at measures of the controlled variable itself. And all the variables used in determining the power law – A, C, V and R-- are measures of the controlled variable – the pattern of movement – itself.

RM: I knew this PCT analysis of the power law would not go down well with Alex, who has apparently been involved in power law research for some time. But I was rather surprised to find that it went down just as badly with people who are ostensibly fans of PCT. There were several lines of attack directed at my analysis. One was based on mathematics, saying that my derivation was wrong for dimensional or parametric or whatever reasons. Another was based on what might be called domain conflation, saying that though equations 1 and 2 were mathematically correct, they were incompatible with physics and/or biology. And, finally, it was implied that my analysis was incorrect because the spreadsheet I used to demonstrate the analysis had errors in it.

RM: Indeed, the spreadsheet had an error (failure to divide a derivative by dt) but this error had no effect on the computations of the power law demonstrated by the spreadsheet. I have attached the corrected spreadsheet for those of you who are interested in seeing that, indeed, all curved movements – regardless of how produced – result in a power law relationship between curvature and velocity, with the power coefficient being close to .33 for the relationship between log R and log V and close to .67 for the relationship between log C and log A. When log D is included in the regression the power coefficient is always exactly .33 for log R and .67 for log (C) and the R^2 value of the regression is always 1.0 (as it should be if equations 1 and 2 are correct). You can demonstrate this to yourself by pressing the “Random Pattern” button, which generates a new random movement pattern each time it is pressed and see what happens to the value of b when only log R or log C is included in the regression and what happens when log D is also included.

RM: Again, I am really sorry if this analysis shows that a whole line of research is based on results that were forced by the research design. But this analysis at least shows why PCT has such a problem getting accepted by conventional psychologists. The problem is that PCT doesn’t explain the phenomena of interest to conventional psychologists – phenomenon like the power law. It explains the phenomenon of control. Those who know PCT only as a theory assume that it’s a theory that explains the results of all research findings. And it does, I suppose. But more often than not what it explains is that the results of conventional research are not what they seem; they are side effects of control that tell you very little about the behavior under study.

RM: I believe the lesson of this PCT analysis of the power law is very important. The lesson is that before conventional psychologists (and neurophysiologists and many fans of PCT) can correctly apply PCT they have to learn to look at behavior through control theory glasses. Phenomena phirst!

Best regards

Rick

PowerLawRegression071.30.xlsm (638 KB)

···

–

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

[wmansell]

So are you saying that in the same way as we know mathematically that a static circle’s circumference is 2[pi]r, that a moving point on any curved line has the mathematical relationship you describe? Is the search for the biology and physics of the power law as elusive as looking for why a circle, once drawn by someone, has the relationship 2[pi]r between its circumference and radius? This doesn’t seem quite right because that mathematics for a circle only applies to a circle and not other shapes, whereas this law seems to apply to any shape drawn at any velocity. I couldn’t get Alex’s MatLab program to work so I still haven’t had the chance to try to produce my own squiggle to try to flout the ‘law’!
Warren

···

On Thu, Aug 4, 2016 at 11:38 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.04.1540)]

Martin Taylor (2016.07.21.23.23)–

MT: This discussion is a waste of time, and moreover likely to prove a strong deterrent for any serious scientist who might think that CSGnet could be a serious place to discuss real PCT problems.

RM: I believe this discussion has been anything but a waste of time. It is indeed likely to prove a deterrent to scientists (if any are listening) who don’t understand the nature of control in living systems – scientists who refuse to look at behavior through control theory glasses. But it may attract those who have the courage to peek through control theory glasses and see that for over 100 years behavioral scientists have been studying mostly irrelevant side effects of control. It takes courage to look through control theory glasses because doing so reveals the “nightmare” that Powers was referring to in his 1978 Psychological Review paper when he said: “The nightmare of
any experimenter is to realize too late that his results were forced by his
experimental design and do not actually pertain to behavior. This nightmare has
a good chance of becoming a reality for a number of behavioral
scientists”. Looking at the research on the power law through control theory glasses reveals this nightmare with mathematical precision.

MT: The problem Alex initially posed is an interesting one. It would be nice if someone would try to address it.

RM: I did address it but you, Bruce, Warren and Alex certainly didn’t appreciate the way I addressed it. So, for the benefit of a couple of people who asked me in private emails to review my analysis – and also to let me see if I am the only one who thinks that I have correctly addressed, from a PCT perspective, the problem Alex posed-- I will give a synopsis of how I addressed the problem Alex posed.

RM: Here is the problem Alex initially posed (to CSGNet on July 6):

Alex: Any ideas why or how “the control of perception” may give rise to this power law constraining geometry and kinematics in humans, and now in fruit fly larvae?
http://biorxiv.org/content/early/2016/07/05/062166

RM: The pointer is to a recent paper by Alex and two other authors which describes the finding of “The velocity-curvature power law in Drosophila larval locomotion”. The first figure in that paper nicely illustrates what was found:

RM: The finding is that the relationship between angular velocity, A(t), and curvature, C(t), at each instant (t) during movements made over time (the blue squiggle) is the same power function (the equation in the upper right) for humans tracing a line (the hand with pencil) and Drosophila larvae following a path (the worm). The “same power function” means that a power function equation (in the upper right of the figure) fits the human and larval movement data nearly perfectly. The fit of a power function to the data is determined using linear regression with the log of the C(t) values as the predictor variable and the log of the A(t) values as the criterion. The resulting R^2 values are quite high, typically greater than .92 (the C(t) values account for about 92% of the variance in the A(t) values).The coefficient, b, that gives the best fit is around .67 for humans and .75 for the larvae. While the coefficient for the larvae is higher than that for the human, it is about the same as the coefficient found for humans making hand movements in water rather than in air.

RM: Research on the power law goes back at least to 1983 and a paper by Lacquaniti, Terzuolo & Viviani (Acta psychologica, 54, 115-130). Because they were finding that the b coefficient of the power law relating C(t) to A(t) for human movement was always close to .67, they proposed the “two-thirds power law” which say the “true” power function relating curvature to angular velocity in human movement is:

A(t) = kC(t)^2/3

RM: The power law is sometimes expressed as a relationship between tangential (rather than angular) velocity, V(t),and radius of curvature (Rather than curvature), R(t),in which case the power law follows the one-third power law:

V(t) = kR(t)^1/3

RM: The change in exponent results from the fact that V(t) and R(t) are proportional to A(t) and C(t) as follows:

V(t) = A(t)/R(t) and R(t) = 1/C(t)

RM: So what is the PCT explanation of this finding of a consistent 2/3 power law relationship between curvature, C(t), and angular velocity, A(t) or the equivalent consistent 1/3 power law relationship between radius of curvature, R(t), and tangential velocity, V(t)?The researchers in this area believe that the power law reflects a biological constraint on how curved movements are produced. That is, curvature, C(t) or R(t), is seen as an independent variable that constrains how quickly the movement around the curve, A(t) or V(t), can be produced. And the models that have been developed to produce movements that follow the power law have been caused-output models that are developed to cause outputs – instantaneous movement velocity, A(t) or V(t) – that obey the constraint that is imposed by the curvature through which the movement is taking place.

RM: The first step in my PCT analysis of the power law was to develop a PCT model of a person intentionally making squiggly drawing movements in X,Y space, like those in the figure above. The PCT model is shown below:

RM: I didn’t expect the movement pattern produced by this model to fit a power law because I was assuming that the power law had something to do with either the nature of the feedback connection of output to controlled input (k.f in the diagram) or the nature the output function itself (o.x and o.y). But in order to test this I had to obtain measures of A(t) and C(t) or V(t) and R(t) to see how well a power function fits the relationship between thee variables. I was able to find computational formulas for V(t) and R(t) in a paper by Gribble & Ostry (1996, Journal of Neurophysiology, 76, 2853-2860). The computational formulas are below:

RM: Since these are formulas for the instantaneous values of the variables the implication is that what is being computed is V(t) and R(t). Alex then gave me the formulas for deriving A(t) and C(t) from these measures of V(t) and R(t):

A(t) =V(t)/R(t) and C(t)=1/R(t)

RM: I then ran a log-log regression on the movement pattern generated by the model and was surprised to find that the power law fit all of them pretty well and the best fit b coefficient was always close to 1/3 for the regression of log R(t) on log V(t) and 2/3 for the regression of log C(t) on log A(t). This made me think that the observed power law may just be a property of curved movement itself and may have nothing to do with how the movement is generated. So I generated a few different curved movements from equations (rather than from models of behavior) and found that the power law held for these as well.

RM: The idea that the power law might just be a property of curved movement led me to wonder whether there was a mathematical power relationship between the variables used in studies of the power law: between R(t) and V(t) and between A(t) and C(t). After all, both variables are measures of the same movement pattern at the same instant. So I looked at the formulas for V and R (above) and noticed that V² occurs in the numerator of R. That is:

V = (X² + Y² )^1/2

R = [(X² + Y² )^3/2]/|X.dotY.2dot-X.2dotY.dot|

Since V² = (X² + Y² )

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

R = V³ |/ |X.dotY.2dot-X.2dotY.dot|

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

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

RM: So the variable V is mathematically in a 1/3 power relationship to R, the very power relationship found in power law studies that is presumed to tell something about how the instantaneous tangential movement (V) around a curve (R) is generated.

RM: Substituting the appropriate formulas for A and C into equation (1) we get the following power relationship between A and C:

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

RM: So the variable A is mathematically in a 2/3 power relationship to C, again the very power relationship found in power law studies that is presumed to tell something about how the instantaneous angular movement (A) around a curve (C) is generated.

RM: The variable D in equations (1) and (2) accounts for the fact that the b coefficient found in power law studies varies around 1/3 (for V as a function of R) or 2/3 (for A as a function of C) for the different movement patterns. Since these studies use only log R or log C as predictors the b coefficient found by the regression will differ from 1/3 or 2/3 depending on the nature of the variance in D that exists in the pattern of movement. When log D is included in the regression, the b coefficient for R or C is always exactly 1/3 (.33) or 2/3 (.67), respectively. (The spreadsheet demonstrating this fact is attached).

RM: So my analysis of the power law, based on PCT, is that the power law is simply a property of curved movement and reveals nothing about how they are produced. It doesn’t matter whether the movement pattern is produced intentionally (is a controlled variable, as it is in my PCT model above and when people intentionally draw squiggles) or a side effect of controlling for something else (as in the case fly larva movement paths). It is based on PCT because I was led to this realization by producing a PCT model of curved movement production. That model showed that any curved movement produced by the model will follow a power law – regardless of how it was produced (in terms of the parameters of the output function).

RM: The implication of equations (1) and (2) is clear; the power law research, which has been going on for over 40 years, is the nightmare Powers warned of: the results were forced by the research design (measuring two related aspects of the same movement pattern) and do not actually pertain to behavior (how the movement pattern was produced). The nightmare exists because researchers took one variable (curvature) to be an environmental variable that is a cause of the other, a behavioral variable (velocity of movement). So the nightmare exists because of a failure to understand that behavior is a control (not a causal) process. Intentionally produced curved movement is a controlled variable.

RM: At any instant the state of a controlled variable, q.i, is a combined result of the effect of system output and environmental disturbances: q.i = q.o+d. So there is no way to learn about the nature of the output, q.o, that produced the observed controlled variable, q.i, by just looking at measures of the controlled variable itself. And all the variables used in determining the power law – A, C, V and R-- are measures of the controlled variable – the pattern of movement – itself.

RM: I knew this PCT analysis of the power law would not go down well with Alex, who has apparently been involved in power law research for some time. But I was rather surprised to find that it went down just as badly with people who are ostensibly fans of PCT. There were several lines of attack directed at my analysis. One was based on mathematics, saying that my derivation was wrong for dimensional or parametric or whatever reasons. Another was based on what might be called domain conflation, saying that though equations 1 and 2 were mathematically correct, they were incompatible with physics and/or biology. And, finally, it was implied that my analysis was incorrect because the spreadsheet I used to demonstrate the analysis had errors in it.

RM: Indeed, the spreadsheet had an error (failure to divide a derivative by dt) but this error had no effect on the computations of the power law demonstrated by the spreadsheet. I have attached the corrected spreadsheet for those of you who are interested in seeing that, indeed, all curved movements – regardless of how produced – result in a power law relationship between curvature and velocity, with the power coefficient being close to .33 for the relationship between log R and log V and close to .67 for the relationship between log C and log A. When log D is included in the regression the power coefficient is always exactly .33 for log R and .67 for log (C) and the R^2 value of the regression is always 1.0 (as it should be if equations 1 and 2 are correct). You can demonstrate this to yourself by pressing the “Random Pattern” button, which generates a new random movement pattern each time it is pressed and see what happens to the value of b when only log R or log C is included in the regression and what happens when log D is also included.

RM: Again, I am really sorry if this analysis shows that a whole line of research is based on results that were forced by the research design. But this analysis at least shows why PCT has such a problem getting accepted by conventional psychologists. The problem is that PCT doesn’t explain the phenomena of interest to conventional psychologists – phenomenon like the power law. It explains the phenomenon of control. Those who know PCT only as a theory assume that it’s a theory that explains the results of all research findings. And it does, I suppose. But more often than not what it explains is that the results of conventional research are not what they seem; they are side effects of control that tell you very little about the behavior under study.

RM: I believe the lesson of this PCT analysis of the power law is very important. The lesson is that before conventional psychologists (and neurophysiologists and many fans of PCT) can correctly apply PCT they have to learn to look at behavior through control theory glasses. Phenomena phirst!

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

–

Dr Warren Mansell
Reader in Clinical Psychology

School of Health Sciences
2nd Floor Zochonis Building
University of Manchester
Oxford Road
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Website: http://www.psych-sci.manchester.ac.uk/staff/131406

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

[wmansell]

Rick, I have the greatest respect for you, but you are presenting this work as though the issues have been dealt with, but my impression is that they all still stand. I think my biggest problem is to believe that a mathematical relationship exists between velocity and curvature that somehow lies outside physics. To me, motion is intrinsically bound up with forces applied to masses that have surfaces in three dimensions, and any mathematical relationship that exists, exists because of these more fundamental properties of moving objects. I think I agree with you that the law could be independent of the biology and nervous system of an organism but just not independent of physics.

Warren

···

On 4 Aug 2016, at 23:38, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.04.1540)]

Martin Taylor (2016.07.21.23.23)–

MT: This discussion is a waste of time, and moreover likely to prove a strong deterrent for any serious scientist who might think that CSGnet could be a serious place to discuss real PCT problems.

RM: I believe this discussion has been anything but a waste of time. It is indeed likely to prove a deterrent to scientists (if any are listening) who don’t understand the nature of control in living systems – scientists who refuse to look at behavior through control theory glasses. But it may attract those who have the courage to peek through control theory glasses and see that for over 100 years behavioral scientists have been studying mostly irrelevant side effects of control. It takes courage to look through control theory glasses because doing so reveals the “nightmare” that Powers was referring to in his 1978 Psychological Review paper when he said: “The nightmare of
any experimenter is to realize too late that his results were forced by his
experimental design and do not actually pertain to behavior. This nightmare has
a good chance of becoming a reality for a number of behavioral
scientists”. Looking at the research on the power law through control theory glasses reveals this nightmare with mathematical precision.

MT: The problem Alex initially posed is an interesting one. It would be nice if someone would try to address it.

RM: I did address it but you, Bruce, Warren and Alex certainly didn’t appreciate the way I addressed it. So, for the benefit of a couple of people who asked me in private emails to review my analysis – and also to let me see if I am the only one who thinks that I have correctly addressed, from a PCT perspective, the problem Alex posed-- I will give a synopsis of how I addressed the problem Alex posed.

RM: Here is the problem Alex initially posed (to CSGNet on July 6):

Alex: Any ideas why or how “the control of perception” may give rise to this power law constraining geometry and kinematics in humans, and now in fruit fly larvae?
http://biorxiv.org/content/early/2016/07/05/062166

RM: The pointer is to a recent paper by Alex and two other authors which describes the finding of “The velocity-curvature power law in Drosophila larval locomotion”. The first figure in that paper nic
ely illustrates what was found:

<image.png>

RM: The finding is that the relationship between angular velocity, A(t), and curvature, C(t), at each instant (t) during movements made over time (the blue squiggle) is the same power function (the equation in the upper right) for humans tracing a line (the hand with pencil) and Drosophila larvae following a path (the worm). The “same power function” means that a power function equation (in the upper right of the figure) fits the human and larval movement data nearly perfectly. The fit of a power function to the data is determined using linear regression with the log of the C(t) values as the predictor variable and the log of the A(t) values as the cri
terion. The resulting R^2 values are quite high, typically greater than .92 (the C(t) values account for about 92% of the variance in the A(t) values).The coefficient, b, that gives the best fit is around .67 for humans and .75 for the larvae. While the coefficient for the larvae is higher than that for the human, it is about the same as the coefficient found for humans making hand movements in water rather than in air.

RM: Research on the power law goes back at least to 1983 and a paper by Lacquaniti, Terzuolo & Viviani (Acta psychologica, 54, 115-130). Because they were finding that the b coefficient of the power law relating C(t) to A(t) for human movement was always close to .67, they proposed the “two-thirds power law” which say the “true” power function relating curvature to angular velocity in human movement is:

A(t) = kC(t)^2/3

RM: The power law is sometimes expressed as a relationship between tangential (rather than angular) velocity, V(t),and radius of curvature (Rather than curvature), R(t),in which case the power law follows the one-third power law:

V(t) = kR(t)^1/3

RM: The change in exponent results from the fact that V(t) and R(t) are proportional to A(t) and C(t) as follows:

V(t) = A(t)/R(t) and R(t) = 1/C(t)

RM: So what is the PCT explanation of this finding of a consistent 2/3 power law relationship between curvature, C(t
), and angular velocity, A(t) or the equivalent consistent 1/3 power law relationship between radius of curvature, R(t), and tangential velocity, V(t)?The researchers in this area believe that the power law reflects a biological constraint on how curved movements are produced. That is, curvature, C(t) or R(t), is seen as an independent variable that constrains how quickly the movement around the curve, A(t) or V(t), can be produced. And the models that have been developed to produce movements that follow the power law have been caused-output models that are developed to cause outputs – instantaneous movement velocity, A(t) or V(t) – that obey the constraint that is imposed by the curvature through which the movement is taking place.

RM: The first step in my PCT analysis of the power law was to develop a PCT model of a person intentionally making squiggly drawing movements in X,Y space, like those in the figure above. The PCT model is shown be
low:

<image.png>

RM: I didn’t expect the movement pattern produced by this model to fit a power law because I was assuming that the power law had something to do with either the nature of the feedback connection of output to controlled input (k.f in the diagram) or the nature the output function itself (o.x and o.y). But in order to test this I had to obtain measures of A(t) and C(t) or V(t) and R(t) to see how well a power function fits the relationship between thee variables. I was able to find computational formulas for V(t) and R(t) in a paper by Gribble & Ostry (1996, Journal of Neurophysiology, 76, 2853-2860). The computational formulas are below:

<image.png>

RM: Since these are formulas for the instantaneous values of the variables the implication is that what is being computed is V(t) and R(t). Alex then gave me the formulas for deriving A(t) and C
(t) from these measures of V(t) and R(t):

A(t) =V(t)/R(t) and C(t)=1/R(t)

RM: I then ran a log-log regression on the movement pattern generated by the model and was surprised to find that the power law fit all of them pretty well and the best fit b coefficient was always close to 1/3 for the regression of log R(t) on log V(t) and 2/3 for the regression of log C(t) on log A(t). This made me think that the observed power law may just be a property of curved movement itself and may have nothing to do with how the movement is generated. So I generated a few different curved movements from equations (rather than from models of behavior) and found that
the power law held for these as well.

RM: The idea that the power law might just be a property of curved movement led me to wonder whether there was a mathematical power relationship between the variables used in studies of the power law: between R(t) and V(t) and between A(t) and C(t). After all, both variables are measures of the same movement pattern at the same instant. So I looked at the formulas for V and R (above) and noticed that V² occurs in the numerator of R. That is:

V = (X² + Y² )^1/2

R = [(X² + Y² )^3/2]/|X.dotY.2dot-X.2dotY.dot|

Since V² = (X² + Y² )

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

R = V³ |/ |X.dotY.2dot-X.2dotY.dot|

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

where D =

X.dotY.2dot-X.2dotY.dot|

RM: So the variable V is mathematically in a 1/3 power relationship to R, the very power relationship found in power law studies that is presumed to tell something about how the instantaneous tangential movement (V) around a curve (R) is generated.

RM: Subst
ituting the appropriate formulas for A and C into equation (1) we get the following power relationship between A and C:

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

RM: So the variable A is mathematically in a 2/3 power relationship to C, again the very power relationship found in power law studies that is presumed to tell something about how the instantaneous angular movement (A) around a curve (C) is generated.

RM: The variable D in equations (1) and (2) accounts for the fact that the b coefficient found in power law studies varies around 1/3 (for V as a function of R) or 2/3 (for A as a function of C) for the different movement patterns. Since these studies use only log R or log C
as predictors the b coefficient found by the regression will differ from 1/3 or 2/3 depending on the nature of the variance in D that exists in the pattern of movement. When log D is included in the regression, the b coefficient for R or C is always exactly 1/3 (.33) or 2/3 (.67), respectively. (The spreadsheet demonstrating this fact is attached).

RM: So my analysi
s of the power law, based on PCT, is that the power law is simply a property of curved movement and reveals nothing about how they are produced. It doesn’t matter whether the movement pattern is produced intentionally (is a controlled variable, as it is in my PCT model above and when people intentionally draw squiggles) or a side effect of controlling for something else (as in the case fly larva movement paths). It is based on PCT because I was led to this realization by producing a PCT model of curved movement production. That model showed that any curved movement produced by the model will follow a power law – regardless of how it was produced (in terms of the parameters of the output function).

RM: The implication of equations (1) and (2) is clear; the power law research, which has been going on for ov
er 40 years, is the nightmare Powers warned of: the results were forced by the research design (measuring two related aspects of the same movement pattern) and do not actually pertain to behavior (how the movement pattern was produced). The nightmare exists because researchers took one variable (curvature) to be an environmental variable that is a cause of the other, a behavioral variable (velocity of movement). So the nightmare exists because of a failure to understand that behavior is a control (not a causal) process. Intentionally produced curved movement is a controlled variable.

RM: At any instant the state of a controlled variable, q.i, is a combined result of the effect of system output and environmental disturbances: q.i = q.
o+d. So there is no way to learn about the nature of the output, q.o, that produced the observed controlled variable, q.i, by just looking at measures of the controlled variable itself. And all the variables used in determining the power law – A, C, V and R-- are measures of the controlled variable – the pattern of movement – itself.

RM: I knew this PCT analysis of the power law would not go down well with Alex, who has apparently been involved in power law research for some time. But I was rather surprised to find that it went down just as badly with people who are ostensibly fans of PCT. There were several lines of attack directed at my analysis. One was based on mathematics, saying that my derivation was wrong for dimensional or parametric or whatever reasons. Another was based on what might be called domain conflation, saying that though equations 1 and 2 were mathematically correct, they were incompatible with physics and/or biology. And, finally, it was implied that my analysis was incorrect because the spreadsheet I used to demonstrate the analysis had errors in it.

RM: Indeed, the spreadsheet had an error (failure to divide a derivative by dt) but this error had no effect on the computations of the power law demonstrated by the spreadsheet. I have attached the corrected spreadsheet for those of you who are interested in seeing that, indeed, all curved movements – regardless of how produced – result in a power law relationship between curvature and velocity, with the power coefficient being close to .33 for the relationship between log R and log V and close to .67 for the relationship between log C and log A. When log D is included in the regression the power coefficient is always exactly .33 for log R and .67 for log (C) and the R^2 value of the regression is always 1.0 (as it should be if equations 1 and 2 are correct). You can demonstrate this to yourself by pressing the “Random Pattern” button, which generates a new random movement pattern each time it is pressed and see what happens to the value of b when only log R or log C is included in the regression and what happens when log D is also included.

RM: Agai
n, I am really sorry if this analysis shows that a whole line of research is based on results that were forced by the research design. But this analysis at least shows why PCT has such a problem getting accepted by conventional psychologists. The problem is that PCT doesn’t explain the phenomena of interest to conventional psychologists – phenomenon like the power law. It explains the phenomenon of control. Those who know PCT only as a theory assume that it’s a theory that explains the results of all research findings. And it does, I suppose. But more often than not what it explains is that the results of conventional research are not what they seem; they are side effects of control that tell you very little about the behavior under study.

RM: I believe the lesson of this PCT analysis of the power law is very important. The lesson is that before conventional psychologists (and neurophysiologists and many fans of PCT) can correctly apply PCT they have to learn to look at behavior through control theory glasses. Phenomena phirst!

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

<PowerLawRegression07.30.xlsm>

[MartinT]

[Martin Taylor 2016.08.04.23.03]

Rick, I have the greatest respect for you, but you are presenting this work as though the issues have been dealt with, but my impression is that they all still stand. I think my biggest problem is to believe that a mathematical relationship exists between velocity and curvature that somehow lies outside physics. To me, motion is intrinsically bound up with forces applied to masses that have surfaces in three dimensions, and any mathematical relationship that exists, exists because of these more fundamental properties of moving objects. I think I agree with you that the law could be independent of the biology and nervous system of an organism but just not independent of physics.
Warren

Warren, I think the central problem is to answer the question Rick usually claims to be the core question of PCT research: to find the controlled variable.

We know that people can choose to go slow on the straights and fast around curves, or equally slowly on all parts of a twisty track if they want, but under many conditions, especially in experiments, they don't do that. They move faster on the straights and shallow curves and slower on the sharp curves, by an amount that often is remarkably close to the power law that describes the shapes of curves in terms of R and dθ/ds.

Obviously the reference values for velocity perception are not the same under these different conditions and may well not even be derived from the same higher-level controlled perception(s). What perception is being controlled when they do and when they don't conform to the 1/3 power law, and is the difference when they and other organisms conform to a 1/4 power law or no power law at all one of perception or of the effects of force on the mass being moved, with the same perception being controlled in both power-law cases? Why is the transition rate from sharp to shallow curves different from the transition rate from shallow to sharp curves? Can that difference provide a clue to what perception is being controlled?

I have no answers, but I think these are the kinds of questions that need to be addressed, rather than spending many lines of text on rewriting the parametric equations that describe spatial curvature, or on repeating explanations as to why that is not a useful approach to the problem. We need hypotheses about what perception(s) is/are being controlled, and suggestions as to how to disturb them so as to allow the TCV to be useful.

Martin

[rsmarken]

[From Rick Marken (2016.08.04.2230)]

···

On Thu, Aug 4, 2016 at 5:46 PM, Warren Mansell wmansell@gmail.com wrote:

WM: So are you saying that in the same way as we know mathematically that a static circle’s circumference is 2[pi]r, that a moving point on any curved line has the mathematical relationship you describe?

RM: Exactly!

WM: Is the search for the biology and physics of the power law as elusive as looking for why a circle, once drawn by someone, has the relationship 2[pi]r between its circumference and radius?

RM: No, as you stated it these are quite different things. The power relationship between A and C (let’s just stick with that one) is a mathematical relationship just like that between the circumference and radius of a circle.

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

^and

c = pi*r²

RM: Searching for the biology and/or physics underlying either of these laws would be silly (not elusive). But searching for the reason why either of these relationships exist would be quite sensible. It’s what mathematicians do.

WM: This doesn’t seem quite right because that mathematics for a circle only applies to a circle and not other shapes, whereas this law seems to apply to any shape drawn at any velocity.

RM: It may not seem sensible to you but both are mathematical facts.

WM: I couldn’t get Alex’s MatLab program to work so I still haven’t had the chance to try to produce my own squiggle to try to flout the ‘law’!

RM: What do you expect to find when you produce your own squiggle movements? What I expect to find (actually, what I know I’ll find) is that every movement pattern you make – making curved lines rapidly or slowly, doing it free hand or holding a weight, whatever – will show a power law relationship between A and C with a power coefficient close to .67. And if log D is included in the regression the power coefficient of C will be exactly .67, the power coefficient of D will be exactly .33 and R^2 will be 1.0 for every movement pattern you make!

Best

Rick

Warren

–

On Thu, Aug 4, 2016 at 11:38 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.04.1540)]

Martin Taylor (2016.07.21.23.23)–

MT: This discussion is a waste of time, and moreover likely to prove a strong deterrent for any serious scientist who might think that CSGnet could be a serious place to discuss real PCT problems.

RM: I believe this discussion has been anything but a waste of time. It is indeed likely to prove a deterrent to scientists (if any are listening) who don’t understand the nature of control in living systems – scientists who refuse to look at behavior through control theory glasses. But it may attract those who have the courage to peek through control theory glasses and see that for over 100 years behavioral scientists have been studying mostly irrelevant side effects of control. It takes courage to look through control theory glasses because doing so reveals the “nightmare” that Powers was referring to in his 1978 Psychological Review paper when he said: “The nightmare of
any experimenter is to realize too late that his results were forced by his
experimental design and do not actually pertain to behavior. This nightmare has
a good chance of becoming a reality for a number of behavioral
scientists”. Looking at the research on the power law through control theory glasses reveals this nightmare with mathematical precision.

MT: The problem Alex initially posed is an interesting one. It would be nice if someone would try to address it.

RM: I did address it but you, Bruce, Warren and Alex certainly didn’t appreciate the way I addressed it. So, for the benefit of a couple of people who asked me in private emails to review my analysis – and also to let me see if I am the only one who thinks that I have correctly addressed, from a PCT perspective, the problem Alex posed-- I will give a synopsis of how I addressed the problem Alex posed.

RM: Here is the problem Alex initially posed (to CSGNet on July 6):

Alex: Any ideas why or how “the control of perception” may give rise to this power law constraining geometry and kinematics in humans, and now in fruit fly larvae?
http://biorxiv.org/content/early/2016/07/05/062166

RM: The pointer is to a recent paper by Alex and two other authors which describes the finding of “The velocity-curvature power law in Drosophila larval locomotion”. The first figure in that paper nicely illustrates what was found:

RM: The finding is that the relationship between angular velocity, A(t), and curvature, C(t), at each instant (t) during movements made over time (the blue squiggle) is the same power function (the equation in the upper right) for humans tracing a line (the hand with pencil) and Drosophila larvae following a path (the worm). The “same power function” means that a power function equation (in the upper right of the figure) fits the human and larval movement data nearly perfectly. The fit of a power function to the data is determined using linear regression with the log of the C(t) values as the predictor variable and the log of the A(t) values as the criterion. The resulting R^2 values are quite high, typically greater than .92 (the C(t) values account for about 92% of the variance in the A(t) values).The coefficient, b, that gives the best fit is around .67 for humans and .75 for the larvae. While the coefficient for the larvae is higher than that for the human, it is about the same as the coefficient found for humans making hand movements in water rather than in air.

RM: Research on the power law goes back at least to 1983 and a paper by Lacquaniti, Terzuolo & Viviani (Acta psychologica, 54, 115-130). Because they were finding that the b coefficient of the power law relating C(t) to A(t) for human movement was always close to .67, they proposed the “two-thirds power law” which say the “true” power function relating curvature to angular velocity in human movement is:

A(t) = kC(t)^2/3

RM: The power law is sometimes expressed as a relationship between tangential (rather than angular) velocity, V(t),and radius of curvature (Rather than curvature), R(t),in which case the power law follows the one-third power law:

V(t) = kR(t)^1/3

RM: The change in exponent results from the fact that V(t) and R(t) are proportional to A(t) and C(t) as follows:

V(t) = A(t)/R(t) and R(t) = 1/C(t)

RM: So what is the PCT explanation of this finding of a consistent 2/3 power law relationship between curvature, C(t), and angular velocity, A(t) or the equivalent consistent 1/3 power law relationship between radius of curvature, R(t), and tangential velocity, V(t)?The researchers in this area believe that the power law reflects a biological constraint on how curved movements are produced. That is, curvature, C(t) or R(t), is seen as an independent variable that constrains how quickly the movement around the curve, A(t) or V(t), can be produced. And the models that have been developed to produce movements that follow the power law have been caused-output models that are developed to cause outputs – instantaneous movement velocity, A(t) or V(t) – that obey the constraint that is imposed by the curvature through which the movement is taking place.

RM: The first step in my PCT analysis of the power law was to develop a PCT model of a person intentionally making squiggly drawing movements in X,Y space, like those in the figure above. The PCT model is shown below:

RM: I didn’t expect the movement pattern produced by this model to fit a power law because I was assuming that the power law had something to do with either the nature of the feedback connection of output to controlled input (k.f in the diagram) or the nature the output function itself (o.x and o.y). But in order to test this I had to obtain measures of A(t) and C(t) or V(t) and R(t) to see how well a power function fits the relationship between thee variables. I was able to find computational formulas for V(t) and R(t) in a paper by Gribble & Ostry (1996, Journal of Neurophysiology, 76, 2853-2860). The computational formulas are below:

RM: Since these are formulas for the instantaneous values of the variables the implication is that what is being computed is V(t) and R(t). Alex then gave me the formulas for deriving A(t) and C(t) from these measures of V(t) and R(t):

A(t) =V(t)/R(t) and C(t)=1/R(t)

RM: I then ran a log-log regression on the movement pattern generated by the model and was surprised to find that the power law fit all of them pretty well and the best fit b coefficient was always close to 1/3 for the regression of log R(t) on log V(t) and 2/3 for the regression of log C(t) on log A(t). This made me think that the observed power law may just be a property of curved movement itself and may have nothing to do with how the movement is generated. So I generated a few different curved movements from equations (rather than from models of behavior) and found that the power law held for these as well.

RM: The idea that the power law might just be a property of curved movement led me to wonder whether there was a mathematical power relationship between the variables used in studies of the power law: between R(t) and V(t) and between A(t) and C(t). After all, both variables are measures of the same movement pattern at the same instant. So I looked at the formulas for V and R (above) and noticed that V² occurs in the numerator of R. That is:

V = (X² + Y² )^1/2

R = [(X² + Y² )^3/2]/|X.dotY.2dot-X.2dotY.dot|

Since V² = (X² + Y² )

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

R = V³ |/ |X.dotY.2dot-X.2dotY.dot|

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

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

RM: So the variable V is mathematically in a 1/3 power relationship to R, the very power relationship found in power law studies that is presumed to tell something about how the instantaneous tangential movement (V) around a curve (R) is generated.

RM: Substituting the appropriate formulas for A and C into equation (1) we get the following power relationship between A and C:

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

RM: So the variable A is mathematically in a 2/3 power relationship to C, again the very power relationship found in power law studies that is presumed to tell something about how the instantaneous angular movement (A) around a curve (C) is generated.

RM: The variable D in equations (1) and (2) accounts for the fact that the b coefficient found in power law studies varies around 1/3 (for V as a function of R) or 2/3 (for A as a function of C) for the different movement patterns. Since these studies use only log R or log C as predictors the b coefficient found by the regression will differ from 1/3 or 2/3 depending on the nature of the variance in D that exists in the pattern of movement. When log D is included in the regression, the b coefficient for R or C is always exactly 1/3 (.33) or 2/3 (.67), respectively. (The spreadsheet demonstrating this fact is attached).

RM: So my analysis of the power law, based on PCT, is that the power law is simply a property of curved movement and reveals nothing about how they are produced. It doesn’t matter whether the movement pattern is produced intentionally (is a controlled variable, as it is in my PCT model above and when people intentionally draw squiggles) or a side effect of controlling for something else (as in the case fly larva movement paths). It is based on PCT because I was led to this realization by producing a PCT model of curved movement production. That model showed that any curved movement produced by the model will follow a power law – regardless of how it was produced (in terms of the parameters of the output function).

RM: The implication of equations (1) and (2) is clear; the power law research, which has been going on for over 40 years, is the nightmare Powers warned of: the results were forced by the research design (measuring two related aspects of the same movement pattern) and do not actually pertain to behavior (how the movement pattern was produced). The nightmare exists because researchers took one variable (curvature) to be an environmental variable that is a cause of the other, a behavioral variable (velocity of movement). So the nightmare exists because of a failure to understand that behavior is a control (not a causal) process. Intentionally produced curved movement is a controlled variable.

RM: At any instant the state of a controlled variable, q.i, is a combined result of the effect of system output and environmental disturbances: q.i = q.o+d. So there is no way to learn about the nature of the output, q.o, that produced the observed controlled variable, q.i, by just looking at measures of the controlled variable itself. And all the variables used in determining the power law – A, C, V and R-- are measures of the controlled variable – the pattern of movement – itself.

RM: I knew this PCT analysis of the power law would not go down well with Alex, who has apparently been involved in power law research for some time. But I was rather surprised to find that it went down just as badly with people who are ostensibly fans of PCT. There were several lines of attack directed at my analysis. One was based on mathematics, saying that my derivation was wrong for dimensional or parametric or whatever reasons. Another was based on what might be called domain conflation, saying that though equations 1 and 2 were mathematically correct, they were incompatible with physics and/or biology. And, finally, it was implied that my analysis was incorrect because the spreadsheet I used to demonstrate the analysis had errors in it.

RM: Indeed, the spreadsheet had an error (failure to divide a derivative by dt) but this error had no effect on the computations of the power law demonstrated by the spreadsheet. I have attached the corrected spreadsheet for those of you who are interested in seeing that, indeed, all curved movements – regardless of how produced – result in a power law relationship between curvature and velocity, with the power coefficient being close to .33 for the relationship between log R and log V and close to .67 for the relationship between log C and log A. When log D is included in the regression the power coefficient is always exactly .33 for log R and .67 for log (C) and the R^2 value of the regression is always 1.0 (as it should be if equations 1 and 2 are correct). You can demonstrate this to yourself by pressing the “Random Pattern” button, which generates a new random movement pattern each time it is pressed and see what happens to the value of b when only log R or log C is included in the regression and what happens when log D is also included.

RM: Again, I am really sorry if this analysis shows that a whole line of research is based on results that were forced by the research design. But this analysis at least shows why PCT has such a problem getting accepted by conventional psychologists. The problem is that PCT doesn’t explain the phenomena of interest to conventional psychologists – phenomenon like the power law. It explains the phenomenon of control. Those who know PCT only as a theory assume that it’s a theory that explains the results of all research findings. And it does, I suppose. But more often than not what it explains is that the results of conventional research are not what they seem; they are side effects of control that tell you very little about the behavior under study.

RM: I believe the lesson of this PCT analysis of the power law is very important. The lesson is that before conventional psychologists (and neurophysiologists and many fans of PCT) can correctly apply PCT they have to learn to look at behavior through control theory glasses. Phenomena phirst!

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

–
Dr Warren Mansell
Reader in Clinical Psychology

School of Health Sciences
2nd Floor Zochonis Building
University of Manchester
Oxford Road
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Website: http://www.psych-sci.manchester.ac.uk/staff/131406

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

[rsmarken]

[From Rick Marken (2016.08.04.2330)]

···

On Thu, Aug 4, 2016 at 6:12 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Rick, I have the greatest respect for you, but you are presenting this work as though the issues have been dealt with, but my impression is that they all still stand.

RM: OK, how about telling me what you think the power law shows? How would you produce a model that produces the behavior in the power law studies (involving hand drawing of squiggles, for example)? What do you think is wrong with my PCT model of squiggle drawing? Since my PCT model produces hand movements that follow the power law, why don’t you consider it an explanation of the power law? The explanation being that any two dimensional movement pattern, be it a controlled variable or just a variable produced by an equation, will show the power law. The only way to show that this PCT explanation of the power law is wrong is to produce a two-dimensional movement pattern that does not follow the power law. If you can produce such a movement pattern I’ll agree that my PCT explanation of the power law is wrong. But I’m quite confident that such a movement pattern is as non-existent as a circle with a ratio of circumference to diameter that is not equal to 3.14159…

WM: I think my biggest problem is to believe that a mathematical relationship exists between velocity and curvature that somehow lies outside physics.

RM: It’s because physics has nothing to do with it, any more than physics has to do with the value of pi. The power law is based on measures of two different aspects of curved movement: C and A. There was not necessarily any physics involved in creating that curved pattern. If the movement pattern was a variable controlled by a control system then the observed pattern is the net result of many physical forces acting on the movements – the system’s output forces and environmental disturbance forces. What one sees as the movement pattern is what is specified by the varying internal references of the control system. The forces that combined to produce these movements are completely invisible in the movements themselves.

RM: I think you continue to see A and C as two independent aspects of the observed movement pattern, with A being a reflection of the output that produces the movement through the curve,C. But A and C are not independent of each other, as per my mathematical derivation. The non-independence of A and C is true whether the movement pattern is a controlled variable or an uncontrolled result.

RM: The power law researchers were deceived into thinking that A and C were independent measures of the curve by their causal view of behavior; they viewed A as a measure of behavior that is caused by a measure of environmental constraint (stimulus). So the power law researchers have been trying to explain the illusion of a causal connection between C and A because they are looking at behavior through causal theory rather than control theory glasses!

WM: To me, motion is intrinsically bound up with forces applied to masses that have surfaces in three dimensions, and any mathematical relationship that exists, exists because of these more fundamental properties of moving objects.

RM: This is true when you move an object in the real world. But the path traced out by the object will follow a power law, not because of the physics but because of the mathematical relationship between A and C. If the exact same pattern of movement were produced by an equation you would get the same power law. Hopefully you will be convinced that this is true if you send me some movement patterns that you think should not follow the power law, yet do.

WM: I think I agree with you that the law could be independent of the biology and nervous system of an organism but just not independent of physics.

RM: Well, let’s get some movement patterns to analyze and we’ll see. Actually, another good test would be to see if you can distinguish real physical movement from simulated movement (like the “Random Pattern” ones in my spreadsheet) in terms of the observed power law. I predict that it will be impossible to tell them apart in terms of the power law coefficient or degree of fit of the power law to the data.

Best regards

Rick

Warren

On 4 Aug 2016, at 23:38, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.04.1540)]

Martin Taylor (2016.07.21.23.23)–

MT: This discussion is a waste of time, and moreover likely to prove a strong deterrent for any serious scientist who might think that CSGnet could be a serious place to discuss real PCT problems.

RM: I believe this discussion has been anything but a waste of time. It is indeed likely to prove a deterrent to scientists (if any are listening) who don’t understand the nature of control in living systems – scientists who refuse to look at behavior through control theory glasses. But it may attract those who have the courage to peek through control theory glasses and see that for over 100 years behavioral scientists have been studying mostly irrelevant side effects of control. It takes courage to look through control theory glasses because doing so reveals the “nightmare” that Powers was referring to in his 1978 Psychological Review paper when he said: “The nightmare of
any experimenter is to realize too late that his results were forced by his
experimental design and do not actually pertain to behavior. This nightmare has
a good chance of becoming a reality for a number of behavioral
scientists”. Looking at the research on the power law through control theory glasses reveals this nightmare with mathematical precision.

MT: The problem Alex initially posed is an interesting one. It would be nice if someone would try to address it.

RM: I did address it but you, Bruce, Warren and Alex certainly didn’t appreciate the way I addressed it. So, for the benefit of a couple of people who asked me in private emails to review my analysis – and also to let me see if I am the only one who thinks that I have correctly addressed, from a PCT perspective, the problem Alex posed-- I will give a synopsis of how I addressed the problem Alex posed.

RM: Here is the problem Alex initially posed (to CSGNet on July 6):

Alex: Any ideas why or how “the control of perception” may give rise to this power law constraining geometry and kinematics in humans, and now in fruit fly larvae?
http://biorxiv.org/content/early/2016/07/05/062166

RM: The pointer is to a recent paper by Alex and two other authors which describes the finding of “The velocity-curvature power law in Drosophila larval locomotion”. The first figure in that paper nic
ely illustrates what was found:

<image.png>

RM: The finding is that the relationship between angular velocity, A(t), and curvature, C(t), at each instant (t) during movements made over time (the blue squiggle) is the same power function (the equation in the upper right) for humans tracing a line (the hand with pencil) and Drosophila larvae following a path (the worm). The “same power function” means that a power function equation (in the upper right of the figure) fits the human and larval movement data nearly perfectly. The fit of a power function to the data is determined using linear regression with the log of the C(t) values as the predictor variable and the log of the A(t) values as the cri
terion. The resulting R^2 values are quite high, typically greater than .92 (the C(t) values account for about 92% of the variance in the A(t) values).The coefficient, b, that gives the best fit is around .67 for humans and .75 for the larvae. While the coefficient for the larvae is higher than that for the human, it is about the same as the coefficient found for humans making hand movements in water rather than in air.

RM: Research on the power law goes back at least to 1983 and a paper by Lacquaniti, Terzuolo & Viviani (Acta psychologica, 54, 115-130). Because they were finding that the b coefficient of the power law relating C(t) to A(t) for human movement was always close to .67, they proposed the “two-thirds power law” which say the “true” power function relating curvature to angular velocity in human movement is:

A(t) = kC(t)^2/3

RM: The power law is sometimes expressed as a relationship between tangential (rather than angular) velocity, V(t),and radius of curvature (Rather than curvature), R(t),in which case the power law follows the one-third power law:

V(t) = kR(t)^1/3

RM: The change in exponent results from the fact that V(t) and R(t) are proportional to A(t) and C(t) as follows:

V(t) = A(t)/R(t) and R(t) = 1/C(t)

RM: So what is the PCT explanation of this finding of a consistent 2/3 power law relationship between curvature, C(t
), and angular velocity, A(t) or the equivalent consistent 1/3 power law relationship between radius of curvature, R(t), and tangential velocity, V(t)?The researchers in this area believe that the power law reflects a biological constraint on how curved movements are produced. That is, curvature, C(t) or R(t), is seen as an independent variable that constrains how quickly the movement around the curve, A(t) or V(t), can be produced. And the models that have been developed to produce movements that follow the power law have been caused-output models that are developed to cause outputs – instantaneous movement velocity, A(t) or V(t) – that obey the constraint that is imposed by the curvature through which the movement is taking place.

RM: The first step in my PCT analysis of the power law was to develop a PCT model of a person intentionally making squiggly drawing movements in X,Y space, like those in the figure above. The PCT model is shown be
low:

<image.png>

RM: I didn’t expect the movement pattern produced by this model to fit a power law because I was assuming that the power law had something to do with either the nature of the feedback connection of output to controlled input (k.f in the diagram) or the nature the output function itself (o.x and o.y). But in order to test this I had to obtain measures of A(t) and C(t) or V(t) and R(t) to see how well a power function fits the relationship between thee variables. I was able to find computational formulas for V(t) and R(t) in a paper by Gribble & Ostry (1996, Journal of Neurophysiology, 76, 2853-2860). The computational formulas are below:

<image.png>

RM: Since these are formulas for the instantaneous values of the variables the implication is that what is being computed is V(t) and R(t). Alex then gave me the formulas for deriving A(t) and C
(t) from these measures of V(t) and R(t):

A(t) =V(t)/R(t) and C(t)=1/R(t)

RM: I then ran a log-log regression on the movement pattern generated by the model and was surprised to find that the power law fit all of them pretty well and the best fit b coefficient was always close to 1/3 for the regression of log R(t) on log V(t) and 2/3 for the regression of log C(t) on log A(t). This made me think that the observed power law may just be a property of curved movement itself and may have nothing to do with how the movement is generated. So I generated a few different curved movements from equations (rather than from models of behavior) and found that
the power law held for these as well.

RM: The idea that the power law might just be a property of curved movement led me to wonder whether there was a mathematical power relationship between the variables used in studies of the power law: between R(t) and V(t) and between A(t) and C(t). After all, both variables are measures of the same movement pattern at the same instant. So I looked at the formulas for V and R (above) and noticed that V² occurs in the numerator of R. That is:

V = (X² + Y² )^1/2

R = [(X² + Y² )^3/2]/|X.dotY.2dot-X.2dotY.dot|

Since V² = (X² + Y² )

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

R = V³ |/ |X.dotY.2dot-X.2dotY.dot|

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

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

RM: So the variable V is mathematically in a 1/3 power relationship to R, the very power relationship found in power law studies that is presumed to tell something about how the instantaneous tangential movement (V) around a curve (R) is generated.

RM: Subst
ituting the appropriate formulas for A and C into equation (1) we get the following power relationship between A and C:

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

RM: So the variable A is mathematically in a 2/3 power relationship to C, again the very power relationship found in power law studies that is presumed to tell something about how the instantaneous angular movement (A) around a curve (C) is generated.

RM: The variable D in equations (1) and (2) accounts for the fact that the b coefficient found in power law studies varies around 1/3 (for V as a function of R) or 2/3 (for A as a function of C) for the different movement patterns. Since these studies use only log R or log C
as predictors the b coefficient found by the regression will differ from 1/3 or 2/3 depending on the nature of the variance in D that exists in the pattern of movement. When log D is included in the regression, the b coefficient for R or C is always exactly 1/3 (.33) or 2/3 (.67), respectively. (The spreadsheet demonstrating this fact is attached).

RM: So my analysi
s of the power law, based on PCT, is that the power law is simply a property of curved movement and reveals nothing about how they are produced. It doesn’t matter whether the movement pattern is produced intentionally (is a controlled variable, as it is in my PCT model above and when people intentionally draw squiggles) or a side effect of controlling for something else (as in the case fly larva movement paths). It is based on PCT because I was led to this realization by producing a PCT model of curved movement production. That model showed that any curved movement produced by the model will follow a power law – regardless of how it was produced (in terms of the parameters of the output function).

RM: The implication of equations (1) and (2) is clear; the power law research, which has been going on for ov
er 40 years, is the nightmare Powers warned of: the results were forced by the research design (measuring two related aspects of the same movement pattern) and do not actually pertain to behavior (how the movement pattern was produced). The nightmare exists because researchers took one variable (curvature) to be an environmental variable that is a cause of the other, a behavioral variable (velocity of movement). So the nightmare exists because of a failure to understand that behavior is a control (not a causal) process. Intentionally produced curved movement is a controlled variable.

RM: At any instant the state of a controlled variable, q.i, is a combined result of the effect of system output and environmental disturbances: q.i = q.
o+d. So there is no way to learn about the nature of the output, q.o, that produced the observed controlled variable, q.i, by just looking at measures of the controlled variable itself. And all the variables used in determining the power law – A, C, V and R-- are measures of the controlled variable – the pattern of movement – itself.

RM: I knew this PCT analysis of the power law would not go down well with Alex, who has apparently been involved in power law research for some time. But I was rather surprised to find that it went down just as badly with people who are ostensibly fans of PCT. There were several lines of attack directed at my analysis. One was based on mathematics, saying that my derivation was wrong for dimensional or parametric or whatever reasons. Another was based on what might be called domain conflation, saying that though equations 1 and 2 were mathematically correct, they were incompatible with physics and/or biology. And, finally, it was implied that my analysis was incorrect because the spreadsheet I used to demonstrate the analysis had errors in it.

RM: Indeed, the spreadsheet had an error (failure to divide a derivative by dt) but this error had no effect on the computations of the power law demonstrated by the spreadsheet. I have attached the corrected spreadsheet for those of you who are interested in seeing that, indeed, all curved movements – regardless of how produced – result in a power law relationship between curvature and velocity, with the power coefficient being close to .33 for the relationship between log R and log V and close to .67 for the relationship between log C and log A. When log D is included in the regression the power coefficient is always exactly .33 for log R and .67 for log (C) and the R^2 value of the regression is always 1.0 (as it should be if equations 1 and 2 are correct). You can demonstrate this to yourself by pressing the “Random Pattern” button, which generates a new random movement pattern each time it is pressed and see what happens to the value of b when only log R or log C is included in the regression and what happens when log D is also included.

RM: Agai
n, I am really sorry if this analysis shows that a whole line of research is based on results that were forced by the research design. But this analysis at least shows why PCT has such a problem getting accepted by conventional psychologists. The problem is that PCT doesn’t explain the phenomena of interest to conventional psychologists – phenomenon like the power law. It explains the phenomenon of control. Those who know PCT only as a theory assume that it’s a theory that explains the results of all research findings. And it does, I suppose. But more often than not what it explains is that the results of conventional research are not what they seem; they are side effects of control that tell you very little about the behavior under study.

RM: I believe the lesson of this PCT analysis of the power law is very important. The lesson is that before conventional psychologists (and neurophysiologists and many fans of PCT) can correctly apply PCT they have to learn to look at behavior through control theory glasses. Phenomena phirst!

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

<PowerLawRegression07.30.xlsm>

–
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

[wmansell]

Hi Rick, I sm very limited by skill, technology and time, but I will eventually produce a model! My current hypothesis is that the power law only follows if one is trying to maximise the overall distance travelled over the time period the pattern was made. If this is not maximised then one is free to slow down at shallow curves and speed up at steep ones. A PCT model could demonstrate this.

Warren

···

On Thu, Aug 4, 2016 at 6:12 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Rick, I have the greatest respect for you, but you are presenting this work as though the issues have been dealt with, but my impression is that they all still stand.

RM: OK, how about telling me what you think the power law shows? How would you produce a model that produces the behavior in the power law studies (involving hand drawing of squiggles, for example)? What do you think is wrong with my PCT model of squiggle drawing? Since my PCT model produces hand movements that follow the power law, why don’t you consider it an explanation of the power law? The explanation being that any two dimensional movement pattern, be it a controlled variable or just a variable produced by an equation, will show the power law. The only way to show that this PCT explanation of the power law is wrong is to produce a two-dimensional movement pattern that does not follow the power law. If you can produce such a movement pattern I’ll agree that my PCT explanation of the power law is wrong. But I’m quite confident that such a movement pattern is as non-existent as a circle with a ratio of circumference to diameter that is not equal to 3.14159…

WM: I think my biggest problem is to believe that a mathematical relationship exists between velocity and curvature that somehow lies outside physics.

RM: It’s because physics has nothing to do with it, any more than physics has to do with the value of pi. The power law is based on measures of two different aspects of curved movement: C and A. There was not necessarily any physics involved in creating that curved pattern. If the movement pattern was a variable controlled by a control system then the observed pattern is the net result of many physical forces acting on the movements – the system’s output forces and environmental disturbance forces. What one sees as the movement pattern is what is specified by the varying internal references of the control system. The forces that combined to produce these movements are completely invisible in the movements themselves.

RM: I think you continue to see A and C as two independent aspects of the observed movement pattern, with A being a reflection of the output that produces the movement through the curve,C. But A and C are not independent of each other, as per my mathematical derivation. The non-independence of A and C is true whether the movement pattern is a controlled variable or an uncontrolled result.

RM: The power law researchers were deceived into thinking that A and C were independent measures of the curve by their causal view of behavior; they viewed A as a measure of behavior that is caused by a measure of environmental constraint (stimulus). So the power law researchers have been trying to explain the illusion of a causal connection between C and A because they are looking at behavior through causal theory rather than control theory glasses!

WM: To me, motion is intrinsically bound up with forces applied to masses that have surfaces in three dimensions, and any mathematical relationship that exists, exists because of these more fundamental properties of moving objects.

RM: This is true when you move an object in the real world. But the path traced out by the object will follow a power law, not because of the physics but because of the mathematical relationship between A and C. If the exact same pattern of movement were produced by an equation you would get the same power law. Hopefully you will be convinced that this is true if you send me some movement patterns that you think should not follow the power law, yet do.

WM: I think I agree with you that the law could be independent of the biology and nervous system of an organism but just not independent of physics.

RM: Well, let’s get some movement patterns to analyze and we’ll see. Actually, another good test would be to see if you can distinguish real physical movement from simulated movement (like the “Random Pattern” ones in my spreadsheet) in terms of the observed power law. I predict that it will be impossible to tell them apart in terms of the power law coefficient or degree of fit of the power law to the data.

Best regards

Rick

Warren

On 4 Aug 2016, at 23:38, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.04.1540)]

Martin Taylor (2016.07.21.23.23)–

MT: This discussion is a waste of time, and moreover likely to prove a strong deterrent for any serious scientist who might think that CSGnet could be a serious place to discuss real PCT problems.

RM: I believe this discussion has been anything but a waste of time. It is indeed likely to prove a deterrent to scientists (if any are listening) who don’t understand the nature of control in living systems – scientists who refuse to look at behavior through control theory glasses. But it may attract those who have the courage to peek through control theory glasses and see that for over 100 years behavioral scientists have been studying mostly irrelevant side effects of control. It takes courage to look through control theory glasses because doing so reveals the “nightmare” that Powers was referring to in his 1978 Psychological Review paper when he said: “The nightmare of
any experimenter is to realize too late that his results were forced by his
experimental design and do not actually pertain to behavior. This nightmare has
a good chance of becoming a reality for a number of behavioral
scientists”. Looking at the research on the power law through control theory glasses reveals this nightmare with mathematical precision.

MT: The problem Alex initially posed is an interesting one. It would be nice if someone would try to address it.

RM: I did address it but you, Bruce, Warren and Alex certainly didn’t appreciate the way I addressed it. So, for the benefit of a couple of people who asked me in private emails to review my analysis – and also to let me see if I am the only one who thinks that I have correctly addressed, from a PCT perspective, the problem Alex posed-- I will give a synopsis of how I addressed the problem Alex posed.

RM: Here is the problem Alex initially posed (to CSGNet on July 6):

Alex: Any ideas why or how “the control of perception” may give rise to this power law constraining geometry and kinematics in humans, and now in fruit fly larvae?
http://biorxiv.org/content/early/2016/07/05/062166

RM: The pointer is to a recent paper by Alex and two other authors which describes the finding of “The velocity-curvature power law in Drosophila larval locomotion”. The first figure in that paper nic
ely illustrates what was found:

<image.png>

RM: The finding is that the relationship between angular velocity, A(t), and curvature, C(t), at each instant (t) during movements made over time (the blue squiggle) is the same power function (the equation in the upper right) for humans tracing a line (the hand with pencil) and Drosophila larvae following a path (the worm). The “same power function” means that a power function equation (in the upper right of the figure) fits the human and larval movement data nearly perfectly. The fit of a power function to the data is determined using linear regression with the log of the C(t) values as the predictor variable and the log of the A(t) values as the cri
terion. The resulting R^2 values are quite high, typically greater than .92 (the C(t) values account for about 92% of the variance in the A(t) values).The coefficient, b, that gives the best fit is around .67 for humans and .75 for the larvae. While the coefficient for the larvae is higher than that for the human, it is about the same as the coefficient found for humans making hand movements in water rather than in air.

RM: Research on the power law goes back at least to 1983 and a paper by Lacquaniti, Terzuolo & Viviani (Acta psychologica, 54, 115-130). Because they were finding that the b coefficient of the power law relating C(t) to A(t) for human movement was always close to .67, they proposed the “two-thirds power law” which say the “true” power function relating curvature to angular velocity in human movement is:

A(t) = kC(t)^2/3

RM: The power law is sometimes expressed as a relationship between tangential (rather than angular) velocity, V(t),and radius of curvature (Rather than curvature), R(t),in which case the power law follows the one-third power law:

V(t) = kR(t)^1/3

RM: The change in exponent results from the fact that V(t) and R(t) are proportional to A(t) and C(t) as follows:

V(t) = A(t)/R(t) and R(t) = 1/C(t)

RM: So what is the PCT explanation of this finding of a consistent 2/3 power law relationship between curvature, C(t
), and angular velocity, A(t) or the equivalent consistent 1/3 power law relationship between radius of curvature, R(t), and tangential velocity, V(t)?The researchers in this area believe that the power law reflects a biological constraint on how curved movements are produced. That is, curvature, C(t) or R(t), is seen as an independent variable that constrains how quickly the movement around the curve, A(t) or V(t), can be produced. And the models that have been developed to produce movements that follow the power law have been caused-output models that are developed to cause outputs – instantaneous movement velocity, A(t) or V(t) – that obey the constraint that is imposed by the curvature through which the movement is taking place.

RM: The first step in my PCT analysis of the power law was to develop a PCT model of a person intentionally making squiggly drawing movements in X,Y space, like those in the figure above. The PCT model is shown be
low:

<image.png>

RM: I didn’t expect the movement pattern produced by this model to fit a power law because I was assuming that the power law had something to do with either the nature of the feedback connection of output to controlled input (k.f in the diagram) or the nature the output function itself (o.x and o.y). But in order to test this I had to obtain measures of A(t) and C(t) or V(t) and R(t) to see how well a power function fits the relationship between thee variables. I was able to find computational formulas for V(t) and R(t) in a paper by Gribble & Ostry (1996, Journal of Neurophysiology, 76, 2853-2860). The computational formulas are below:

<image.png>

RM: Since these are formulas for the instantaneous values of the variables the implication is that what is being computed is V(t) and R(t). Alex then gave me the formulas for deriving A(t) and C
(t) from these measures of V(t) and R(t):

A(t) =V(t)/R(t) and C(t)=1/R(t)

RM: I then ran a log-log regression on the movement pattern generated by the model and was surprised to find that the power law fit all of them pretty well and the best fit b coefficient was always close to 1/3 for the regression of log R(t) on log V(t) and 2/3 for the regression of log C(t) on log A(t). This made me think that the observed power law may just be a property of curved movement itself and may have nothing to do with how the movement is generated. So I generated a few different curved movements from equations (rather than from models of behavior) and found that
the power law held for these as well.

RM: The idea that the power law might just be a property of curved movement led me to wonder whether there was a mathematical power relationship between the variables used in studies of the power law: between R(t) and V(t) and between A(t) and C(t). After all, both variables are measures of the same movement pattern at the same instant. So I looked at the formulas for V and R (above) and noticed that V² occurs in the numerator of R. That is:

V = (X² + Y² )^1/2

R = [(X² + Y² )^3/2]/|X.dotY.2dot-X.2dotY.dot|

Since V² = (X² + Y² )

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

R = V³ |/ |X.dotY.2dot-X.2dotY.dot|

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

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

RM: So the variable V is mathematically in a 1/3 power relationship to R, the very power relationship found in power law studies that is presumed to tell something about how the instantaneous tangential movement (V) around a curve (R) is generated.

RM: Subst
ituting the appropriate formulas for A and C into equation (1) we get the following power relationship between A and C:

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

RM: So the variable A is mathematically in a 2/3 power relationship to C, again the very power relationship found in power law studies that is presumed to tell something about how the instantaneous angular movement (A) around a curve (C) is generated.

RM: The variable D in equations (1) and (2) accounts for the fact that the b coefficient found in power law studies varies around 1/3 (for V as a function of R) or 2/3 (for A as a function of C) for the different movement patterns. Since these studies use only log R or log C
as predictors the b coefficient found by the regression will differ from 1/3 or 2/3 depending on the nature of the variance in D that exists in the pattern of movement. When log D is included in the regression, the b coefficient for R or C is always exactly 1/3 (.33) or 2/3 (.67), respectively. (The spreadsheet demonstrating this fact is attached).

RM: So my analysi
s of the power law, based on PCT, is that the power law is simply a property of curved movement and reveals nothing about how they are produced. It doesn’t matter whether the movement pattern is produced intentionally (is a controlled variable, as it is in my PCT model above and when people intentionally draw squiggles) or a side effect of controlling for something else (as in the case fly larva movement paths). It is based on PCT because I was led to this realization by producing a PCT model of curved movement production. That model showed that any curved movement produced by the model will follow a power law – regardless of how it was produced (in terms of the parameters of the output function).

RM: The implication of equations (1) and (2) is clear; the power law research, which has been going on for ov
er 40 years, is the nightmare Powers warned of: the results were forced by the research design (measuring two related aspects of the same movement pattern) and do not actually pertain to behavior (how the movement pattern was produced). The nightmare exists because researchers took one variable (curvature) to be an environmental variable that is a cause of the other, a behavioral variable (velocity of movement). So the nightmare exists because of a failure to understand that behavior is a control (not a causal) process. Intentionally produced curved movement is a controlled variable.

RM: At any instant the state of a controlled variable, q.i, is a combined result of the effect of system output and environmental disturbances: q.i = q.
o+d. So there is no way to learn about the nature of the output, q.o, that produced the observed controlled variable, q.i, by just looking at measures of the controlled variable itself. And all the variables used in determining the power law – A, C, V and R-- are measures of the controlled variable – the pattern of movement – itself.

RM: I knew this PCT analysis of the power law would not go down well with Alex, who has apparently been involved in power law research for some time. But I was rather surprised to find that it went down just as badly with people who are ostensibly fans of PCT. There were several lines of attack directed at my analysis. One was based on mathematics, saying that my derivation was wrong for dimensional or parametric or whatever reasons. Another was based on what might be called domain conflation, saying that though equations 1 and 2 were mathematically correct, they were incompatible with physics and/or biology. And, finally, it was implied that my analysis was incorrect because the spreadsheet I used to demonstrate the analysis had errors in it.

RM: Indeed, the spreadsheet had an error (failure to divide a derivative by dt) but this error had no effect on the computations of the power law demonstrated by the spreadsheet. I have attached the corrected spreadsheet for those of you who are interested in seeing that, indeed, all curved movements – regardless of how produced – result in a power law relationship between curvature and velocity, with the power coefficient being close to .33 for the relationship between log R and log V and close to .67 for the relationship between log C and log A. When log D is included in the regression the power coefficient is always exactly .33 for log R and .67 for log (C) and the R^2 value of the regression is always 1.0 (as it should be if equations 1 and 2 are correct). You can demonstrate this to yourself by pressing the “Random Pattern” button, which generates a new random movement pattern each time it is pressed and see what happens to the value of b when only log R or log C is included in the regression and what happens when log D is also included.

RM: Agai
n, I am really sorry if this analysis shows that a whole line of research is based on results that were forced by the research design. But this analysis at least shows why PCT has such a problem getting accepted by conventional psychologists. The problem is that PCT doesn’t explain the phenomena of interest to conventional psychologists – phenomenon like the power law. It explains the phenomenon of control. Those who know PCT only as a theory assume that it’s a theory that explains the results of all research findings. And it does, I suppose. But more often than not what it explains is that the results of conventional research are not what they seem; they are side effects of control that tell you very little about the behavior under study.

RM: I believe the lesson of this PCT analysis of the power law is very important. The lesson is that before conventional psychologists (and neurophysiologists and many fans of PCT) can correctly apply PCT they have to learn to look at behavior through control theory glasses. Phenomena phirst!

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

<PowerLawRegression07.30.xlsm>

–
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

[Matti]

[From MK (2016.08.05.1330 CET)]

"The fact that Marken was repeatedly told this, interpreted it to mean that others were jealous of his precision, and continued to produce experimental "results" of the same sort along with bold claims of their predictive power, makes him a crank."

<https://urldefense.proofpoint.com/v2/url?u=http-3A__lesswrong.com_lw_20y_rationality-5Fquotes-5Fapril-5F2010_1ugo&d=CwMFaQ&c=8hUWFZcy2Z-Za5rBPlktOQ&r=-dJBNItYEMOLt6aj_KjGi2LMO_Q8QB-ZzxIZIF8DGyQ&m=ekaeoVXGit6e77dvooOQvVzs182Q3otJl8eA75o08To&s=ycop6mLU5zZZvIkEaAs_LkucFrikQVPKbVijndeWLJ4&e=>http://lesswrong.com/lw/20y/rationality_quotes_april_2010/1ugo
<https://urldefense.proofpoint.com/v2/url?u=http-3A__lesswrong.com_lw_14v_the-5Fusefulness-5Fof-5Fcorrelations_11iu_&d=CwMFaQ&c=8hUWFZcy2Z-Za5rBPlktOQ&r=-dJBNItYEMOLt6aj_KjGi2LMO_Q8QB-ZzxIZIF8DGyQ&m=ekaeoVXGit6e77dvooOQvVzs182Q3otJl8eA75o08To&s=pgh7v44iibybBRDBUa24F9NGryYa7_99JcO4DezQzZ4&e=>http://lesswrong.com/lw/14v/the_usefulness_of_correlations/11iu/

M

[_Erling_Jorgensen]

[From Erling Jorgensen (2016.08.05 0800 EDT)]

Rick Marken (2016.08.04.1540)

Mathematics is definitely not my forte, but if I followed Bruce Abbott’s recent analysis (2016-07-30-0935 EDT), I have one comment to make.

In Rick’s post & derivation he lists the following:

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

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

This is the part that confuses me. It seems as though the right hand side of the equation is multiplying the velocity of X by the acceleration of Y, and then subtracting from that the product of the acceleration of X by the velocity of Y.

If that is the case, then not all the Velocity terms have gotten on the left hand side of the upper equation. And I would expect that using it as is would lead to meaningless results.

I’ll have to defer to those who know the maths much better than I.

All the best,

Erling

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[MartinT]

[Martin Taylor 2016.08.05.10.06]

Erling, Rick's mathematics is OK. If you want to know where the bit

that confuses you comes from, look at “Curvature” in Wikipedia,
where the standard formula is provided. It is Rick’s “D” over a
formal “velocity” denumerator. You need only elementary algebra to
get from there to Rick’s formula. If you want to go a bit deeper
into why this is the formula, various web sites will tell you, such
as , which also
introduces the formal quasi-time variable “t” that leads to the
formal quasi-velocity variable.
The problem is and always has been that this purely formal
intermediary “velocity” variable is primarily introduced to make the
mathematical construct of “curvature” more easily visualized, but
Rick takes it as representing the speed of anything that in “real
life” follows the curve. The effect is rather like saying that
because you are lying on a river bank to watch the stream flow by,
you are therefore monitoring the financial flows in every specific
commercial bank from the Family Bank of Podunk to the Bank of
America. The word “velocity” is used in both cases, but in one it is
only a formal parameter for describing shape that eventually
vanishes (as Bruce and I explicitly demonstrated), whereas in the
other it describes whether you can run fast enough to get to the
Family Bank of Podunk before it closes.
Parenthetically, it has always seemed to me that a lot of
philosophical argument over the ages has been based on similar uses
of one word to convey two quite different meanings, We don’t need
that kind of argument on CSGnet, which is supposedly devoted to the
advancement of PCT research.
Martin

···

On 2016/08/5 8:12 AM, Erling Jorgensen
wrote:

[From Erling Jorgensen (2016.08.05 0800 EDT)]

    >           Rick Marken

(2016.08.04.1540)

      Mathematics is definitely not my forte,

but if I followed Bruce Abbott’s recent analysis
(2016-07-30-0935 EDT), I have one comment to make.

      In Rick's post & derivation he lists

the following:

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

          where

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

          This

is the part that confuses me. It seems as though the
right hand side of the equation is multiplying the
velocity of X by the acceleration of Y, and then
subtracting from that the product of the acceleration of X
by the velocity of Y.

          If

that is the case, then not all the Velocity terms have
gotten on the left hand side of the upper equation. And I
would expect that using it as is would lead to meaningless
results.

          I'll

have to defer to those who know the maths much better than
I.

          All

the best,

        Erling

http://mathworld.wolfram.com/Curvature.html

[Abbott]

[From Bruce Abbott (2016.08.05.1055 EDT)]

Erling Jorgensen (2016.08.05 0800 EDT)] –

Rick Marken (2016.08.04.1540)

RM: Mathematics is definitely not my forte, but if I followed Bruce Abbott’s recent analysis (2016-07-30-0935 EDT), I have one comment to make.

EJ: In Rick’s post & derivation he lists the following:

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

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

This is the part that confuses me. It seems as though the right hand side of the equation is multiplying the velocity of X by the acceleration of Y, and then subtracting from that the product of the acceleration of X by the velocity of Y.

If that is the case, then not all the Velocity terms have gotten on the left hand side of the upper equation. And I would expect that using it as is would lead to meaningless results.

I’ll have to defer to those who know the maths much better than I.

BA: If you cube both sides of Rick’s equation you get

V³ = D*R. Dividing both side by D gives

V³/D = R. This is the equation for computing the radius of curvature. The units are as follows:

Length³/time³ divided by length²/time³ = length

Thus R is expressed in length units, as required for a radius.

The problem is not to be found in Rick’s equation for V, but in Rick’s failure to understand the fact that the denominator of the equation for R makes R independent of velocity. To see this, imagine a point moving around a circle of radius R at velocity V. As the point moves around the circle, V is constant and D is also constant. Thus R is constant as well, and the computation of R yields the actual radius of the circle.

We now double the velocity of the point. V is now twice as great, but so is D, so R remains the same. Plug any value of V into the equation for R and you will get the same R. Conclusion: the point can move around the circle at any velocity you choose, and R will remain constant, yielding the actual radius of the circle. What Rick calls “D” eliminates the effect of changes in velocity on the computation of R. We could vary V moment by moment (speeding up or slowing down the motion of the point around the circle) and R would not change. Rick has asserted that the radius of a circle somehow constrains the point to move at a given velocity V, but obviously it does not. The point can move at any rate, but the effect of the denominator D will be to make R constant so long as the actual curvature of the line at a given point remains the same.

Now let’s work from the other direction. Imagine that the point is following a curve whose radius is changing (an ellipse, and doing so at a constant velocity. Although V is constant, Rick’s denominator D will be changing with the curvature. Therefore R will change to reflect the changing curvature. So now we have a constant V but a changing R. But Rick asserts that V must be determined by R.

We can have both V and R changing as the point moves along a squiggle and the result will be the same. R will change to reflect changes in curvature and V will change as the person drawing the squiggle varies the speed of the pencil. At any given point, the velocity will have some value that, when plugged into the formula for R will yield the value of R at that point. Alternatively, you could plug the values of R and D into Rick’s formula and obtain the current velocity at that point.

The insight Rick lacks is that the values of V and R can change independently across observations as the point moves, even though V is a function of R at any given point. Across observations, V and R can change independently of one another.

Thus Alex’s original question: Why do motions produced by organisms often follow the power law – a particular relationship between V and R that emerges as a point moves along a given path? From the PCT perspective this translates to “what variable or variables are being controlled during that motion, and what is the specific organization of the control hierarchy that results in the observed conformity to the power law?”

Bruce

[rsmarken]

[From Rick Marken (2016.08.05.0900)]

···

On Fri, Aug 5, 2016 at 2:29 AM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I sm very limited by skill, technology and time, but I will eventually produce a model!

RM: That would be nice. But in the meantime how about telling me what’s wrong with mine? It’s clearly a PCT model and it explains the power law finding perfectly. So what’s the problem?

WM: My current hypothesis is that the power law only follows if one is trying to maximise the overall distance travelled over the time period the pattern was made.

RM: But we already know that that isn’t the case since the power law follows for movement patterns that are completely unintentional (produced by a set of equations that have no goals for anything about the movement pattern, such as the goal of maximizing the overall distance traveled over a time period).

WM: If this is not maximised then one is free to slow down at shallow curves and speed up at steep ones. A PCT model could demonstrate this.

RM: But there is nothing to explain. The power law holds whether or not one slows down at shallow curves and speeds up at steep ones. This is a fact that was found by the power law researchers themselves. Here’s the power law data for ellipses drawn at two different speeds:

RM: Note that the power coefficient estimates (and the R^2 values) are the same for the ellipses that are drawn slowly (.6 Hz) and more quickly (.8 Hz). The movement around the same shallow and steep curves of the ellipse is different in the two cases but the 1/3 power law holds in both cases (because the power coefficient was based on a regression of log(R) on log (V)).

RM: My equations and the accompanying spreadsheet demonstration shows that you can’t help but find a power law relationship between V and R or A and C for ANY movement pattern (except a straight line) produced in ANY way when you regress log(R) on log (V) or log (C) on log (A) and the power coefficient will be close to .33 for the regression of log(R) on log (V) and close to .67 or the regression of log(C) on log (A).

Best

Rick

Warren

On Friday, August 5, 2016, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.04.2330)]

–
Dr Warren Mansell
Reader in Clinical Psychology

School of Health Sciences
2nd Floor Zochonis Building
University of Manchester
Oxford Road
Manchester M13 9PL
Email: warren.mansell@manchester.ac.uk

Tel: +44 (0) 161 275 8589

Website: http://www.psych-sci.manchester.ac.uk/staff/131406

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 Thu, Aug 4, 2016 at 6:12 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Rick, I have the greatest respect for you, but you are presenting this work as though the issues have been dealt with, but my impression is that they all still stand.

RM: OK, how about telling me what you think the power law shows? How would you produce a model that produces the behavior in the power law studies (involving hand drawing of squiggles, for example)? What do you think is wrong with my PCT model of squiggle drawing? Since my PCT model produces hand movements that follow the power law, why don’t you consider it an explanation of the power law? The explanation being that any two dimensional movement pattern, be it a controlled variable or just a variable produced by an equation, will show the power law. The only way to show that this PCT explanation of the power law is wrong is to produce a two-dimensional movement pattern that does not follow the power law. If you can produce such a movement pattern I’ll agree that my PCT explanation of the power law is wrong. But I’m quite confident that such a movement pattern is as non-existent as a circle with a ratio of circumference to diameter that is not equal to 3.14159…

WM: I think my biggest problem is to believe that a mathematical relationship exists between velocity and curvature that somehow lies outside physics.

RM: It’s because physics has nothing to do with it, any more than physics has to do with the value of pi. The power law is based on measures of two different aspects of curved movement: C and A. There was not necessarily any physics involved in creating that curved pattern. If the movement pattern was a variable controlled by a control system then the observed pattern is the net result of many physical forces acting on the movements – the system’s output forces and environmental disturbance forces. What one sees as the movement pattern is what is specified by the varying internal references of the control system. The forces that combined to produce these movements are completely invisible in the movements themselves.

RM: I think you continue to see A and C as two independent aspects of the observed movement pattern, with A being a reflection of the output that produces the movement through the curve,C. But A and C are not independent of each other, as per my mathematical derivation. The non-independence of A and C is true whether the movement pattern is a controlled variable or an uncontrolled result.

RM: The power law researchers were deceived into thinking that A and C were independent measures of the curve by their causal view of behavior; they viewed A as a measure of behavior that is caused by a measure of environmental constraint (stimulus). So the power law researchers have been trying to explain the illusion of a causal connection between C and A because they are looking at behavior through causal theory rather than control theory glasses!

WM: To me, motion is intrinsically bound up with forces applied to masses that have surfaces in three dimensions, and any mathematical relationship that exists, exists because of these more fundamental properties of moving objects.

RM: This is true when you move an object in the real world. But the path traced out by the object will follow a power law, not because of the physics but because of the mathematical relationship between A and C. If the exact same pattern of movement were produced by an equation you would get the same power law. Hopefully you will be convinced that this is true if you send me some movement patterns that you think should not follow the power law, yet do.

WM: I think I agree with you that the law could be independent of the biology and nervous system of an organism but just not independent of physics.

RM: Well, let’s get some movement patterns to analyze and we’ll see. Actually, another good test would be to see if you can distinguish real physical movement from simulated movement (like the “Random Pattern” ones in my spreadsheet) in terms of the observed power law. I predict that it will be impossible to tell them apart in terms of the power law coefficient or degree of fit of the power law to the data.

Best regards

Rick

Warren

On 4 Aug 2016, at 23:38, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.08.04.1540)]

Martin Taylor (2016.07.21.23.23)–

MT: This discussion is a waste of time, and moreover likely to prove a strong deterrent for any serious scientist who might think that CSGnet could be a serious place to discuss real PCT problems.

RM: I believe this discussion has been anything but a waste of time. It is indeed likely to prove a deterrent to scientists (if any are listening) who don’t understand the nature of control in living systems – scientists who refuse to look at behavior through control theory glasses. But it may attract those who have the courage to peek through control theory glasses and see that for over 100 years behavioral scientists have been studying mostly irrelevant side effects of control. It takes courage to look through control theory glasses because doing so reveals the “nightmare” that Powers was referring to in his 1978 Psychological Review paper when he said: “The nightmare of
any experimenter is to realize too late that his results were forced by his
experimental design and do not actually pertain to behavior. This nightmare has
a good chance of becoming a reality for a number of behavioral
scientists”. Looking at the research on the power law through control theory glasses reveals this nightmare with mathematical precision.

MT: The problem Alex initially posed is an interesting one. It would be nice if someone would try to address it.

RM: I did address it but you, Bruce, Warren and Alex certainly didn’t appreciate the way I addressed it. So, for the benefit of a couple of people who asked me in private emails to review my analysis – and also to let me see if I am the only one who thinks that I have correctly addressed, from a PCT perspective, the problem Alex posed-- I will give a synopsis of how I addressed the problem Alex posed.

RM: Here is the problem Alex initially posed (to CSGNet on July 6):

Alex: Any ideas why or how “the control of perception” may give rise to this power law constraining geometry and kinematics in humans, and now in fruit fly larvae?
http://biorxiv.org/content/early/2016/07/05/062166

RM: The pointer is to a recent paper by Alex and two other authors which describes the finding of “The velocity-curvature power law in Drosophila larval locomotion”. The first figure in that paper nic
ely illustrates what was found:

<image.png>

RM: The finding is that the relationship between angular velocity, A(t), and curvature, C(t), at each instant (t) during movements made over time (the blue squiggle) is the same power function (the equation in the upper right) for humans tracing a line (the hand with pencil) and Drosophila larvae following a path (the worm). The “same power function” means that a power function equation (in the upper right of the figure) fits the human and larval movement data nearly perfectly. The fit of a power function to the data is determined using linear regression with the log of the C(t) values as the predictor variable and the log of the A(t) values as the cri
terion. The resulting R^2 values are quite high, typically greater than .92 (the C(t) values account for about 92% of the variance in the A(t) values).The coefficient, b, that gives the best fit is around .67 for humans and .75 for the larvae. While the coefficient for the larvae is higher than that for the human, it is about the same as the coefficient found for humans making hand movements in water rather than in air.

RM: Research on the power law goes back at least to 1983 and a paper by Lacquaniti, Terzuolo & Viviani (Acta psychologica, 54, 115-130). Because they were finding that the b coefficient of the power law relating C(t) to A(t) for human movement was always close to .67, they proposed the “two-thirds power law” which say the “true” power function relating curvature to angular velocity in human movement is:

A(t) = kC(t)^2/3

RM: The power law is sometimes expressed as a relationship between tangential (rather than angular) velocity, V(t),and radius of curvature (Rather than curvature), R(t),in which case the power law follows the one-third power law:

V(t) = kR(t)^1/3

RM: The change in exponent results from the fact that V(t) and R(t) are proportional to A(t) and C(t) as follows:

V(t) = A(t)/R(t) and R(t) = 1/C(t)

RM: So what is the PCT explanation of this finding of a consistent 2/3 power law relationship between curvature, C(t
), and angular velocity, A(t) or the equivalent consistent 1/3 power law relationship between radius of curvature, R(t), and tangential velocity, V(t)?The researchers in this area believe that the power law reflects a biological constraint on how curved movements are produced. That is, curvature, C(t) or R(t), is seen as an independent variable that constrains how quickly the movement around the curve, A(t) or V(t), can be produced. And the models that have been developed to produce movements that follow the power law have been caused-output models that are developed to cause outputs – instantaneous movement velocity, A(t) or V(t) – that obey the constraint that is imposed by the curvature through which the movement is taking place.

RM: The first step in my PCT analysis of the power law was to develop a PCT model of a person intentionally making squiggly drawing movements in X,Y space, like those in the figure above. The PCT model is shown be
low:

<image.png>

RM: I didn’t expect the movement pattern produced by this model to fit a power law because I was assuming that the power law had something to do with either the nature of the feedback connection of output to controlled input (k.f in the diagram) or the nature the output function itself (o.x and o.y). But in order to test this I had to obtain measures of A(t) and C(t) or V(t) and R(t) to see how well a power function fits the relationship between thee variables. I was able to find computational formulas for V(t) and R(t) in a paper by Gribble & Ostry (1996, Journal of Neurophysiology, 76, 2853-2860). The computational formulas are below:

<image.png>

RM: Since these are formulas for the instantaneous values of the variables the implication is that what is being computed is V(t) and R(t). Alex then gave me the formulas for deriving A(t) and C
(t) from these measures of V(t) and R(t):

A(t) =V(t)/R(t) and C(t)=1/R(t)

RM: I then ran a log-log regression on the movement pattern generated by the model and was surprised to find that the power law fit all of them pretty well and the best fit b coefficient was always close to 1/3 for the regression of log R(t) on log V(t) and 2/3 for the regression of log C(t) on log A(t). This made me think that the observed power law may just be a property of curved movement itself and may have nothing to do with how the movement is generated. So I generated a few different curved movements from equations (rather than from models of behavior) and found that
the power law held for these as well.

RM: The idea that the power law might just be a property of curved movement led me to wonder whether there was a mathematical power relationship between the variables used in studies of the power law: between R(t) and V(t) and between A(t) and C(t). After all, both variables are measures of the same movement pattern at the same instant. So I looked at the formulas for V and R (above) and noticed that V² occurs in the numerator of R. That is:

V = (X² + Y² )^1/2

R = [(X² + Y² )^3/2]/|X.dotY.2dot-X.2dotY.dot|

Since V² = (X² + Y² )

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

R = V³ |/ |X.dotY.2dot-X.2dotY.dot|

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

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

RM: So the variable V is mathematically in a 1/3 power relationship to R, the very power relationship found in power law studies that is presumed to tell something about how the instantaneous tangential movement (V) around a curve (R) is generated.

RM: Subst
ituting the appropriate formulas for A and C into equation (1) we get the following power relationship between A and C:

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

RM: So the variable A is mathematically in a 2/3 power relationship to C, again the very power relationship found in power law studies that is presumed to tell something about how the instantaneous angular movement (A) around a curve (C) is generated.

RM: The variable D in equations (1) and (2) accounts for the fact that the b coefficient found in power law studies varies around 1/3 (for V as a function of R) or 2/3 (for A as a function of C) for the different movement patterns. Since these studies use only log R or log C
as predictors the b coefficient found by the regression will differ from 1/3 or 2/3 depending on the nature of the variance in D that exists in the pattern of movement. When log D is included in the regression, the b coefficient for R or C is always exactly 1/3 (.33) or 2/3 (.67), respectively. (The spreadsheet demonstrating this fact is attached).

RM: So my analysi
s of the power law, based on PCT, is that the power law is simply a property of curved movement and reveals nothing about how they are produced. It doesn’t matter whether the movement pattern is produced intentionally (is a controlled variable, as it is in my PCT model above and when people intentionally draw squiggles) or a side effect of controlling for something else (as in the case fly larva movement paths). It is based on PCT because I was led to this realization by producing a PCT model of curved movement production. That model showed that any curved movement produced by the model will follow a power law – regardless of how it was produced (in terms of the parameters of the output function).

RM: The implication of equations (1) and (2) is clear; the power law research, which has been going on for ov
er 40 years, is the nightmare Powers warned of: the results were forced by the research design (measuring two related aspects of the same movement pattern) and do not actually pertain to behavior (how the movement pattern was produced). The nightmare exists because researchers took one variable (curvature) to be an environmental variable that is a cause of the other, a behavioral variable (velocity of movement). So the nightmare exists because of a failure to understand that behavior is a control (not a causal) process. Intentionally produced curved movement is a controlled variable.

RM: At any instant the state of a controlled variable, q.i, is a combined result of the effect of system output and environmental disturbances: q.i = q.
o+d. So there is no way to learn about the nature of the output, q.o, that produced the observed controlled variable, q.i, by just looking at measures of the controlled variable itself. And all the variables used in determining the power law – A, C, V and R-- are measures of the controlled variable – the pattern of movement – itself.

RM: I knew this PCT analysis of the power law would not go down well with Alex, who has apparently been involved in power law research for some time. But I was rather surprised to find that it went down just as badly with people who are ostensibly fans of PCT. There were several lines of attack directed at my analysis. One was based on mathematics, saying that my derivation was wrong for dimensional or parametric or whatever reasons. Another was based on what might be called domain conflation, saying that though equations 1 and 2 were mathematically correct, they were incompatible with physics and/or biology. And, finally, it was implied that my analysis was incorrect because the spreadsheet I used to demonstrate the analysis had errors in it.

RM: Indeed, the spreadsheet had an error (failure to divide a derivative by dt) but this error had no effect on the computations of the power law demonstrated by the spreadsheet. I have attached the corrected spreadsheet for those of you who are interested in seeing that, indeed, all curved movements – regardless of how produced – result in a power law relationship between curvature and velocity, with the power coefficient being close to .33 for the relationship between log R and log V and close to .67 for the relationship between log C and log A. When log D is included in the regression the power coefficient is always exactly .33 for log R and .67 for log (C) and the R^2 value of the regression is always 1.0 (as it should be if equations 1 and 2 are correct). You can demonstrate this to yourself by pressing the “Random Pattern” button, which generates a new random movement pattern each time it is pressed and see what happens to the value of b when only log R or log C is included in the regression and what happens when log D is also included.

RM: Agai
n, I am really sorry if this analysis shows that a whole line of research is based on results that were forced by the research design. But this analysis at least shows why PCT has such a problem getting accepted by conventional psychologists. The problem is that PCT doesn’t explain the phenomena of interest to conventional psychologists – phenomenon like the power law. It explains the phenomenon of control. Those who know PCT only as a theory assume that it’s a theory that explains the results of all research findings. And it does, I suppose. But more often than not what it explains is that the results of conventional research are not what they seem; they are side effects of control that tell you very little about the behavior under study.

RM: I believe the lesson of this PCT analysis of the power law is very important. The lesson is that before conventional psychologists (and neurophysiologists and many fans of PCT) can correctly apply PCT they have to learn to look at behavior through control theory glasses. Phenomena phirst!

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

<PowerLawRegression07.30.xlsm>

–
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

[rsmarken]

[From Rick Marken (2016.08.05.0905)]

···

RM: And yet, it turns.

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

MK (2016.08.05.1330 CET)

“The fact that Marken was repeatedly told this, interpreted it to mean that others were jealous of his precision, and continued to produce experimental “results” of the same sort along with bold claims of their predictive power, makes him a crank.”

http://lesswrong.com/lw/20y/rationality_quotes_april_2010/1ugo
http://lesswrong.com/lw/14v/the_usefulness_of_correlations/11iu/

M

[rsmarken]

[From Rick Marken (2016.08.05.0940)]

···

Martin Taylor (2016.08.04.23.03)–

MT: Warren, I think the central problem is to answer the question Rick usually claims to be the core question of PCT research: to find the controlled variable.

RM: Exactly.

MT: What perception is being controlled when they do and when they don’t conform to the 1/3 power law, and is the difference when they and other organisms conform to a 1/4 power law or no power law at all one of perception or of the effects of force on the mass being moved, with the same perception being controlled in both power-law cases? Why is the transition rate from sharp to shallow curves different from the transition rate from shallow to sharp curves? Can that difference provide a clue to what perception is being controlled?

RM: Actually, my model already does that, under the assumption that in hand movements its a perception of the X,Y position of the hand that is controlled relative to a varying reference. The model shows that the coefficient of the power law that is observed depends only on the nature of the pattern produced (and there is never a case where a pattern of movement does not follow a power law). So anything that affects the detailed shape of the movements produced will affect the value of the exponent of the power law. So testing to see how well the behavior of a control system fits the power law tells you very little about how the behavior was produced.

RM: A far more productive direction for research on hand movement would be to identify more of the variables that are under control when these movements are made, and how (and whether) control of these variables can be shown to be related to each other. In other words, stop wasting time on power law research and start doing PCT research – the kind suggested by Powers in his “Cybernetic Model for Research in Human Development” paper in LCS I; research aimed at identifying the controlled variables involved in the behavior of interest, how those variables are controlled and why. In other words, start studying living organisms as the autonomous, purposeful systems that they are. Just jump off the causal research train and head for freedom;-)

MT: I have no answers, but I think these are the kinds of questions that need to be addressed, rather than spending many lines of text on rewriting the parametric equations that describe spatial curvature, or on repeating explanations as to why that is not a useful approach to the problem. We need hypotheses about what perception(s) is/are being controlled, and suggestions as to how to disturb them so as to allow the TCV to be useful.

RM: I agree completely!

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

[_Erling_Jorgensen]

[From Erling Jorgensen (2016.08.05 1220 EDT)]

Bruce Abbott (2016.08.05.1055 EDT)]

Hi Bruce,

I have appreciated your comments, both your analysis back on 7/30/2016 & with these clarifying remarks.

Erling Jorgensen (2016.08.05 0800 EDT)] –

EJ: In Rick’s post & derivation he lists the following:

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

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

This is the part that confuses me. It seems as though the right hand side of the equation is multiplying the velocity of X by the acceleration of Y, and then subtracting from that the product of the acceleration of X by the velocity of Y.

If that is the case, then not all the Velocity terms have gotten on the left hand side of the upper equation. And I would expect that using it as is would lead to meaningless results.

BA: If you cube both sides of Rick’s equation you get

V³ = D*R. Dividing both side by D gives

V³/D = R. This is the equation for computing the radius of curvature.

EJ: Here, you are getting all the Velocity terms on the same side of the equation, as I raised above.

BA: The problem is not to be found in Rick’s equation for V, but in Rick’s failure to understand the fact that the denominator of the equation for R makes R independent of velocity.

EJ: If I understand this correctly, the Radius here is being scaled by Velocity, which when divided into Velocity-cubed, yields the proper units for length.

BA: Plug any value of V into the equation for R and you will get the same R. Conclusion: the point can move around the circle at any velocity you choose, and R will remain constant, yielding the actual radius of the circle. What Rick calls “D� eliminates the effect of changes in velocity on the computation of R.

EJ: So it seems as though Rick is busy proving a tautology. I continue to believe one cannot get meaningful conclusions from that continued reassertion.

BA: The insight Rick lacks is that the values of V and R can change independently across observations as the point moves, even though V is a function of R at any given point. Across observations, V and R can change independently of one another.

BA: Thus Alex’s original question: Why do motions produced by organisms often follow the power law – a particular relationship between V and R that emerges aas a point moves along a given path?

EJ: So Alex is noticing a stability emerge where it should not be. To those of us versed in PCT, that always makes us suspect a control process may be at work, as you say. Thanks for the clarifications.

All the best,

Erling

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[MartinT]

Â

Hi Bruce,

          I have appreciated your

comments, both your analysis back on 7/30/2016 & with
these clarifying remarks.Â

Â

···

Erling
Jorgensen (2016.08.05 0800 EDT)] –

Â

          >>EJ: In

Rick’s post & derivation he lists the following:

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

          where D

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

          >>This is the part

that confuses me. It seems as though the right hand side
of the equation is multiplying the velocity of X by the
acceleration of Y, and then subtracting from that the
product of the acceleration of X by the velocity of Y.Â

          >>If that is the case,

then not all the Velocity terms have gotten on the left
hand side of the upper equation. And I would expect that
using it as is would lead to meaningless results.Â

Â

          >BA: If you cube both

sides of Rick’s equation you get

V³
= D*R. Dividing both side by D gives

V³ /D
= R. This is the equation for computing the radius of
curvature.Â

Â

          EJ:Â  Here, you are getting

all the Velocity terms on the same side of the equation,
as I raised above.Â

Â

          >BA: The problem is not

to be found in Rick’s equation for V, but in Rick’s
failure to understand the fact that the denominator of the
equation for R makes R independent of velocity.Â

Â

          EJ:Â  If I understand this

correctly, the Radius here is being scaled by Velocity,
which when divided into Velocity-cubed, yields the proper
units for length.Â

Â

          >BA: Plug any value of

V into the equation for R and you will get the same R.Â
Conclusion: the point can move around the circle at any
velocity you choose, and R will remain constant, yielding
the actual radius of the circle. What Rick calls “D�
eliminates the effect of changes in velocity on the
computation of R.Â

Â

          EJ:Â  So it seems as though

Rick is busy proving a tautology. I continue to believe
one cannot get meaningful conclusions from that
continued reassertion.Â

Â

BA: The insight Rick lacks is
that the values of V and R can change independently across
observations as the point moves, even though V is a
function of R at any given point. Across observations, V
and R can change independently of one another.

          >BA: Thus Alex’s

original question: Why do motions produced by organisms
often follow the power law – a particular relationship
between V and R that emerges as a point moves along a
given path?Â

Â

          EJ:Â  So Alex is noticing a

stability emerge where it should not be. To those of us
versed in PCT, that always makes us suspect a control
process may be at work, as you say. Thanks for the
clarifications.Â

Â

[rsmarken]

[From Rick Marken (2016.08.05.1020)]

···

Erling Jorgensen (2016.08.05 0800 EDT)–

EJ: Mathematics is definitely not my forte, but if I followed Bruce Abbott’s recent analysis (2016-07-30-0935 EDT), I have one comment to make.

EJ: In Rick’s post & derivation he lists the following:

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

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

EJ: This is the part that confuses me. It seems as though the right hand side of the equation is multiplying the velocity of X by the acceleration of Y, and then subtracting from that the product of the acceleration of X by the velocity of Y.

If that is the case, then not all the Velocity terms have gotten on the left hand side of the upper equation. And I would expect that using it as is would lead to meaningless results.

RM: The term you find confusing is the denominator of the computational formula for R, the measure of curvature used in studies of the power law. Indeed, the formulas for V and R that I posted are the formulas used to compute the instantaneous velocity (V) and curvature (R) of the movement in power law studies. All I did was show that the computation of V is related to the computation of R by the formula:

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

RM: This shows that the V values that are computed for a movement pattern are mathematically related to R to the 1/3 power. But they are also related to D to the 1/3 power. This means that when you do a regression analysis to find the power coefficient for R, the value of the coefficient will be close to 1/3, but will “miss” to the extent that the unaccounted variance in D causes the regression to come up with a slightly different estimate of the power coefficient.

RM: I developed the spreadsheet to see how the unaccounted variance in D affects the estimate of the power coefficient. I did this by calculating V and R for different movement patterns and then running regressions with just R as a predictor (actually, log R) and with both log D and log R as predictors. The result is that the estimate of the power coefficient with just log R as a predictor is always close to .33 but can go as low as .2 or as high as .5. But when log D is included in the regression the power coefficient for R is always exactly .33. This shows that the variations in the power law coefficient that are observed (like the difference in the coefficient for movements made in air versus water) are a function of the type of movement pattern made and not on how the movement pattern is made.

RM: Bruce and Martin’s criticisms of my derivation of the the formula you find confusing:

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

RM: are attempts to make the power law mean something that is doesn’t. Their criticisms are all about finding the true meaning of the variables V and R. My analysis is based on an analysis of the relationship between variables that power law researchers are actually, measuring; what they compute in their spreadsheets. The equations I posted for V and R are the computational formulas that are used in power law research. The researchers take the string of X,Y values that are the positions of the movement over time, and compute a value of V and R (based on these X,Y values) for each time sample; and they do this using the equations for V and R that I posted. I do the same thing in my spreadsheet. All I did was show that the computational formulas for V and R are not measures of independent features of the data. In fact, they are related as:

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

Best

Rick

EJ: I’ll have to defer to those who know the maths much better than I.

–
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

[rsmarken]

[From Rick Marken (2016.08.05.1040)]

···

Bruce Abbott (2016.08.05.1055 EDT)

BA: The problem is not to be found in Rick’s equation for V, but in Rick’s failure to understand the fact that the denominator of the equation for R makes R independent of velocity.Â

RM: So you’re saying that R is independent of V? Looks to me like V is completely dependent on V and D per:

V = D^1/3Â *R^1/3Â Â

RM: And when I regress log(D) and log(R) on log (V) I find the coefficients of log(D) and log (R) to be exactly .33 and the R^2 value to be 1.0. And wouldn’t independence of V and R imply that there are movement patterns where there is 0.0 correlation between log (V) and log (R) (implying a power coefficient of 0)?And yet I have never seen such a movement pattern (Except for a perfectly straight line). So until you can show me a movement pattern where there is no correlation between log(V) and log (R) – that is, a movement pattern that doesn’t follow the power law – I’ll believe that the equation above is the reason why people have observed the power law. 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.Â

Best

Rick

Â

To see this, imagine a point moving around a circle of radius R at velocity V. As the point moves around the circle, V is constant and D is also constant. Thus R is constant as well, and the computation of R yields the actual radius of the circle.

We now double the velocity of the point. V is now twice as great, but so is D, so R remains the same. Plug any value of V into the equation for R and you will get the same R. Conclusion: the point can move around the circle at any velocity you choose, and R will remain constant, yielding the actual radius of the circle. What Rick calls “D� eliminates the effect of changes in velocity on the computation of R. We could vary V moment by moment (speeding up or slowing down the motion of the point around the circle) and R would not change. Rick has asserted that the radius of a circle somehow constrains the point to move at a given velocity V, but obviously it does not. The point can move at any rate, but the effect of the denominator D will be to make R constant so long as the actual curvature of the line at a given point remains the same.

Now let’s work from the other direction. Imagine that the point is following a curve whose radius is changing (an ellipse, and doing so at a constant velocity. Although V is constant, Rick’s denominator D will be changing with the curvature. Therefore R will change to reflect the changing curvature. So now we have a constant V but a changing R. But Rick asserts that V must be determined by R.

We can have both V and R changing as the point moves along a squiggle and the result will be the same. R will change to reflect changes in curvature and V will change as the person drawing the squiggle varies the speed of the pencil. At any given point, the velocity will have some value that, when plugged into the formula for R will yield the value of R at that point. Alternatively, you could plug the values of R and D into Rick’s formula and obtain the current velocity at that point.

The insight Rick lacks is that the values of V and R can change independently across observations as the point moves, even though V is a function of R at any given point. Across observations, V and R can change independently of one another.

Thus Alex’s original question: Why do motions produced by organisms often follow the power law – a particular relationship between V andd R that emerges as a point moves along a given path? From the PCT perspective this translates to “what variable or variables are being controlled during that motion, and what is the specific organization of the control hierarchy that results in the observed conformity to the power law?â€?

Bruce

Â

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

[rsmarken]

[From Rick Marken (2016.08.05.1115)]

···

Martin Taylor (2016.08.05.12.50)–

          EJ:  So Alex is noticing a

stability emerge where it should not be. To those of us
versed in PCT, that always makes us suspect a control
process may be at work, as you say. Thanks for the
clarifications.

MT: That was exactly Alex’s original question a month ago: “* Any ideas
why or how “the control of perception” may give rise to this power
law constraining geometry and kinematics in humans, and now in
fruit fly larvae* ?” The quest for an answer has unfortunately
been drowned out in the intervening month by an essentially
irrelevant discussion about this tautology, contained in at least
three threads.

RM: Actually it was my answer to this question that has been drowned out by irrelevant mathematical discussions of why my answer is wrong. But first of all I should point out that the question itself is of the “when did you stop beating your wife?” variety. When Alex asked "Any ideas why or how “the control of perception” may give rise to this power law constraining geometry and kinematics in humans, and now in fruit fly larvae? he implied that he knew that the power law “constrains geometry and kinematics in humans and the fruit fly larvae”. My answer to his question shows that the power law does no such thing.

RM: I answered the factual part of Alex’s question: “Any ideas why or how “the control of perception” may give rise to this power law”. This question is factual because the power law is a fact and Alex is asking for a PCT explanation of this fact. The idea that the power law “constrains geometry and kinematics in humans and the fruit fly larvae” is not a fact; it is an interpretation of a fact. And my PCT analysis of the power law shows that this interpretation is incorrect.

RM: Since my answer to Alex’s question starts with my PCT model of hand movement perhaps you could start by telling me what’s wrong with that model.

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 (2016.08.05.1055 EDT)]

Rick Marken (2016.08.05.1040) –

Bruce Abbott (2016.08.05.1055 EDT)

BA: The problem is not to be found in Rick’s equation for V, but in Rick’s failure to understand the fact that the denominator of the equation for R makes R independent of velocity.

RM: So you’re saying that R is independent of V? Looks to me like V is completely dependent on V and D per:

V = D1/3 *R1/3

RM: And when I regress log(D) and log(R) on log (V) I find the coefficients of log(D) and log (R) to be exactly .33 and the R^2 value to be 1.0. And wouldn’t independence of V and R imply that there are movement patterns where there is 0.0 correlation between log (V) and log (R) (implying a power coefficient of 0)?And yet I have never seen such a movement pattern (Except for a perfectly straight line). So until you can show me a movement pattern where there is no correlation between log(V) and log (R) – that is, a movement pattern that doesn’t follow the power law – I’ll believe that the equation above is the reason why people have observed the power law. 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. To keep things simple, I have a point moving around a circle as a function of the angle subtended by a line connecting the point on the circumference to the center of the circle (0 degrees = horizontal with the point to the right of the origin). As the number of degrees increases, the point moves counterclockwise around the circle. In the spreadsheet, I track V and R while increasing the number of degrees from 0 to 45 in one-degree steps. For a given radius, V is constant and R as estimated from the changes in the point’s velocity and acceleration is also constant.

BA: The position of the point is computed from the central angle as follows: X = radiuscos(angle); Y = radiussin(angle). The spreadsheet sine and cosine functions require the angle to be specified in radians, so angle in degrees is converted to radians before these functions are called. R is computed from the formula R = V³/D.

BA: The number in cell B2 is the radius of the circle. With an arbitrary curve we would not know this value; the formula for R serves to estimate it. I’ve set the radius to 10 cm initially, but you can change it to other values to explore how changing the circle’s radius alters the values of V and the estimated radius R.

BA: I tried radius values of 1, 10, and 20, 30, and 40 cm. Increasing the radius increases V, and R changes to approximate the actual radius of the circle. The relationship between V and R is as follows:

V = 0.017453* R cm/degree

Note that V is in units of cm/degree, i.e., number of cm traveled along the circumference per degree of angle. It is not velocity computed by dividing distance by time.

However, if we change the angle at the rate of one degree/second, then we have V = cm/degree * degrees/second = cm/second, the velocity or speed of the point around the circle. In this case then,

V = 0.017453*R cm/second

As you change the value of the radius in cell B2, note how this change affects V and R. As the radius of the circle increases, the number of centimeters traversed by the point also increases, because the point must complete a full circle by the time the angle has reached 360 degrees, and the circumference gets larger as the radius increases. As the radius increases, the velocity also increases, but this increase is linear, not a power function.

If you want to vary the velocity of the point in cm/second, just use a different number of degrees/second revolution rate. That velocity can be any value you want; what is important is the number of cm traversed along the circumference per degree of angle, not the velocity in cm/sec.

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.

Put THAT in your pipe and smoke it!

Bruce

Circle R.xlsx (20.7 KB)