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

[From Rick Marken (2016.07.24.1220)]

Martin Taylor (16.07.20.21.19)

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

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

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

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

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

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

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

Testing the Curvature- Velocity (110 Bytes)

Testing the Curvature- Velocity1 (111 Bytes)

PowerLawRegression.xlsm (577 KB)

···

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

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

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

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

And from there it’s a couple steps to:

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

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

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

V = D1/3 *R1/3

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

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

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

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

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

Best regards

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

V = D1/3 *R1/3

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

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

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

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

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

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

[From Fred Nickols (2016.07.24.1534 ET)]

Rick et al:

I have been following this thread as closely as I can. I don’t pretend to understand any of the math but I can follow the discussion (or at least I think I can).

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

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

Fred Nickols

···

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

[From Rick Marken (2016.07.24.1220)]

Martin Taylor (16.07.20.21.19)

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

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

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

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

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

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

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

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

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

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

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

And from there it’s a couple steps to:

V = D1/3 *R1/3

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

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

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

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

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

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

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

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

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

V = D1/3 *R1/3

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

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

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

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

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

Best regards

Rick

Richard S. Marken

“The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves.” – William T. Powers

[From
Fred Nickols (2016.07.24.1534 ET)]

``````        Rick
``````

et al:

``````        I
``````

have been following this thread as closely as I can. I
don’t pretend to understand any of the math but I can follow
the discussion (or at least I think I can).

``````        It
``````

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

Re Testing the Curvature- Velo (109 Bytes)

Re Testing the Curvature- Velo1 (110 Bytes)

Re Testing the Curvature- Velo2 (110 Bytes)

Re Testing the Curvature- Velo3 (110 Bytes)

Re Testing the Curvature- Velo4 (110 Bytes)

Re Testing the Curvature- Velo5 (110 Bytes)

Re Testing the Curvature- Velo6 (110 Bytes)

···

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

[From Rick Marken (2016.07.24.1220)]

Martin Taylor (16.07.20.21.19)

``````            MT: If "V" is a true velocity, then
``````

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

``````            MT: I'm leaving the rest of my
``````

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

``````            RM:  The power law is tested using
``````

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

``````            RM: The top figure shows the two
``````

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

``````            RM: To do the regression of log(R) on
``````

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

``````          and the value of R (curvature) is
``````

computed at each point as follows:

``````          RM: These are formulas I used for
``````

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

``````                RM: This graph is from Wann, Nimmo-Smith & Wing
``````

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

``````                RM: The fact that a power law (with a coefficient
``````

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

``````                RM: Looking at the formulas for V and R I noticed
``````

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

``````                  R
``````

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

``````                  And
``````

from there it’s a couple steps to:

``````                        V
``````

= D1/3 *R1/3

``````                        where
``````

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

``````                      RM: So the math shows
``````

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

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

``````                        where
``````

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

``````                      RM: The research on the
``````

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

``````                      RM: But here is an example
``````

of a movement pattern that results in a
perfect power law relationship between V and R
with a power coefficient of exactly 1/3.

``````                RM: This movement pattern was
``````

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

``````                RM: A regression analysis of this
``````

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

``````                RM: These results show that
``````

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

V = D1/3 *R1/3

``````                RM: And this equation can
``````

be found by simply observing that the equation that
defines V is part of the equation that defines R.

``````                RM: So the big question is
``````

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

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

``````                RM: The power law research
``````

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

``````                RM: I'm attaching the
``````

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

Best regards

Rick

``````                                        Richard
``````

S. Marken

``````                                          "The childhood of the
``````

human race is far from
over. We have a long way
to go before most people
will understand that what
they do for others is just
as important to their
well-being as what they do
for themselves." –
William T. Powers

[From Rick Marken (2015.07.24.2210)]

Re Testing the Curvature- Velo7 (110 Bytes)

Re Testing the Curvature- Velo8 (110 Bytes)

Re Testing the Curvature- Velo9 (110 Bytes)

Re Testing the Curvature- Velo10 (111 Bytes)

Re Testing the Curvature- Velo11 (111 Bytes)

Re Testing the Curvature- Velo12 (111 Bytes)

Re Testing the Curvature- Velo13 (111 Bytes)

···

Martin Taylor (2016.07.25.00.11)–

``````        FN: It
``````

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

``````MT: It would be nice if that had some chance of being true, but it
``````

doesn’t, since Rick continues not to address the question originally
posed, as Alex and I have repeatedly pointed out. I suppose it’s no
use repeating for the umpteenth time the what Rick calls “V”
(velocity) has no relation to the velocity for which the power law
is a puzzle.

RM: But I did deal with that. I gave references to two papers that use the measure of velocity described by the equation:

RM: If you know of some other papers that use what you consider to be the correct measure of velocity then give me the references and the computational formula and I’ll re-do the analysis using that measure of velocity.

Best

Rick

``````MT: Rick's V is simply a long understood property of smooth
``````

curves that has no significance for PCT whatever. Any controller
that produces smooth trajectories will produce curves that have the
power-law property computed by Rick, as has been known for many

``````  RM: So the big question is why did all the
``````

researchers in this area fail to see either that this mathematical
relationship between V and R exists or, if they did know that it
exists, why did they fail to see its implication, which is that
any observed relationship between V and R is determined by math,
not by anything about how the movement is generated?

``````"Velocity" is actually quite a misnomer for the quantity Rick labels
``````

“V”, since its computation requires no speed or distance-time
relationship. It’s just a property of shapes. But although I realize
that this point will not influence Rick’s belief, it might be worth
making yet again.

``````Rick asked:

Right from the beginning, and from references in a paper Alex
``````

linked, that beginning is over 100 years ago, all the people
studying the problem have known this relationship between what Rick
calls “V” and R. That’s been the starting point for the research on
the question Alex posed to us. Why does the same power law apply to
actual along-track speed under some circumstances, and why does a
different power law apply under other circumstances?

``````The question Alex raised and that nobody has addressed (I tried a
``````

suggestion that I no longer believe in a early message in this
thread) can be posed equivalently both as a power law relating
along-track speed (linear velocity measured in distance units per
time unit) to curvature (the sharper the curve, the slower people
and other organisms go around it), and as angular velocity (change
in tangent angle per second) as a function of curvature (the plots
in the article Alex first linked).

``````Rick's V is neither. His V is change in tangent angle per unit
``````

distance travelled along the curve. No speed (velocity) involved at
all. In his derivation using the dot notation, one set of dots
refers to time derivatives, the other, which he equated with the
first, actually refers to space derivatives. When the along-track
speed is defined to be fixed at 1 exactly, the two are, of course
numerically the same. But when along-track speed variation is the
core of the problem, they aren’t.

``````Alex implicitly (or explicitly?) posed two interesting questions, to
``````

PCT, in that if PCT can answer it when at least three decades of
non-PCT research have failed, it would be a public coup that might
make a few more researchers seriously interested in the
ramifications of perceptual control. The two questions are (1) why
the 1/3 (or 2/3) power law holds in so many different conditions,
and (2) why it changes to a 1/4 (or 3/4) power law when the motion
is in a viscous medium such as water or when the organism is a
crawling fly larva (the subject in Alex’s own paper, the first one
he linked). The mere fact that question (2) has to be asked shows
that the 1/3 power law is not a property of the trajectory, as Rick
keeps claiming.

``````One might add a third question, which would be why slowly rambling
``````

organisms often don’t follow a power law at all, but I think the
answer to that one is intuitively clear: it’s because if they are
going slowly, they don’t need go even slower for sharp curves. I can
walk around a mountain hairpin turn at the same speed I walk along
the straights, but a car going at highway speeds along the straights
has to slow down around the hairpin turn.

``````Martin
``````
``````        I
``````

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

``````        Fred
``````

Nickols

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

[From Rick Marken (2016.07.24.1220)]

Martin Taylor (16.07.20.21.19)

``````            MT: If "V" is a true velocity, then
``````

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

``````            MT: I'm leaving the rest of my
``````

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

``````            RM:  The power law is tested using
``````

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

``````            RM: The top figure shows the two
``````

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

``````            RM: To do the regression of log(R) on
``````

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

``````          and the value of R (curvature) is
``````

computed at each point as follows:

``````          RM: These are formulas I used for
``````

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

``````                RM: This graph is from Wann, Nimmo-Smith & Wing
``````

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

``````                RM: The fact that a power law (with a coefficient
``````

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

``````                RM: Looking at the formulas for V and R I noticed
``````

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

``````                  R
``````

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

``````                  And
``````

from there it’s a couple steps to:

``````                        V
``````

= D1/3 *R1/3

``````                        where
``````

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

``````                      RM: So the math shows
``````

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

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

``````                        where
``````

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

``````                      RM: The research on the
``````

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

``````                      RM: But here is an example
``````

of a movement pattern that results in a
perfect power law relationship between V and R
with a power coefficient of exactly 1/3.

``````                RM: This movement pattern was
``````

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

``````                RM: A regression analysis of this
``````

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

``````                RM: These results show that
``````

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

V = D1/3 *R1/3

``````                RM: And this equation can
``````

be found by simply observing that the equation that
defines V is part of the equation that defines R.

``````                RM: So the big question is
``````

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

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

``````                RM: The power law research
``````

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

``````                RM: I'm attaching the
``````

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

Best regards

Rick

``````                                        Richard
``````

S. Marken

``````                                          "The childhood of the
``````

human race is far from
over. We have a long way
to go before most people
will understand that what
they do for others is just
as important to their
well-being as what they do
for themselves." –
William T. Powers

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From Rick Marken (2016.07.25.1020)

···

Martin Taylor (2016.07.25.00.11)

``````MT: The question Alex raised and that nobody has addressed ...can be posed equivalently both as a power law relating
``````

along-track speed (linear velocity measured in distance units per
time unit) to curvature (the sharper the curve, the slower people
and other organisms go around it), and as angular velocity (change
in tangent angle per second) as a function of curvature (the plots
in the article Alex first linked).

RM: Now I see one of the things you are confused about. Some studies express the power law in terms of V and R. Recent papers tend to express it in terms of the variables you like, A (angular velocity) and C (curvature). As Alex pointed out to me, A = V/R and C = 1/R. So plugging these equations for A and C into the expression relating R to V:

V = D1/3+V1/3

where

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

we get the relationship between A and C:

A= D1/3+C2/3

When you do a log - log regression of C on A you get a b value close to .67 (2/3) rather than close to .33 (1/3) as you do when you regress log (R) on log (V). And, indeed, this is what Alex found in his study. Here’s the results of Alex’s analysis of the path of a fruit fly larva. The path is on the left; the log-log regression of C on A is on the right.

I get very similar results when I do a log-log regression of C on A for the little squiggle movement I drew. My results are below; the movement pattern is on the left, the log log regression of C on A on the right.

RM: The power coefficient is .80 for the human path, compared to .83 for the larva above. The R^2 values are about the same as well; .91 for me, .94 for the larva. So you’ll be happy to know that I behave pretty much like a fly larva;-) If you include the D term in a multiple regression you find that the coefficients of log D and log C are .33 and .67, respectively, as predicted by the equation for the relationship between C and A ( A= D1/3+C2/3).

RM: I think the only thing for you to do now is to develop some other explanations of why I’m all wrong. I’m sure you can do it! After all, if I move like a fly larva I probably think like one too;-)

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

[From Rick Marken (2015.07.25.1040)
RM: Fly larva brain strikes again! I said:

RM: As Alex pointed out to me, A = V/R and C = 1/R. So plugging these equations for A and C into the expression relating R to V:
V = D1/3+V1/3

RM: Of course, that should be:
V = D1/3+R1/3

RM: From which we get:
A= D1/3+C2/3
Best
Rick

···

--
Richard S. Marken
"The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves." -- William T. Powers

[Martin Taylor 2016.07.25.13.13]

``````Maybe the following will be self-evident when it is pointed out, but
``````

it often seems to be ignored. If, say, an equation says that a mass
is equal to a distance, or a voltage is equal to a velocity, there’s
something wrong with the equation. Yet we often see equations in
which similarly impossible equalities are asserted. So I thought it
might be useful to offer a quick guide to dimensional analysis.

``````We start with a set of basic dimensions. They are basic in the sense
``````

than none can be derived from operations on the others. One possible
set is distance, time, and mass, which we can symbolize as L, T, and
M, respectively. If these are taken as the basic dimensions, then
Force cannot be basic, since F=Ma, and acceleration is based on
distance and time There are other basic dimensions such as
electromagnetic ones, but L, T, and M will do for my purpose in this
message.

``````It is impossible to add a mass to a time and get anything sensible,
``````

or to add a distance to an area, an area being one distance
multiplied by another. Taking X and Y to be different dimensions
(basic or compound), we have the following rules.

``````X + X = X (adding a distance to a distance yields a distance; adding
``````

an area to an area yields an area)

``````X * X = X<sup>2</sup> (multiplying a distance by a distance yields
``````

an area; multiplying X by itself n times yields Xn.)

``````X / X = 0 (dividing a distance by a distance give a dimensionless
``````

quantity, just a number)

``````X+Y = BAD (you can't add a distance to a mass)

X * Y = XY  (you can multiply a distance by a mass)

X / Y = XY<sup>-1</sup> (you can divide a distance by a time; the
``````

result is called speed or velocity)

``````An equation:

X = X is plausible

X = Y is not plausible

Differentiation and integration.

dX/dY = XY<sup>-1</sup>    . (a differential is of the same dimension as
``````

its base quantity)

``````d<sup>2</sup>X/dy<sup>2</sup> = XY<sup>-1</sup>Y<sup>-1</sup> = XY<sup>-2</sup>.

Integral(X dY) = XY (the integral can be seen as the area under a
``````

graph of X in terms of Y, and dY has the dimension of Y)

``````Logs and exponentiation

log(X) = X (the logarithm of a distance is a distance)

exp(X) = X (because exp(log(v)) = v)

Any compound such as X<sup>a</sup>Y<sup>b</sup> can be substituted
``````

for X and Y above, where X and Y are different compounds in any one
equation.

``````The addition rules above work for subtraction as well as addition.

Some examples:

An angle measured in radians can be described by the ratio of the
``````

arc of a circle to the radius of the circle, so it has dimensions
L/L = 0. It is a pure number. So is an angle measured in degrees
(but its scale is different when you use the equation with measured
variables). So is the rate at which a clock gains or loses (seconds
per day has dimension T/T = 0).

``````A velocity is distance per unit time, or L/T = LT<sup>-1</sup>.

An angular velocity is a number per unit time of T<sup>-1</sup>.

The equation for the radius of a circle is R = sqrt(x<sup>2</sup> +
``````

y2) which dimensionally is (L2 + L2)-2
= (L2)-2 = L. The equation is plausible.

``````Non-integer exponents are allowed.

>X'*Y''-X.''*Y.dot|<sup>1/3</sup>
* R<sup>1/3</sup> is (LT<sup>-1</sup>*LT<sup>-2</sup> - LT<sup>-2</sup>*LT<sup>-1</sup>)<sup>1/3</sup>*L<sup>1/3</sup>
=(L<sup>2</sup>T<sup>-3</sup>* - L<sup>2</sup>T<sup>-3</sup>)<sup>1/3</sup>*L1<sup>/3</sup>
= (L<sup>2</sup>T<sup>-</sup><sup>3</sup>)<sup>1/3</sup>*L<sup>1/3</sup>
``````

= L/T which is a linear velocity.

``````dθ/ds = L<sup>-1</sup> (angular change per unit path length, if θ is
``````

an angle and s is a distance such as a portion of a curve)

``````dθ/dt = T<sup>-1</sup> (angular velocity, angular change per unit
``````

time).

``````d<sup>2</sup>s/dt<sup>2</sup> (where s is a distance) yields LT<sup>-2</sup>    ,
``````

so force has dimension MLT-2 because F=ma.

``````And so forth.

Anyone can use dimensional analysis to check the plausibility of
``````

their equations. You don’t have to be a mathematician. If the
dimensions on opposite sides of an equal sign differ, the equation
is wrong. It could be wrong even if the dimensions are the same, but
the plausibility check is often a good place to start. You can also
often use dimensional analysis to check whether you are starting
your problem solution with a plausible set of variables. It’s a
quick and easy way to find some easily made errors.

``Martin``

[From Rick Marken (2016.07.26. 1340)]

···

Martin Taylor (2016.07.25.13.13)–]

``````MT: If, say, an equation says that a mass
``````

is equal to a distance, or a voltage is equal to a velocity, there’s
something wrong with the equation. Yet we often see equations in
which similarly impossible equalities are asserted. So I thought it
might be useful to offer a quick guide to dimensional analysis.

RM: I’ll take this as your latest attempt to show that my analysis is incorrect. Since you are the one who says that my analysis doesn’t pass dimensional analysis muster I think it behooves you to show that this is true. I don’t see any dimensional issue at all.

RM: My analysis shows that there is a mathematical relationship between measures of curvature (measured as R or C) and angular velocity (measured as V or A) of the form:

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

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

RM: These derivations are based on the computational formulas for V and R given in Gribble & Ostry (1996) J. Neurophysiology, 76(5), 2853-2860. Here is the section of their paper that gives the formulas for V and R:

RM: Clearly, the velocities (X.dot and Y.dot) and accelerations (X.2dot and Y.2dot) in the equations for both V and R are the *same variables *. This means that it is perfectly legitimate for substitute V2 = X.dot2 + Y.dot2 into the numerator of the equation for R resulting in

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

which reduces to:

R = V1/3 /|X.dotY.2dot-X.2dotY.dot|

and solving for V gives:

V = |X.dotY.2dot-X.2dotY.dot|1/3 *R1/3

RM: So simple math shows that there is a 1/3 power relationship between R and V (and a 2/3 power relationship between C and A).

RM: Empirical studies of the power law aim to determine the actual coefficient of the power relationship between R and V (or between C and A). This coefficient is called b and the result of these efforts (up to 1996) are summarized by Gribble and Ostry as follows:

RM: The fact that b always turns out to be close to 1/3 for the power relationship between R and V and close to 2/3 for the power relationship between C and A is taken to be a reflection of some bio-mechanical law or constraint. But the equations for V as a function of R and for A as a function of C suggests that thus simply be a mathematical property of movements along a curve. The deviations from 1/3 for the V/R and 2/3 for the A/C power coefficient (b) could be the result of different variations in the D variable for different movement patterns, variations that are not taken into account in the power law research. So the idea that the power “law” reflects some bio-mechanical fact about how movement is produced is an example of what in PCT we call a “behavioral illusion”.

RM: My simulations show that different movement patterns result in different estimates of b but the estimates of b are always close to 1/3 for the V/R and 2/3 for the A/C power function, the same as what is found empirically. When the D variable is taken into account, using multiple regression, the b coefficient for regressing R on V is always 1/3, the b coefficient for regressing C on A is always 2/3 and the coefficient for D in either regression is always 1/3. And the multiple regression that includes the D variable always pick up 100% of the variance in V (or A), which is strong computational evidence that the equations I derived for the relationships between V and R and A and C are correct, with no dimensional problems.

RM: In an earlier post you say the following:

MT: Alex implicitly (or explicitly?) posed two interesting questions, to which an effective answer might really be a significant advance for PCT, in that if PCT can answer it when at least three decades of non-PCT research have failed, it would be a public coup that might make a few more researchers seriously interested in the ramifications of perceptual control. The two questions are (1) why the 1/3 (or 2/3) power law holds in so many different conditions, and (2) why it changes to a 1/4 (or 3/4) power law when the motion is in a viscous medium such as water or when the organism is a crawling fly larva (the subject in Alex’s own paper, the first one he linked). The mere fact that question (2) has to be asked shows that the 1/3 power law is not a property of the trajectory, as Rick keeps claiming.

RM: These are very good questions and I have already given the PCT-based answers to both. The answer to (1) – why the power law coefficients are always 1/3 or 2/3 – is because:

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

RM: This finding is based on PCT because PCT starts with the understanding that behavior, like drawing curved lines, is a control process. The curved lines that are produced are controlled variables, not caused outputs. Since the output used to produce controlled variables occurs in a closed loop it cannot be “seen” in the behavior of the controlled variable itself. So an understanding of PCT (and of the nature of control) leads to the conclusion that neither R (relative to V) nor C (relative to A) can possibly be measures of the output that produced the observed behavior (curved movement reflected in V or A). The explanation of the power law must lie somewhere other than in how the behavior (controlled variable) was produced and I found it by looking at the behavior (the curved movement, the controlled variable) itself.

RM: The answer to (2) is that the viscosity of the medium in which curved movements are produced affects the feedback connection between output (muscle force) and controlled result (curved movement). I have used a PCT model to produce curved movements in simulations of both high and low viscosity media (viscosity being simulated by the value of the feedback function coefficient, 1.0 for low and .5 for high) and found that the b coefficient of the power law for movement patterns in low viscosity media was typically close to 1/3 while that for movement patterns in high viscosity media was typically close to 1/4.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, this means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

Warren

···

Martin Taylor (2016.07.25.13.13)–]

``````MT: If, say, an equation says that a mass
``````

is equal to a distance, or a voltage is equal to a velocity, there’s
something wrong with the equation. Yet we often see equations in
which similarly impossible equalities are asserted. So I thought it
might be useful to offer a quick guide to dimensional analysis.

RM: I’ll take this as your latest attempt to show that my analysis is incorrect. Since you are the one who says that my analysis doesn’t pass dimensional analysis muster I think it behooves you to show that this is true. I don’t see any dimensional issue at all.

RM: My analysis shows that there is a mathematical relationship between measures of curvature (measured as R or C) and angular velocity (measured as V or A) of the form:

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

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

RM: These derivations are based on the computational formulas for V and R given in Gribble & Ostry (1996) J. Neurophysiology, 76(5), 2853-2860. Here is the section of their paper that gives the formulas for V and R:

<image.png>

RM: Clearly, the velocities (X.dot and Y.dot) and accelerations (X.2dot and Y.2dot) in the equations for both V and R are the *same variables *. This means that it is perfectly legitimate for substitute V2 = X.dot2 + Y.dot2 into the numerator of the equation for R resulting in

which reduces to:

R = V1/3 /|X.dotY.2dot-X.2dotY.dot|

and solving for V gives:

V = |X.dotY.2dot-X.2dotY.dot|1/3 *R1/3

RM: So simple math shows that there is a 1/3 power relationship between R and V (and a 2/3 power relationship between C and A).

RM: Empirical studies of the power law aim to determine the actual coefficient of the power relationship between R and V (or between C and A). This coefficient is called b and the result of these efforts (up to 1996) are summarized by Gribble and Ostry as follows:

<image.png>

RM: The fact that b always turns out to be close to 1/3 for the power relationship between R and V and close to 2/3 for the power relationship between C and A is taken to be a reflection of some bio-mechanical law or constraint. But the equations for V as a function of R and for A as a function of C suggests that thus simply be a mathematical property of movements along a curve. The deviations from 1/3 f
or the V/R and 2/3 for the A/C power coefficient (b) could be the result of different variations in the D variable for different movement patterns, variations that are not taken into account in the power law research. So the idea that the power “law” reflects some bio-mechanical fact about how movement is produced is an example of what in PCT we call a “behavioral illusion”.

RM: My simulations show that different movement patterns result in different estimates of b but the estimates of b are always close to 1/3 for the V/R and 2/3 for the A/C power function, the same as what is found empirically. When the D variable is taken into account, using multiple regression, the b coefficient for regressing R on V is always 1/3, the b coefficient for regressing C on A is always 2/3 and the coefficient for D in either regression is always 1/3. And the multiple regression that includes the D variable always pick up 100% of the variance in V (or A), which is strong computational evidence that the equations I derived for the relationships between V and R and A and C are correct, with no dimensional problems.

RM: In an earlier post you say the following:

[From Rick Marken (2016.07.26.1455)]

···

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, this means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

Mmm, one more thing…

RM: C = 1/R

···

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

[From Rick Marken (2016.07.26.1455)]

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, this means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

[From Rick Marken (2016.07.26.1535)]

···

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

WM: Mmm, one more thing…

RM: C = 1/R

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

Best

Rick

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

[From Rick Marken (2016.07.26.1455)]

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, this means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

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

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

Warren

···

On Tue, Jul 26, 2016 at 11:36 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.26.1535)]

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

WM: Mmm, one more thing…

RM: C = 1/R

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

[From Rick Marken (2016.07.26.1455)]

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, this means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

[Martin Taylor 2016.07.27.09.42]

[From Rick Marken (2016.07.26. 1340)]

``````No. As I used your revised analysis in my tutorial as an example of
``````

a correct dimensional equation, and separately noted that you had
fixed the problem, I don’t see why you say this.

``````I did (and do) say you have a different kind of dimensional problem
``````

in that you deal with the derivatives of the curvature analysis as
though they were time derivatives rather than space derivatives. The
two approaches give the same numerical result if the speed is unity,
so often it doesn’t matter. But sometimes it does matter. I thought
you were making the mistake because you were relying either on a
misreading of the Wikipedia article on Curvature or because your
mathematical filial assistant had used the same example. But now I
see that neither was your source, and your source also used time
derivatives. But Gribble and Ostry were not treating their V as a
fixed function of R, as you do. Alex pointed out the problem with
doing this very early in the curvature series of threads. I’ll try
to make his point again, coming at it from a different angle.

``````So far, no mathematical problem.

No it doesn't. Look at the expression you wind up with. "2dot" means
``````

acceleration, and acceleration, which appears twice in this
expression, is a completely free quantity (or rather, it’s a
function of R and V). Where does it come from? It comes from one of
the expressions for curvature, Gribble and Ostry’s equation 9 that
you so kindly reproduced. So, instead of saying that your V =

dx/dt*d2y/dt2 - d2X/dt2*dy/dt|1/3
is a constant, you should note its dependence on V and R, thus
destroying the simple proportionality of V and R1/3.

``````Gribble and Ostry have it right apart from the misleading impression
``````

that the derivatives need to be with respect to time, but you
misinterpret your mathematically reorganization of their basic
equations because you take this function of R to be a constant
multiplier of R1/3 . Dimensionally that would be wrong
since V is L/T and R1/3 is L1/3 , unless the
“constant” had dimension L2/3 /T. And such an odd
dimension should make one suspicious that something funny was going
on.

``````Excellent. Maybe it's even the right answer. But it does give the
``````

lie to your long-standing claim that the exponent must always be
1/3. I was thinking along the same lines, but I wasn’t thinking of
the feedback function coeffiicient. I was thinking of the different
integral-differential effects of force on mass in a viscous versus a
friction free environment. Could you show the model?

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

Re Testing the Curvature- Velo14 (111 Bytes)

···

Martin Taylor (2016.07.25.13.13)–]

``````            MT: If, say, an equation says that
``````

a mass is equal to a distance, or a voltage is equal to
a velocity, there’s something wrong with the equation.
Yet we often see equations in which similarly impossible
equalities are asserted. So I thought it might be useful
to offer a quick guide to dimensional analysis.

``````          RM: I'll take this as your latest attempt to show that
``````

my analysis is incorrect. Since you are the one who says
that my analysis doesn’t pass dimensional analysis muster
I think it behooves you to show that this is true. I don’t
see any dimensional issue at all.

``````          RM: My analysis shows that there is a mathematical
``````

relationship between measures of curvature (measured as R
or C) and angular velocity (measured as V or A) of the
form:

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

``````              where A = V/R , C = 1/R and
``````

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

``````              RM: These derivations are based
``````

on the computational formulas for V and R given in Gribble
& Ostry (1996) J. Neurophysiology, 76(5), 2853-2860.
Here is the section of their paper that gives the formulas
for V and R:

``````           RM:  Clearly, the velocities (X.dot and
``````

Y.dot) and accelerations (X.2dot and Y.2dot) in the
equations for both V and R are
the *same variables * . This means that it is perfectly legitimate
for substitute V2 = X.dot2
+ Y.dot2 into
the numerator of the equation for R resulting in

``````        R
``````

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

``````        which
``````

reduces to:

``````                    R
``````

= V1/3 /|X.dotY.2dot-X.2dotY.dot|

``````        and
``````

solving for V gives:

``````        V
``````

= |X.dotY.2dot-X.2dotY.dot|1/3 *R1/3

``````        RM:
``````

So simple math shows that there is a 1/3 power relationship
between R and V (and a 2/3 power relationship between C and
A).

``````        RM: The answer to (2) is that the viscosity of the medium
``````

in which curved movements are produced affects the feedback
connection between output (muscle force) and controlled
result (curved movement). I have used a PCT model to produce
curved movements in simulations of both high and low
viscosity media (viscosity being simulated by the value of
the feedback function coefficient, 1.0 for low and .5 for
high) and found that the b coefficient
of the power law for movement patterns in low viscosity
media was typically close to 1/3 while that for movement
patterns in high viscosity media was typically close to 1/4.

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

[From Rick Marken (2016.07.27.0930)]

···

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

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

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

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

RM: where D is a variable.

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

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

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

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

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

Best regards

Rick

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

Warren

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

On Tue, Jul 26, 2016 at 11:36 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.26.1535)]

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

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

WM: Mmm, one more thing…

RM: C = 1/R

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

[From Rick Marken (2016.07.26.1455)]

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, this means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

I am a bit worried you are experiencing the behavioral illusion. Are you not assuming that the trace of a larva is the purpose of the larva? Surely the larva cannot plot its movements in X and Y dimensions from an observer’s perspective like in your model? Surely the little larva is controlling for some perception much simpler from it own perspective? Like forward motion on the retina? Then it might control for more intense food smells and veer off in one direction from this path. But it doesn’t have a reference perception of it’s aerial X and Y coordinates does it? Do we necessarily? And other animals?

Warren

···

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

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

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

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

RM: where D is a variable.

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

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

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

<image.png>

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

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

Best regards

Rick

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

Warren

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

On Tue, Jul 26, 2016 at 11:36 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.26.1535)]

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

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

WM: Mmm, one more thing…

RM: C = 1/R

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

[From Rick Marken (2016.07.26.1455)]

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

WM: Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, this means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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

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

Best

Rick

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

Hi Rick, following on from this I have a colleague who can plug in a video and get X and Y coordinates tabulated by time. Shall I send him some videos of agents in the crowd demo to see if they obey the power law at various parameter values? This might lead to a PCT explanation of the power law without plugging the movement pattern into the reference value. The agents in the crowd demo don’t have reference values for their motion, only their distances from various objects, right?

All the best,

Warren

···

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

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

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

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

RM: where D is a variable.

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

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

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

<image.png>

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

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

Best regards

Rick

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

Warren

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

On Tue, Jul 26, 2016 at 11:36 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.26.1535)]

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

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

WM: Mmm, one more thing…

RM: C = 1/R

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

[From Rick Marken (2016.07.26.1455)]

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

On Tue, Jul 26, 2016 at 2:24 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I am Colombo. So radius is inversely proportional to curvature? If so, this means that as the curvature increases then angular velocity increases but on-track velocity decreases? Don’t they therefore meet in the middle or have some sort of pay off against one another?

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

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From Rick Marken (2016.07.27.1040]

···

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Best regards

Rick

Warren

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

[From Rick Marken (2016.07.27.0930)]

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

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

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

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

RM: where D is a variable.

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

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

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

<image.png>

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

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

Best regards

Rick

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

Warren

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

On Tue, Jul 26, 2016 at 11:36 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.26.1535)]

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

WM: Mmm, one more thing…

RM: C = 1/R

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

Best

Rick

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

Tel: +44 (0) 161 275 8589

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

Available Now

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

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

[From Rick Marken (2016.07.26.1455)]

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

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

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

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

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From Rick Marken (2016.07.27.1115)]

···

Martin Taylor (2016.07.27.09.42)

``````MT: No it doesn't. Look at the expression you wind up with. "2dot" means
``````

acceleration, and acceleration, which appears twice in this
expression, is a completely free quantity (or rather, it’s a
function of R and V). Where does it come from? It comes from one of
the expressions for curvature, Gribble and Ostry’s equation 9 that
you so kindly reproduced. So, instead of saying that your V =
|dx/dt*d2y/dt2 - d2X/dt2*dy/dt|1/3
is a constant, you should note its dependence on V and R, thus
destroying the simple proportionality of V and R1/3.

RM: I never said that V = |dx/dt*d2y/dt2 - d2X/dt2*dy/dt|1/3. is a constant. Here is my equation for V:

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

RM: The term |dx/dt*d2y/dt2 - d2X/dt2*dy/dt| is a variable; I called it D because it’s easier to write:

V = D1/3.+ R1/3

``````MT: Excellent. Maybe it's even the right answer. But it does give the
``````

lie to your long-standing claim that the exponent must always be
1/3.

RM: Geez, Martin. I’ve been saying over and over (last time was earlier this morning) that you will get varying estimates of the power coefficient if you regress just log(R) on log(V) (or log (C) on log (A)) leaving out log (D).

``````MT: I was thinking along the same lines, but I wasn't thinking of
``````

the feedback function coeffiicient. I was thinking of the different
integral-differential effects of force on mass in a viscous versus a
friction free environment. Could you show the model.

RM: I’m quite sure we were not thinking along the same lines. You still seem to think the variations in the observed value of the power coefficient shows something about how the movement pattern is produced. My derivation shows that it doesn’t; it is a result of only doing the regression with log (R) or log (C) as the predictor variable and only shows something about the nature of the curve itself. I will eventually get the model to you. I’ve got some other real work that I should be doing. But this is the most important project that I am working on at the moment.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

``````        RM:
``````

So simple math shows that there is a 1/3 power relationship
between R and V (and a 2/3 power relationship between C and
A).

``````        RM: The answer to (2) is that the viscosity of the medium
``````

in which curved movements are produced affects the feedback
connection between output (muscle force) and controlled
result (curved movement).

[Martin Taylor 2016.07.27.14.26]

[From Rick Marken (2016.07.27.1115)]

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

you?

``````You have the multiplication sign in your answer to Warren [From Rick
``````

Marken (2016.07.27.1040]. The difference is important, because D
varies with both V and R and the could affect the exponent of R
which wouldn’t happen with addition. Regardless of that, you simply
can’t call it an equation that presents V as a function of R when
the expression for V includes terms in dV/dt .

``````You just can't.

It's a differential equation that must be solved before you can
``````

write an expression for V in terms of R.

``````And even if you solved that differential equation, you would have an
``````

answer that referred to the shape and not the movement, because
Gribble and Ostry’s equations, with which you start, actually work
only for derivatives with respect to s, the arc length, and are not
valid for derivatives with respect to t (time) unless dt/ds = 1.
Since you didn’t start, as I had thought likely, with the Wikipedia
article on Curvature, I suggest you read it, taking particular note
of the condition of speed being constant at one unit. The
(for G+O’s first expression) and “Local expressions” (for their
second expression).

``````Martin
``````
···

Martin Taylor (2016.07.27.09.42)

``````            MT: No it doesn't. Look at the expression you
``````

wind up with. “2dot” means acceleration, and
acceleration, which appears twice in this expression, is
a completely free quantity (or rather, it’s a function
of R and V). Where does it come from? It comes from one
of the expressions for curvature, Gribble and Ostry’s
equation 9 that you so kindly reproduced. So, instead of
saying that your V = |dx/dt*d2y/dt2
- d2X/dt2*dy/dt|1/3 is
a constant, you should note its dependence on V and R,
thus destroying the simple proportionality of V and R1/3.

RM: I never said that V = |dx/dt*d2y/dt2 -
d2X/dt2*dy/dt|1/3. is a
constant. Here is my equation for V:

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

``````                      RM:
``````

So simple math shows that there is a 1/3 power
relationship between R and V (and a 2/3 power
relationship between C and A).

RM: The term |dx/dt*d2y/dt2 - d2X/dt2 *dy/dt|
is a variable; I called it D because it’s easier to write:

V = D1/3.+ R1/3