The Power Law: Why Rick's Derivation is Wrong

[From Bruce Abbott (2016.07.30.0935 EDT)]

I’ve gone back to Rick’s post of 7/24 because it contains a clear statement of his derivation of the relationship between velocity and the radius of curvature, which he believes to be a fixed relationship that holds for “all two-dimensional movement patterns so long as D is taken into account.� I will show that Rick’s conclusion is based on a misunderstanding of the equation for the radius of curvature and explain (in plain English) why it is wrong. My analysis begins at the end of Rick’s post copied below:

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: 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:

image484.png

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:

image00520.png

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: 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.

BA: The reason previous researchers did not see the relationship Rick discovered by means of a simple rearrangement of mathematical terms is that Rick’s rearrangement does not show what he thinks it shows. To establish this, I begin by reproducing the two equations Rick uses in his derivation. The first is this:

Inline image 1

V is the velocity of a moving point along a curve, such as a car going around a bend in the road. V would be the speed shown on the car’s speedometer. The above equation is for motion in two dimensions, X and Y. The dots over X and Y indicate that these are the first derivatives of X and Y with respect to time, or in other words, velocities in the X and Y directions. The equation is simply the Pythagorean Theorem in which X.dot and Y.dot are the sides of a right triangle (the horizontal and vertical components of the velocity) and V is the hypotenuse. V is the instantaneous velocity along the curve at a given time T.

Dimensional analysis. X.dot and Y.dot are speeds along the X and Y dimensions, measured, say, in meters/second. Squaring each gives meters squared/seconds squared for each. As the units are the same for both, we can add them. Taking the square root gives V in meters/second.

The second equation used in Rick’s derivation is this one:

Inline image 2

This is the equation for the radius of curvature. It is the radius of a circle whose curvature matches that of the line being followed, at a given point on the line.

The numerator of this equation is simply the equation for velocity V, raised to the third power. (The portion inside the parentheses is raised to the ½ power, which is the same as taking its square root. The result is then cubed. So we can rewrite the numerator as V3.

In the denominator we have the velocities of the point along the X and Y dimensions (X.dot and Y.dot) and the accelerations of the point along those same dimensions. With respect to acceleration, picture a point moving horizontally at some speed X.dot. Vertical speed (Y.dot) is zero. If the point is following a circle, however, then the point must begin to accelerate in the vertical (Y) direction and decelerate in the horizontal (X) direction. When the point reaches the right edge of the circle it will be traveling vertically at some speed but its horizontal velocity will be zero. The denominator takes these velocities and accelerations into account.

Because V in the numerator is derived by adding the squares of X.dot and Y.dot and then taking the square root, V cannot be negative. As we want R, the radius of curvature, to be a positive number, we remove any negative value in the denominator by taking its absolute value. Rick refers to this quantity as “D� in his analysis.

Dimensional analysis. Recall that V is the instantaneous velocity of a point along the curve and that we are assuming units of meters/second for this velocity. V cubed will this be in units of meters3/seconds3.

In the denominator, we multiply a velocity by an acceleration. If velocity is in meters/second, then acceleration is in meters/second/second, or meters/second2. Multiplying meters/second by meters/second2 given meters2/second3. Thus both X.dot times Y.dot.dot and X.dot.dot times Y.dot are in these same units and can be added together. Taking the absolute value gives a positive value in meters2/second3.

Dividing the numerator by the denominator gives meters3/seconds3 divided by meters2/second3, which leaves the dimension in meters (seconds cubed cancel and meters cubed-squared = meters). A unit length is just what is wanted for a radius.

So, what do we have? The first equation gives the instantaneous tangential velocity (speed) of a point along a curve at time T, in meters/second. The second equation gives the instantaneous radius of curvature, in meters, of the path the point is following at that same time T (corresponding to a given position along the path that has been reached at time T). ***This equation compensates for the velocity of the point (in meters/second) so that the value of R is independent of the velocity.*** It will give the same radius of curvature for any velocity > 0, for any given curvature.

Rick’s mistake. Rick noticed that the numerator of the equation for R is equivalent to V cubed. He then rearranges the equation to solve for V and discovers that the value of V is essentially determined by the value of R:Â

V = D1/3 *R1/3 where “Dâ€? is the denominator of the equation for R. Â

But what this equation tells us is the value of V at the instant that radius R was being computed. A moment later V could be higher or lower, and so would D (because D compensates for changes in V to give a radius of curvature that is independent of speed). Because Rick does not understand what D represents, he concludes that velocity is a function of R and that this relation will hold no matter what shape of figure is being followed by the moving point.

But in fact R is independent of V. Consequently, V can be any value whatsoever no matter how sharp or gentle the curve, in so far at the math is concerned. Yet empirically, biological organisms do tend to adjust their speeds depending on the radius of the curve, slowing for sharper curves and speeding up for more gentle ones. The empirical relationship has been found to follow the power law with beta = 1/3 in most cases (but with significant exceptions, such as when tracing a figure in a viscous medium. The question is, how can we account for this relationship?

Bruce

[From Bruce Abbott (2016.07.30.1630 EDT)]

Martin Taylor 2016.07.30.13.22 –

[From Bruce Abbott (2016.07.30.0935 EDT)]

I’ve gone back to Rick’s post of 7/24 because it contains a clear statement of his derivation of the relationship between velocity and the radius of curvature, which he believes to be a fixed relationship that holds for “all two-dimensional movement patterns so long as D is taken into account.� I will show that Rick’s conclusion is based on a misunderstanding of the equation for the radius of curvature and explain (in plain English) why it is wrong. My analysis begins at the end of Rick’s post copied below:

MT: Thanks for this. It’s all clear except for why one needs V in order to compute curvature. In fact, one doesn’t and one shouldn’t. To introduce V is a convenience that might help some people understand what is going on. Unfortunately it also seems to confuse some people. See below.

BA: The second equation used in Rick’s derivation is this one:

image00255.png

BA: This is the equation for the radius of curvature. It is the radius of a circle whose curvature matches that of the line being followed, at a given point on the line.

MT: Let’s rewrite this in Leibnitz’s notation rather than Newton’s “dotty” notation.

image00351.jpg

BA: The numerator of this equation is simply the equation for velocity V, raised to the third power. (The portion inside the parentheses is raised to the ½ power, which is the same as taking its square root. The result is then cubed. So we can rewrite the numerator as V3.

MT: Or we can do so explicitly if we recognize that V is ds/dt, and that dz/dt = dz/dsds/dt, so that the numerator derivatives become dx/dsds/dt, etc.

BA: By the Chain Rule, right? (I’m just learning calculus . . .)

In this version, “s” is position along the curve, the parameter in any parametric representation of the curve. As I explained yesterday [Martin Taylor 2016.07.29.15.08], only if v is specified can t be used as a parameter in representing the curve, because only if v is known does t uniquely specify a point along the curve with respect to your starting point.

ATT00003.jpg

From which the terms in V (ds/dt) cancel out.

image00353.jpg

BA:   Thanks for that, Martin.  I stayed with velocities and accelerations as functions of time because I thought it made the exposition clearer. (We are used to thinking of velocity as distance/time, for example, and our usual numerical simulations of the motion of a point along a path plot position as a function of dt, the discrete time-step of the simulation.)   For those who may be trying to follow along, as you show above, these time-based measures actually are used to give the positions arrived at along the curve at each successive instant during the point’s movement. For example, dx/ds is the change in position along the X-axis (dx) per change in distance along the curve (ds). The units are thus distance/distance (e.g., meters/meter), but the dimensional analysis ends up with R expressed as distance (meters) just as when dt rather than ds was used.

Bruce

[From Rick Marken (2016.07.30.1410)]

···

Martin Taylor (2016.07.30.13.22)–

MT: Now follows Alex's original question, which Rick continues to

obfuscate (see [From Rick
Marken (2016.07.30.1030)], in which he still conflates the V
parameter of the curve shape description with the velocity that an
organism might choose to use at any point on the curve.

RM: So the spreadsheet is an obfuscation? The variable (it’s not a parameter of anything) is the V variable measured in all studies of the power law. What is Alex’s original question that the spreadsheet obfuscates? And how does it obfuscate it?

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 2016.07.30.17.48]

[From Rick Marken (2016.07.30.1410)]

No it isn't. It's the parameter that links time to the position

along the curve in the parametric description of the curve.

Several of us have tried over the last two weeks to make it easy for

you in all sorts of different ways, mathematical and anecdotal and
by detailing data, showing that your analysis doesn’t conform to the
facts because you are mixing up the parametric velocity for
describing curvature with the actual velocity of an object that
moves along the curve. Your only response has been to keep touting
your spreadsheet demonstration of the tautological connection
between the parametric V and the curvature. Nobody disputes that
relationship, but it’s not interesting to anyone except possibly a
geometer.

Here's another try to make it even easier for you to understand the

problem.

George is driving along a twisty mountain road with many scenic

viewpoints and no traffic. George doesn’t stop but slows down to
look whenever a gap in the trees allows him to see the view.
George’s speed is unaffected by the radius of curvature of the road
because he goes quicker when he can’t see through the trees and
slower when there’s a gap with a good view. Some of those gaps are
on straightaways, some on the curves. Whatever the curvature, he
always goes slowly enough to be safe around the sharpest curves, so
as to be sure he doesn’t miss small gaps that might have really good
views. Sometimes the tree screen is thickest on the curves, and then
he would be going his (slow) fastest.

Alex wants to know whether PCT can explain why under most

experimental conditions the subjects (human or otherwise) don’t
behave like George, but do in fact move along the track with a
velocity that is close to a power function of the local radius of
curvature. Is it a coincidence or is it significant that the power
under many experimental conditions is close to the 0.33 power that
applies to the parametric V? He would also like to know the PCT
explanation for conditions under which (like George) they don’t
conform to any power law, or under which the numeric value of the
power changes.

Does that make Alex's question easier to understand?

Martin
···

Martin Taylor (2016.07.30.13.22)–

            MT: Now

follows Alex’s original question, which Rick continues
to obfuscate (see [From Rick Marken (2016.07.30.1030)], in
which he still conflates the V parameter of the curve
shape description with the velocity that an organism
might choose to use at any point on the curve.

          RM: So the spreadsheet is an obfuscation? The variable

(it’s not a parameter of anything) is the V variable
measured in all studies of the power law.

          What is Alex's original question that the spreadsheet

obfuscates? And how does it obfuscate it?

Best

Rick

[From Rick Marken (2016.07.30.1550)]

···

Martin Taylor (2016.07.30.17.48)–

MT: No it isn't. It's the parameter that links time to the position

along the curve in the parametric description of the curve.

RM: Then what is the variable called “V” that is measured in studies of the power law? All the variables in the spreadsheet demonstration – R, V, C and A – are the variables measured in every study of the power law. The spreadsheet just duplicates the calculations made in these studies. If you’ve think some other method was used to measure these variables then let me know and I’ll but those measures into the spreadsheet.

MT: Alex wants to know whether PCT can explain why under most

experimental conditions the subjects (human or otherwise) don’t
behave like George, but do in fact move along the track with a
velocity that is close to a power function of the local radius of
curvature.

RM: That’s what I thought; he want’s to know if PCT can explain the power law results. I have provided that explanation. He didn’t like it, nor did you or those several other people who tried over the last two weeks to make it easy for me to understand that my PCT explanation was wrong. So now I have presented the spreadsheet demonstration to show how the PCT explanation works. I would like to know how the spreadsheet “obfuscates” Alex’s question. I think it makes the question and answer very clear.

MT: Is it a coincidence or is it significant that the power

under many experimental conditions is close to the 0.33 power that
applies to the parametric V? He would also like to know the PCT
explanation for conditions under which (like George) they don’t
conform to any power law, or under which the numeric value of the
power changes.

MT: Does that make Alex’s question easier to understand?

RM: I already understood that to be Alex’s question and I have answered it in spades. My spreadsheet reproduces the main results obtained in all the different experiments on the power law. And it does it based on the PCT observation that what is being observed in these experiments is a side effect of the mathematical fact that:

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

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

RM: The PCT model accounts for the data, not for imaginings about what the data say.

Best

          RM: So the spreadsheet is an obfuscation? The variable

(it’s not a parameter of anything) is the V variable
measured in all studies of the power law.

          RM: What is Alex's original question that the spreadsheet

obfuscates? And how does it obfuscate it?

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 2016.07.30.23/54]

Just what Alex asked about initially. It's something quite different

from your parameter mapping “V”.
No, your "V’ isn’t. Your V is simply the mapping of t onto s in the
computation of curvature. The V asked about and measured in studies
of the power law is unrelated to that mapping. It is the actual
speed of movement of the output of an actual control system
controlling for something. We don’t know what perception is being
controlled, but we do know that the result is often not a power law
with an exponent 0.33. You demonstrated that yourself
[From Rick Marken (2016.07.26. 1340)] when you simulated a model
that you haven’t yet published, but you said about it:
--------start quote-----
RM(07.26.13.40]:
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 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. -------end quote-----
MT: Reading that, it sounds as though you measured the along-track
velocity that is measured in most studies of the power law, which is
why I responded “Excellent”. But it’s not the V in your math
derivation, for which the exponent is necessarily exactly 0.33 when
the equation is complete (i.e. not dropping your “D” factor).
Your explanation says quite explicitly that George cannot behave
as my description of his sightseeing driving said he behaved.
George’s driving speed has no fixed relation to the local curvature
of the road, as it is determined by the density of the tree screen
the most of the time obscures the view. Your explanation says that
George’s speed is always exactly determined by the local curvature
of the road.
Your explanation says quite explicitly that your own PCT model that
you used in your simulation cited above could not produce the data
it did produce.
So far as I know, you have offered only one “PCT explanation” in the
form of the simulation that showed a way that the exponent might
change from 1/3 to 1/4. Apart from that, all you have done is
reconfigure the two expressions that describe curvature. I’m sorry
that nothing anyone has said has been easy enough for you to
understand this very simple fact.
It has no relation to the question being asked because the V
variable has no relation to the V By mixing up the parametric representation of a curve with what an
independent control system may do when following the curve, treating
those two quite independent constructs as though they were one and
the same.
No. Your answer has consistently been that the data are wrong, and
the power function is always 1/3, because that’s a geometric
property of all smooth curves.
I don’t know what that simple geometrical fact has to do with PCT,
but it does seem to say you programmed your “viscosity” model wrong,
and that nobody could possibly drive like my George, or walk at the
same speed on a straight road as on a twisty one. When a “model”
produces results in direct contradiction to observations (such as
when you “proved” to me [From Rick Marken (2016.07.20.1130)] that
people and other organisms go faster around curves the sharper they
are), I tend to believe the observations rather than the model.
The point is not simply that your model isn’t a “PCT model” but a
geometric description, but that it contradicts data it is supposed
to explain. Most people, when they produce a model that directly
contradicts the data, would try to find out what is wrong with the
model rather than simply repeat over and over again, like DJT, that
they are right and everyone else (and the data) are wrong.
You even say occasionally that the power law is just a property of
curves, and that is correct. Bruce and I both showed you that the
1/3 power law is consistent with any velocity or variation of
velocity. Since you know that it is a property of all smooth curves,
that should have been enough without any urging from us to tell you
that your “V” has nothing to do with the velocity of actual things
that follow the curve.
What I really don’t understand is why, having shown that you know
this, you then say that nobody can vary their speed at will along
different parts of a curved track because the possible speed is
strictly determined by the curvature. If that were so, why would we
ever have a sign for the recommended speed around a road curve? Why
would there be any point in having Grand Prix road races if all the
cars have to go the same speed everywhere on the track?
Martin

···

[From Rick Marken (2016.07.30.1550)]

Martin Taylor (2016.07.30.17.48)–

                        RM: So the spreadsheet is an obfuscation?

The variable (it’s not a parameter of
anything) is the V variable measured in all
studies of the power law.

            MT: No it isn't. It's the parameter that links

time to the position along the curve in the parametric
description of the curve.

          RM: Then what is the variable called "V" that is

measured in studies of the power law?

          All the variables in the spreadsheet demonstration  --

R, V, C and A – are the variables measured in every study
of the power law.

b

          The spreadsheet just duplicates the calculations made

in these studies. If you’ve think some other method was
used to measure these variables then let me know and I’ll
but those measures into the spreadsheet.

                        RM: What is Alex's original question that

the spreadsheet obfuscates? And how does it
obfuscate it?

            MT: Alex wants to know whether PCT

can explain why under most experimental conditions the
subjects (human or otherwise) don’t behave like George,
but do in fact move along the track with a velocity that
is close to a power function of the local radius of
curvature.

          RM: That's what I thought; he want's to know if PCT can

explain the power law results. I have provided that
explanation.

          He didn't like it, nor did you or those several other

people who tried over the last two weeks to make it easy
for me to understand that my PCT explanation was wrong.

          So now I have presented the spreadsheet demonstration

to show how the PCT explanation works.

          I would like to know how the spreadsheet "obfuscates"

Alex’s question.

I think it makes the question and answer very clear.

            MT: Is it a coincidence or is it

significant that the power under many experimental
conditions is close to the 0.33 power that applies to
the parametric V? He would also like to know the PCT
explanation for conditions under which (like George)
they don’t conform to any power law, or under which the
numeric value of the power changes.

            MT: Does that make Alex's question easier to understand?
          RM: I already understood that to be Alex's question

and I have answered it in spades.

          My spreadsheet reproduces the main results obtained in

all the different experiments on the power law. And it
does it based on the PCT observation that what is being
observed in these experiments is a side effect of the
mathematical fact that:

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

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

          RM: The PCT model accounts for the data, not for

imaginings about what the data say.

Martin, you have put it very clearly and helpfully to Rick, and this is exactly how I see it too.

Warren

···

[From Rick Marken (2016.07.30.1550)]

Martin Taylor (2016.07.30.17.48)–

                        RM: So the spreadsheet is an obfuscation?

The variable (it’s not a parameter of
anything) is the V variable measured in all
studies of the power law.

            MT: No it isn't. It's the parameter that links

time to the position along the curve in the parametric
description of the curve.

          RM: Then what is the variable called "V" that is

measured in studies of the power law?

          All the variables in the spreadsheet demonstration  --

R, V, C and A – are the variables measured in every study
of the power law.

b

          The spreadsheet just duplicates the calculations made

in these studies. If you’ve think some other method was
used to measure these variables then let me know and I’ll
but those measures into the spreadsheet.

                        RM: What is Alex's original question that

the spreadsheet obfuscates? And how does it
obfuscate it?

            MT: Alex wants to know whether PCT

can explain why under most experimental conditions the
subjects (human or otherwise) don’t behave like George,
but do in fact move along the track with a velocity that
is close to a power function of the local radius of
curvature.

          RM: That's what I thought; he want's to know if PCT can

explain the power law results. I have provided that
explanation.

          He didn't like it, nor did you or those several other

people who tried over the last two weeks to make it easy
for me to understand that my PCT explanation was wrong.

          So now I have presented the spreadsheet demonstration

to show how the PCT explanation works.

          I would like to know how the spreadsheet "obfuscates"

Alex’s question.

I think it makes the question and answer very clear.

            MT: Is it a coincidence or is it

significant that the power under many experimental
conditions is close to the 0.33 power that applies to
the parametric V? He would also like to know the PCT
explanation for conditions under which (like George)
they don’t conform to any power law, or under which the
numeric value of the power changes.

            MT: Does that make Alex's question easier to understand?
          RM: I already understood that to be Alex's question

and I have answered it in spades.

          My spreadsheet reproduces the main results obtained in

all the different experiments on the power law. And it
does it based on the PCT observation that what is being
observed in these experiments is a side effect of the
mathematical fact that:

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

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

          RM: The PCT model accounts for the data, not for

imaginings about what the data say.

[From Rick Marken (2016.07.31.1730)]

···

Martin Taylor (2016.07.30.23/54)–

MT: Just what Alex asked about initially. It's something quite different

from your parameter mapping “V”.

RM: Alex asked how PCT would account for the observed power relationship between V and R (measured as per the formulas given in the Gribble/Ostry and many other papers) and A and C (measured by the formulas Alex gave me that are simple functions of V and R).

MT: Your explanation says quite explicitly that George _cannot_ behave

as my description of his sightseeing driving said he behaved.

RM: The power law has nothing to do with your example of George driving around curves. What is observed (the data collected) in power law studies is a 2D (and sometimes 3D) trace of a temporal pattern of movements. In terms of a control diagram,what is observed are variations in q.i, the controlled variable, over time. Since q.i = q.o + d, what is being observed (the data) is variations in a controlled result of variations in output, q.o, compensating for disturbances, d. If the temporal movement pattern were the path of a driver, like George, negotiating the race course at Monte Carlo, the data would consist of the path of the car – q.i – and not the outputs, q.o, (steering movements, accelerator presses, etc) that are compensating for disturbances, d, (centrifugal force, road friction, etc) to produce that path.

RM: When you observe q.i only, you can’t tell anything about how outputs, q.o, combined with disturbances, d, to produce the resulting variations in q.i. The basic mistake of the power law research is the assumption that you can learn about q.o from q.i alone. The power law itself is an observed relationship between two measures of temporal variations in q.i, either V and R or A and C. Regardless of how these variables are measured, you cannot tell anything about how q.i was produced by observing the relationship between any measures of q.i.

RM: For example, you couldn’t tell what George did to produce the observed temporal pattern of movements through the course at Monte Carlo by just looking at measures of this temporal pattern (like V,R,A and C) or relationships between these measures. This is why I have ignored your (and Bruce’s) suggestions about the “correct” way to measure V and R (or A and C). It’s not just because you are both wrong about how these variables are measured in the power law research; it’s because even your suggested measures would reveal nothing about how observed variations in q.i were produced.

MT: So far as I know, you have offered only one "PCT explanation" in the

form of the simulation that showed a way that the exponent might
change from 1/3 to 1/4.

RM: The PCT explanation of the observed power law is that it is a mathematical characteristic of any observed temporal pattern of movement (except movement in a perfectly straight line) and reveals nothing about how that pattern of movement was produced. The same power law relationship between V and R or A and C will be observed for a particular pattern of movement whether it is produced by a living organism or an equation, on purpose or by accident, on a train or on a plane, in a car or near and far (lots of Dr. Seuss this trip;-), etc.

RM: The change from an exponent of 1/3 to 1/4 with a change in the feedback function is just a small piece of this explanation; it shows why the change in exponent might have been observed. The change occurs because of a slight change in the shape of the elliptical movement pattern. This change in shape could be caused by a change in the feedback connection between q.o and q.i for an organism controlling for making elliptical movements in different mediums but it could also be produced by the motion of a poorly aligned windmill on land or under water.

MT: It has no relation to the question being asked because the V

variable has no relation to the V

RM: The spreadsheet reproduces exactly what is done in research on the power law. It takes a pattern of movement, computes V, R, A and C from that pattern. And then does a log-log regression of R on V and of A on C to find the coefficient of the power law relationship between these variables.

RM: What the spreadsheet proves is that the coefficient of the power law relationship depends only on the pattern of movement itself and not on how that pattern was produced. The same range of power coefficients observed in studies of temporal patterns of movement made by living organisms is found for power coefficients of temporal patterns of movement made by equations.

MT: By mixing up the parametric representation of a curve with what an

independent control system may do when following the curve, treating
those two quite independent constructs as though they were one and
the same.

RM: See my explanation of the spreadsheet in my comment above. There is no mixing up of a parametric representation of a curve with anything. The spreadsheet does exactly what is done in researchon the power law. This can be seen by going to the spreadsheet tab labeled “Regression”. Columns B and C of that worksheet contain the temporal variations in the x,y coordinates of the movement pattern being analyzed. Columns D, E, J and K compute the variables R, V, C and A for each point on the waveform. The other columns take the logs of these variables (as well as of variable D which is computed in column F). The regression analyses are computed using the LINEST() function in columns U, V and W.

RM: These regressions on the logs of R, V, C, A give the estimates of the power coefficients relating R to V and C to A, exactly as is done in all studies of the power law. The fact that these regressions give coefficients close to the ones found in power law research, even when the pattern of movement analyzed was not produced by an organism (as is the particularly interesting case when you press the “Random Pattern” button in tab “Main”, which inserts a mathematically generated squiggle pattern into columns B and C of tab "Regression) is evidence that the power coefficients found in power law research tell you nothing about how the patterns were produced.

RM: The fact that all the variance in the movement pattern is accounted for when log (D) is included in the analysis (R^2 = 1.0) and that the power coefficient of log(R) is always exactly .33, the power coefficient of log(C) is always exactly .67 and the coefficient of log(D) is always .33 shows that my derivation of the mathematical relationship between these variables is correct: V = D^1/3+R^1/3 and A = D^1/3+R^2/3. And it shows that the variations in the power coefficient that is found in research on the power law when only R or C is included in the regression analysis is a results of differences in the variance of D for the different movement patterns.

MT: No. Your answer has consistently been that the data are wrong, and

the power function is always 1/3, because that’s a geometric
property of all smooth curves.

RM: Not quite. The data are never wrong. The power coefficient is always .33 (for the relationship between R and V) but ONLY when D is also included in the analysis. When the regression is only of R on V you get variation in the value of the power coefficient for different movement patterns (as demonstrated by the spreadsheet, especially when using the “Random Pattern” generator). So the variation in the power coefficient does depend on the geometry of the movement pattern – but the variation is always around the value .33 (for R on V) and around .67 (for C on A).

MT: When a "model"

produces results in direct contradiction to observations (such as
when you “proved” to me [From Rick Marken (2016.07.20.1130)] that
people and other organisms go faster around curves the sharper they
are), I tend to believe the observations rather than the model.

RM: The power law is the observation and it shows that angular velocity (measured as V or A) increases with curvature (measured as R or C), though at a decreasing rate. I never said that people don’t slow down through curves; but the slowing down (which is actually a reduction in the force output of the engine) is an output that compensates for force disturbances that would throw the car off the road (curve).The result is that the car stays on the road; and, as I said above, if you did a power law analysis of the resulting path of the car you would find that V = D^1/3+R^1/3 and A = D^1/3+C^2/3. Your mistake is taking the observed measure of velocity through a curved movement pattern as a measure of output, q.o, rather than as what it is: a measure of the state of a controlled variable, q.i.

MT: The point is not simply that your model isn't a "PCT model" but a

geometric description, but that it contradicts data it is supposed
to explain.

RM: The data that I explain is the power law. And, as you will see from the spreadsheet once you understand it, my explanation is perfectly consistent with the data.

MT: You even say occasionally that the power law is just a property of

curves, and that is correct.

RM: Yes, that is exactly what I say. And demonstrate to be true with the spreadsheet.

MT: Bruce and I both showed you that the

1/3 power law is consistent with any velocity or variation of
velocity. Since you know that it is a property of all smooth curves,
that should have been enough without any urging from us to tell you
that your “V” has nothing to do with the velocity of actual things
that follow the curve.

RM: Actually, since you knew this, I thought it should have been enough, without any explanation from me, to convince you that the power law had nothing to do with how a temporal movement pattern is produced.

MT: What I really don't understand is why, having shown that you know

this, you then say that nobody can vary their speed at will along
different parts of a curved track because the possible speed is
strictly determined by the curvature.

RM: But I never said this. Informally we say that people vary their speed around a turn. But what they actually do is vary the force, q.o, produced by the engine to compensate for the centripetal and other forces, d, that affect your path as you round a curve. When your force output is varied properly (by proper variation of accelerator and steering wheel position) you stay on the intended path, q.i.

RM: Again, your mistake is taking measures of q.i as indications of q.o, the very mistake power law researchers are making. They take measures of the instantaneous velocity of the observed movement, q.i, as indications of the outputs, q.o, that are being used to produce the curved movements along the path (measured as curvature). This is a mistake that could only be made by people who don’t understand the nature of control.

MT: If that were so, why would we

ever have a sign for the recommended speed around a road curve?

RM: So that you can be prepared to reduce the force output of the engine before the force of the curve takes you off the road. And if you did go off the road, that path too would follow the power law.

MT: Why

would there be any point in having Grand Prix road races if all the
cars have to go the same speed everywhere on the track?

RM: My analysis of the power law for movement patterns does not require that the movement be made at constant speed: V (and A) are variables.

          RM: Then what is the variable called "V" that is

measured in studies of the power law?

          RM: So now I have presented the spreadsheet demonstration

to show how the PCT explanation works.

          RM: I would like to know how the spreadsheet "obfuscates"

Alex’s question.

            MT: Is it a coincidence or is it

significant that the power under many experimental
conditions is close to the 0.33 power that applies to
the parametric V? …

          RM: I already understood that to be Alex's question

and I have answered it in spades.

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 (2016.07.31.1745)]

···

On Sun, Jul 31, 2016 at 12:31 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Martin, you have put it very clearly and helpfully to Rick, and this is exactly how I see it too.

RM: Clearly I didn’t find Martin’s post as clear and helpful as you had hoped;-)

Best

Rick

Warren

On 31 Jul 2016, at 05:39, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2016.07.30.23/54]

[From Rick Marken (2016.07.30.1550)]

Just what Alex asked about initially. It's something quite different

from your parameter mapping “V”.

No, your "V' isn't. Your V is simply the mapping of t onto s in the

computation of curvature. The V asked about and measured in studies
of the power law is unrelated to that mapping. It is the actual
speed of movement of the output of an actual control system
controlling for something. We don’t know what perception is being
controlled, but we do know that the result is often not a power law
with an exponent 0.33. You demonstrated that yourself
[From Rick Marken (2016.07.26. 1340)] when you simulated a model
that you haven’t yet published, but you said about it:

--------start quote-----

RM(07.26.13.40]:

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.

-------end quote-----

MT: Reading that, it sounds as though you measured the along-track

velocity that is measured in most studies of the power law, which is
why I responded “Excellent”. But it’s not the V in your math
derivation, for which the exponent is necessarily exactly 0.33 when
the equation is complete (i.e. not dropping your “D” factor).

Your explanation says quite explicitly that George _cannot_ behave

as my description of his sightseeing driving said he behaved.
George’s driving speed has no fixed relation to the local curvature
of the road, as it is determined by the density of the tree screen
the most of the time obscures the view. Your explanation says that
George’s speed is always exactly determined by the local curvature
of the road.

Your explanation says quite explicitly that your own PCT model that

you used in your simulation cited above could not produce the data
it did produce.

So far as I know, you have offered only one "PCT explanation" in the

form of the simulation that showed a way that the exponent might
change from 1/3 to 1/4. Apart from that, all you have done is
reconfigure the two expressions that describe curvature. I’m sorry
that nothing anyone has said has been easy enough for you to
understand this very simple fact.

It has no relation to the question being asked because the V

variable has no relation to the V

By mixing up the parametric representation of a curve with what an

independent control system may do when following the curve, treating
those two quite independent constructs as though they were one and
the same.

No. Your answer has consistently been that the data are wrong, and

the power function is always 1/3, because that’s a geometric
property of all smooth curves.

I don't know what that simple geometrical fact has to do with PCT,

but it does seem to say you programmed your “viscosity” model wrong,
and that nobody could possibly drive like my George, or walk at the
same speed on a straight road as on a twisty one. When a “model”
produces results in direct contradiction to observations (such as
when you “proved” to me [From Rick Marken (2016.07.20.1130)] that
people and other organisms go faster around curves the sharper they
are), I tend to believe the observations rather than the model.

The point is not simply that your model isn't a "PCT model" but a

geometric description, but that it contradicts data it is supposed
to explain. Most people, when they produce a model that directly
contradicts the data, would try to find out what is wrong with the
model rather than simply repeat over and over again, like DJT, that
they are right and everyone else (and the data) are wrong.

You even say occasionally that the power law is just a property of

curves, and that is correct. Bruce and I both showed you that the
1/3 power law is consistent with any velocity or variation of
velocity. Since you know that it is a property of all smooth curves,
that should have been enough without any urging from us to tell you
that your “V” has nothing to do with the velocity of actual things
that follow the curve.

What I really don't understand is why, having shown that you know

this, you then say that nobody can vary their speed at will along
different parts of a curved track because the possible speed is
strictly determined by the curvature. If that were so, why would we
ever have a sign for the recommended speed around a road curve? Why
would there be any point in having Grand Prix road races if all the
cars have to go the same speed everywhere on the track?

Martin


Richard S. Marken

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

Martin Taylor (2016.07.30.17.48)–

            MT: No it isn't. It's the parameter that links

time to the position along the curve in the parametric
description of the curve.

          RM: Then what is the variable called "V" that is

measured in studies of the power law?

                        RM: So the spreadsheet is an obfuscation?

The variable (it’s not a parameter of
anything) is the V variable measured in all
studies of the power law.

          All the variables in the spreadsheet demonstration  --

R, V, C and A – are the variables measured in every study
of the power law.

          The spreadsheet just duplicates the calculations made

in these studies. If you’ve think some other method was
used to measure these variables then let me know and I’ll
but those measures into the spreadsheet.

            MT: Alex wants to know whether PCT

can explain why under most experimental conditions the
subjects (human or otherwise) don’t behave like George,
but do in fact move along the track with a velocity that
is close to a power function of the local radius of
curvature.

          RM: That's what I thought; he want's to know if PCT can

explain the power law results. I have provided that
explanation.

                        RM: What is Alex's original question that

the spreadsheet obfuscates? And how does it
obfuscate it?

          He didn't like it, nor did you or those several other

people who tried over the last two weeks to make it easy
for me to understand that my PCT explanation was wrong.

          So now I have presented the spreadsheet demonstration

to show how the PCT explanation works.

          I would like to know how the spreadsheet "obfuscates"

Alex’s question.

I think it makes the question and answer very clear.

            MT: Is it a coincidence or is it

significant that the power under many experimental
conditions is close to the 0.33 power that applies to
the parametric V? He would also like to know the PCT
explanation for conditions under which (like George)
they don’t conform to any power law, or under which the
numeric value of the power changes.

            MT: Does that make Alex's question easier to understand?
          RM: I already understood that to be Alex's question

and I have answered it in spades.

          My spreadsheet reproduces the main results obtained in

all the different experiments on the power law. And it
does it based on the PCT observation that what is being
observed in these experiments is a side effect of the
mathematical fact that:

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

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

          RM: The PCT model accounts for the data, not for

imaginings about what the data say.

Hi Rick, I agree with everything you say about the fundamental issue of trying to infer output from controlled input regardless of whether your equation is correct. But to me, that is the first step. The next step is to hypothesise or model a plausible control system(s) with CV(s) that reveal the power law, rather than using the power law-conforming movements observed as your reference values of a controlled perception…

···

On Mon, Aug 1, 2016 at 1:45 AM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.31.1745)]

On Sun, Jul 31, 2016 at 12:31 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Martin, you have put it very clearly and helpfully to Rick, and this is exactly how I see it too.

RM: Clearly I didn’t find Martin’s post as clear and helpful as you had hoped;-)

Best

Rick

Warren

On 31 Jul 2016, at 05:39, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2016.07.30.23/54]

[From Rick Marken (2016.07.30.1550)]

Just what Alex asked about initially. It's something quite different

from your parameter mapping “V”.

No, your "V' isn't. Your V is simply the mapping of t onto s in the

computation of curvature. The V asked about and measured in studies
of the power law is unrelated to that mapping. It is the actual
speed of movement of the output of an actual control system
controlling for something. We don’t know what perception is being
controlled, but we do know that the result is often not a power law
with an exponent 0.33. You demonstrated that yourself
[From Rick Marken (2016.07.26. 1340)] when you simulated a model
that you haven’t yet published, but you said about it:

--------start quote-----

RM(07.26.13.40]:

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.

-------end quote-----

MT: Reading that, it sounds as though you measured the along-track

velocity that is measured in most studies of the power law, which is
why I responded “Excellent”. But it’s not the V in your math
derivation, for which the exponent is necessarily exactly 0.33 when
the equation is complete (i.e. not dropping your “D” factor).

Your explanation says quite explicitly that George _cannot_ behave

as my description of his sightseeing driving said he behaved.
George’s driving speed has no fixed relation to the local curvature
of the road, as it is determined by the density of the tree screen
the most of the time obscures the view. Your explanation says that
George’s speed is always exactly determined by the local curvature
of the road.

Your explanation says quite explicitly that your own PCT model that

you used in your simulation cited above could not produce the data
it did produce.

So far as I know, you have offered only one "PCT explanation" in the

form of the simulation that showed a way that the exponent might
change from 1/3 to 1/4. Apart from that, all you have done is
reconfigure the two expressions that describe curvature. I’m sorry
that nothing anyone has said has been easy enough for you to
understand this very simple fact.

It has no relation to the question being asked because the V

variable has no relation to the V

By mixing up the parametric representation of a curve with what an

independent control system may do when following the curve, treating
those two quite independent constructs as though they were one and
the same.

No. Your answer has consistently been that the data are wrong, and

the power function is always 1/3, because that’s a geometric
property of all smooth curves.

I don't know what that simple geometrical fact has to do with PCT,

but it does seem to say you programmed your “viscosity” model wrong,
and that nobody could possibly drive like my George, or walk at the
same speed on a straight road as on a twisty one. When a “model”
produces results in direct contradiction to observations (such as
when you “proved” to me [From Rick Marken (2016.07.20.1130)] that
people and other organisms go faster around curves the sharper they
are), I tend to believe the observations rather than the model.

The point is not simply that your model isn't a "PCT model" but a

geometric description, but that it contradicts data it is supposed
to explain. Most people, when they produce a model that directly
contradicts the data, would try to find out what is wrong with the
model rather than simply repeat over and over again, like DJT, that
they are right and everyone else (and the data) are wrong.

You even say occasionally that the power law is just a property of

curves, and that is correct. Bruce and I both showed you that the
1/3 power law is consistent with any velocity or variation of
velocity. Since you know that it is a property of all smooth curves,
that should have been enough without any urging from us to tell you
that your “V” has nothing to do with the velocity of actual things
that follow the curve.

What I really don't understand is why, having shown that you know

this, you then say that nobody can vary their speed at will along
different parts of a curved track because the possible speed is
strictly determined by the curvature. If that were so, why would we
ever have a sign for the recommended speed around a road curve? Why
would there be any point in having Grand Prix road races if all the
cars have to go the same speed everywhere on the track?

Martin


Richard S. Marken

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

Martin Taylor (2016.07.30.17.48)–

            MT: No it isn't. It's the parameter that links

time to the position along the curve in the parametric
description of the curve.

          RM: Then what is the variable called "V" that is

measured in studies of the power law?

                        RM: So the spreadsheet is an obfuscation?

The variable (it’s not a parameter of
anything) is the V variable measured in all
studies of the power law.

          All the variables in the spreadsheet demonstration  --

R, V, C and A – are the variables measured in every study
of the power law.

          The spreadsheet just duplicates the calculations made

in these studies. If you’ve think some other method was
used to measure these variables then let me know and I’ll
but those measures into the spreadsheet.

            MT: Alex wants to know whether PCT

can explain why under most experimental conditions the
subjects (human or otherwise) don’t behave like George,
but do in fact move along the track with a velocity that
is close to a power function of the local radius of
curvature.

          RM: That's what I thought; he want's to know if PCT can

explain the power law results. I have provided that
explanation.

                        RM: What is Alex's original question that

the spreadsheet obfuscates? And how does it
obfuscate it?

          He didn't like it, nor did you or those several other

people who tried over the last two weeks to make it easy
for me to understand that my PCT explanation was wrong.

          So now I have presented the spreadsheet demonstration

to show how the PCT explanation works.

          I would like to know how the spreadsheet "obfuscates"

Alex’s question.

I think it makes the question and answer very clear.

            MT: Is it a coincidence or is it

significant that the power under many experimental
conditions is close to the 0.33 power that applies to
the parametric V? He would also like to know the PCT
explanation for conditions under which (like George)
they don’t conform to any power law, or under which the
numeric value of the power changes.

            MT: Does that make Alex's question easier to understand?
          RM: I already understood that to be Alex's question

and I have answered it in spades.

          My spreadsheet reproduces the main results obtained in

all the different experiments on the power law. And it
does it based on the PCT observation that what is being
observed in these experiments is a side effect of the
mathematical fact that:

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

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

          RM: The PCT model accounts for the data, not for

imaginings about what the data say.

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

Tel: +44 (0) 161 275 8589

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

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

Available Now

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

[From Rick Marken (2016.07.31.1800)]

···

On Sun, Jul 31, 2016 at 5:52 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I agree with everything you say about the fundamental issue of trying to infer output from controlled input regardless of whether your equation is correct. But to me, that is the first step. The next step is to hypothesise or model a plausible control system(s) with CV(s) that reveal the power law, rather than using the power law-conforming movements observed as your reference values of a controlled perception…

RM: My analysis, as implemented in the spreadsheet, shows that all movements are power law conforming movements; there is no such thing as a power law un-conforming movement (except for a perfectly straight line). Thus, I’ve already presented the control model that can produce movement patterns that conform to the power law. Any model, control model or not, that can make movements over time will be power law conforming. The spreadsheet proves it. Of course, I could be proved wrong if someone could show me a power law un-conforming movement!!

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 Mon, Aug 1, 2016 at 1:45 AM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.31.1745)]


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

Tel: +44 (0) 161 275 8589

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

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

Available Now

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

On Sun, Jul 31, 2016 at 12:31 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Martin, you have put it very clearly and helpfully to Rick, and this is exactly how I see it too.

RM: Clearly I didn’t find Martin’s post as clear and helpful as you had hoped;-)

Best

Rick

Warren

On 31 Jul 2016, at 05:39, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2016.07.30.23/54]

[From Rick Marken (2016.07.30.1550)]

Just what Alex asked about initially. It's something quite different

from your parameter mapping “V”.

No, your "V' isn't. Your V is simply the mapping of t onto s in the

computation of curvature. The V asked about and measured in studies
of the power law is unrelated to that mapping. It is the actual
speed of movement of the output of an actual control system
controlling for something. We don’t know what perception is being
controlled, but we do know that the result is often not a power law
with an exponent 0.33. You demonstrated that yourself
[From Rick Marken (2016.07.26. 1340)] when you simulated a model
that you haven’t yet published, but you said about it:

--------start quote-----

RM(07.26.13.40]:

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.

-------end quote-----

MT: Reading that, it sounds as though you measured the along-track

velocity that is measured in most studies of the power law, which is
why I responded “Excellent”. But it’s not the V in your math
derivation, for which the exponent is necessarily exactly 0.33 when
the equation is complete (i.e. not dropping your “D” factor).

Your explanation says quite explicitly that George _cannot_ behave

as my description of his sightseeing driving said he behaved.
George’s driving speed has no fixed relation to the local curvature
of the road, as it is determined by the density of the tree screen
the most of the time obscures the view. Your explanation says that
George’s speed is always exactly determined by the local curvature
of the road.

Your explanation says quite explicitly that your own PCT model that

you used in your simulation cited above could not produce the data
it did produce.

So far as I know, you have offered only one "PCT explanation" in the

form of the simulation that showed a way that the exponent might
change from 1/3 to 1/4. Apart from that, all you have done is
reconfigure the two expressions that describe curvature. I’m sorry
that nothing anyone has said has been easy enough for you to
understand this very simple fact.

It has no relation to the question being asked because the V

variable has no relation to the V

By mixing up the parametric representation of a curve with what an

independent control system may do when following the curve, treating
those two quite independent constructs as though they were one and
the same.

No. Your answer has consistently been that the data are wrong, and

the power function is always 1/3, because that’s a geometric
property of all smooth curves.

I don't know what that simple geometrical fact has to do with PCT,

but it does seem to say you programmed your “viscosity” model wrong,
and that nobody could possibly drive like my George, or walk at the
same speed on a straight road as on a twisty one. When a “model”
produces results in direct contradiction to observations (such as
when you “proved” to me [From Rick Marken (2016.07.20.1130)] that
people and other organisms go faster around curves the sharper they
are), I tend to believe the observations rather than the model.

The point is not simply that your model isn't a "PCT model" but a

geometric description, but that it contradicts data it is supposed
to explain. Most people, when they produce a model that directly
contradicts the data, would try to find out what is wrong with the
model rather than simply repeat over and over again, like DJT, that
they are right and everyone else (and the data) are wrong.

You even say occasionally that the power law is just a property of

curves, and that is correct. Bruce and I both showed you that the
1/3 power law is consistent with any velocity or variation of
velocity. Since you know that it is a property of all smooth curves,
that should have been enough without any urging from us to tell you
that your “V” has nothing to do with the velocity of actual things
that follow the curve.

What I really don't understand is why, having shown that you know

this, you then say that nobody can vary their speed at will along
different parts of a curved track because the possible speed is
strictly determined by the curvature. If that were so, why would we
ever have a sign for the recommended speed around a road curve? Why
would there be any point in having Grand Prix road races if all the
cars have to go the same speed everywhere on the track?

Martin


Richard S. Marken

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

Martin Taylor (2016.07.30.17.48)–

            MT: No it isn't. It's the parameter that links

time to the position along the curve in the parametric
description of the curve.

          RM: Then what is the variable called "V" that is

measured in studies of the power law?

                        RM: So the spreadsheet is an obfuscation?

The variable (it’s not a parameter of
anything) is the V variable measured in all
studies of the power law.

          All the variables in the spreadsheet demonstration  --

R, V, C and A – are the variables measured in every study
of the power law.

          The spreadsheet just duplicates the calculations made

in these studies. If you’ve think some other method was
used to measure these variables then let me know and I’ll
but those measures into the spreadsheet.

            MT: Alex wants to know whether PCT

can explain why under most experimental conditions the
subjects (human or otherwise) don’t behave like George,
but do in fact move along the track with a velocity that
is close to a power function of the local radius of
curvature.

          RM: That's what I thought; he want's to know if PCT can

explain the power law results. I have provided that
explanation.

                        RM: What is Alex's original question that

the spreadsheet obfuscates? And how does it
obfuscate it?

          He didn't like it, nor did you or those several other

people who tried over the last two weeks to make it easy
for me to understand that my PCT explanation was wrong.

          So now I have presented the spreadsheet demonstration

to show how the PCT explanation works.

          I would like to know how the spreadsheet "obfuscates"

Alex’s question.

I think it makes the question and answer very clear.

            MT: Is it a coincidence or is it

significant that the power under many experimental
conditions is close to the 0.33 power that applies to
the parametric V? He would also like to know the PCT
explanation for conditions under which (like George)
they don’t conform to any power law, or under which the
numeric value of the power changes.

            MT: Does that make Alex's question easier to understand?
          RM: I already understood that to be Alex's question

and I have answered it in spades.

          My spreadsheet reproduces the main results obtained in

all the different experiments on the power law. And it
does it based on the PCT observation that what is being
observed in these experiments is a side effect of the
mathematical fact that:

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

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

          RM: The PCT model accounts for the data, not for

imaginings about what the data say.

I am keen to try if I can get some software that converts mouse movement to X and Y coordinates over time…

···

On Sun, Jul 31, 2016 at 5:52 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I agree with everything you say about the fundamental issue of trying to infer output from controlled input regardless of whether your equation is correct. But to me, that is the first step. The next step is to hypothesise or model a plausible control system(s) with CV(s) that reveal the power law, rather than using the power law-conforming movements observed as your reference values of a controlled perception…

RM: My analysis, as implemented in the spreadsheet, shows that all movements are power law conforming movements; there is no such thing as a power law un-conforming movement (except for a perfectly straight line). Thus, I’ve already presented the control model that can produce movement patterns that conform to the power law. Any model, control model or not, that can make movements over time will be power law conforming. The spreadsheet proves it. Of course, I could be proved wrong if someone could show me a power law un-conforming movement!!

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 Mon, Aug 1, 2016 at 1:45 AM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.31.1745)]


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

Tel: +44 (0) 161 275 8589

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

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

Available Now

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

On Sun, Jul 31, 2016 at 12:31 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Martin, you have put it very clearly and helpfully to Rick, and this is exactly how I see it too.

RM: Clearly I didn’t find Martin’s post as clear and helpful as you had hoped;-)

Best

Rick

Warren

On 31 Jul 2016, at 05:39, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2016.07.30.23/54]

[From Rick Marken (2016.07.30.1550)]

Just what Alex asked about initially. It's something quite different

from your parameter mapping “V”.

No, your "V' isn't. Your V is simply the mapping of t onto s in the

computation of curvature. The V asked about and measured in studies
of the power law is unrelated to that mapping. It is the actual
speed of movement of the output of an actual control system
controlling for something. We don’t know what perception is being
controlled, but we do know that the result is often not a power law
with an exponent 0.33. You demonstrated that yourself
[From Rick Marken (2016.07.26. 1340)] when you simulated a model
that you haven’t yet published, but you said about it:

--------start quote-----

RM(07.26.13.40]:

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.

-------end quote-----

MT: Reading that, it sounds as though you measured the along-track

velocity that is measured in most studies of the power law, which is
why I responded “Excellent”. But it’s not the V in your math
derivation, for which the exponent is necessarily exactly 0.33 when
the equation is complete (i.e. not dropping your “D” factor).

Your explanation says quite explicitly that George _cannot_ behave

as my description of his sightseeing driving said he behaved.
George’s driving speed has no fixed relation to the local curvature
of the road, as it is determined by the density of the tree screen
the most of the time obscures the view. Your explanation says that
George’s speed is always exactly determined by the local curvature
of the road.

Your explanation says quite explicitly that your own PCT model that

you used in your simulation cited above could not produce the data
it did produce.

So far as I know, you have offered only one "PCT explanation" in the

form of the simulation that showed a way that the exponent might
change from 1/3 to 1/4. Apart from that, all you have done is
reconfigure the two expressions that describe curvature. I’m sorry
that nothing anyone has said has been easy enough for you to
understand this very simple fact.

It has no relation to the question being asked because the V

variable has no relation to the V

By mixing up the parametric representation of a curve with what an

independent control system may do when following the curve, treating
those two quite independent constructs as though they were one and
the same.

No. Your answer has consistently been that the data are wrong, and

the power function is always 1/3, because that’s a geometric
property of all smooth curves.

I don't know what that simple geometrical fact has to do with PCT,

but it does seem to say you programmed your “viscosity” model wrong,
and that nobody could possibly drive like my George, or walk at the
same speed on a straight road as on a twisty one. When a “model”
produces results in direct contradiction to observations (such as
when you “proved” to me [From Rick Marken (2016.07.20.1130)] that
people and other organisms go faster around curves the sharper they
are), I tend to believe the observations rather than the model.

The point is not simply that your model isn't a "PCT model" but a

geometric description, but that it contradicts data it is supposed
to explain. Most people, when they produce a model that directly
contradicts the data, would try to find out what is wrong with the
model rather than simply repeat over and over again, like DJT, that
they are right and everyone else (and the data) are wrong.

You even say occasionally that the power law is just a property of

curves, and that is correct. Bruce and I both showed you that the
1/3 power law is consistent with any velocity or variation of
velocity. Since you know that it is a property of all smooth curves,
that should have been enough without any urging from us to tell you
that your “V” has nothing to do with the velocity of actual things
that follow the curve.

What I really don't understand is why, having shown that you know

this, you then say that nobody can vary their speed at will along
different parts of a curved track because the possible speed is
strictly determined by the curvature. If that were so, why would we
ever have a sign for the recommended speed around a road curve? Why
would there be any point in having Grand Prix road races if all the
cars have to go the same speed everywhere on the track?

Martin


Richard S. Marken

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

Martin Taylor (2016.07.30.17.48)–

            MT: No it isn't. It's the parameter that links

time to the position along the curve in the parametric
description of the curve.

          RM: Then what is the variable called "V" that is

measured in studies of the power law?

                        RM: So the spreadsheet is an obfuscation?

The variable (it’s not a parameter of
anything) is the V variable measured in all
studies of the power law.

          All the variables in the spreadsheet demonstration  --

R, V, C and A – are the variables measured in every study
of the power law.

          The spreadsheet just duplicates the calculations made

in these studies. If you’ve think some other method was
used to measure these variables then let me know and I’ll
but those measures into the spreadsheet.

            MT: Alex wants to know whether PCT

can explain why under most experimental conditions the
subjects (human or otherwise) don’t behave like George,
but do in fact move along the track with a velocity that
is close to a power function of the local radius of
curvature.

          RM: That's what I thought; he want's to know if PCT can

explain the power law results. I have provided that
explanation.

                        RM: What is Alex's original question that

the spreadsheet obfuscates? And how does it
obfuscate it?

          He didn't like it, nor did you or those several other

people who tried over the last two weeks to make it easy
for me to understand that my PCT explanation was wrong.

          So now I have presented the spreadsheet demonstration

to show how the PCT explanation works.

          I would like to know how the spreadsheet "obfuscates"

Alex’s question.

I think it makes the question and answer very clear.

            MT: Is it a coincidence or is it

significant that the power under many experimental
conditions is close to the 0.33 power that applies to
the parametric V? He would also like to know the PCT
explanation for conditions under which (like George)
they don’t conform to any power law, or under which the
numeric value of the power changes.

            MT: Does that make Alex's question easier to understand?
          RM: I already understood that to be Alex's question

and I have answered it in spades.

          My spreadsheet reproduces the main results obtained in

all the different experiments on the power law. And it
does it based on the PCT observation that what is being
observed in these experiments is a side effect of the
mathematical fact that:

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

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

          RM: The PCT model accounts for the data, not for

imaginings about what the data say.

[Martin Taylor 2016.07.21.23.23]

[From Rick Marken (2016.07.31.1730)]

I suppose you read their paper as carefully as you read my postings.

I have heard the expression "invincible ignorance", but I never

expected to see an example in real life. Well, live and learn.

I give up. Mathematics, verbal explanations, simplifications... I

can think of nothing further to try to explain to you that fooling
around with rearrangements of the parametric equations that describe
curvature is never going to tell you anything about the velocities
with which people and other organisms move along their curved
trajectories, though they will tell you that the trajectories they
follow have properties common to all curves. The “V” you use is part
of the equations that describe curves, nothing more.

But since I have pointed this out many, many, times in different

ways, I realize that there’s no more I can do, so I resign. This
discussion is a waste of time, and moreover likely to prove a strong
deterrent for any serious scientist who might think that CSGnet
could be a serious place to discuss real PCT problems. Trying to
teach the fundamental analysis of parametrically represented curves
is NOT what I want to do on this list, nor is it something that
ought to be necessary to do more than once here, and I won’t do it
any more.

The problem Alex initially posed is an interesting one. It would be

nice if someone would try to address it.

Martin
···

Martin Taylor (2016.07.30.23/54)–

            MT: Just what Alex asked about initially. It's

something quite different from your parameter mapping
“V”.

          RM: Alex asked how PCT would account for the observed

power relationship between V and R (measured as per the
formulas given in the Gribble/Ostry and many other papers)

                        RM: Then what is the variable called "V"

that is measured in studies of the power
law?

[From
Bruce Abbott (2016.07.30.0935 EDT)]

Â

        I’ve

gone back to Rick’s post of 7/24 because it contains a clear
statement of his derivation of the relationship between
velocity and the radius of curvature, which he believes to
be a fixed relationship that holds for “all two-dimensional
movement patterns so long as D is taken into account.� I
will show that Rick’s conclusion is based on a
misunderstanding of the equation for the radius of curvature
and explain (in plain English) why it is wrong. My analysis
begins at the end of Rick’s post copied below:

Re The Power Law Why Rick’s D (108 Bytes)

Re The Power Law Why Rick’s D1 (109 Bytes)

Re The Power Law Why Rick’s D2 (109 Bytes)

(Attachment RbyT.jpg is missing)

(Attachment RbyS andT.jpg is missing)

(Attachment RbyS.jpg is missing)

···

Rick Marken (2016.07.24.1220) –

Â

Martin Taylor  (16.07.20.21.19)

Â

          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…

Â

          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.

Â

          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:

Â

        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.Â

              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.Â

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

that V =
 X.dot2+Y.dot ,
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 asÂ

                R =

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

                And

from there it’s a couple steps to:Â

                      V

= D1/3Â *R1/3

Â

                      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:Â

log
(V) = a * log (D)
 +
b * log (R)

Â

                      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.Â

Â

                    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. Â

Â

. . . . .

              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. Â

Â

RM:Â 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. AndÂ
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.

                BA:Â 

The reason previous researchers did not see the
relationship Rick discovered by means of a simple
rearrangement of mathematical terms is that Rick’s
rearrangement does not show what he thinks it shows.
To establish this, I begin by reproducing the two
equations Rick uses in his derivation. The first is
this:

                V

is the velocity of a moving point along a curve,
such as a car going around a bend in the road. V
would be the speed shown on the car’s speedometer.Â
The above equation is for motion in two dimensions,
X and Y. The dots over X and Y indicate that these
are the first derivatives of X and Y with respect to
time, or in other words, velocities in the X and Y
directions. The equation is simply the Pythagorean
Theorem in which X.dot and Y.dot are the sides of a
right triangle (the horizontal and vertical
components of the velocity) and V is the
hypotenuse. V is the instantaneous velocity along
the curve at a given time T.

** Dimensional
analysis** .Â
X.dot and Y.dot are speeds along the X and Y
dimensions, measured, say, in meters/second.Â
Squaring each gives meters squared/seconds squared
for each. As the units are the same for both, we
can add them. Taking the square root gives V in
meters/second.

                The

second equation used in Rick’s derivation is this
one:

                This

is the equation for the radius of curvature. It is
the radius of a circle whose curvature matches that
of the line being followed, at a given point on the
line.

                The

numerator of this equation is simply the equation
for velocity V, raised to the third power. (The
portion inside the parentheses is raised to the ½
power, which is the same as taking its square root.Â
The result is then cubed. So we can rewrite the
numerator as V3.

                So,

what do we have? The first equation gives the
instantaneous tangential velocity (speed) of a point
along a curve at time T, in meters/second. The
second equation gives the instantaneous radius of
curvature, in meters, of the path the point is
following at that same time T (corresponding to a
given position along the path that has been reached
at time T). *** This equation compensates for
the velocity of the point (in meters/second) so
that the value of R is independent of the
velocity.*** Â It will give the same radius
of curvature for any velocity > 0, for any given
curvature.

** Rick’s
mistake** .Â
Rick noticed that the numerator of the equation for
R is equivalent to V cubed. He then rearranges the
equation to solve for V and discovers that the value
of V is essentially determined by the value of R:Â

V = D1/3 *R1/3
where “Dâ€? is the denominator of the equation for R. Â

              But what this equation tells

us is the value of V at the instant that radius R was
being computed.

              A moment later V could be

higher or lower, and so would D (because D compensates
for changes in V to give a radius of curvature that is
independent of speed).

              Because Rick does not

understand what D represents, he concludes that
velocity is a function of R and that this relation
will hold no matter what shape of figure is being
followed by the moving point.

              But in fact R is independent

of V. Consequently, V can be any value whatsoever no
matter how sharp or gentle the curve, in so far at the
math is concerned.

              Yet empirically, biological

organisms do tend to adjust their speeds depending on
the radius of the curve, slowing for sharper curves
and speeding up for more gentle ones. The empirical
relationship has been found to follow the power law
with beta = 1/3 in most cases (but with significant
exceptions, such as when tracing a figure in a viscous
medium. The question is, how can we account for this
relationship?

Bruce

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