Behavioral Illusions: The Basis of a Scientific Revolution

[amatic]

RM: What Bill corrected me on was my saying (or implying) that there was no causal connection from from q.d to q.o via the organism ona closed-loop system. He noted that there certainly was and he described each of the causal functions along the way – q.d (along with q.o) causes q.i and qi causes q.o. So I accepted his correction and now understand that there certainly is an actual causal connection from q.d to q.o in a closed loop control system.

Good. I see now you understand it is a mistake to define the behavioral illusion as “the appearance of a causal relationship between behavioral variables when such a relationship does not, in fact, exist”. M&S (2017).

RM: This helped me better understand the S-R behavioral illusion described in the 1978 paper. The illusion results from the fact that the observed relationship between q.d and q.o when studying a closed loop control system is not the actual causal relationship between q…d and q.o that exists in such systems. In fact, the observed relationship between q.d and q.o is the inverse of the actual, causal feedback connection from q.o to q.i. So the observed functional relationship between q.d and q.o does not reflect the causal relationship that actually exists between q.d and q.o in such systems; what you are seeing is the inverse of the feedback function relating q.o to q.i. That may be nonsense to you but it’s PCT-sense to me.

It sure is nonsense. To me, it looks like you have a deep misunderstanding of the mathematics of control loops. Less errors than before, though, so maybe you did learn something from me. You still insist on using your own little invented language to describe relationships in the feedback loop. Those are just functions, not “causal functions”. You mix “relationships”, “functions”, and “connections”, between variables, and those are very different things. A relationship between two variables can be a function or not. There is no such thing as “inverse connection”. Instead of inventing strange terms, it would be better use standard terminology from mathematics and classical control theory, like Bill did.

RM: The fact that log (V) = 1/3 * log( R) +1/3 * log (D) is not a scientific theory, it is a mathematical fact that is proved (not tested, as is done with scientific theories) by following the rules of algebra to find the relationship between the computational formulae for V and R.

It is a correct formula, that is not the issue. Your theory is that because of that equation, the power law is a statistical artifact, and not a real phenomenon. Other people say that because of that equation, the 2/3 power law appears if the brain is planning a constant affine velocity trajectory. Their theory is refutable - you apply a disturbance and measure if the affine velocity is still constant; or require the person to follow a non-constant affine velocity trajectory, etc.

How does one test your theory? Or are you saying it is not a scientific theory?

RM: No I don’t agree with it. FInd all the quotes you like. But you can see from the results of the regression analysis that no such assumption is needed. The regression doesn’t care why R and D covary; it only cares about how much they covary.

You can do regression between two identical variables, for all regression cares. Garbage in, garbage out, as always in statistics. Interpretation is everything.

The problem with using D as a predictor is that you already use velocity (the first derivative of position) to calculate D, and then use D as a predictor of velocity.

[amatic]

AM: So, I’ve asked, what if the velocity of a particle moving along a curve can be predicted perfectly from the regression equation that does NOT include the variable D? Would that be good enough evidence?

RM: No. It is expected if the covariance between D and R is precisely 0.

What if D and R have a convariance =/= 0, and you could still predict the velocity of the particle perfectly, just from regression parameters and curvature, without using the variable D?

[amatic]

RM: But I now agree with Bill and Winston and I will try to be optimistic about you and anyone else convinced by your arguments

What exactly do you find wrong with the kinds of arguments I’ve presented in these two topics?

Literature review, definition of the behavioral illusion from the original paper?
Simulated analog computer model of the behavioral illusion?
Model of rubber band demo with knot position control demonstrating that the shape of the figure, even though it is a side effect of the properties of the control loop, still depends of the parameters of the control loop?
Model of the step response showing that the shape of the velocity profile depends on the properties of the control loop, even though it is a side effect of control of position?
Model of position control in tracking fast ellipses showing that position is not the controlled variable, as the model does not behave the same as the human?
Color-coded plots of path and velocity?

and believe that all of you will eventually “get” PCT in a way that will allow you to do the kind of research that Bill and I claim to be the only right way to study the behavior of living control systems.

I’m not so optimistic that you will eventually get PCT and that you will start doing the kind of research Bill and I claim to be correct.

[rsmarken]

Hi Adam

RM: What Bill corrected me on was my saying (or implying) that there was no causal connection from from q.d to q.o via the organism ona closed-loop system. …

AM: Good. I see now you understand it is a mistake to define the behavioral illusion as “the appearance of a causal relationship between behavioral variables when such a relationship does not, in fact, exist”. M&S (2017).

RM: Actually, in the context of the power law argument to which M&S 2017 was a reply that was exactly the right way to say it because the illusion in the case of the power law (among some of the testyest competitors, anyway) is that there is a causal relationship between curvature and speed of movement.

RM: But you are right that describing the illusion Powers described in the 1978 paper as " “the appearance of a causal relationship between behavioral variables when such a relationship does not, in fact, exist” would not be correct. The way to describe that illusion is that it seems like you are seeing the forward causal relationship between disturbance and output when, in fact, you are seeing the inverse of the backward causal relationship between output and controlled variable.

RM: This helped me better understand the S-R behavioral illusion described in the 1978 paper. The illusion results from the fact that the observed relationship between q.d and q.o when studying a closed loop control system is not the actual causal relationship between q…d and q.o that exists in such systems. In fact, the observed relationship between q.d and q.o is the inverse of the actual, causal feedback connection from q.o to q.i. So the observed functional relationship between q.d and q.o does not reflect the causal relationship that actually exists between q.d and q.o in such systems; what you are seeing is the inverse of the feedback function relating q.o to q.i. That may be nonsense to you but it’s PCT-sense to me.

AM: It sure is nonsense.

RM: As you wish.

RM: The fact that log (V) = 1/3 * log( R) +1/3 * log (D) is not a scientific theory, it is a mathematical fact that is proved (not tested, as is done with scientific theories) by following the rules of algebra to find the relationship between the computational formulae for V and R.

AM: It is a correct formula, that is not the issue. Your theory is that because of that equation, the power law is a statistical artifact, and not a real phenomenon.

RM: No, that is not my “theory”. Something can be a real phenomenon and a statistical artifact. The power “law” is a real phenomenon; anyone can see it for themselves. But what is seen is a statistical artifact that results from the failure to include the variable D in the regression.

AM: Other people say that because of that equation, the 2/3 power law appears if the brain is planning a constant affine velocity trajectory.

RM: I know (said in the way only Mike Birbiglia can say it).

AM: Their theory is refutable - you apply a disturbance and measure if the affine velocity is still constant; or require the person to follow a non-constant affine velocity trajectory, etc.

RM: But we already know that it is not the constancy of affine velocity that results in the power law; it’s the lack of correlation between affine velocity and curvature that results in the power law. I suppose you could disturb this correlation – the disturbance would be a control system that tries to keep affine velocity constant. That would be a good study to do, actually.

AM: How does one test your theory? Or are you saying it is not a scientific theory?

RM: Mine is not a theory; it is just a description of behavior that is seen as “output”. That is my fundamental criticism of the power law research; they don’t see power law behavior as a side effect of control; a side effect hat is easily explained as a statistical artifact. My theory is that the outputs you see are side effects of the agent controlling perceptions – proprioceptive perceptions, visual perceptions, higher level perceptions of transitions, configurations, sequences, etc. . I presented my based, initial theory a couple years ago and it was resoundingly (and rather rudely) rejected. So I’m out of the power law theorizing business. I’m just trying to keep you from continuing down a blind alley.

RM: No I don’t agree with it. FInd all the quotes you like. But you can see from the results of the regression analysis that no such assumption is needed. The regression doesn’t care why R and D covary; it only cares about how much they covary.

AM: You can do regression between two identical variables, for all regression cares. Garbage in, garbage out, as always in statistics. Interpretation is everything.

AM: The problem with using D as a predictor is that you already use velocity (the first derivative of position) to calculate D, and then use D as a predictor of velocity.

RM: So did the first people to study the power law back in the 1980s know that V is a function of both R and D (or that A is a function of both C and D) and, based on their interpretation of the situation, decide to leave D out of the regression analysis?

Best

Rick

[rsmarken]

Hi Adam

RM: But I now agree with Bill and Winston and I will try to be optimistic about you and anyone else convinced by your arguments

AM: What exactly do you find wrong with the kinds of arguments I’ve presented in these two topics?

AM: Literature review, definition of the behavioral illusion from the original paper?
Simulated analog computer model of the behavioral illusion?
Model of rubber band demo with knot position control demonstrating that the shape of the figure, even though it is a side effect of the properties of the control loop, still depends of the parameters of the control loop?
Model of the step response showing that the shape of the velocity profile depends on the properties of the control loop, even though it is a side effect of control of position?
Model of position control in tracking fast ellipses showing that position is not the controlled variable, as the model does not behave the same as the human?
Color-coded plots of path and velocity?

RM: Why not organize these things into a paper and present it at the IAPCT meeting! We could use some presentations of research at the meeting.

RM: and believe that all of you will eventually “get” PCT in a way that will allow you to do the kind of research that Bill and I claim to be the only right way to study the behavior of living control systems.

AM: I’m not so optimistic that you will eventually get PCT and that you will start doing the kind of research Bill and I claim to be correct.

RM: Touche. As you know, I forgot everything I knew about PCT when Bill passed away (indeed, became the enemy of PCT by some accounts) so it’s good that you have picked up the torch.

Best

Rick

[amatic]

RM: No, that is not my “theory”. Something can be a real phenomenon and a statistical artifact.

No, it cannot. Name some other examples that are real phenomena and also statistical artifacts.

RM: But what is seen is a statistical artifact that results from the failure to include the variable D in the regression.

Well, that is your theory then. You prefer to call it a hypothesis? The speed-curvature power law is a statistical artifact that results from the failure to include the variable D in the regression.

Are you saying it is not possible to test weather something is a statistical artifact, but it just has to be because of the algebra?

I’ll just back up a second to establish the terms.

v is the velocity vector, with components x’ and y’
a is the acceleration vector with components x’’ and y’’
D is the area closed by a and v, calculated as the magnitude of the cross product of a and v, as in (1), or by components of a and v, as in (2)

(1) D = | a x v |
(2) D = | y’‘x’ - x’‘y’ |

You are saying that the area D closed by vectors a and v, should be included in the regression analysis between V (speed, magnitude of v, one side of the parallelogram) and C, even though D is directly determined by V?

If a is has constant magnitude, the blue side of the parallelogram is always the same. All of the variance in D is going to come from variance in V. Agreed?

[amatic]

RM: As you know, I forgot everything I knew about PCT when Bill passed away (indeed, became the enemy of PCT by some accounts)

You did not forget, you never really knew the mathematical aspects of control theory, from what I can see in the archives and from what you are writing now. Bill spent some 30-40 years before you met him
working with analog computers and designing control systems.
You never went down that path, so while Bill was alive, he would nicely and patiently correct you and direct you with the more difficult parts of the math.
It would do you good to study classical control theory, maybe from an old textbook. It seems process control has similar terminology to PCT, maybe that would be a good start.

[rsmarken]

Hi Adam

RM: No, that is not my “theory”. Something can be a real phenomenon and a statistical artifact.

AM: No, it cannot. Name some other examples that are real phenomena and also statistical artifacts.

RM: The observed relationship between monthly ice cream sales and murder rate. And all other spurious correlations, like that between curvature and velocity of movement.

RM: But what is seen is a statistical artifact that results from the failure to include the variable D in the regression.

AM: Well, that is your theory then. You prefer to call it a hypothesis? The speed-curvature power law is a statistical artifact that results from the failure to include the variable D in the regression.

RM: I’s neither a theory nor a hypothesis; it is a mathematical fact.

AM: Are you saying it is not possible to test weather something is a statistical artifact, but it just has to be because of the algebra?

RM: No.

AM: I’ll just back up a second to establish the terms.

v is the velocity vector, with components x’ and y’
a is the acceleration vector with components x’’ and y’’
D is the area closed by a and v, calculated as the magnitude of the cross product of a and v, as in (1), or by components of a and v, as in (2)

(1) D = | a x v |
(2) D = | y’‘x’ - x’‘y’ |

You are saying that the area D closed by vectors a and v, should be included in the regression analysis between V (speed, magnitude of v, one side of the parallelogram) and C, even though D is directly determined by V?
If a is has constant magnitude, the blue side of the parallelogram is always the same. All of the variance in D is going to come from variance in V. Agreed?

RM: I think the problem is that you and your cohorts are looking at the power law as a physics problem using methods borrowed from the behavioral sciences, such as linear regression, to study it. From the point of view of regression analysis, each instant in a trajectory is a “case” and each case is associated with the values of a criterion variable and one or more predictor variable. The analysis doesn’t care how the variables are computed or the order in which they occur in the list of cases (so temporal order is irrelevant to regression analysis of data that occurs over time, such as movement trajectory data).

RM: In a conventional power “law” analysis of a movement trajectory the values of only two variables are entered into the regression – log V or A as the criterion variable and log R or C as the predictor variable. The result is often that there is a strong linear relationship between the log of the variables indicating a power relationship and the coefficient of that relationship is typically in the range from .2 to .35, averaging about .3 (for R vs V) or .61 to .8, averaging about .7 (for C vs A).

RM: What I and apparently everyone else who studies the power law knows is that the mathematical relationship between the measures of curvature (R or C) and speed (V or A) used in these analysis is a power law relationship of the form:

log (V) = 1/3* log( R) + 1/3* log(D)

or

log (A) = 2/3* log( C) + 1/3* log(D)

RM: When I discovered this and pointed it out to the power law people I thought for sure that they would experience the nightmare Powers talked about in the 1978 paper (realizing too late that their results – the power “law” – were forced by their experimental design ) or would at least be a little embarrassed and start studying movement as a control phenomenon; this because the 1/3 and 2/3 power law are right there in the mathematical equations relating the variables they use to measure speed and curvature.

RM: But they remained completely unfazed, perhaps because there was a second variable, D, that could be used as a Trump card (a Trump card in this situation is not the same as a trump card in bridge; it’s named after the President of the US who is skilled at the use of irrelevant distractions to take attention away from his failures). The variable D could be used to deflect attention away from the fact that the exact 1/2 and 2/3 power laws were what you would always find if the variable D were included in the analysis.

RM: SO with D as their Trump card they march forward, with their MPGA (Make PCT Great Again) hats, wrecking PCT as effectively as Trump is wrecking America. And they’ve managed to get at least 40% of the PCT crowd behind them. And I am their Deep State, getting slapped around like an FBI agent investigating Russian Interference in the US election.

RM: So I will politely drop out of this conversation since I have lost all conviction and don’t’ care to continue to suffer the abuse of those full of passionate intensity. But not before answering Adam’s last post.

[rsmarken]

Hi Adam

RM: As you know, I forgot everything I knew about PCT when Bill passed away (indeed, became the enemy of PCT by some accounts)

AM: You did not forget, you never really knew the mathematical aspects of control theory, from what I can see in the archives and from what you are writing now. Bill spent some 30-40 years before you met him
working with analog computers and designing control systems.
You never went down that path, so while Bill was alive, he would nicely and patiently correct you and direct you with the more difficult parts of the math.
It would do you good to study classical control theory, maybe from an old textbook. It seems process control has similar terminology to PCT, maybe that would be a good start.

RM: A deep knowledge of classical control theory is not necessary in order to understand PCT. Indeed, it can get in the way. Early in the life of CSGNet we had a control engineer take an interest in PCT and he couldn’t (or wouldn’t) get it at all. I think Bill’s attempt to get the guy to “get” PCT is somewhere in the archives. You seem to be able to go through those pretty well so I’ll let you find it. It’s probably from 1990.

RM: And I was in on the discussions between Bill and John Flach, a psychologist who knows the mathematics of control theory far better than Bill did – Bill struggled to learn Laplace transforms to try to keep up with him – and all that math didn’t help him get PCT; indeed, the interaction led to the work shown in the “adaptation without adaptation” chapter of LCSIII – work that Flach didn’t get at all.

RM: And i don’t remember Bill patiently correcting and directing me with the more difficult parts of the math. But if he did it wasn’t often and that’s the kind of correction that was particularly useful. The things people don’t get about PCT are not related to the math, they are about how the model maps to behavior. That’s why Bill was unable to get many (any?) experts in classical control theory to understand how control theory should be applied to behavior.

RM: And when it comes to Bill correcting people on the net, I think you should take a look at his posts to others besides me. As I recall, Bill’s main complaints about me were about my tone, not about my understanding of PCT. If you want to see Bill complaining about people’s understanding of PCT – people who now claim to be experts in PCT – you might take a look at Bill’s posts to Martin Taylor regarding Martin’s ideas about there being “information in perception about the disturbance”, or about his belief in affordances or about there being an entity called a CEV.

RM: But I’m tired of the insults and the condescending comments from you. And I really don’t want to do this on my own anymore. I used to have Bill but now there is no one – at least no one on this net – who is willing or able to take my “side” of this argument. So I’ll leave at least for a while and let you carry on as you wish. But while you’re telling people how little Bill thought of my understanding of PCT you might point them to Bill’s preface to MIND READINGS to see what Bill said he thought of my work. Of course, that’s probably Fake News, right?

Best

Rick

[amatic]

You get terribly dramatic when it comes to you being wrong.

First I was the Spanish inquisition, forcing you Mr. Galilei to admit falsehoods. Now I’m Trump slapping and you, the honest FBI agent, for doing his job. You are dramatically leaving the scene, head bowed down. Someone yells “You are a poster child for the Dunning-Krugger effect!”. You turn around, eyes closed, showing your tongue, and exit.

I think you are leaving because you cannot face the fact that you could be wrong. That would mean that, horror of horrors, Martin Taylor was right! Alex was right for calling bullshit! Everyone else on the net who disagrees with you on the topic of the power law was right, and you were really the enemy of PCT.

It probably is not easy to deal with those possibilities, but you made your own bed.

RM: The observed relationship between monthly ice cream sales and murder rate. And all other spurious correlations, like that between curvature and velocity of movement.

A spurious correlation is not a statistical artifact. Ice cream sales and murder rates can be really correlated in some cases. The correlation is spurious because they are not causally related.

Instantaneous curvature and velocity are also correlated in some cases and are not causally related. No one ever said they were causally related, you have misread or misunderstood that. Stretching the definition, you could say that it is a spurious correlation, if you would show the real mechanism producing the correlation, and all the causal factors - the controlled variables, input and functions, etc.

A statistical artifact, by definition, is not a real phenomenon. The correlation itself is not reflecting a real correlation. Empirically, fast hand movement in elliptical trajectories shows a real correlation between speed and curvature, as many people have been showing for many years. I’ve also put up here a little plot of speed and curvature, color coded, that shows how people move slower in the curved parts, and faster in the flat parts. You said you can see that, yes, people do really move slower in the more curved parts.

You’ve confused two terms in statistics, a spurious correlation and a statistical artifact. What does that say about your knowledge of statistics?

There are statistical problems with power laws, but the fundamental problem of the speed-curvature power law is - why do people move slower in curved parts? Empirically, before any statistics, you can see it in the plots, they do move slower in the curved parts, if the average velocity is high enough. That is what all the “power law people” are studying. What are the mechanisms of tracing lines, of drawing letters, of making rhythmical hand movements, etc. Whatever the mechanisms is, it has to slow down in curved parts, and go faster in the other ones.

The perspective of PCT is to search for perceptual variables that are maintained, and the control loop that produces the slowing down as a side effect.

AM: Well, that is your theory then. You prefer to call it a hypothesis? The speed-curvature power law is a statistical artifact that results from the failure to include the variable D in the regression.

RM: I’s neither a theory nor a hypothesis; it is a mathematical fact.

Calling some phenomenon a statistical artifact is an interpretation, a hypothesis, not a fact. The equation you mention is a mathematical fact, but different people have different interpretations of that equation, so that is the debatable part.

RM: What I and apparently everyone else who studies the power law knows is that the mathematical relationship between the measures of curvature (R or C) and speed (V or A) used in these analysis is a power law relationship of the form:
RM: log (V) = 1/3* log( R) + 1/3* log(D)

That is a correct equation. If you rearrange the equation, you get D = V^3 / R. What you have done is discovered a formula for calculating D from speed and the radius of curvature and rearranged it. If you further rearrange it and play with algebra, you would get that D = | a x v |.

The jump you make between this formula and calling the empirically found correlation a statistical artifact is the problematic part.

RM: this because the 1/3 and 2/3 power law are right there in the mathematical equations relating the variables they use to measure speed and curvature .

This part.

The power law is not in the mathematical equations relating speed and curvature. It is in the equation relating speed, curvature and affine velocity. There is a big difference.

A mathematical equation that relates speed and curvature would have to have just speed and curvature in. V = f( C ). This is not the case. Speed and curvature are not related mathematically, they can vary independently, just like you can drive your car faster or slower in the curves, or draw lines with any speed you please (if it is slow enough). In principle, a particle can move with any speed trough any curvature. And still, the formula you mentioned would be valid, because that is just a formula for calculating affine velocity from speed and curvature.

[amatic]

RM: A deep knowledge of classical control theory is not necessary in order to understand PCT.

Oh, I’m not saying you don’t have a deep knowledge of classical control theory. I’m saying you don’t have the BASIC knowledge of classical control theory, and the math that is needed to understand the behavioral illusion. If it makes you happy, it looks like that neither Martin Taylor had it.

Not fake news, here is Bill correcting you two bozos on how to derive the equations of the behavioral illusion in 2010! [Beyond the Fringe (was An Opportunity for PCT PR) - #26 by Bill_Powers1]

That is just a small example. In the previous topic on the behavioral illusion, where I followed the 1978 paper derivation, you made the same kinds of mistakes, just like in this topic.

[amatic]

You made exactly the same type of error in the previous topic where you wrote that qi is related by the feedback function to qo. This would imply qi = f(qo), which is not true. qi = f1(qd) + f2(qo).

Now you are saying that because R = V^3 / D, speed is related to radius of curvature. Again not true, because the term “related” implies that V = f ( R ). All that formula says is that if you had any the two variables calculated or measured, you could find the third one.

RM: I used to have Bill but now there is no one – at least no one on this net – who is willing or able to take my “side” of this argument

You have no one on your side because you are wrong. In your papers on the power law and behavioral illusion, as I’ve demonstrated, you’ve been wrong on the definition of the behavioral illusion. You’ve been wrong on the issue of side effects reflecting properties of the control loop. Now you incorrectly defined the term statistical artifact. And those are just the terms in the TITLE of your papers. You are also wrong about speed and curvature being mathematically related.

Bill would have been deep on the opposite side of the power law argument, and he would have cut it long before you could publish a paper. For the derivation of the behavioral illusion equations, Bill gave you a B. For this nonsense with the power law, you’d be flunked back to high school.

Here is what Bill said about one “power law experiment” : [From Bill Powers (930428.0700)]

“In JEP-Human Perception and Performance, there was a good control-theory experiment:
Viviani, P. and Stucchi, N. Behavioral movements look uniform: evidence of perceptual-motor interactions (JEP-HPP 18 #3, 603-623 (August 1992). Here the authors presented subjects with spots of light moving in ellipses and “scribbles” on a PC screen, and had them press the “>” or “<” key to make the motion look uniform (as many trials as needed). The key altered an exponent in a theoretical expression used to relate tangential velocity to radius of curvature in the model. The correlation of the formula with an exponent of 2/3 (used as a generative model) with the subjects’ adjustments of the exponent was 0.896, slope = 0.336, intercept 0.090. This is just the kind of experiment a PCTer would do to explore hypotheses about what a subject is perceiving. By giving the subject control over the perception in a specified dimension, the experiment allows the subject to bring the perception to a specified state – here, uniformity of motion – and thus reveals a possible controlled variable (at the “transition” level?). The authors didn’t explain what they were doing in that way, but this is clearly a good PCT experiment. Even the correlation was respectable, if not outstanding (the formula was rather arbitrary, so it should be possible to improve the correlation considerably by looking carefully at the way the formula misrepresented the data).
There is a world of difference between the kinds of experiments reported in J Exp. Soc. Psych and the two described above (and between the two described above and most of the others in JEP).”

AM: So, there is a dot moving on the screen, according to a speed-curvature power law with an adjustable exponent. Meaning - subjects adjust the amount of slowing down in the curves. The idea is to see which exponent results in the perception of uniform motion. There are no issues of “speed and curvature being related” for that particle. Instead of a particle, put someone’s hand moving according to the same equation. No difference in the math. “This is just the kind of experiment a PCTer would do”.

It is not everyone else, it is you.

[rsmarken]

Hi Adam

RM: I’ll reply to this because I think it is relevant to my point of view on power law research:

AM: Here is what Bill said about one “power law experiment” : [From Bill Powers (930428.0700)]

BP: “In JEP-Human Perception and Performance, there was a good control-theory experiment:
Viviani, P. and Stucchi, N. Behavioral movements look uniform: evidence of perceptual-motor interactions (JEP-HPP 18 #3, 603-623 (August 1992). Here the authors presented subjects with spots of light moving in ellipses and “scribbles” on a PC screen, and had them press the “>” or “<” key to make the motion look uniform (as many trials as needed). The key altered an exponent in a theoretical expression used to relate tangential velocity to radius of curvature in the model. The correlation of the formula with an exponent of 2/3 (used as a generative model) with the subjects’ adjustments of the exponent was 0.896, slope = 0.336, intercept 0.090. This is just the kind of experiment a PCTer would do to explore hypotheses about what a subject is perceiving. By giving the subject control over the perception in a specified dimension, the experiment allows the subject to bring the perception to a specified state – here, uniformity of motion – and thus reveals a possible controlled variable (at the “transition” level?). The authors didn’t explain what they were doing in that way, but this is clearly a good PCT experiment. Even the correlation was respectable, if not outstanding (the formula was rather arbitrary, so it should be possible to improve the correlation considerably by looking carefully at the way the formula misrepresented the data).
There is a world of difference between the kinds of experiments reported in J Exp. Soc. Psych and the two described above (and between the two described above and most of the others in JEP).”

RM: Yes, I know about the Viviani & Stucchi (1992) experiment and, in an effort to end our M&S 2018 reply on a positive note, I referred to it as an example of the kind of experiment one might do to study control of movement. Bill found this paper in April 1993, while searching through current issues of various psychology journals for examples of psychological research that looked like they didn’t need to be “deposited in the wastebasket” (the phrase he used in his Foreword to MIND READINGS to describe what the work described in that book suggested should be done with “a whole segment of the scientific literature”).

RM: Bill found the Viviani/Stucchi paper in JEP: HPP and saw that it looked like a good control theory experiment. Indeed, it looks a lot like a test for the controlled variable. It was different than most other conventional experiments because the researchers seemed to be aware of the fact that producing a motor movement (like moving the finger around in an elliptical or squiggly trajectory) was equivalent to producing a perception (in this case, of a spot of light moving around in an elliptical or squiggly trajectory). The hypotheses in the experiment was like a hypothesis in the TCV: that the the perception (and, presumably, production) involved control of a perception of a power relationship between curvature and velocity that had a coefficient – beta – was close to .33. The “disturbance” to this movement was the initial setting of beta that was ,. to .33 and the subject could compensate for this disturbance by pressing the > or < keys to increase of decrease beta.

RM: The results impressed Bill because he seems to have found a high correlation (.896) “of the formula with an exponent of 2/3 [did he mean 1/3?] with the subjects’ adjustments of the exponent”. I couldn’t find that correlation reported in the paper and his description of it makes no sense to me. But he certainly wasn’t impressed by the research because of the assumption of a power law relationship between curvature and velocity with a particular exponent. You can see this in the quote above where Bill says that “…the formula [used to generate spot movement] was rather arbitrary, so it should be possible to improve the correlation considerably by looking carefully at the way the formula misrepresented the data”. I think that can be interpreted as saying that the hypothesis that a 2/3 (or 1/3) power relationship between speed and curvature is what is being controlled when controlling for uniform movement should be rejected and a new hypothesis tested.

RM: So while the Viviani/Stucchi study is as good an example of a PCT- like study as one is likely to find in the conventional literature, it could be considerably improved if carried out with an understanding of how control works in living systems. So this study would be a great place for power law researchers to start developing research aimed at understanding the perceptual variable(s) involved in the control of movement. As I said in the conclusion to M&S 2018:

… once the controlled perceptual input variable is identified it should be possible to build a simple control system model that explains the movement behavior produced by living systems as it occurs under many different circumstances, as was the case with our object interception model noted above. To paraphrase Powers’ conclusion to his 1978 Psychological Review paper (p. 434): for a thousand unconnected empirical generalizations about movement behavior that are based on superficial similarities between features of movement trajectories, we here substitute one general underlying principle: control of input.

Best regards

Rick

[amatic]

The point is, Bill was working in an observatory. He knew all about orbital mechanics, power laws, speeds, angular speeds, curvatures, etc. He saw nothing wrong mathematically with a point on a screen moving in an elliptic trajectory, having different amounts of slowing down in the curved parts, and having this motion described by a power law.

If you want to generate a power-law path, you simply calculate the speed to be the cube root of curvature or whatever, and keep staying on the path. Nothing to it for a computer.

You, on the other hand, claim that “curvature and speed are mathematically related”, which is utter nonsense; does not come out of the formula for calculating D, and Bill would have caught the error right in the beginning.

[rsmarken]

Hi Adam

AM: The point is, Bill was working in an observatory. He knew all about orbital mechanics, power laws, speeds, angular speeds, curvatures, etc. He saw nothing wrong mathematically with a point on a screen moving in an elliptic trajectory, having different amounts of slowing down in the curved parts, and having this motion described by a power law.

RM: Nor do I. But that’s irrelevant to why Bill thought the Viviani/Stucchi experiment was like a control theory experiment. It was like a control theory experiment because… actually, why don’t you tell me why Bill thought it was like a control theory experiment.

AM: If you want to generate a power-law path, you simply calculate the speed to be the cube root of curvature or whatever, and keep staying on the path. Nothing to it for a computer.

RM: Even more nothing nowadays with computers that are nearly 30 years newer. Could you send me the computer algorithm they used to generate the spot movement? I’d like to see what they look like.

AM: You, on the other hand, claim that “curvature and speed are mathematically related”, which is utter nonsense;

RM: And yet they are mathematically related: V = R^1/3 * D^1/3 and A = C^2/3 * D^1/3. There is a no-nonsense, mathematical (power) relationship between speed (V, A) and curvature (R, C).

AM: does not come out of the formula for calculating D, and Bill would have caught the error right in the beginning.

RM: There is no error. If you use regression analysis to analyze the spot trajectories used by Viviani and Stucchi I think you will find the beta value they used to generate each one if you just regressed R on V or C on A. But if you include D in the regression you would find that the power coefficient (beta) of R regressed on C to be exactly 1/3 and the power coefficient of C regressed on A to be exactly 2/3 with R^2 values of 1.0 for both regressions. At least I’m pretty sure that that’s what you would find. Try it and see. Or I’ll try it if you like; just send me the Viviani/Stucchi algorithm for generating the spot trajectories.

RM: But, again, Bill’s evaluation of the Viviani/Stucchi experiment as being like a control theory experiment had nothing to do with how the spot movements were produced or whether the algorithm that produced them was based on a power law formula (remember, Bill did say that “the formula [used to generate spot movement] was rather arbitrary”). Bill saw the Viviani/Stucchi experiment as control theory-like for reasons that had nothing to do with the power “law”. If you can figure out what that reason was you may be on your way to becoming a researcher who knows the proper way to study the behavior of living control systems.

Best

Rick

[amatic]

Here is some python code for generating a trajectory with angular velocity equal to curvature raised to arbitrary exponent, with possibility to average speed or total time.

def get_times(ds, C, target_time, target_beta):
  dt = ds * C ** (1- target_beta)
  T = cumsum(dt)
  k = target_time / T[-1]
  t =  k * T 
  return t

def retrack(x, y, t, target_beta, target_time=None):
  if (target_time == None): target_time = t[-1]
  r = analyze(x, y, t) # get arc-length ds and curvature C
  ts = get_times(r.ds, r.C, target_time, target_beta)  # get target power law exponent, and target duration
  d = analyze(r.x, r.y, ts) # using new time vector, resample to const dt
  return d

rest of the code: Google Colab

In short, you take the time variable of an arbitrary trajectory, then set the time between each to points such that speed equals the desired value. Then resample the whole trajectory into equally spaced time intervales.

RM: Nor do I. But that’s irrelevant to why Bill thought the Viviani/Stucchi experiment was like a control experiment.

Uniformity of movement as a potential controlled variable. Now, back to bold. It is very relevant to the topic of the power law that you don’t see the problem that a dot of light is moving according to a power law, and yet you do see a problem when a human hand is moving, approximately, according to the same formula.

Why is that? The formulas don’t know what made the trajectory, if it was generated by a computer or by a human.

RM: And yet they are mathematically related: V = R^1/3 * D^1/3 and A = C^2/3 * D^1/3. There is a no-nonsense, mathematical (power) relationship between speed (V, A) and curvature (R, C).

The exact analogy of this is saying that two sides of a right triangle are mathematically related. If you have sides a and b closing the right angle, and the side c as the hypothenuse, Pythagoras’ theorem is not relating two of the sides, it is relating three of them. c² = a² + b². Any two sides are not mathematically related, and neither are speed and curvature. Instead, speed, curvature and D are related, and we have time as the additional variable. If you look at the code above, you can see something like:

V = sqrt(xvel**2.0 + yvel**2.0)
D = abs(yacc * xvel - xacc * yvel)
R = (V**3.0) / D

Written in that order (in function analyze). First, calculate speed from x and y components of velocity (xvel, yvel). Then calculate D as a cross-product of acceleration (xacc, yacc) and velocity (xvel, yvel). Then calculate radius of curvature according to the formula you mention. These are all lists of scalar values, with time as the fourth implicit variable.

RM: There is no error. If you use regression analysis to analyze the spot trajectories used by Viviani and Stucchi I think you will find the beta value they used to generate each one if you just regressed R on V or C on A. But if you include D in the regression you would find that the power coefficient (beta) of R regressed on C to be exactly 1/3 and the power coefficient of C regressed on A to be exactly 2/3 with R^2 values of 1.0 for both regressions. At least I’m pretty sure that that’s what you would find. Try it and see. Or I’ll try it if you like; just send me the Viviani/Stucchi algorithm for generating the spot trajectories.

There is no error in the raw calculating process. For any trajectory, given your procedure, I get the same results. No problem there. I have the issue with your interpretation of that procedure.

Should the omitted variable bias improve the prediction of the criterion variables from the predictors and the regression parameters?

RM: If you can figure out what that reason was you may be on your way to becoming a researcher who knows the proper way to study the behavior of living control systems.

That is condescending. You are implying that you know the proper way to study them, and that you can judge whether I know it. Now I could call you Master Marken, or whatever, to mock the implication, but that doesn’t really lead us anywhere.

[amatic]

Here is an illustration of the error of claiming that two variables are mathematically related, while in the formula there are three variables.

link to code

Variables a and b are lists of 100 items, completely random scalar values, in the range of 0 to 100, generated by this code:

a = np.random.random(100) * 100 
b = np.random.random(100) * 100

The correlation between them is near zero, and so is the r2. There is no statistical procedure that will make them mathematically related.

Now I introduce a third variable c with the following code:

c = 3 * a + 0.5 * b

Then I use linear regression with predictors a, and c, and criterion variable b.
regr.fit( df[['a', 'c']], df["b"])

This gives me back coefficients -6 and 2 for the predictor variables, which is exactly the formula for c rearranged, b = -6a + 2c. I get the same result for any list of random variables a and b. I also get the correct coefficients in all the cases I’ve tested.

This did not make a and b mathematically related. They are still completely random variables, and simply introducing a formula to calculate c does not change the randomness and does not create a mathematical relationship between a and b.

The same procedure, without fail, always returns the coefficient 0.33 for the ellipse speed, radius of curvature and D. The formula R = V^3 / D does not make speed and radius of curvature mathematically related.

[rsmarken]

Hi Adam

@amatic

AM: Here is some python code for generating a trajectory with angular velocity equal to curvature raised to arbitrary exponent, with possibility to average speed or total time.

def get_times(ds, C, target_time, target_beta):
  dt = ds * C ** (1- target_beta)
  T = cumsum(dt)
  k = target_time / T[-1]
  t =  k * T 
  return t

def retrack(x, y, t, target_beta, target_time=None):
  if (target_time == None): target_time = t[-1]
  r = analyze(x, y, t) # get arc-length ds and curvature C
  ts = get_times(r.ds, r.C, target_time, target_beta)  # get target power law exponent, and target duration
  d = analyze(r.x, r.y, ts) # using new time vector, resample to const dt
  return d

RM: I’m afraid I don’t read Python and I’d rather not take the time to learn right now. Could you just give me some relatively simple pseudocode that will vary the x,y coordinates of a spot moving in a squiggle pattern over time according to a power law; V = R^1/3 or A = C^1/3 into the time? Or better yet could you just send me several columns of x,y coordinates of a spot moving in a squiggle pattern according to a power law with several different beta values? Or even better still would be both. I would like to see what the movement looks like with different coefficients and do the regression analysis on the trajectories.

RM: Nor do I. But that’s irrelevant to why Bill thought the Viviani/Stucchi experiment was like a control experiment.

AM: Uniformity of movement as a potential controlled variable.

RM: I would say they were testing to see what variable is being controlled when a person is controlling a perception of uniform motion. Their hypothesis was that subjects control the value of beta in a power law equation relating speed to curvature. The assumption was that subjects would see uniform movement (which I presume means constant speed movement) when beta is .33. The disturbance to the hypothetical controlled variable was the initial setting of beta. Subjects corrected for this disturbance (if they needed to) and brought their perception of the movement back to the reference state of “uniform”, by pressing the “>” or “<” keys, in order to get beta = .33.

AM: Now, back to bold. It is very relevant to the topic of the power law that you don’t see the problem that a dot of light is moving according to a power law, and yet you do see a problem when a human hand is moving, approximately, according to the same formula.

AM: Why is that? The formulas don’t know what made the trajectory, if it was generated by a computer or by a human.

RM: The power law has nothing to do with why the Viviani/Stucchi study makes sense from a control theory perspective and other power law studies don’t. The Viviani/Stucchi study actually tests for a controlled variable; other power law studies don’t. The fact that the hypothesized controlled variable in the Viviani/Stucchi study is a power coefficient is beside the point. At least it is treated as a hypothesis about a variable aspect of the movement that might be controlled. Power law studies just look at features of the movement itself under the assumption that this will tell them something about how the movement is produced; it won’t.

RM: And yet they are mathematically related: V = R^1/3 * D^1/3 and A = C^2/3 * D^1/3. There is a no-nonsense, mathematical (power) relationship between speed (V, A) and curvature (R, C).

:AM: The exact analogy of this is saying that two sides of a right triangle are mathematically related.

RM: It just struck me as a pretty amazing coincidence that in the equations above the power exponents of R and C --1/3 and 2/3, respectively – are exactly the exponents that have been proposed as those that characterize the power “law”; a law that, I believe, is presumed to reflect properties of nature (behavior), not of mathematics. Indeed, in many articles on the power law the title often says that the article is about the “1/3 power law” or the “2/3 power law”.

AM: Should the omitted variable bias improve the prediction of the criterion variables from the predictors and the regression parameters?

RM: Yes, unless all of the variance in the criterion variable (V or A) is accounted for by the included predictor (R or C). This would be the case, for example, in a perfect ellipse, which follows the power “law” exactly.

RM: If you can figure out what that reason was you may be on your way to becoming a researcher who knows the proper way to study the behavior of living control systems.

AM: That is condescending.

RM: You are absolutely right and I apologize. I didn’t intend it to be condescending but I should have known that it might be taken that way. Civil behavior is all about knowing what the unintended consequences of one’s behavior might be and trying to avoid them. So I am sorry.

I would really appreciate it if you could get that data that I asked for to me, if possible – the time varying x.y coordinates of the Viviant/Stucchi power law generated spot squiggle movements. And the pseudocode too?

Best

Rick

[amatic]

RM: Or better yet could you just send me several columns of x,y coordinates of a spot moving in a squiggle pattern according to a power law with several different beta values?

Here is an excel file with the squiggles: power_law_squiggles.xlsx

Here is the code (still in python): code

All the squiggles look like this:

The time variable is in the first column, with dt of 1/60 seconds, equal for all squiggles. Next columns are in pairs, x and y, with beta calculated for A=kC^beta, so that at beta=1, the speed is constant. The equations are not taken from Viviani and Stucchi paper, Alex and I derived them, but I assume they should be pretty similar, since they give correct results. The geometry or the shape of each squiggle is exactly the same, and the speeds of the dot are different. In regression analysis, with just logC as predictor and logA as criterion, the coefficient is very nearly the one set in the retrack function as “target beta”. That is, regression analysis, without taking D into account, returns precisely the beta that was used to generate the trajectory.

When adding log D to the regression analysis as a predictor, the coefficient for log C is 0.67, and for logD is 0.15, for all trajectories. I would interpret this result as a statistical artifact - the same one as in my post above, with random a and b)

RM: The power law has nothing to do with why the Viviani/Stucchi study makes sense from a control theory perspective and other power law studies don’t. The Viviani/Stucchi study actually tests for a controlled variable; other power law studies don’t. The fact that the hypothesized controlled variable in the Viviani/Stucchi study is a power coefficient is beside the point. At least it is treated as a hypothesis about a variable aspect of the movement that might be controlled. Power law studies just look at features of the movement itself under the assumption that this will tell them something about how the movement is produced; it won’t

First bold - agreed, the power law itself does not have much to do with controlled variables, but the valuable aspect of this study, and most other power law studies, is finding relatively strong laws in behavior. A correlation of 0.9 is not commonly seen in life sciences (“respectable, if not outstanding”). There is another study by Viviani testing position and velocity control model in 2d pursuit tracking. Viviani et al(1987): Following Powers (1978), one could, in fact, argue that the subject’s primary goal is not to produce a specified overt behavior (i.e., a specific trajectory) but rather to minimize the discrepancy between the actual and ideal value of the position error
link to pdf

I think there is another paper, but with ellipses and the same model. So, for the second bold, it is not true that people assume that the power law itself tells them something about how the behavior is produced. The power law is a descriptive model of behavior with a relatively narrow range, only fast movements; although sometimes the generality is a overstated. Still, it is perfectly clear that it is a description of behavior with an unknown explanation. People make all kinds of models that are supposed to be generating movements that fit some power law.

RM: It just struck me as a pretty amazing coincidence that in the equations above the power exponents of R and C --1/3 and 2/3, respectively – are exactly the exponents that have been proposed as those that characterize the power “law”; a law that, I believe, is presumed to reflect properties of nature (behavior), not of mathematics.

It is a pretty strange coincidence! It doesn’t look so amazing when you see that people can move at low speeds with absolutely no power law, or that for shapes that are not ellipses, when traced in fast and smooth manner, the exponent is reliably a very different value from the value found in ellipses and squiggles. It does not have to be 1/3 or 2/3. The exponent does reflect something about the behaving system, some kind of a limitation in control and production of speed, coupled with environment properties. This was determined by, again, Viviani, with drawing ellipses in water vs in air; or in children drawing vs adults drawing.

RM: Yes, unless all of the variance in the criterion variable (V or A) is accounted for by the included predictor (R or C). This would be the case, for example, in a perfect ellipse, which follows the power “law” exactly.

Not quite right - you can check with the squiggles - regression of logA and logC should give very nearly the exponent in the column name, and the r2 is near 1 for all of them, so there is no room or need for improvement. Same thing with ellipses.

[amatic]

AM: When adding log D to the regression analysis as a predictor, the coefficient for log C is 0.67, and for logD is 0.15, for all trajectories.

On second try, double checking, the coefficient for log (D) is 0.33 (using log base 10 in all cases), as it should be according to log(A) = 0.67 * log( C ) + 0.33 * log(D), or A = C^2/3 D ^1/3

AM: Should the omitted variable bias improve the prediction of the criterion variables from the predictors and the regression parameters?

RM: Yes, unless all of the variance in the criterion variable (V or A) is accounted for by the included predictor (R or C). This would be the case, for example, in a perfect ellipse, which follows the power “law” exactly.

Maybe not the squiggles because they have a near perfect power law law fit. What would be a good test case for this? Empirical data with not so great r² ? Are you saying that the prediction of A from C from the power law fit should be improved when using a different beta (calculated from multiple regression)?

Or that the prediction of A should be improved when using the coefficients from multiple regression from C and D? If yes, then why even do the multiple regression, both coefficients are known from the formula?