Behavioral Illusions: The Basis of a Scientific Revolution

Fundamentals
101

RM: Those who made or were convinced by arguments like Adam’s weren’t going to do the kind of research that I (and Bill) claim to be the only right way to study the behavior of living control systems anyway.

I provided modeling studies, along with source code for all models, and empirical data from human subjects. That is a pretty solid base. Actually, you yourself were at some point convinced by the arguments I made about what is the behavioral illusion, and whether side effects tell you something about the behaving system or not.

When you realized some broader consequences of those facts - one being that be behavioral illusions have nothing to do with the speed-curvature power law, and therefore the title of your paper makes no sense - then you went back on the agreement. No good reason other than not being able to take the consequences of having made a mistake. At least that is how it looks from my perspective - you did not give any other reason for your changes in opinion.

There is still the issue of the statistical artifacts, of course, but that doesn’t have anything to do with PCT or control theory in general.

The word salads you make about the math of movement are pretty bad. Now you speak about “physically independent vs mathematically independent”, and you had another silly idea that “speed is apparently changing with curvature in a wheel”. You know you are not very good at math, you’ve said so yourself, and you’ve demonstrated it in the last few topics by arguing on and on about “causal functions” and “non-causal functions” and all the other inventions, while failing in basic, BASIC static analysis of a control loop, even negating existence of the feedback quantity!

It’s really sad to read that nonsense. How did you manage to work in PCT for 40 years and not learn quasi-static analysis? And the relationships between variables in the loop? And what is the difference between a function and a relationship?

RM And that’s because they all have a vested interest of some kind in doing research the conventional way. So it’s really useless to keep arguing about this. As they say, you can’t get a person to understand something when their [paycheck, prestige, reputation] depends on their not understanding it.

Well, you’re writing a book about PCT research, so your “reputation” could be affected if your statements or entire papers are proven to be wrong. You would be very resistant to that just because you want to keep your reputation, not because of any real certainty in the claims about the power law or behavioral illusions.

102

Hi Sorry for delays, I have been (and will be) busy with teaching in workdays and with grandchildren in weekends. And the discussion has continued but, however, I will send this message which I started many days ago.

RK: No, you will travel the same curve, but faster. [EP: about driving a car with steering wheel in a stable position.]

EP: Good, just that I started to suspect in my previous message.

RK: Yes, in principle direction and speed can vary independently of each other. But most processes tend to produce a correlation of high speed with low curvature, for various reasons, some of which I described in my previous message, and none of which have to do with a power law relationship being either planned or a controlled variable.

EP: That is very interesting. (But is it really “most” or “many”?)

[EP: About the equation A = D1∕3C2∕3]:

RK: For a fixed D, mathematically you have a power law relating A and C. But this is just a mathematical fact that says nothing about the mechanism, nor about whether D does or does not vary.

RM: This is precisely the point we made in M & S 2017.

EP: Really? I had understood that this point – that for fixed (stable) D you have a power law relating A and C – was made by Pollick & Sapiro 1997? “Constant affine velocity predicts the 1/3 power law” as they said it in their title.

RK: If D is not known to be fixed, there need be no power law between A and C. The “classical” power law of A being proportional to C2/3 is mathematically equivalent to the claim that D remains constant.

EP: So, could the “classical” research question of so called power law research perhaps be reformulated: Why D is constant sometimes and why it is not constant in other times?

EP: I have also understood that the point made in M & S 2017 was that independently of the variation of D you have a power law relating A and C – with a “true value” of 1/3 or 2/3, right? If so, then according to Richard’s dictum above it is simply wrong. (RK: “If D is not known to be fixed, there need be no power law between A and C.”)

EP: D is an “affine velocity” and it can be calculated from A and C. It thus contains the A velocity. Now if you add D as a predictor to a regression analysis predicting A, you certainly get a high correlations in every case because you are actually demonstrating the correlation between velocity and velocity, right?

EP: You call it Omitted Variable Bias if there is NOT D as a predictor. I think adding D as a predictor could be called Doubled Variable Bias – or manipulation of the results against good scientific practice.

EP: For me it seems that you are here following a principle: “Theory first! Never mind the data”. Strange, isn’t it?

Eetu

103

RM: This is a great question. The answer is related to the fact that correlation does not imply causality because a “third variable” may be at least partially responsible for the observed correlation. In this case we are dealing with a mathematical rather than a correlational relationship between variables. We know from the math that

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

or in linear form:

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

RM: This equation describes the true, mathematical relationship between measures of A, C and D that are calculated from an actual trajectory.

RM: When we do a regression that looks only for the relationship between measures of C and A you are ignoring the known contribution of measures of D – the third variable – to the variance of the measures of A in that trajectory. So taking the empirical relationship between C and A as a reflection of the actual relationship between C and A is like taking the empirical relationship between yearly sales of organic food ($OF) and the yearly number of individuals diagnosed with autism (AD) at face value. There is a strong linear relationship between $OF and AD but this relationship depends on a third variable – the year in which $OF and AD were measured. Both $OF and AD increase over the years. When this third variable is taken into account in a regression that includes the third variable as a predictor, the apparent relationship between $OF and AD almost completely disappears.

RM: In the case of the relationship between A and C, the observed relationship between the measured values of these variables for any particular trajectory will depend on the unobserved third variable, D. When this third variable, D, is included in the regression there is always exactly a 2/3 power relationship between A and C and a 1/3 power relationship between A and D.

RM: So the fact that A = C^2/3 * D^1/3 for all curved trajectories means that the observed relationship between A and C alone with depend on how D relates to C in each of those trajectories. That is, the nature of the relationship between A and C depends on characteristics of the trajectory itself and not on how the trajectory was produced.

104

The variable D is the magnitude of the cross product of the velocity and the acceleration vectors:

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Geometrically, D is the area of the parallelogram closed between the two vectors. It is completely determined by the acceleration and velocity vectors, it cannot be an independent predictor.

RM . That is, the nature of the relationship between A and C depends on characteristics of the trajectory itself and not on how the trajectory was produced.

This is correct, but does not follow from your analysis.

RM: In the case of the relationship between A and C, the observed relationship between the measured values of these variables for any particular trajectory will depend on the unobserved third variable, D. When this third variable, D, is included in the regression there is always exactly a 2/3 power relationship between A and C and a 1/3 power relationship between A and D.

How do you calculate D independently of curvature and velocity?

What evidence would you consider as proof that your thesis about D is not correct? Maybe accurately predicting V from C and regression offset and exponent, without including D in the analyisis?

105

Hi Adam

RM: Those who made or were convinced by arguments like Adam’s weren’t going to do the kind of research that I (and Bill) claim to be the only right way to study the behavior of living control systems anyway.

AM: I provided modeling studies, along with source code for all models, and empirical data from human subjects. That is a pretty solid base.

RM: Maybe I missed it but I didn’t see any control models tested. But I shouldn’t have said what I said above. I once had a talk with Bill when he was visiting me back in the mid 1990s and I said I believed that a person who will remain unnamed was apparently never going to “get” PCT and start doing PCT based research. Bill scolded me for saying that because he viewed such a belief as a reference for that result: he said I was controlling for the guy never getting it. I didn’t agree with Bill because I felt like I really wanted the guy to get it and was just disappointed; he was a very smart guy; as smart as you.

RM: But now I think that, whether Bill was right or not, there is no percentage in believing that what you want to happen, won’t. Bill was the eternal optimist; like Winston Churchill I think Bill just didn’t see any use in being anything else. Of course, it turns out that I was right; the guy never did get PCT - at least, not in a way that would lead him to do PCT research. I hope it wasn’t by belief that made that happen.

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 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. My apologies for believing otherwise.

AM: It’s really sad to read that nonsense. How did you manage to work in PCT for 40 years and not learn quasi-static analysis? And the relationships between variables in the loop? And what is the difference between a function and a relationship?

RM: My only excuse for my continued nonsense and my failure to understand all those things that you say I should understand is that I never heard a complaint about them from Bill. In fact, he seemed to like my work…a lot. Not that he never criticized my work. But when he did, I learned from the criticism and went on to do right.

AM: Well, you’re writing a book about PCT research, so your “reputation” could be affected if your statements or entire papers are proven to be wrong. You would be very resistant to that just because you want to keep your reputation, not because of any real certainty in the claims about the power law or behavioral illusions.

RM: I have no “reputation” that I care to protect. I just have a record of discussion, research and publication that is out there for people to take or leave. Some people have taken it and become PCT researchers themselves, some people have taken it the wrong way, so to speak, and most people have left it. I’m thankful for the small number of people in that first group.

RM: When things I’ve said or published turn out to be wrong – that is, when I can see that I have, indeed, erred – I try to learn from my mistakes. But I don’t think I’m wrong about the power law/behavioral illusion. I know that almost everyone who has paid any attention at all to that discussion thinks I’m wrong. Indeed, I’m pretty sure that most people involved in PCT think that I did some OK work on PCT many years ago but that I no longer understand PCT very well at all. I date the onset of my PCT dementia to May 24, 2013, the day Bob Dylan turn 72 (oh, and the day Bill Powers died).

RM: My main reason for continuing these discussions, writing books and doing research based on PCT is to try to get people to do research based on an understanding of organisms as perceptual control systems. I believe the science of purposeful behavior doesn’t have a chance of going “mainstream” until there is a considerable empirical foundation on which to build that science. So that is why I keep at it – not to preserve reputation but to try to get more people doing PCT research.

Best

Rick

106

Hi Adam:

AM: How do you calculate D independently of curvature and velocity?

RM: I don’t understand what you mean? I just calculate it from the first and second derivatives of X and Y.

RM: Maybe what you mean is that D is not independent of R since R = V^3/D. But that’s no problem for the regression analysis. The predictor variables in the regression equation don’t need to be independent (uncorrelated); indeed, the multiple regression analysis takes the correlation between all predictor variables into account when solving for the B weights and intercept constant.

AM: What evidence would you consider as proof that your thesis about D is not correct? Maybe accurately predicting V from C and regression offset and exponent, without including D in the analysis?

RM: Yes, I actually did that. Here’s the output of a spreadsheet calculation that shows this:

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RM: The column called “Expected Beta” contains the calculation of the power exponent for a regression of R on V (cell containing .23) and C on A (cell containing .77). These are predictions of the power exponent before the regression is done. It is a prediction of what Beta will be based only on a measure of the variance of the curvature (var (R) or var (C )) and the covariance between curvature and D (cov (R,D) or cov (C,D)) for the trajectory under study (the squiggle in the upper left).

RM: The Expected Beta for the power relationship between R and V is calculated as

Beta = 1/3+1/3*cov(R,D)/var( R)

And the Expected Beta for the power relationship between C and A is calculated as

Beta = 2/3 + 1/3*cov(C,D)/var(C )

RM: Note that the Expected power exponents, Betas, calculated from just measures of curvature and D correspond exactly to the power exponents found when curvature (R or C) is regressed on speed (V or A).

RM: Also notice that when D is included in the regression analysis the beta weights for curvature are exactly what is expected from the “power law”: .33 for R regressed on V and .67 for C regressed on A. And notice also that the R^2 in these cases is 1.0. The data fits the mathematical model exactly! The “power law” is clearly a statistical artifact of the mathematical relationship between measures of speed and curvature. To quote from Powers (1978) :

BP: The nightmare of any experimenter is to realize too late that his results were forced by his experimental design and do not actually pertain to behavior. This nightmare has a good chance of becoming a reality for a number of behavioral scientists.

Best

Rick

107

AM: What evidence would you consider as proof that your thesis about D is not correct? Maybe accurately predicting V from C and regression offset and exponent, without including D in the analysis?

RM: Yes, I actually did that.

I asked: what evidence would you consider as proof that your thesis about D is NOT correct? That is, that the power law is not a statistical artifact, resulting form omitting the third predictor variable D. A scientific theory or a hypothesis needs to be refutable in principle.

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?
If not, what other test can be used?

108

RM: My only excuse for my continued nonsense and my failure to understand all those things that you say I should understand is that I never heard a complaint about them from Bill. In fact, he seemed to like my work…a lot. Not that he never criticized my work. But when he did, I learned from the criticism and went on to do right.

He corrected you on the issue of causality in the behavioral illusion, quite recently, and I’ve posted the quote in the previous behavioral illusion thread. Your first reaction was quite reasonable, along the lines of “it is a difficult subject, we all have areas to learn”. Great. Then two post later, you start all that causal function nonsense.

As far as I saw in the archives, he also corrected your steady state analysis multiple times, sometimes directly related to the behavioral illusion. And yet, you did not learn from those corrections, and you proceeded to say things like the feedback quantity is not a real entity, or something along those lines.

I hope that is bad memory and not lying from your side.

RM: Maybe I missed it but I didn’t see any control models tested.

All the arguments I made in this thread and the previous one about the behavioral illusion are based on models of the control loop, written in python and with source code linked in the post, as well as plots of variables coming directly from those models. And steady-state analysis repeating the steps of the 1978 paper, with some renaming of variables

About the power law, I’ve posted some behavioral data, and also a test of a position control model in the same situation - and its failure to reproduce human behavior in tracking a target that is moving on an elliptical path. It produces a path that is much larger than that of the target (for high speed), it doesn’t have the same lead and follow dynamics, etc.

109

RM: I don’t understand what you mean? I just calculate it from the first and second derivatives of X and Y.

Maybe what you mean is that D is not independent of R since R = V^3/D. But that’s no problem for the regression analysis. The predictor variables in the regression equation don’t need to be independent (uncorrelated);

They don’t need to be uncorrelated, but all the predictor variables do need to be independent in the sense that they can, in principle, vary without affecting other predictor variables. Do you agree with this, or should I look for textbook quotes or something?

110

Hi Adam

AM: What evidence would you consider as proof that your thesis about D is not correct? Maybe accurately predicting V from C and regression offset and exponent, without including D in the analysis?

RM: Yes, I actually did that.

AM: I asked: what evidence would you consider as proof that your thesis about D is NOT correct? That is, that the power law is not a statistical artifact, resulting form omitting the third predictor variable D. A scientific theory or a hypothesis needs to be refutable in principle.

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: 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.

AM: If not, what other test can be used?

RM: The only test that will test my “thesis” about D is to show that it is algebraically not true that log (V) = 1/3 * log( R) +1/3 * log (D).

Best

Rick

111

Hi again

RM: Maybe what you mean is that D is not independent of R since R = V^3/D. But that’s no problem for the regression analysis. The predictor variables in the regression equation don’t need to be independent (uncorrelated);

AM: They don’t need to be uncorrelated, but all the predictor variables do need to be independent in the sense that they can, in principle, vary without affecting other predictor variables. Do you agree with this, or should I look for textbook quotes or something?

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.

Best

Rick

112

HI Adam:

RM: My only excuse for my continued nonsense and my failure to understand all those things that you say I should understand is that I never heard a complaint about them from Bill. In fact, he seemed to like my work…a lot. Not that he never criticized my work. But when he did, I learned from the criticism and went on to do right.

AM: He corrected you on the issue of causality in the behavioral illusion, quite recently, and I’ve posted the quote in the previous behavioral illusion thread. Your first reaction was quite reasonable, along the lines of “it is a difficult subject, we all have areas to learn”. Great. Then two post later, you start all that causal function nonsense.

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.

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: About the power law, I’ve posted some behavioral data, and also a test of a position control model in the same situation - and its failure to reproduce human behavior in tracking a target that is moving on an elliptical path. It produces a path that is much larger than that of the target (for high speed), it doesn’t have the same lead and follow dynamics, etc.

RM: Well, I look forward to your paper describing your modeling efforts.

Best

Rick

113

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.

114

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?

115

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.

116

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? :wink:

Best

Rick

117

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

118

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.

image

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?

119

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.

120

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.

image

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.