# Bogus mathematics, (was Re: L'état de P CT, c'est moi (was ...))

[From Rick Marken (2018.08.17.04:48)]

[From Erling Jorgensen (2018.08.15 2345 EDT)]

RM: The fact that the power law is an illusion can be determined without any

knowledge of statistics.

EJ:Â I like how you insert the word âfactâ? about what is indeed a proposal

still being contested.

RM: I said that this fact “can be determined”. That is a conditional phrase. And it is correct; you can (if you want to) determine that the power law is an illusion without any of the statistical analysis that I provided. The statistical analysis just shows why something close to a power relationship between curvature and velocity is so regularly found using regression analysis.

RM:Â Once you know that, you know that the power law is an unintended side

effect of this controlling.Â

EJ:Â Yes, once you have presumed it as a fact, then you have constructed a

âknowingâ that it must be an unintended side effect.

RM: I did not presume anything as a fact. The “that” in my statement above referred to this statement: . “All you have to know is that movement – the changing positions of the wrist as the arm moves, for example – is a controlled result of action”. In other words, what I said was: once you know that the changing position of the movement is a controlled variable you know that the power law is an unintended side effect of controlling this variable".Â

Â

EJ:Â But there is another way to go about it, the way Bill demonstrated with

his Little Man V2 model.Â He showed that a very simplified control model could

generate what appear to be sophisticated calculations, as a by-product, or

side effect, of the working of the model.

RM: As did we. Marken and Shaffer (2017) found that the movement trajectories produced by the object interception model – movement trajectories that were not created with the aim of producing a power law relationship between velocity and curvature --Â conformed to a power law that was equivalent to that found byÂ Zago et al. (2016) for the movements of fruit flies.

EJ:Â If you read the rest of my post and my previous one, you will know I am

aiming at that very thing.Â I believe I am doing it with a little less hubris.

RM: Hubris?!?!Â

Â

EJ: However, following Bill, any alternate model will still need to generate

behavior that is akin to power-law data, without specifically controlling for

that outcome.

RM: As I noted, we (Dennis and I) have already done that with our control model of object interception. AndÂ power law researchers have already done it with their open-loop model of movement (see Gribble and Ostry, 1996, Table I). It’s easy to use a model to produce movements that follow the power law; it’s a little more interesting toÂ produce such movements in a disturbance-prone environment, as we did in Marken & Shaffer (2018, Figure 1). But it would certainly be great if you would develop a model of movement control and see if it produces a power law as a side effect. (I’ll tell you in advance that it will always produce something close to a power law, the degree of deviation from the power law being proportional to the covariance between the curvature and affine velocity of the movement trajectory).

Best

Rick

···

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

RM: (if you want to) determine that the power law is an illusion without any of the statistical analysis that I provided. The statistical analysis just shows why something close to a power relationship between curvature and velocity is so regularly found using regression analysis.

AM:

Maoz et al are using pure noise to show that if there is a lot of noise in the data, the power law could be a statistical artifact. They also add noise to generated data to do the same thing. Their conclusion is that noise is important to take into account. If you generate noise-free data, or if you smooth it correctly, any correlation between speed and velocity is just that - a correlation between speed and velocity.

It is a side effect of controlling something, definitely, but side effects are not behavioral illusions.

[From Bill Powers (930920.1130 MDT)]

BP: The “behavioral illusion” that I have talked about is the relationship between the disturbing variable and the output action of the control system, (…) The appearance is that a distal stimulus acts generally on the senses of the organism to cause a response, a change in output action. This is an illusion when it omits an actual controlled variable. (…)

There’s no a priori way of proving that a behavioral illusion is in effect. The only way to prove that there is an illusion is to demonstrate that there is in fact a controlled variable being affected as above. If there is no controlled variable found, then there is no illusion and the S-R interpretation may be correct. PCT doesn’t automatically prove that no S-R connections exist.

AM: Also, in your Steven’s Power Law text, you note:

RM: The illusion is that an observed relationship between
environmental inputs and behavioral outputs reflect characteristics of the system itself when it
actually reflects properties of the feedback connection between the system’s output and a
controlled perceptual input.

AM:

So… curvature is not a stimulus, velocity is not a response… No response illusion.

No controlled variable found - no way to prove the illusion.

And again - I’m not talking about the power law being a side effect of controlling something, everyone already knows that - it is just that the power law is not the behavioral illusion. Velocity profiles in point-to-point movements are not an example of the behavioral illusion, they are a side effect of controlling position, and they do reflect properties of the behaving system. For example, if the output gain was higher, or delays shorter, the profiles would look differently. Why would you ever want to extend the definition to side effect of controlling something?

You’ve claimed in a few places that trajectory is a controlled variable, and also you’ve claimed that position is a controlled variable, or something like “time varying position”, then you suggest maybe it is affine velocity, but then again you go back to claiming it is position. If you even want to begin claiming that something is a behavioral illusion (the response illusion), you’d first need to find a controlled variable.

[Rick Marken 2018-08-18_17:07:20]Â

AM: Maoz et al are using pure noise to show that if there is a lot of noise in the data, the power law could be a statistical artifact. They also add noise to generated data to do the same thing. Their conclusion is that noise is important to take into account. If you generate noise-free data, or if you smooth it correctly, any correlation between speed and velocity is just that - a correlation between speed and velocity.Â

RM: I see their results somewhat differently. The first result they describe is for “100 Monte-Carlo simulations, each drawing a time series of 1,000,000 normally distributed points”, which are random 2-D movement trajectories. The important finding here is that the average beta value for these random trajectories (using the usual regression analysis that omits the affine velocity variable)Â is -.28! This value, as they note, is pretty close toÂ -.33, which is their mathematically derived coefficient (beta) relating curvature to velocity. What this finding means is that randomly produced trajectories tend to have power coefficients relating curvature to velocity that are close to -.33. I found the same thing when I was first working on this power law, sometimes generating random trajectories and computing beta.Â I found that randomly generated trajectories tend to have a power coefficient relating curvature (measured as R rather than 1/R) to velocity that is close to .33.

RM: This finding should have led Maoz et alÂ to see that the power law is a statistical artifact. Apparently they did see the implications of this finding but then pulled back from that horrifying brink and saidÂ “We
do not suggest that the power-law…is
a bogus phenomenon…Yet
our results do suggest caution when carrying out experiments that either aim to
verify the power-law or assume its existence”. So they are assuming that the power law is a real phenomenon that can be detected in human movement data, but only when steps are taken to ensure that there isn’t too much “noise” in the data. The “noise”, however, is the correlation between affine velocity and curvature that obscures the power law when the affine velocity variable is omitted from the regression analysis. Thus, “noise” is defined under the assumption that the power law is a real phenomenon that corresponds to the power relationshipÂ between velocity and curvature in the equation that defines velocity as a power function of both curvature and affine velocity. So, as we said in the our rebutta,l their “noise” analysis is based on the assumption that their theory of the cause of the power law is true; that the power law is being generated by processes in the actor and not as an artifact of omitting the affine velocity variable from the regression analysisÂ used to determine whether a movement corresponds to the power law.Â

AM: It is a side effect of controlling something, definitely, but side effects are not behavioral illusions.

RM: That is true. The power law is a side effect of control that results from the mathematical relationship between the measurements of velocity and curvature. Side-effects are behavioral illusions only when they are not seen as side effects. That is, they are a behavioral illusion when they are seen as observations that tell us something about how the observed behavior was produced.

[From Bill Powers (930920.1130 MDT)]

BP: The “behavioral illusion” that I have talked about is the relationship between the disturbing variable and the output action of the control system, (…)

RM: Yes, this implies that there are other behavioral illusions that he has not talked about. And, indeed, BillÂ talks about other behavioral illusions – such as the illusion of selection by consequences --Â in other places.

BP: The appearance is that a distal stimulus acts generally on the senses of the organism to cause a response, a change in output action. This is an illusion when it omits an actual controlled variable. (…)

RM: Exactly! It’s the omission of the controlled variable that results in all the behavioral illusions that occur when observing the behavior of a living control system.Â

Â

AM: There’s no a priori way of proving that a behavioral illusion is in effect. The only way to prove that there is an illusion is to demonstrate that there is in fact a controlled variable being affected as above. If there is no controlled variable found, then there is no illusion and the S-R interpretation may be correct. PCT doesn’t automatically prove that no S-R connections exist.Â

RM: The main thing PCT says about S-R relationships is “If you want to understand the behavior of living systems stop looking for S-R relationships and start looking for controlled variables”.

Â

AM: So… curvature is not a stimulus, velocity is not a response… No response illusion. No controlled variable found - no way to prove the illusion.Â

RM: You don’t “prove” or “disprove” behavioral illusions. The power law is not a result of the same factors as the S-R illusion described in Powers (1978). But both are illusions because they are both side effects of control that are taken to reveal something about the mechanisms that produce the observed behavior.Â

AM: And again - I’m not talking about the power law being a side effect of controlling something, everyone already knows that - it is just that the power law is not the behavioral illusion. Velocity profiles in point-to-point movements are not an example of the behavioral illusion, they are a side effect of controlling position, and they do reflect properties of the behaving system. For example, if the output gain was higher, or delays shorter, the profiles would look differently. Why would you ever want to extend the definition to side effect of controlling something?

AM: You’ve claimed in a few places that trajectory is a controlled variable, and also you’ve claimed that position is a controlled variable, or something like “time varying position”, then you suggest maybe it is affine velocity, but then again you go back to claiming it is position. If you even want to begin claiming that something is a behavioral illusion (the response illusion), you’d first need to find a controlled variable.

Â RM: Just go ahead and do whatever work you think is good. I’d love to see it when you have something to show.

Best

Rick

···

On Sat, Aug 18, 2018 at 12:22 AM Adam Matic csgnet@lists.illinois.edu wrote:

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

RM: This finding should have led Maoz et al to see that the power law is a statistical artifact. Apparently they did see the implications of this finding but then pulled back from that horrifying brink and said “We
do not suggest that the power-law…is
a bogus phenomenon…Yet
our results do suggest caution when carrying out experiments that either aim to
verify the power-law or assume its existence”. So they are assuming that the power law is a real phenomenon that can be detected in human movement data,

AM:

They say: “We do not suggest that the power-law, which stems from analysis of human data, is a
bogus phenomenon, resulting only from measurement noise.” They are accepting that a point can move around a curve with different velocities, in such a regular way that the correlation between speed and velocity follows a power law with some exponent. They are testing, for example, constant tangential speed trajectories around an elliptical path, with a beta of 0. These trajectories were generated to conform to a speed-curvature power law. Those are, by definition, power law trajectories.

The noise they speak about was adding a shift in x and y to each point (illustrated in one of their plots).

They are pointing to possible artifacts in empirical data. What you’re saying sounds like “a correlation between speed and curvature doesn’t reflect the real correlation between speed and curvature even when we are analyzing computer generated data, where instantaneous speed was explicitly set to be equal to curvature raised to a some power.”

RM: The power law is a side effect of control that results from the mathematical relationship between the measurements of velocity and curvature. Side-effects are behavioral illusions only when they are not seen as side effects. That is, they are a behavioral illusion when they are seen as observations that tell us something about how the observed behavior was produced.

AM:

Side effect of control of what? “Side effect of control” is quite vague as in not telling much. “Observations that tell us something” is also dangerously vague, but in the opposite way, because many times “something” can be seen even in side effects.

Side effects are, in my view, just unintended effects. People don’t really intend to move with bell shaped velocity profiles, and don’t intent to slow down in curves. They intend to move their hand from position A to position B, or draw a shape. Under specific conditions, the side effect just happens because the system is organised in some specific way. Under some conditions, people can’t help but move in those ways, even if they intend otherwise.

If some side effects are found to hold consistently, then they can tell us something about the behaving system - another reason I don’t agree with expanding the definition of the behavioral illusion to side effects. They still don’t tell us what is the controlled variable, but for example, they can point to things that are not controlled variables - like trajectory (in both cases).

BP: There’s no a priori way of proving that a behavioral illusion is in effect. The only way to prove that there is an illusion is to demonstrate that there is in fact a controlled variable being affected as above. If there is no controlled variable found, then there is no illusion and the S-R interpretation may be correct. PCT doesn’t automatically prove that no S-R connections exist.

RM: You don’t “prove” or “disprove” behavioral illusions.

AM: That was Bill’s quote on proving illusions up there, not mine. I’m still not in favor of calling other illusions “the behavioral illusion”, I’ll reserve that for what you call the SR illusion.

Best,

[Rick Marken 2018-08-21_14:53:13]

I think the only chance we have of making any progress on this is if you would tell me what your model of voluntary movement is. I have proposed a model of movement as being the result of temporal variations in the actor’s reference specification for the position of the movement. No one seemed to like that model, based on the idea that the model was a put up job since the variations in the reference signal that I used in the model themselves followed a power law. But IÂ believe that my model is the correct PCT model of movement. I think we show this in Figure 1 of Marken and Shaffer (2018). In that Figure we show that the varying position of the cursor passes the test for being a controlled variable – controlled in a temporally varying reference state. So the model that explains this behavior has to be a control model with a reference signal that varies in approximately the same way as does the cursor position. I estimated the model’s reference signal variation by simply taking it to be the average of the observed cursor movements at each instant in the trajectory. But there are more sophisticated ways to do this (see Powers (1989) Measurement of volition, in Hershberger (Ed) Volitional Action: Conation and Control, pp. 315-333). But the important thing is that this model accounts for the movement data in Figure 1 of Marken and Shaffer (2018). What is your model of movement?Â

RM: This finding should have led Maoz et alÂ to see that the power law is a statistical artifact…

AM: They say: “We do not suggest that the power-law, which stems from analysis of human data, is a
bogus phenomenon, resulting only from measurement noise.”

Â

RM Yes, but they showed that the power law is, indeed, bogus in the sense that it is a statistical artifact of regressing curvature on velocity while omitting affine velocity. If they included affine velocity in a multiple regression analysis they would fin d that the power law holds for all trajectories because theÂ mathematical relationship between curvature and velocity is a power law.

AM: The noise they speak about was adding a shift in x and y to each point (illustrated in one of their plots).Â

RM: This is “noise” only under the assumption that the true result of producing the movement is either the power law or it is not; the power law is treated as the “signal” that is to be detected in this noise. But regression analysis that is used to detect the power law can’t detect it because the power law is a mathematical property of all curved trajectories. The only thing that obscures this mathematical property of all trajectories is the correlation between curvature and the variable omitted from the regression, affine velocity.Â

Â

AM: Side effect of control of what?

RM The instantaneous position of the moved entity – such as the cursor in Figure 1 of Marken and Shaffer (2018).Â

Â

AM: Side effects are, in my view, just unintended effects.

Â RM: Yes, they are unintended side effects if intentional (control) behavior, such as the relationship between disturbance and output when a variable is being kept under control.Â

AM: If some side effects are found to hold consistently, then they can tell us something about the behaving system

RM: That is not true of the side-effects of control. Side effects, such as the relationship between disturbance and output, which is seen as a relationship between S and R when the controlled variable is unseen or ignored, tell us next to nothing about the behaving system. That was the point of Powers (1978) and why the subtitle of that paper was “Some spadework at the foundations of scientific psychology.” Bill shows that the relationship between disturbance and output – the apparentÂ causal or S-R relationship that is the basis of scientific research on psychology – actually tells you very little if anything abort the organism,Â

BP: There’s no a priori way of proving that a behavioral illusion is in effect. The only way to prove that there is an illusion is to demonstrate that there is in fact a controlled variable being affected as above. If there is no controlled variable found, then there is no illusion and the S-R interpretation may be correct. PCT doesn’t automatically prove that no S-R connections exist.Â

RM: You don’t “prove” or “disprove” behavioral illusions.Â

AM: That was Bill’s quote on proving illusions up there, not mine. I’m still not in favor of calling other illusions “the behavioral illusion”, I’ll reserve that for what you call the SR illusion.

RM: Yes, my statement there is wrong. You can “prove” or “disprove” behavioral illusions. You can disprove the S-R illusion – that is, show that S really causes R – by showing that there is no controlled variable that is affected by both S and R. You can disprove the power law illusion in the same way – by showing that there is no controlled variable involved. But for the power law the controlled variable that has to be shown to not exist would not be one affected by both curvature and velocity (since there is none) because, as you’ve said, the power law is not taken to be a causal relationship between these variables. The power law illusion is that the power law reflects physiological and/or physical “constraints” involved in the production of movement. But once you see that the instantaneous position of the moved entity is a controlled variable you see that there is no way for these constraints to consistently affect curvature and velocity so that the result is a power law.

BestÂ

Rick

···

On Sun, Aug 19, 2018 at 7:11 PM Adam Matic csgnet@lists.illinois.edu wrote:

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

RM: I think the only chance we have of making any progress on this is if you would tell me what your model of voluntary movement is.

AM:

The problem I’m currently trying to figure out is more narrow than that, there could be many possible controlled variables in all forms of voluntary movement. Specifically, I’m interested in the mechanisms for drawing shapes, tracing lines and tracking predictable targets (as opposed to tracing random-moving targets). Don’t have much to say yet, though, still doing experiments and models.

RM: I have proposed a model of movement as being the result of temporal variations in the actor’s reference specification for the position of the movement. No one seemed to like that model, based on the idea that the model was a put up job since the variations in the reference signal that I used in the model themselves followed a power law. But I believe that my model is the correct PCT model of movement. I think we show this in Figure 1 of Marken and Shaffer (2018). In that Figure we show that the varying position of the cursor passes the test for being a controlled variable – controlled in a temporally varying reference state. So the model that explains this behavior has to be a control model with a reference signal that varies in approximately the same way as does the cursor position. I estimated the model’s reference signal variation by simply taking it to be the average of the observed cursor movements at each instant in the trajectory. But there are more sophisticated ways to do this (see Powers (1989) Measurement of volition, in Hershberger (Ed) Volitional Action: Conation and Control, pp. 315-333). But the important thing is that this model accounts for the movement data in Figure 1 of Marken and Shaffer (2018). What is your model of movement?

AM:

The critique of the already power-law trajectory in the reference stands (you can also call you model an open-loop trajectory control, because trajectory is not controlled in closed loop, only position is) - but here is different angle. There is no problem with position being a controlled variable in tracking random-moving targets, even slow moving targets that go around known paths. On the other hand, It is quite easy to show that position control fails when you have a faster moving target - make the reference position change faster in going around an ellipse. Humans can track the target quite successfully, while the model cannot. So, that would mean humans don’t just track position, and there are other variables involved.

RM Yes, but they showed that the power law is, indeed, bogus in the sense that it is a statistical artifact of regressing curvature on velocity while omitting affine velocity. If they included affine velocity in a multiple regression analysis they would fin d that the power law holds for all trajectories because the mathematical relationship between curvature and velocity is a power law.

RM: This is “noise” only under the assumption that the true result of producing the movement is either the power law or it is not; the power law is treated as the “signal” that is to be detected in this noise. But regression analysis that is used to detect the power law can’t detect it because the power law is a mathematical property of all curved trajectories. The only thing that obscures this mathematical property of all trajectories is the correlation between curvature and the variable omitted from the regression, affine velocity.

AM:

The point of the paper is that the addition of noise gives you the false impression that curvature and velocity are related, while in the noiseless trajectory, they may not be. The power law is not treated as the signal to be detected in the noise, but rather as a statistical artifact that comes out because of noise, while the real trajectory might not have a consistent relationship between curvature and velocity. Did you notice that Tamar Flash is a coauthor on both the noise paper and the reappraisal paper. She didn’t quite agree with your interpretation of her math. Did you notice all those non-power law trajectories in the reappraisal paper (planets and all that)?

But forgetting the calculations - do you deny that purely mathematical points can have trajectories in which there is no relationship between curvature and velocity?

AM: Side effects are, in my view, just unintended effects.

RM: Yes, they are unintended side effects if intentional (control) behavior, such as the relationship between disturbance and output when a variable is being kept under control.

AM:

Yeah, you can say it like that. You could also say that the correlation between disturbance and output is kinda the main effect of control, not a side effect. If control is good, and the controlled variable stable, that is exactly what you find.

AM: If some side effects are found to hold consistently, then they can tell us something about the behaving system

RM: That is not true of the side-effects of control. Side effects, such as the relationship between disturbance and output, which is seen as a relationship between S and R when the controlled variable is unseen or ignored, tell us next to nothing about the behaving system. That was the point of Powers (1978) and why the subtitle of that paper was “Some spadework at the foundations of scientific psychology.” Bill shows that the relationship between disturbance and output – the apparent causal or S-R relationship that is the basis of scientific research on psychology – actually tells you very little if anything abort the organism,

AM:

Absolutely agreed on the SR illusion. An apparent behavioral law is just telling you something about the experimental conditions and the feedback path. SR correlations (disturbance-output) don’t tell you nothing about the organism. Other types of side effects can tell you something.

RM: But once you see that the instantaneous position of the moved entity is a controlled variable you see that there is no way for these constraints to consistently affect curvature and velocity so that the result is a power law.

AM:

Well, try running the model faster by giving it a faster reference. Look how much better is your control than that of the position-control model.

Also, look at time-series data of angular velocity and curvature, or tangential speed and radius of curvature in published papers. You don’t see an obvious correlation?

Best,

[Rick Marken 2018-08-23_18:12:40]

RM: I think the only chance we have of making any progress on this is if you would tell me what your model of voluntary movement is.

AM:

The problem I’m currently trying to figure out is more narrow than that, there could be many possible controlled variables in all forms of voluntary movement. Specifically, I’m interested in the mechanisms for drawing shapes, tracing lines and tracking predictable targets (as opposed to tracing random-moving targets). Don’t have much to say yet, though, still doing experiments and models.

RM: Sounds great. I look forward to seeing what you get.

RM: I have proposed a model of movement as being the result of temporal variations in the actor’s reference specification for the position of the movement…

AM: The critique of the already power-law trajectory in the reference stands (you can also call you model an open-loop trajectory control, because trajectory is not controlled in closed loop, only position is) - but here is different angle. There is no problem with position being a controlled variable in tracking random-moving targets, even slow moving targets that go around known paths. On the other hand, It is quite easy to show that position control fails when you have a faster moving target - make the reference position change faster in going around an ellipse. Humans can track the target quite successfully, while the model cannot. So, that would mean humans don’t just track position, and there are other variables involved.Â

RM: Yes, they are higher level variables that are controlled by means of controlling position. What you are finding is what isÂ found in the Demonstration of Fourth Order control in Powers (1960), which is reprinted on p. 31 of LCS I. The fourth order perception is a circular target movement in that demo; it’s equivalent to the elliptical target movement in your experiment. It looks like your research is moving in the right direction.

Â

RM: This is “noise” only under the assumption that the true result of producing the movement is either the power law or it is not; the power law is treated as the “signal” that is to be detected in this noise. But regression analysis that is used to detect the power law can’t detect it because the power law is a mathematical property of all curved trajectories. The only thing that obscures this mathematical property of all trajectories is the correlation between curvature and the variable omitted from the regression, affine velocity.Â

AM: The point of the paper is that the addition of noise gives you the false impression that curvature and velocity are related, while in the noiseless trajectory, they may not be.

RM: Yes, that’s what noise does; it leads to false alarms. They are still thinking of the regression analysis used to determine whether a trajectory follows the power law as a signal detection analysis, which assumes that the power law is a real characteristic of the movements produced by living systems. But it is not. The “noise” added byÂ Maoz et al simply produces a different trajectory but one where the correlation between curvature and affine velocity is still relatively low so that the “power coefficient” in an omitted variable regression (one that leaves out affine velocity) is still close to the power law value (-1/3 in their case).Â

AM: The power law is not treated as the signal to be detected in the noise, but rather as a statistical artifact that comes out because of noise, while the real trajectory might not have a consistent relationship between curvature and velocity.

RM: The idea is that there is a “real trajectory” (one not affected by noise) that may or may not follow the power law. This means that they are looking at this as a signal detection problem.Â

AM: Did you notice that Tamar Flash is a coauthor on both the noise paper and the reappraisal paper. She didn’t quite agree with your interpretation of her math.

Â RM: That’s fair. I didn’t agree with her interpretation of her math.

AM: Did you notice all those non-power law trajectories in the reappraisal paper (planets and all that)?

RM: Of course. Did you notice the non-power law trajectories in our rebuttal to the reappraisal paper.Â It turns out that you get both power law and non-power law trajectories produced by both living and non living systems. It is impossible to say whether a movement trajectory was produced by a living or non-living system based on whether or not the trajectory follows the power law. That’s because whether or not a trajectory follows the power law depends on characteristics of the trajectory itself and has nothing to do with how it was produced.Â This is a point we made in Marken & Shaffer (2017) by noting that the trajectories of both the pursuers (living control system) and helicopters (not living control systems) produce power law conforming trajectories.Â

AM: But forgetting the calculations - do you deny that purely mathematical points can have trajectories in which there is no relationship between curvature and velocity?

RM: Of course I don’t deny it. There are many trajectories that would result in no correlation between curvature and velocity. A perfect circular trajectory is one.

AM: Side effects are, in my view, just unintended effects.

Â RM: Yes, they are unintended side effects if intentional (control) behavior, such as the relationship between disturbance and output when a variable is being kept under control.Â

AM: Yeah, you can say it like that. You could also say that the correlation between disturbance and output is kinda the main effect of control, not a side effect. If control is good, and the controlled variable stable, that is exactly what you find.Â

RM:A correlation between disturbance and output is seen only when the disturbance is the only influence on the controlled variable exerted by the environment (no other disturbances are having an effect on the controlledÂ variable
), the output is the only influence on theÂ
controlled variableÂ exerted by the organism (there is no unobserved output that is affecting the controlled variable), the feedback function connecting output toÂ controlled variable
Â is constant, and theÂ
controlled variableÂ is being maintained in a constant reference state. Note that the only thing common to all of these requirements is the existence of a controlled variable. So I would say that the correlation between disturbance and output is a side-effect of the existence of a controlled variable; that is, it is a side effect of control. So understanding why we observe a correlation between disturbance and output requires that we know about the existence of a controlled variable. And once we know what the controlled variable is, all possible side effects of controlling this variable can be readily deduced.Â

Â

AM: Well, try running the model faster by giving it a faster reference. Look how much better is your control than that of the position-control model.Â

Â RM: I will look at this. My prediction is that this is true only when you are making a regular movement trajectory pattern (like an ellipse) but not when you are moving in a “random” or arbitrary squiggle movement trajectory pattern.Â

AM: Also, look at time-series data of angular velocity and curvature, or tangential speed and radius of curvature in published papers. You don’t see an obvious correlation?

RM: Of course I do. And the correlation gets close to being perfect as the trajectory approaches one, like an ellipse, where affine velocity is constant.Â

RM: Again, please keep me posted on the status of your research and relevant publications. I think you are doing things that could, indeed, advance the study of living control systems.Â

Best

Rick

Â

···

On Wed, Aug 22, 2018 at 2:20 PM Adam Matic csgnet@lists.illinois.edu wrote:

Best,

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

RM: Yes, that’s what noise does; it leads to false alarms. They are still thinking of the regression analysis used to determine whether a trajectory follows the power law as a signal detection analysis, which assumes that the power law is a real characteristic of the movements produced by living systems.

AM:

You’re mixing two things here - the existence of the power law in human movement and the existence of the power law in computer-generated trajectories. They don’t need to assume that generated trajectories follow the power law or not - those trajectories were generated to have a specific relationship between speed and velocity. Imagine a point along a curve and setting its velocity at each point equal to some multiple of curvature, or curvature to some exponent. There is no assuming, they generate it.

You should try some other formulas for curvature and velocity, line this osculating circle thing. To calculate curvature at each point, take the previous point and the next point, as if they form a triangle, then find r: https://math.stackexchange.com/questions/133638/how-does-this-equation-to-find-the-radius-from-3-points-actually-work

That will be the radius of curvature profile.

RM: But it is not. The “noise” added by Maoz et al simply produces a different trajectory but one where the correlation between curvature and affine velocity is still relatively low so that the “power coefficient” in an omitted variable regression (one that leaves out affine velocity) is still close to the power law value (-1/3 in their case).

AM:

The noise produces a “jittery” trajectory, as you can see in their illustration. If it there is any correlation, it is an artifact of the noise. And they are not doing omitted variable regression - affine velocity is dependent both on curvature and speed, and is cannot be an independent predictor. They have one formula like the one in your paper, but a completely different dataset and a completely different interpretation.

RM: That’s fair. I didn’t agree with her interpretation of her math.

AM: Oh, right. Sorry Mr. Fields medalist.

RM: Did you notice the non-power law trajectories in our rebuttal to the reappraisal paper. It turns out that you get both power law and non-power law trajectories produced by both living and non living systems. It is impossible to say whether a movement trajectory was produced by a living or non-living system based on whether or not the trajectory follows the power law.

AM:

If you look into the literature, you’ll find that no one ever claimed that it is exclusively produced by biological systems. In fact, many non-biological examples were given, like low pass filters, orthogonal sine waves, etc. Have you read anything before writing your paper?

RM: That’s because whether or not a trajectory follows the power law depends on characteristics of the trajectory itself and has nothing to do with how it was produced. This is a point we made in Marken & Shaffer (2017) by noting that the trajectories of both the pursuers (living control system) and helicopters (not living control systems) produce power law conforming trajectories.

AM:

I don’t see any meaning in the first sentence. Yes, you only look at the trajectory to see if curvature and velocity are correlated. You don’t need to know how it was produced to analyze it.

RM: So I would say that the correlation between disturbance and output is a side-effect of the existence of a controlled variable; that is, it is a side effect of control. So understanding why we observe a correlation between disturbance and output requires that we know about the existence of a controlled variable. And once we know what the controlled variable is, all possible side effects of controlling this variable can be readily deduced.

AM:

How about reaction time? The minimum time of reaction is set by the physical limits of the particular control system, like delays, integral lags, etc. That is a side effect of controlling a specific variable, that tells you a lot about a system. Could hint to the level in the hierarchy where the perception is happening, and such interesting things.

It cannot go lower than it does, because of the physical limitations of the system, so you consistently get the same minimum reaction time to the same stimuli. Same with speed and curvature in human movement. They have a consistent relationship under certain experimental conditions.

Best,

[Rick Marken 2018-08-26_15:19:21]

RM:…Â The “noise” added byÂ Maoz et al simply produces a different trajectory but one where the correlation between curvature and affine velocity is still relatively low so that the “power coefficient” in an omitted variable regression (one that leaves out affine velocity) is still close to the power law value (-1/3 in their case).Â

AM:Â The noise produces a “jittery” trajectory, as you can see in their illustration. If it there is any correlation, it is an artifact of the noise.

Â RM: There is some degree of correlation between curvature and affine velocity in all movement trajectories. This correlation varied somewhat over the normally distributed samples trajectories analyzed by Moaz et al. but was surprisingly consistent. When they added noise to non-power law trajectories – trajectories with power coefficients that were essentially zero rather than -1/3 – they found that you would get power law conforming trajectories – trajectories with power coefficients that are close to -1/3 – with very little added noise. This means that with very little noise added to a trajectory you can change the correlation between curvature and affine velocity from 1.0, which results in a power coefficient of 0, to 0.0, which results in a power coefficient of -1/3. These variations in the observed power coefficient are due only to variations in mathematical characteristicsÂ of the movement trajectories themselves. The characteristic of a trajectory that makes the difference in terms of whether or not it conforms to the power law is the correlation between curvature and affine velocity.Â

AM: And they are not doing omitted variable regression

RM: Actually, they did. Whenever you regress curvature on velocity, omitting affine velocity as a predictor, you are doing an omitted variable regression. In other words, power law researchers always do omitted variable regression to see if a trajectory conforms to the power law.Â

RM: What is more important is that Moaz et al did an omitted variable bias (OVB) analysis, just as we did (Marken and Shaffer, 2017). Their analysis, like ours, showed that a regression analysis that includes only curvature as a predictor of velocity, omitting affine velocity as a predictor, will find a power coefficient that deviates from the power law coefficient (-1/3, 1/3 or 2/3, depending on how curvature and velocity are measured) by an amount proportional to the correlation between curvature and affine velocity.Â

RM: The Moaz et al OVB analysis is given in their equation (6):

RM: This is equivalent to equation (12) in Marken and Shaffer (2017):

RM: TheÂ d (delta)Â in our equation (12) is equivalent toÂ x/3Â (x/3) in their equation (6), whereÂ xÂ (x) is equal toÂ

Covariance [log(curvature), log(affine velocity)]/ Variance [log (curvature)]

RM:Â bâ€™obsÂ (beta’.obs) in our equation (12) is equivalent toÂ bÂ (beta) in their equation (6). We call oursÂ Â bâ€™obsÂ (beta’.obs) because it is theÂ power coefficient --Â bÂ (beta) – that is observed when one does a regression of curvature on velocity while omitting affine velocity, which is the variable we called D and that they callÂ aÂ (alpha).Â Finally,Â btrueÂ (beta.true) in our equation (12) is equivalent to -1/3 in their equation (6). We called itÂ btrueÂ (beta.true) because it is the coefficient of curvature in the formula that gives the linear relationship between curvature and velocity. ForÂ Marken & Shaffer (2017)Â this formula was:Â

log (V) =1/3log (R)Â +1/3 log(D)Â

and for Moaz et al (using equivalentÂ names for the variables) it was

log (V) = -1/3log (1/R)Â +1/3 log(D)Â Â Â

RM: So in our OVB analysis,Â btrueÂ (beta.true) is 1/3 and in the Moaz et al OVB analysisÂ btrueÂ (beta.true) is -1/3.Â

RM: So Moaz et al found exactly what we found: When affine velocity is left out of the regression analysis that is used to determine whether or not a movement conforms to the power law, the power coefficient relating curvature to velocity that is found by this omitted variable regression will deviate from the power law value – the mathematically “true” value of -1/3, 1/3 or 2/3, depending on how velocity and curvature are measured – by an amount proportional to the covariance between curvature and affine velocity (per equation (6) in Moaz et al and equation (12) in Marken and Shaffer (2017).

RM: This is what I mean when I say that whether or not you find that a movement trajectory follows the power law depends on the nature of the trajectory itself and tells you nothing about how that trajectory was produced. Trajectories where the covariance between curvature and affine velocity is close to zero will be found to conform to the power law; trajectories where the covariance between curvature and affine velocity is high will be found to deviate from the power law by an amount proportional to the size of this covariance.

AM: - affine velocity is dependent both on curvature and speed, and is cannot be an independent predictor.

RM: Predictor variables do not have to be independent (uncorrelated with each other) in multiple regression analysis.Indeed, they virtually always dependent on each other to some extent.Â When you do a multiple regression analysis on movement trajectories using both curvature and affine velocity as predictors of velocity you find that the regression coefficients for these variable are exactly the ones in the mathematical equation relating velocity to curvature. That is, if you use V as the criterion variable, and 1/R and D as the two predictor variables in a multiple regression analysis, then the resultÂ will be a regression equation that corresponds exactly to the Moaz et al version of the equation relating curvature (1/R) and affine velocity (D) to velocity:

log (V) = -1/3log (1/R)Â +1/3 log(D)Â Â Â

RM: That is, the regression coefficients will be exactly -1/3 and 1/3 for 1/R and D, respectively, and R^2 will be 1.0.Â

Â

AM: They have one formula like the one in your paper, but a completely different dataset and a completely different interpretation.Â

RM: They got the same results with their dataset as we got with ours. And they gave a completely different interpretation of their results than we did because people don’t like finding out that they have been making a huge mistake in their research. This aversion to believing one is wrong is what kept the US in the war in Vietnam; it’s what keeps Trump supporters Trump, it’s why power law researchers were angry at my PCT interpretation of the power lawÂ and it’s why behavioral scientists in general don’t like PCT. And it’s all perfectly understandable in terms of PCT.

RM: That’s fair. I didn’t agree with her interpretation of her math.

AM: Oh, right. Sorry Mr. Fields medalist.Â

Â

RM: Did you notice the non-power law trajectories in our rebuttal to the reappraisal paper.Â It turns out that you get both power law and non-power law trajectories produced by both living and non living systems…

Â AM: If you look into the literature, you’ll find that no one ever claimed that it is exclusively produced by biological systems…

RM: I would think that that alone would suggest that the power law is not an interesting finding regarding the behavior of organismsÂ

RM: That’s because whether or not a trajectory follows the power law depends on characteristics of the trajectory itself and has nothing to do with how it was produced…

AM: I don’t see any meaning in the first sentence.

RM: What it means is what I showed in the OVB analysis above.Â

Â

RM: So I would say that the correlation between disturbance and output is a side-effect of the existence of a controlled variable; that is, it is a side effect of control. So understanding why we observe a correlation between disturbance and output requires that we know about the existence of a controlled variable. And once we know what the controlled variable is, all possible side effects of controlling this variable can be readily deduced.Â

AM: How about reaction time? The minimum time of reaction is set by the physical limits of the particular control system, like delays, integral lags, etc. That is a side effect of controlling a specific variable, that tells you a lot about a system. Could hint to the level in the hierarchy where the perception is happening, and such interesting things.Â

RM: Good point. If you want to call reaction time a side effect then it can certainly be an informative one. But I’m not sure I would call reaction time a side effect of control in the same way that the disturbance-output relationship is a side effect. Reaction time, in the form of transport lag and output slowing, is an intrinsic component of the control process; you have to put these timing variables into a control model to make it work. But you don’t put the disturbance-output transfer function into a control model to make it work; the relationship between disturbance and output is truly a side effect of the operation of a control system acting to keep a controlled variable in a reference state.

Best regards

Rick

···

On Sat, Aug 25, 2018 at 12:36 PM Adam Matic csgnet@lists.illinois.edu wrote:

It cannot go lower than it does, because of the physical limitations of the system, so you consistently get the same minimum reaction time to the same stimuli. Same with speed and curvature in human movement. They have a consistent relationship under certain experimental conditions.Â

Best,

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

RM: So Moaz et al found exactly what we found: When affine velocity is left out of the regression analysis that is used to determine whether or not a movement conforms to the power law, the power coefficient relating curvature to velocity that is found by this omitted variable regression will deviate from the power law value – the mathematically “true” value of -1/3, 1/3 or 2/3, depending on how velocity and curvature are measured – by an amount proportional to the covariance between curvature and affine velocity (per equation (6) in Moaz et al and equation (12) in Marken and Shaffer (2017).

(…)

AM:

Yeah, lots of nonsense, I could repeat to you the criticism of OVB from the reapraisal paper, or you can just read it again. Consider it pasted here.

RM: they gave a completely different interpretation of their results than we did because people don’t like finding out that they have been making a huge mistake in their research. This aversion to believing one is wrong is what kept the US in the war in Vietnam; it’s what keeps Trump supporters Trump, it’s why power law researchers were angry at my PCT interpretation of the power law and it’s why behavioral scientists in general don’t like PCT. And it’s all perfectly understandable in terms of PCT.

AM:

You could understand people’s behavior if you knew their goals and values, which you don’t know. You’re just saying “I’m right because no one likes being wrong”. And you’re forgetting all the people who do not have any skin in the power law game, who like PCT, and who still disagree with you. I suppose you assign motivations to them individually.

Your ideas of mathematical relationships are just bad math, not PCT. Your analysis did not reveal a controlled variable, and discovering a controlled variable is required to prove an SR illusion, and is just generally one of the main points of PCT research. So, you did not prove and SR-illusion. You’ve tried expanding the definition of the behavioral illusion to include claims that some side effects can say “something important” about the behaving system, but some side effects do tell you “something important”, so it is not a consistent definition (also, it is different from Bill’s definition).

You’re claiming that measures of velocity and curvature are mathematically related. These calculations are not original, made by “power law researchers”, they are hundreds of years old, going back at least to Newton - apparently, there is a formula for the radius of curvature in his Principia. There is no mention of this third variable necessary to find the real and true relationship between velocity and curvature up until 2017 and you and Shaffer. If that doesn’t get you a Nobel or a Fields, it will get people calling you a crank.

AM: If you look into the literature, you’ll find that no one ever claimed that it is exclusively produced by biological systems…

RM: I would think that that alone would suggest that the power law is not an interesting finding regarding the behavior of organisms

AM: Right, so you kinda skimmed over a couple of papers. Very sloppy and unscholarly.

RM: Good point. If you want to call reaction time a side effect then it can certainly be an informative one. But I’m not sure I would call reaction time a side effect of control in the same way that the disturbance-output relationship is a side effect. Reaction time, in the form of transport lag and output slowing, is an intrinsic component of the control process; you have to put these timing variables into a control model to make it work. But you don’t put the disturbance-output transfer function into a control model to make it work; the relationship between disturbance and output is truly a side effect of the operation of a control system acting to keep a controlled variable in a reference state.

AM:

That is exactly what I’m saying. The power law is a side effect of control very different from a disturbance-output relationship. Mainly because curvature and velocity are both properties of the output trajectory; neither is a stimulus, both are part of the response. Just like velocity profiles in Bill’s example, just like reaction time; or various frequency response profiles, etc, they can tell us something interesting and useful about the control systems in question.

Best,

[Rick Marken 2018-08-27_12:41:16]

RM: So Moaz et al found exactly what we found: When affine velocity is left out of the regression analysis that is used to determine whether or not a movement conforms to the power law, the power coefficient relating curvature to velocity that is found by this omitted variable regression will deviate from the power law value – the mathematically “true” value of -1/3, 1/3 or 2/3, depending on how velocity and curvature are measured – by an amount proportional to the covariance between curvature and affine velocity (per equation (6) in Moaz et al and equation (12) in Marken and Shaffer (2017).

(…)Â

AM:

Yeah, lots of nonsense, I could repeat to you the criticism of OVB from the reapraisal paper, or you can just read it again. Consider it pasted here.Â

Â RM: OK, carry on with your research. Again, I look forward to seeing what you find.Â

BestÂ

Rick

···

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery