Behavioral Illusions: The Basis of a Scientific Revolution

Fundamentals
141

AM: References for velocity in both models are set by the output of the position model.

RM: I believe that is true for Bill’s model and your copy of it but it can’t possibly be true for Viviani’s model, can it:

image ; the reference for

RM: The implicit reference for the difference between velocities is a constant 0.

RM: Is there something wrong with this model?

AM: It works only for squiggles

RM: In what way does it work for squiggles and not work for non-squiggles?

RM The only difference between my model and that of Viviani and Campadelli (other than that mine did compensatory rather than pursuit tracking) is that it controls position relative to a varying reference. I guess Alex thought that the varying reference was “cheating” in some way but I think it was on the right track, as is Viviani/Campadelli model.

AM: I can think of several objections for the varying reference proposal.

AM: (1) A reference is an output of a higher-level control system. If it is not specified how and why the reference forms, that is not a solution but simply transferring the problem to a different place. The problem is still what are the controlled variables at this level that creates a varying reference for the position or distance control level.

RM: Well, it doesn’t seem like doing it merited the level scorn directed at my efforts to model “power law” behavior. And the model with varying reference does show that the changing reference state of the cursor is a controlled variable and that the movement can follow a power law while the actions that produced it don’t. And Bill has suggested ways to recover a changing reference signal from data (see See Powers (1989) Quantitative Measurement of Volition in W. Hershberger (Ed), Volitional Action, North Holland, pp. 315-334. Using this method you could get an estimate of the reference variations over time and use that estimate as the reference input to the model.

AM: (2) It does not work. For the speeds where the power law appears consistently, the position control system does not follow the reference - it creates much bigger shapes than the reference shape, just like tracking fast targets. A quick and dirty solution is to have a smaller reference ellipse, and this works somewhat, but there is again the problem of matching subject speed or rhythm. This can be solved, but still, the solution does not work for shapes other than ellipses.

RM: But if it works at slower speeds at least it’s a start. But it’s your project so do as you see fit (as if you need my permission;-)

Best

Rick

142

RM: I believe that is true for Bill’s model and your copy of it but it can’t possibly be true for Viviani’s model, can it:

Sure it can. Bill’s model can be written in one line as
F = Kp * Kv * C_T_distance - Kv * velocity_of_the_cursor

Viviani’s model is:
A = alpha * C_T_distance + beta * C_T_velocity_difference.
If we put pos_output = vel_reference = alpha * C_T_cistance, then
A = vel_reference + beta * (C_T_velocity_difference)

And of course, beta can be negative.

The controlled variables are the cursor-target difference in position (distance), with the implicit reference of 0, and the cursor-target difference in velocity, with the implicit reference of position_output. Interestingly, in the velocity control loop, the gain beta is an input gain, not an output gain like VelGain, and that makes the tuning of velocity and position control loops independent, like in proportional-derivative control.

RM: In what way does it work for squiggles and not work for non-squiggles?

It works in the sense that it behaves similarly to human subjects in the same tasks of following targets along non-predictable squiggly paths, and it does not work for non-squiggles or “predictable” movements or patterned movements, because it does not behave very similar to human subjects in the same task.

Also in slow movements, if the target is moving along a known path, like an ellipse, subjects will often go in front of the target, while the model, if it is simply following the target, will always stay behind it. This what they call the lead-follow dynamics.

RM: And the model with varying reference does show that the changing reference state of the cursor is a controlled variable and that the movement can follow a power law while the actions that produced it don’t.

What is the controlled variable? Does the reference already contain a correlation between speed and curvature? Actions are the same as the movement in all Bill’s models that I know of, and the speed-curvature power law is found in actions=movements. If the actions are forces, why isn’t there a calculation of accelerations, then an integration between accelerations and velocity, and another between velocity and position?

Though, in a general sense, the position control model is a good start because it has been proven to model human behavior in many experiments with random-moving targets. If we take that model as the starting point, along with the idea of the hierarchy, then, of course, the reference for the position is changing as the output from a higher level. That is almost an apriori start from the “PCT perspective as I see it”.

The question is then, what sort of variables can be controlled so that their output creates a varying reference signal for position control, that further results in actions that follow the power law.

If the reference is set by the modeler, and it already contains the power law, that doesn’t seem like a very general solution. For each movement, say, different speeds of drawing an ellipse, or different shapes, like numbers or letters, the modeler would need to set up a new reference signal that results in a movement similar to the movement of human subjects. Really, the modeler is doing all the controlling when he is setting up the reference signal, and when he is not involved, the loop of the model is open.

143

Hi Adam

RM: Again, I want to publicly admit that I was wrong to say that the power law is a statistical artifact. It is not and I apologize for saying so. My only excuse is that I was seduced into thinking so because of the fact that the coefficient of curvature in the mathematical relationships between measures of curvature (R, C) and speed (V, A) of movement are 1/3 and 2/3, exactly the values that had been proposed for the power law. I still think it’s a weird coincidence that the observed power law values are close to those values but your demonstration convinced me that the usual single variable regression of log curvature on log speed give an accurate measure of the beta value of the actual power relationship between curvature and speed. I still think that the observed power law is an irrelevant side-effect of controlling for the production of curved movement and is, therefore, an example of a behavioral illusion. But it’s not because the power law itself is a statistical artifact.

RM: I believe that is true for Bill’s model and your copy of it but it can’t possibly be true for Viviani’s model, can it:

AM: Sure it can. Bill’s model can be written in one line as
F = Kp * Kv * C_T_distance - Kv * velocity_of_the_cursor

AM: Viviani’s model is:
A = alpha * C_T_distance + beta * C_T_velocity_difference.
If we put pos_output = vel_reference = alpha * C_T_cistance, then
A = vel_reference + beta * (C_T_velocity_difference)

RM: The problem with this for me is that it conceals the architecture required to implement this model. You need two hierarchically related control systems to implement Bill’s model because C T distance in Bill’s model is actually the difference between the perception of C and the reference for C determined by the system itself. The resulting C T distance is then an error signal that becomes the reference for the lower level velocity control system.

RM: In the Viviani system you don’t need a higher level system if C T distance is considered to be the perceived error that becomes the reference for the velocity control system. This would make the Viviani system a hybrid S-R control system with C T distance being the stimulus that causes the reference to the lower level velocity control system as the response. And the velocity control system would be the control system component of this S-R /control system hybrid. This could easily be shown to the the wrong model by simply asking the subject to keep the distance between C and T at some value other than 0. To account for this you would have to have a separate control system controlling C T distance.

RM: In what way does it work for squiggles and not work for non-squiggles?

AM: It works in the sense that it behaves similarly to human subjects in the same tasks of following targets along non-predictable squiggly paths, and it does not work for non-squiggles or “predictable” movements or patterned movements, because it does not behave very similar to human subjects in the same task.

RM: This is a well known phenomenon. Conventional psychologists would solve it by putting prediction into the model; PCT would solve it by assuming that the subject is controlling a higher level perception, which is the observed pattern of movement.

RM: And the model with varying reference does show that the changing reference state of the cursor is a controlled variable and that the movement can follow a power law while the actions that produced it don’t.

AM: What is the controlled variable?

RM: One is the position of the cursor and the higher order one is probably the pattern of movement being produced.

AM: Does the reference already contain a correlation between speed and curvature?

RM: Yes, because there is a correlation between speed and curvature for all curved movements.

AM: Actions are the same as the movement in all Bill’s models that I know of, and the speed-curvature power law is found in actions=movements. If the actions are forces, why isn’t there a calculation of accelerations, then an integration between accelerations and velocity, and another between velocity and position?

RM: This is not really necessary when modeling things like tracking a cursor on a computer screen. The dynamics are absorbed into the output function. I think it would be necessary (and enlightening) if the tracking were done with a joystick whose resistance to applied force could be varied. Then you could presumably show control of joystick position by varying the resistance of the stick (a disturbance to joystick position) and at the same time show control of cursor position by varying a computer generated disturbance to cursor position. What you should be able you could show (and model) is two nested levels of control happening simultaneously. I don’t know if anyone has done this but I bet, with your skill, you could easily do that in your lab.

AM: Though, in a general sense, the position control model is a good start because it has been proven to model human behavior in many experiments with random-moving targets. If we take that model as the starting point, along with the idea of the hierarchy, then, of course, the reference for the position is changing as the output from a higher level. That is almost an apriori start from the “PCT perspective as I see it”.

RM: Completely agree!

AM: The question is then, what sort of variables can be controlled so that their output creates a varying reference signal for position control, that further results in actions that follow the power law.

AM: If the reference is set by the modeler, and it already contains the power law, that doesn’t seem like a very general solution.

RM: But what if that’s what the subject is doing – varying their reference for position in a way that produces trajectories that sometimes come close to a power law? I think the goal should be developing models that match the behavior of the subject, whether the subject’s behavior matches a power law or not.

AM: For each movement, say, different speeds of drawing an ellipse, or different shapes, like numbers or letters, the modeler would need to set up a new reference signal that results in a movement similar to the movement of human subjects. Really, the modeler is doing all the controlling when he is setting up the reference signal, and when he is not involved, the loop of the model is open.

RM: I know how you feel. We want our models to fit the data without putting the data itself into the model. And this is a particular problem for PCT because the controlled results we see often depend on people varying their references in a way that corresponds to the results we see. But I think there are ways to get around this. One is by doing experiments like the ones done by Viviani and Stucchi, where the subject can control a variable without having to produce the outputs that are typically used to control the result. So have the subject control a movement by pressing keys that affect the parameters of the observed movement rather than by moving a finger. Another way is to have the subject produce the result in the usual way but in the face of measurable disturbances and see if the model deals with them as expected if it were controlling a particular higher level variable.

RM: There are ways to do this; at least I trust that there are. Otherwise, PCT is untestable, a possibility up with which I will not put.

Best

Rick

144

Hi Rick,

RM: Again, I want to publicly admit that I was wrong to say that the power law is a statistical artifact. It is not and I apologize for saying so.

EP: Fine, that’s great. I appreciate that. But…

RM: My only excuse is that I was seduced into thinking so because of the fact that the coefficient of curvature in the mathematical relationships between measures of curvature (R, C) and speed (V, A) of movement are 1/3 and 2/3, exactly the values that had been proposed for the power law. I still think it’s a weird coincidence that the observed power law values are close to those values but your demonstration convinced me that the usual single variable regression of log curvature on log speed give an accurate measure of the beta value of the actual power relationship between curvature and speed. I still think that the observed power law is an irrelevant side-effect of controlling for the production of curved movement and is, therefore, an example of a behavioral illusion. But it’s not because the power law itself is a statistical artifact.

EP: …for me it seems like there could still be a problematic belief in the background which perhaps initially caused your erroneous opinion. Do you really still mean that there exists a mathematical relationship of 1/3 and 2/3 between the measures of curvature (R, C) and speed (V, A) of a movement?

EP: There does not exist any general mathematical relationship between curvature and speed – and neither between the measures of them. They are in principle mathematically independent of each other. (Richard Kennaway affirmed this.)

EP: You probably think so because there exists a certain kind of mathematical relationship between the members of the triad: curvature, speed and affine speed? But this relationship is produced by introducing the third member, affine speed, as a kind of a cross product of the two other members.

EP: Think of any independent variables x and y, for example the dimensions of a coordinate system. You cannot calculate or infer one from the other. But if you know a third variable z which happens to be the product of them (x * y), then there is a mathematical relationship and you can always calculate the third variable from the two others.

z = x * y is equivalent with x = z / y

EP: Now it would be reasonable to test for the correlation between x and y using for example linear regression. There can be correlations by chance or by some causal relationships. But it is NOT reasonable to test multiple correlation from x AND z to y because there is that mathematical relationship, which causes full 100% correlation between them.

EP: I is a little bit similar case than if you were interested in dependence between people’s sex and marriage and you added a third variable, bachelorhood (= unmarried male), as a predictor to a multiple correlation analysis and then claimed there must be a mathematical relationship between sex and marriage because you got a really high correlations in your analysis.

EP: I hope Adam, Richard or some other mathematically oriented person would tell whether I am right or wrong here.

145

RM: because of the fact that the coefficient of curvature in the mathematical relationships between measures of curvature (R, C) and speed (V, A) of movement are 1/3 and 2/3, exactly the values that had been proposed for the power law.

See Eetu’s post. Speed and curvature are not mathematically related. The coefficients of 1/3 and 2/3 are only there in the formula relating speed, curvature and D. If you invent another variable Q that you can calculate from A and C, with different coefficients, then multiple regression will give you those coefficients back, just as gives them back with D. The mistake was to use multiple regression with this data.

RM: I still think that the observed power law is an irrelevant side-effect of controlling for the production of curved movement and is, therefore, an example of a behavioral illusion.

Irrelevant - if that is a personal judgment, great, no objections. If it means irrelevant from a scientific perspective, I disagree. The job of a scientist is to find and explain stable, repeatable phenomena, so it is relevant to the extent it is reliably found in human movement.

A side effect - that is my hypothesis too, but it is still unproven. First, controlled variables need to be found, and “production of curved movement” is not a controlled variable, and even if it was, the power law is not consistently found in all curved movements, but only in relatively fast ones.

therefore, an example of a behavioral illusion - Nope. side effects are not behavioral illusions.

RM: The problem with this for me is that it conceals the architecture required to implement this model. You need two hierarchically related control systems to implement Bill’s model because C T distance in Bill’s model is actually the difference between the perception of C and the reference for C determined by the system itself. The resulting C T distance is then an error signal that becomes the reference for the lower level velocity control system.

Bold, no, in Bill’s model the C_T distance is the difference between the cursor position and the target position. e3 := SepRef - (Pos - Target). Then e3 is multiplied by PosGain, and then it is a reference for the velocity control.

Exactly the same thing is happening in Viviani’s model. The output of the position control system is the reference for the velocity control. The only difference is the definition of velocity - in one case it is the rate of change of position of the cursor, in the other case, it is the rate of change of the distance between cursor and target.

AM: Does the reference already contain a correlation between speed and curvature?
RM: Yes, because there is a correlation between speed and curvature for all curved movements.

No. For one, if the speed is constant, there is no correlation. If you had a reference composed of two orthogonal sine waves, you would get an ellipse with the 2/3 exponent power law.

RM: This is not really necessary when modeling things like tracking a cursor on a computer screen. The dynamics are absorbed into the output function.

Sure, if you choose the cursor position on the screen to be the controlled variable. The issue was where you would expect to see the slowing down in curved parts. If cursor movements are the output, the behavior of the system, then that is the variable you look into. That is the variable to be compared with the behavior of human subjects, not the disturbance + cursor position variable.

You were making an unwarranted distinction between movements and actions.

AM: If the reference is set by the modeler, and it already contains the power law, that doesn’t seem like a very general solution.

RM: But what if that’s what the subject is doing – varying their reference for position in a way that produces trajectories that sometimes come close to a power law?

AM: Then the modeler needs to find a control system that produces the varying reference for the position control system. First, the controlled variable, or variables, for that higher level, then all the other parts of the system, like the output function.

146

| amatic
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RM: because of the fact that the coefficient of curvature in the mathematical relationships between measures of curvature (R, C) and speed (V, A) of movement are 1/3 and 2/3, exactly the values that had been proposed for the power law.

AM: See Eetu’s post. Speed and curvature are not mathematically related. The coefficients of 1/3 and 2/3 are only there in the formula relating speed, curvature and D. If you invent another variable Q that you can calculate from A and C, with different coefficients, then multiple regression will give you those coefficients back, just as gives them back with D. The mistake was to use multiple regression with this data.

RM: This is a question for both you and Eetu: Why do you find it so disturbing that I would (probably mistakenly) think that the coefficients (1/3 and 2/3) in an easily derived mathematical formula relating measures of curvature to measures of speed might have something to do with the finding of a 1/3 or 2/3 “power law” relationship between these variables?

Best

Rick

147

It is not a problem that you would think it is an unusual coincidence that some coefficients match. There are several papers exploring exactly the same coincidence and showing that the exact 2/3 coefficient between A and C is present in trajectories with constant D, or D1/3 aka affine velocity. If D is constant, then its cube root is also constant.

If A = D1/3 C2/3, and D is constant, then A = kC2/3., which is exactly the power law for fast ellipses (k=D1/3)

The problem or the “disturbing” part of your post is your use of the term “mathematical relationship between measures of speed and curvature”, the part I put in bold to emphasize. There is no such thing.

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| amatic
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AM: It is not a problem that you would think it is an unusual coincidence that some coefficients match…

AM: The problem or the “disturbing” part of your post is your use of the term “mathematical relationship between measures of speed and curvature”, the part I put in bold to emphasize. There is no such thing.

RM: And why is it important that I recognize that there is no such thing?

149

That is a strange question. One - because the statement is not correct. Incorrect premises lead to incorrect conclusions. You use that incorrect argument later to explain the power law in the reference, and doesn’t fit the story.

And two, you can recognize or not recognize whatever you want. Why do you ask specifically about the mathematical relationship between speed and curvature? Do you think that is a correct statement?

150

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AM: That is a strange question. One - because the statement is not correct. Incorrect premises lead to incorrect conclusions.

RM: What is the incorrect conclusion to which it led?

151

Because:

152

Strange, Discourse deleted rest of the message. Perhaps because of formatting. Let’s try again.

Because:

  1. You claimed that there is a mathematical relationship between curvature and speed just because there is a triad relationship between curvature, speed, and affine velocity. This is the same as if you claimed that there is mathematical relationship between x and y because z = x * y and thus x = z * 1/y. And this is almost the same if you claimed that there is a mathematical relationship between sex and marriage because all bachelors are male.

  2. You tried to prove the existence of that claimed mathematical relationship by adding the affine velocity to multiple regression analysis as a predictor which is almost the same as searching correlation between bachelorhood and sex.

Sorry to say this, but these are such elemental mathematical, logical, and statistical mistakes that they can make it difficult to take seriously anything you say about science. And it can also cast a shadow of doubt and ridiculousness over the whole reputation of PCT. That is why it disturbs me so strongly.

I think the coefficients as such play a minor or no role.

Eetu

153

RM: What is the incorrect conclusion to which it led?

It could lead to all sorts of confusion and incorrect conclusions. For one, if it was true that “speed and curvature are mathematically related”, that would mean you could calculate speed from curvature, or curvature from speed, for any trajectory. This is not true.

Here is one example from few post up:

AM: Does the reference already contain a correlation between speed and curvature?
RM: Yes, because there is a correlation between speed and curvature for all curved movements.

There isn’t a correlation between speed and curvature for all curved movements, neither empirically, nor mathematically. If the reference you used for the elliptical movement already contains the power law - if the reference slows down in the curved parts, then this is a problem for the explanation of how the power law is created.

154

RM: My question was: Why do you find it so disturbing that I would (probably mistakenly) think that the coefficients (1/3 and 2/3) in an easily derived mathematical formula relating measures of curvature to measures of speed might have something to do with the finding of a 1/3 or 2/3 “power law” relationship between these variables?

RM: Your answer is apparently “because the “easily derived” formulas that show a 1/3 and 2/3 power relationship between curvature and speed don’t really show that and the fact that I think they do shows that I am too stupid to be believed about anything else scientific”. This was also the reason Martin Taylor rejected my PCT analysis of the power law in the original Marken and Shaffer (2017) paper.

RM: I guess there’s not much I can do about it. I think both your points 1 and 2 (as well as Martin’s reply to our paper) are wrong. But that’s not really relevant to what I am trying to get at with my question so I’ll answer my own question when I reply to Adam.

Best

Rick

155

Hi Adam

RM: I had spent the last couple days composing a reply to you and Eetu and it seems to have disappeared. And this one just got posted before I finished. So I will just quickly finish it and try to get some work done.

RM: What is the incorrect conclusion to which it led?

AM: Here is one example from few post up:

AM: Does the reference already contain a correlation between speed and curvature?
RM: Yes, because there is a correlation between speed and curvature for all curved movements.
AM: There isn’t a correlation between speed and curvature for all curved movements, neither empirically, nor mathematically.

RM: I think you must mean that there isn’t a power law correlation between speed and curvature for all curved movements. I think there is a correlation between speed and curvature for all curved movements. But I think it’s been shown that there is something close to a power law relationship between speed and curvature for most curved trajectories. Indeed, I just tried to produce a non-power law trajectory by speeding through the steep curves and slowing down through the shallow ones and the power coefficient was .34. (and .66)

RM: Maybe you could study why people have such difficulty making non-power law curves.

RM: I would also suggest a variant of the Viviani/Stucchi experiment to see what people might be controlling when they make elliptical movements. I would have a spot moving in an elliptical trajectory on the screen. The subject would use a slider to affect just the speed of movement (measured as tangential or angular velocity, whichever you think works best) during the movement without affecting curvature.

RM: The speed would also be affected by a slowly and smoothly varying disturbance. So if the subject did nothing the speed of the spot would be varying as the spot moved along the elliptical path. The subject is to move the slider so that the spot appears to be moving at constant velocity. You could use this data to see if there is evidence that speed is being controlled or whether what is being controlled is some relationship between speed and curvature.

Best

Rick

156

If you are getting these posts by email that last email does not contain the complete post. Here it is, just in case.

Hi Adam

RM: I had spent the last couple days composing a reply to you and Eetu and it seems to have disappeared. And this one just got posted before I finished. So I will just quickly finish it and try to get some work done.

RM: What is the incorrect conclusion to which it led?

AM: Here is one example from few post up:

AM: Does the reference already contain a correlation between speed and curvature?
RM: Yes, because there is a correlation between speed and curvature for all curved movements.
AM: There isn’t a correlation between speed and curvature for all curved movements, neither empirically, nor mathematically.

RM: I think you must mean that there isn’t a power law correlation between speed and curvature for all curved movements. I think there is a correlation between speed and curvature for all curved movements. But I think it’s been shown that there is something close to a power law relationship between speed and curvature for most curved trajectories. Indeed, I just tried to produce a non-power law trajectory by speeding through the steep curves and slowing down through the shallow ones and the power coefficient was .34. (and .66)

RM: Maybe you could study why people have such difficulty making non-power law curves.

RM: I would also suggest a variant of the Viviani/Stucchi experiment to see what people might be controlling when they make elliptical movements. I would have a spot moving in an elliptical trajectory on the screen. The subject would use a slider to affect just the speed of movement (measured as tangential or angular velocity, whichever you think works best) during the movement without affecting curvature.

RM: The speed would also be affected by a slowly and smoothly varying disturbance. So if the subject did nothing the speed of the spot would be varying as the spot moved along the elliptical path. The subject is to move the slider so that the spot appears to be moving at constant velocity. You could use this data to see if there is evidence that speed is being controlled or whether what is being controlled is some relationship between speed and curvature.

Best

Rick

157

RM: Indeed, I just tried to produce a non-power law trajectory by speeding through the steep curves and slowing down through the shallow ones and the power coefficient was .34. (and .66)

Try going very slowly at constant speed. If there is noise, smooth the trajectory, but just a bit so it doesn’t change the shape of the curve. A common cutoff for a lowpass filter is from 5-10 Hz.

RM: I would also suggest a variant of the Viviani/Stucchi …

Why do you keep suggesting those half-thought-out experiments to me? If you think it is worthwhile, do it yourself. I’m not asking you to suggest anything. I don’t trust you can make a good suggestion when I don’t see you even understand the problem. You certainly keep saying strange things about the mathematics of movement. You also say strange things about behavior of control loops.

RM: Maybe you could study why people have such difficulty making non-power law curves.

That is a good observation! Maybe you are starting to understand the whole problem of the power law.

158

Hi Adam

RM: I would also suggest a variant of the Viviani/Stucchi …

AM: Why do you keep suggesting those half-thought-out experiments to me?

RM: I’m suggesting ways you might approach understanding the power law from a PCT perspective.

RM: Maybe you could study why people have such difficulty making non-power law curves.

AM: That is a good observation! Maybe you are starting to understand the whole problem of the power law.

RM: I don’t think so. I think the problem of the power law is that it is a blind alley, just like invariant velocity profiles. But I suggest that you continue this conversation with Richard Kennaway, if he will be kind enough to get involved. It’s clearly way over my head. Richard and I did a talk last year in Manchester about behavioral illusions and here are the slides Richard produced for the power law segment:

Kennaway1
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Kennaway2
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RM: If you can get Richard to discuss this with I think you might have better luck with him than with me. He’s a real mathematician, after all.

Best

Rick

159

RM: I’m suggesting ways you might approach understanding the power law from a PCT perspective

I didn’t ask for any suggestions, but worse than giving unsolicited advice is that you keep giving bad advice, experiments you did not think through and did not even try on a single person.

RM: But I suggest that you continue this conversation with Richard Kennaway,

I did not ask for help. Any discussion is welcome, of course. You could also ask him to explain to you why he said that curvature and speed are not mathematically related.

My posts were mostly criticisms of your two papers on “the” or “a” behavioral illusion, and your approach to the study of human movement on the example of the power law. I think I mostly kept my posts civil, sorry if I crossed the line sometimes, but the arguments were logicaly sound, you could see control loop diagrams, empirical data, models, simulations, the whole shabang.

160

Hi Adam

RM: I’m suggesting ways you might approach understanding the power law from a PCT perspective

AM: I didn’t ask for any suggestions, but worse than giving unsolicited advice is that you keep giving bad advice, experiments you did not think through and did not even try on a single person.

RM: I did think them through but I think I understand why you would reject them.

RM: But I suggest that you continue this conversation with Richard Kennaway,

AM: I did not ask for help.

RM: I know. Yes, you are pretty sure of yourself.

AM: Any discussion is welcome, of course. You could also ask him to explain to you why he said that curvature and speed are not mathematically related.

RM: If he said that I would be interested in why he did because I’m pretty sure that that would have been wrong. Perhaps he means that there is no constraint on how curvature and speed are related in a trajectory. Which is true. But there is certainly a mathematical relationship between those variables and that mathematical relationship will show up in a regression analysis when the “third variable” is included in the analysis.

AM: My posts were mostly criticisms of your two papers on “the” or “a” behavioral illusion, and your approach to the study of human movement on the example of the power law. I think I mostly kept my posts civil, sorry if I crossed the line sometimes, but the arguments were logicaly sound, you could see control loop diagrams, empirical data, models, simulations, the whole shabang.

RM: I found your posts far less than civil but I know it’s difficult to know how what you say will affect someone else. But at least one of your insults – calling me a bozo – was reassuring because it was directed at both me and Martin Taylor. It seemed to me that when I was talking to you I was hearing Martin Taylor’s arguments (augmented with some actual modeling and data) about the nature of behavioral illusions and against my arguments about the PCT explanation of the power law. So it was nice to see that you apparently came up with these arguments on your own.

RM: And as far as the substance of your arguments, they may have been logically sound and you certainly presented “…control loop diagrams, empirical data, models, simulations, the whole shabang”. But I didn’t find them very convincing. They seemed to reflect a surprising lack of understanding of the fundamentals of perceptual control theory from someone who has clearly gone through all the control theory literature and even replicated all of Powers’ demos. Though I guess I shouldn’t have been surprised because this is not unprecedented; there have been others who did this – and were very skilled at the mathematics of control, as well – and still came away still seeing behavior from the conventional point of view.

RM: One last thing. I asked Richard Kennaway to make a diagram of the Viviani control model that you see as consistent with PCT. It came out exactly as I expected.

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RM: The dashed line is, of course, the border between system and environment. The difference between this and a PCT model of the same phenomenon (tracking a target, T, with a cursor, C, is small and probably trivial from your point of view. But it is the essence of the difference between PCT and conventional applications of control theory to behavior. And from a psychological perspective, where the goal of research is to understand the nature of the system under study, it is the whole enchilada. And it is why I find your approach to understanding movement unconvincing.

Best

Rick