# Behavioral Illusions: The Basis of a Scientific Revolution

As for your quote, you are mistaking what is irrelevant to the subject with what is irrelevant to the researcher. If you look in that comment on A&H, you will find ‘the invariants are interesting as a check on the model’.

I’ve asked where did you get the silly idea that side effects of control don’t reflect properties of the control system? The discussion of velocity profiles in A&H does not address this question. If you would put different gains to the arm, you would get different velocity profiles. Side effects, such as the Venus drawings, or the exact shape of the velocity profile, do reflect properties of the system that made them.

Any other sources?

Here are some velocity profiles. Two position control systems get a step disturbance at t=2s. The high gain control system has a Ko of 100, for low gain it is 20. The velocity profiles are both calculated as dqo / dt, velocity of output.

The exact shape of the velocity profile is not controlled, the system is only controlling position. The shape of the velocity profile is a side effect of position control and reflects the gain of the position control system, and really reflects many other properties of the system - such as the fact that it is a simulated system that can have instantaneous jumps in velocity, though small. A more realistic simulation system would have an object with mass being moved, maybe some friction etc, and then the velocity profile would resemble a bell. But this one does not simulate mass, and it is visible in the velocity profile.

To spell it out - the exact shape of the velocity profile is not relevant for the control system, it is a side effect of position control but it absolutely reflects system properties.

AM: Still back on square one.

RM: Not really. I think we are just talking past each other. This is all lovely work in this post. It shows that when you know the controlled variable you know what is most important about the behavior.

AM: A researcher has two drawings of Venus from the same rubber band experiment, made by two different subjects who had different gains for position control. No amount of looking at the drawing will show how it was made, if the researcher does not know the controlled variable. The exact pattern was irrelevant to the subjects, all they saw was a knot staying in one position because they moved their hand to keep it there.

RM: Yes. Exactly.

AM: The Venus drawings are definitely a side effect of control of position, and they are not identical. Two different subjects produced two different drawings. Why are there any differences? Because the drawing fidelity reflects subject’s gain.

RM: Probably. But it could also reflect differences in the feedback functions in the two cases. The lower fidelity drawing might have been drawn underwater, for example.

AM: The side effects of control of position are irrelevant to the subject, but are very relevant to the researcher. He can fit two models to two drawings of Venus, providing them with the same disturbance he gave to subjects, and he can find out the gains of the subjects for position control.

RM: It’s not really the side effects per se that are relevant to the researcher. The side effect of S’s behavior (controlling) in your example is the shape of the squiggle produced by variations in S’s output, qo. In this case E is not trying to evaluate the relative gain of 2 S’s by comparing how well the outputs of each S match a particular squiggle shape;. Rather, E is evaluating relative gain by looking at the correlation between variations in qo and variations in qd (or by measuring rms deviation between these two variables). The actual shape of the pattern traced out by qo and qd – the side effect of variations in qo and qd – is irrelevant.

AM: This was fun.

RM: It was indeed! You do such nice work!

AM: Maybe you can point to which variable is the side effect and which variable is not the side effect. I just see one qo per model.

RM: Again, it’s the squiggly shape traced out by variations in qo that is the side effect of S’s controlling; this side effect is irrelevant to both S and E. What is relevant to E in this analysis is the correlation between variations in qo and variations qd (or the rms deviation of qo from qd).

AM: As for your quote, you are mistaking what is irrelevant to the subject with what is irrelevant to the researcher. If you look in that comment on A&H, you will find ‘the invariants are interesting as a check on the model’.

RM: What Bill is saying here is that these invariants are interesting in the same way that the observed behavior of a real subject doing the tracking tasks in your demo are a check on your model of that behavior. Your model should (and will) produce the same squiggle in response to the squiggly disturbance as does the real subject. Indeed, I did exactly this analysis in the “Control Blindness” paper.

RM: Here is a shot of an S controlling the position of the knot in the rubber band game:

RM: It shows the trace of S’s variations in qo (upper left) that compensate for E’s variations in qd (lower right). Most observer’s of a video if this task didn’t notice that S was controlling the position of the knot. Many thought that S was tracing out a kangaroo. Others thought S was tracing out something else. The shape of S’s trace is a side-effect of S’s controlling the perceived position of the knot. Indeed, it’s clear that the pattern traced out by S’s movements are an irrelevant side effect of S’s controlling because what that pattern was was in the eye of the beholder.

RM: I converted the time course of these traces to numbers that could be used in a contorl model and fitted thel model to the data. The results are here:

RM: The correlation between the output of the model (red dots) and the actual movements made by S (green dots) was .98. Since the model fit the pattern traced out by S quite well you could say that the pattern – a side effect of controlling – provided an interesting check on the model. But it’s not really the pattern per se that provided the check on the model; it was the fit of the model’s variations in qo – variations that made the movement pattern as a side effect – to S’s variations in qo that was the actual check on the model.

AM: I’ve asked where did you get the silly idea that side effects of control don’t reflect properties of the control system? The discussion of velocity profiles in A&H does not address this question. If you would put different gains to the arm, you would get different velocity profiles. Side effects, such as the Venus drawings, or the exact shape of the velocity profile, do reflect properties of the system that made them.

RM: The invariant velocity profiles that Bill referred to as “an interesting check on the model” are a check on the model in the same way that S’s output traces are a check on the model of behavior in the rubber band demo. The check was to see whether the Little Man model doing the same tasks as the Ss in the A-H experiment would produce outputs (qo) that are the same as those produced by the real Ss. The only thing about the outputs that is known to us is that they could be converted to invariant velocity profiles. So the fact that the Little Man outputs could be converted to invariant velocity profiles was one way to check on whether the model fit the data – whether the model was controlling in the same way as the real subjects. This check was evidence that the model was controlling the same perceptual variables in the same way as the real subjects.

RM: It is in this sense that the invariant velocity profiles are an “interesting as a check on the model”. The fit of the model to those profiles tells you that the model is controlling the right perceptions in the right way. Of course, there would be some parameter fitting to get a best fit of model to data. And the parameter values that give you the best fit will certainly tell you something about how the controlling works. But what is most important is that in the case of the A-H data we found that a model controlling a hierarchy of certain perceptions will produce outputs that, as a side effect, result in invariant velocity profiles.

RM: The model was evaluated in terms of those side effects because that’s the only measures of performance that were available. We would have gotten the same result – a successful fit of model to data – if the model were fitted to the raw movement data – the data from which the invariant velocity profiles were derived.

RM: PCT modeling is all about finding the right perceptions to control – “right” in the sense that when the model controls these perceptions it behaves just like the real organism.

Best regards

Rick

RM: It’s not really the side effects per se that are relevant to the researcher.

Saying that “side effects are relevant” is maybe not the best choice of words. Saying that something is a side effect should be a conclusion, based on a model that explains the behavior where the main effect is keeping some cv stable, and side effects are other observed regularities.

The relevant thing for any scientist are regular patterns, laws, stable variables, invariances, etc. Some laws are just statistical tendencies and trends. Nothing much to do with them but look for other laws. On the other hand, some laws are fairly strong. They need to be explained.

After the model is found, it is not a behavioral illusion to say that side effects, or regularities that are not the controlled variable, reveal something about the mechanisms of behavior, because they usually do. I think that is one of the mistakes in your behavioral illusion papers from the PCT perspective.

AM: Still back on square one.
RM: Not really. I think we are just talking past each other.

I doesn’t seem like we are. It seems like you are twisting your definition of side effects, and REALLY trying to avoid saying you were wrong. Why is that? What exactly are trying to achieve by not admitting you were wrong?

Or do you still hold to your definition of “a” behavioral illusion as taking side effects to tell you something about the behaving mechanism when they do not?

That definition is wrong. For one, it doesn’t appear anywhere in PCT literature. The behavioral illusion is mistaking the environmental feedback function for the organism function. Perhaps a more general behavioral illusion is taking the lineal causation model as a model of organism behavior. Which would then possibly lead to the behavioral illusion.

And two, in all control systems, quantitative analysis of various side effects reveals something about the system. And how could it not - side effects, just like main effects, depend on the organisation and the properties of the system. You change gains, time constants, functions, etc, you get different behavior for every little change.

RM: It’s not really the side effects per se that are relevant to the researcher. The side effect of S’s behavior (controlling) in your example is the shape of the squiggle produced by variations in S’s output, qo. In this case E is not trying to evaluate the relative gain of 2 S’s by comparing how well the outputs of each S match a particular squiggle shape;. Rather, E is evaluating relative gain by looking at the correlation between variations in qo and variations in qd (or by measuring rms deviation between these two variables). The actual shape of the pattern traced out by qo and qd – the side effect of variations in qo and qd – is irrelevant.

In the picture above, the shape traced by M1 is different than the shape traced by M2. Just by looking at them, you can see there is a difference in the properties of M1 and M2 (if they were both doing the task in the same conditions, not one of the under water). By doing further analysis on qo and other variables, you can approximate gains, time constants, etc…

RM: Again, it’s the squiggly shape traced out by variations in qo that is the side effect of S’s controlling; this side effect is irrelevant to both S and E. What is relevant to E in this analysis is the correlation between variations in qo and variations qd (or the rms deviation of qo from qd).

The shape is a property of qo. An analysis of qo can be an analysis of the shapes traced by the models. You have changed your definition of the side effects from a variable in some model, which you called qo2, to a vague definition of “shape”; and the shape is still revealing properties of the system.

Can’t start a revolution based on wrong definitions.

RM: I am trying to achieve nothing. And, sure enough, nothing has become of nothing.

AM> Or do you still hold to your definition of “a” behavioral illusion as taking side effects to tell you something about the behaving mechanism when they do not?

RM: Yes,I still hold to that definition. I really like it.

AM That definition is wrong.

RM: OK, great. Clearly I don’t think it is, which is why this discussion is over as far as I’m concerned. It’s just like you go your way and I’ll go mine. And I hope that when we meet at the crossroads I’ll see that you have produced some great contributions to PCT. Or perhaps proved it wrong. I would happily sell my soul to the devil to see either of those things. But I’m pretty sure I’ll get neither.

Best

Rick

AM> Or do you still hold to your definition of “a” behavioral illusion as taking side effects to tell you something about the behaving mechanism when they do not ?

RM: Yes,I still hold to that definition. I really like it.

Why don’t you like your first definition? “The appearance of a causal relationship between behavioral variables when such a relationship does not, in fact, exist” (M&S 2017) ?

Do you think that one is wrong somehow? Or do you think it is correct? Why redefine it again in M&S 2018 if it is correct?

RM: OK, great. Clearly I don’t think it is, which is why this discussion is over as far as I’m concerned. It’s just like you go your way and I’ll go mine. And I hope that when we meet at the crossroads I’ll see that you have produced some great contributions to PCT. Or perhaps proved it wrong.

Nope. Not proving PCT wrong. Proving that you have written things incompatible with PCT in your last two papers. PCT is great, it is your papers that contain mistakes that you are not acknowledging nor correcting.

It is what it is. It seems to me you are just posing as some kind of infallible guru instead of engaging in scientific discussion. Not sure if I preffer Baba Marken or Rick Dass…

AM: Or do you still hold to your definition of “a” behavioral illusion as taking side effects to tell you something about the behaving mechanism when they do not ?

RM: Yes,I still hold to that definition. I really like it.

AM: Why don’t you like your first definition? “The appearance of a causal relationship between behavioral variables when such a relationship does not, in fact, exist” (M&S 2017) ?

RM: Because it only defines a special case of a behavioral illusion. I used it in the 2017 paper because one of the many claims about why my PCT explanation of the power law was wrong was because there was actually a causal relationship between curvature and velocity: increased curvature caused decreased velocity.

AM: Do you think that one is wrong somehow?

RM: Yes. In the context in which I said it in the 2017 paper I should have said that the behavioral illusion is the appearance of a causal relationship between behavioral variables when such a relationship is not, in fact, what you are seeing.

AM: Or do you think it is correct? Why redefine it again in M&S 2018 if it is correct?

RM: I don’t think it is correct but it is not wrong in an important way since the power law is not an example of the S-R illusion. Which is why I preferred referring to the power law illusion as “taking side effects [of control] to tell you something about the behaving mechanism when they do not”.

RM: I hope that when we meet at the crossroads I’ll see that you have produced some great contributions to PCT. Or perhaps proved it wrong.

AM: Nope. Not proving PCT wrong. Proving that you have written things incompatible with PCT in your last two papers.

RM: A very peculiar scientific goal but good luck.

AM: PCT is great, it is your papers that contain mistakes that you are not acknowledging nor correcting.

RM: I sure I have made other mistakes in the way I’ve said things. WHat whatever those mistakes were in the power law paper they don’t diminish the demonstrations, in both the 2017 and 2018 papers, that the power law (which isn’t even much of a law) is an irrelevant side effect of control.

AM: It is what it is. It seems to me you are just posing as some kind of infallible guru instead of engaging in scientific discussion. Not sure if I preffer Baba Marken or Rick Dass…

RM: I think you could spend your time more profitably by applying your considerable talents to doing something positive, such as developing your own model of the power law. Show the right way to do things rather than concentrating on all the wrong ways I’ve said things.

Just a suggestion.

Best

Rick

RM: You said::

AM: PCT is great, it is your papers that contain mistakes that you are not acknowledging nor correcting.

RM: I forgot to ask : Why do you think PCT is great? I haven’t seen you or anyone else in your lab use it?

Best

Rick

RM: I forgot to ask : Why do you think PCT is great?

There are many ways it is great. One way I’m looking recently is that PCT unites two big ideas about explanations of organism behavior - taking the perspective of the organism when doing research, trying to find what is relevant to the organism from its perspective; and the second idea of using control theory to make quantitative models of behavior.

RM: I think you could spend your time more profitably by applying your considerable talents to doing something positive, such as developing your own model of the power law. how the right way to do things rather than concentrating on all the wrong ways I’ve said things.

Yes, Master Marken, thank you for the suggestions of how I should spend my time.

My day job is doing research on how people move their hands when they trace figures and such, whole bunch of models already made, some presented in posters, none really good enough for publication, but getting there. At night, I like to do peer review, which in this case includes correcting nonsense you said about PCT and human behavior

Maybe more importantly - correcting mistakes in applying PCT in research of human behavior - like claiming to know what is a side effect before determining the controlled variable.

RM: Which is why I preferred referring to the power law illusion as “taking side effects [of control] to tell you something about the behaving mechanism when they do not”.

Why do you prefer that definition?

Side effects do depend on the behaving mechanisms. There is nothing else to depend on, but the properties of all the elements of the behaving loop, including the environment and the organism. Whatever you change in the loop, side effects will quantitatively change.

AM: At night, I like to do peer review, which in this case includes correcting nonsense you said about PCT and human behavior

RM: I guess I should be glad you’re doing that but somehow I don’t feel like you are doing it to help me out;-)

AM: Maybe more importantly - correcting mistakes in applying PCT in research of human behavior - like claiming to know what is a side effect before determining the controlled variable.

RM: Actually, I never claimed to know what a side effect is before determining the controlled variable. I think if you do your “correcting” more carefully you will see that in both the 2017 and 2018 papers we showed that the power law is found for movements created by a control model of movement. Just as you can’t make a fire without a spark you can’t make a control model without a controlled variable (even if we’re just dancing in the dark;-). So both the movement of the pursuer in the helicopter interception model (2017 paper) and the movement of the cursor in the elliptical movement model (2018 paper) are control models that are controlling perceptual variables (which are described in the papers in which the data is presented).

RM: Which is why I preferred referring to the power law illusion as “taking side effects [of control] to tell you something about the behaving mechanism when they do not”.

AM: Side effects do depend on the behaving mechanisms. There is nothing else to depend on, but the properties of all the elements of the behaving loop, including the environment and the organism. Whatever you change in the loop, side effects will quantitatively change.

RM: Sure. But the point of my statement is that you can’t tell what those mechanisms are by just looking at what happen to be the side effects of control. Of course, if you know they are side effects then you already know the mechanism that produced them – control of perception. So there’s not much more to learn from them.

Best

Rick

RM: Actually, I never claimed to know what a side effect is before determining the controlled variable.

Ok, if you put it like that, you did find controlled variables in helicopter tracking and slow ellipse tracking (drawing?). You then showed that the power law can appear in those kinds of mechanisms. It would be easy to show that under different conditions - different helicopter trajectories, for example - the power law would not appear.

That is what happens with ellipses too - in some cases you can get no power law, consistently; in some cases you always get the power law, even if subjects are trying to follow a constant velocity target, if the average speed is high enough. A good explanation must address both of those cases, or rather the whole range between when the power law happens and when it does not happen.

I correct my criticism - the problem is that you did not find the controlled variable in relevant behaviors. With a designed disturbance, you could make any 2D or 3D movement control model follow a power law. Explanations from helicopters and slow tracking don’t generalize to fast tracking or tracing - the controlled variables are different, and side effects are different.

RM: Sure. But

Sure, but? That is a rather quick jump. Just to make sure - you do see that, after finding the controlled variable, taking side effects of control to tell you something about the behaving mechanism is NOT a behavioral illusion?

RM the point of my statement is that you can’t tell what those mechanisms are by just looking at what happen to be the side effects of control. Of course, if you know they are side effects then you already know the mechanism that produced them – control of perception. So there’s not much more to learn from them.

Oh, there is lots to find out. The controlled variable is just the beginning. Just look how many things you can find out from the basic tracking model when you fit it to a single person’s behavior, and that model only applies to tracking smooth, random targets.

For example - what about step (square, jump) disturbances? You know the controlled variable, but the same model that works in the previous situation now gives, when fitted to same subject behavior, longer delays and lower gains! It also gives a peak of velocity in the beginning of the movement! On the other hand, a second order model with a double integrator in the output matches the velocity profile (bell shaped, just like in the Little Man).

The controlled variable is just one variable - there are a whole bunch of other functions and variables in the loop, all of which can be responsible for some funny effect visible in behavior.

RM the point of my statement is that you can’t tell what those mechanisms are by just looking at what happen to be the side effects of control

That is said from the perspective of knowing what is the controlled variable. I think we shouldn’t even use the term side effect before finding the controlled variable.

If you don’t know the controlled variable - it very well might be related to any of the regular, stable effects seen in the behavior, or none of them. The only way to find out is to disturb them one by one and observe if there is control or not.

So, it is NOT a behavioral illusion to assume that, for example, affine velocity in drawing ellipses is the controlled variable. Or trajectory. Those are valid guesses. It will probably turn out that constancy of affine velocity and other properties of trajectory are side effects of control of things related to accuracy of tracing and average velocity (but not instantaneous velocity), and other properties of the organism, and of the disturbance etc.

RM: Actually, I never claimed to know what a side effect is before determining the controlled variable.

AM: I correct my criticism - the problem is that you did not find the controlled variable in relevant behaviors.

I don’t understand; what’s a relevant behavior?

AM:you do see that, after finding the controlled variable, taking side effects of control to tell you something about the behaving mechanism is NOT a behavioral illusion?

RM: Yes, I do see that it should not longer be called a behavioral illusion. A better term for it would be a “waste of time”.

RM the point of my statement is that you can’t tell what those mechanisms are by just looking at what happen to be the side effects of control…

AM: Oh, there is lots to find out. The controlled variable is just the beginning. Just look how many things you can find out from the basic tracking model when you fit it to a single person’s behavior, and that model only applies to tracking smooth, random targets.

RM: But you are not finding that out from side effects; you are learning it from the person’s ability to control the controlled variable in various circumstances.

AM: For example - what about step (square, jump) disturbances? You know the controlled variable, but the same model that works in the previous situation now gives, when fitted to same subject behavior, longer delays and lower gains!

RM: But you are learning that from measurements of the system’s ability to control the controlled variable(s). What you are describing are the kinds of things manual control theorists want to learn about behavior. There are tons of studies in the manual control literature on, for example, the effect of spectral characteristics of disturbances, time lags, etc on the ability to control. They can find those things out because the manual control theorist knows what variable a person is (or should be) controlling. As I told you before, I think this is the version of control theory you should be looking into. A good reference is the book I review here: https://www.mindreadings.com/BookReview.htm.

AM: The controlled variable is just one variable - there are a whole bunch of other functions and variables in the loop, all of which can be responsible for some funny effect visible in behavior.

RM: I know. But the controlled variable is the one variable that distinguishes PCT from other applications of control theory in psychology. Manual control theorists know what variables people should control and they determine what factors influence how well they control them. PCT theorists want to know what variables people do control so they try to determine what these variables are, how they control them and why.

Best

Rick

RM I don’t understand; what’s a relevant behavior?

Behaviors relevant to the phenomenon of the speed-curvature power law are those where the phenomenon has been consistently found - fast drawing, tracing and tracking, mainly of ellipses where you have the 2/3 exponent, but also other shapes where you can get different exponents depending on the shape and speed.

The problem of the power law can be restated - why can’t people move their hand along a curved path at high constant speed, and instead have to slow down in the curved parts of the path?. Fast movements along curved lines are relevant behaviors.

AM:you do see that, after finding the controlled variable, taking side effects of control to tell you something about the behaving mechanism is NOT a behavioral illusion?

RM: Yes, I do see that it should not longer be called a behavioral illusion. A better term for it would be a “waste of time”.

We agree! It is not a behavioral illusion! Mazel tov!

It might be a blind alley, a red herring, etc. Experiments will tell. For me, any model that produces hand trajectories must also produce the power law when drawing fast, and must also produce accurate traces, just like humans do. For example, if you put the ‘standard’ tracking model to track a fast elliptical target, it will produce a power law without fail, but the path will be much larger than the ‘desired’ path. It is just missing a top level with one or two extra variables.

RM: In the context in which I said it in the 2017 paper I should have said that the behavioral illusion is the appearance of a causal relationship between behavioral variables when such a relationship is not, in fact, what you are seeing .

Missed this one.

I would add - the appearance of a lineal causal relationship (…). What you said might be interpreted as saying there is no causal relationship, but there is still a causal relationship between the stimulus and the response: it goes through the controlled variable, the organism, environment and back to the cv in a loop; i.e. there is really a circular causal relationship, not a lineal one.

AM: Oh, there is lots to find out. The controlled variable is just the beginning. Just look how many things you can find out from the basic tracking model when you fit it to a single person’s behavior, and that model only applies to tracking smooth, random targets.

RM: But you are not finding that out from side effects; you are learning it from the person’s ability to control the controlled variable in various circumstances.

The quantitatively assessed ability is a side effect of various time constants and gains in the loop.

RM What you are describing are the kinds of things manual control theorists want to learn about behavior. There are tons of studies in the manual control literature on, for example, the effect of spectral characteristics of disturbances, time lags, etc on the ability to control. They can find those things out because the manual control theorist knows what variable a person is (or should be) controlling. As I told you before, I think this is the version of control theory you should be looking into.

I did read that book some time ago, and I’ve read the critique that you linked before in this thread. Check out the new edition, they give a nod to PCT in the first chapter. They don’t have a better solution for tracking, IMO.

And PCT is perceptual control theory. Perceptual controlled variables come first, and then the organization of the loop, but the organization is very important. You seem to treat it as an afterthought, as if every loop is basically the same. Maybe you should look in to J.J. Gibsons informational basis of various behaviors, and forget about control theory? (not to belittle Gibson, I think he put forward some very interesting hypotheses that could be tested as controlled variables)

It is an open question in PCT just what is a good model of the step response - Bill thought there might be a two level solution - the top level is ‘turning on’ the lower level and that is why we see a longer lag. Also, you mentioned you were looking into the step response of the Little Man - to see if it can produce bell-shaped velocity profiles. Bill’s NonAdaptive model from LCSIII, also shows a bell-shaped velocity.

RM: I know. But the controlled variable is the one variable that distinguishes PCT from other applications of control theory in psychology. Manual control theorists know what variables people should control and they determine what factors influence how well they control them. PCT theorists want to know what variables people do control so they try to determine what these variables are, how they control them and why .

The how they control is a whole world of models. Baseball catching is different from tracking random targets is different from drawing letters is different from controlling sequences… There can be many different output functions, input functions, hierarchical arrangements, models of the environment… all for the same one or several controlled variables.

RM I don’t understand; what’s a relevant behavior?

AM: Behaviors relevant to the phenomenon of the speed-curvature power law are those where the phenomenon has been consistently found - fast drawing, tracing and tracking, mainly of ellipses where you have the 2/3 exponent, but also other shapes where you can get different exponents depending on the shape and speed.

AM: The problem of the power law can be restated - why can’t people move their hand along a curved path at high constant speed, and instead have to slow down in the curved parts of the path?. Fast movements along curved lines are relevant behaviors.

RM: Here’s my suggestion about how to go about studying this. We already have a reasonably good model of a person moving their hand in three space. It’s the little man program. Given your skills I’m sure you could reproduce this model in a current language and add the appropriate physics, such as the forces on the arm that are created when the hand is moved through varying curves at varying speeds. Then I would compare the model performance to that of people who are asked to do tracing and tracking at various speeds. If the model deviates from the human behavior in any significant ways then you can try to figure out a way to change the model – including, possibly, assumptions about the physics – to get it to fit the data.

RM: I think just looking at superficial characteristics of the behavior of people doing the tracing and tracking at different speeds could be quite misleading. For example, I presume your belief that people slow down their hand movement through curves is based on the power law. But as I (and several other power law researchers) have shown, the exponent of the power law, as derived from regression analysis, depends on the covariance between the included predictor variable (a measure of curvature) and the omitted predictor variable (a measure of affine velocity). The effect of this covariance on the size of the power coefficient is given by:

Where delta is the change in the coefficient of the included variable (I), beta.omit is 1/3 or 2/3 (depending on how velocity and curvature are measured), which is the actual coefficient of the omitted (O) predictor (affine velocity), Cov(I,O) is the covariance between the included and omitted variable and Var(I) is the variance of the omitted variable.

RM: The effect of delta on the coefficient of the power law is given by:

RM: Where beta.obs is the observed power coefficient and beta.true is the actual power coefficient (also 1/3 or 2/3).

RM: The value of beta.obs – the power “law” coefficient – with be close to 1/3 or 2/3 as long as Cov(I,O) – the covariance between curvature and affine velocity – is close to 0.0. In that case it will look like movements slow down when curvature increases. If, however, Cov (I,O) is negative the observed power exponent will be negative (since beta.omit is always the same as beta.true) which would imply that the movement speeds up through curves (like the movement of the planets in their orbits).

RM: The point is that the so-called power law – a measure of a superficial aspect of behavior – gives a very seductive but misleading picture of what the person is doing. The little man model is a better “first approximation” to what a person is doing when they move their hand about. I would highly recommend re-building the little man with what you think are the relevant physics (mass of limb segments, gravity, torque, centripetal, etc) and compare its behavior to that of real people. You should be able to get it to move in exactly the same way as the people do, which would mean that the model movement data should result in the same regression solutions for the power law as for the people. If it doesn’t then try to figure out what has to be changed about the model to get it to fit.

Gezunterheyt

Rick

RM: Here’s my suggestion about how to go about studying this. We already have a reasonably good model of a person moving their hand in three space. It’s the little man program. Given your skills I’m sure you could reproduce this model in a current language and add the appropriate physics, such as the forces on the arm that are created when the hand is moved through varying curves at varying speeds. Then I would compare the model performance to that of people who are asked to do tracing and tracking at various speeds. If the model deviates from the human behavior behavior in any significant ways then you can try to figure out a way to change the model – including, possibly, assumptions about the physics – to get it to fit the data.

The physics is not the problem, the controlled variables are the problem. I do have the LittleMan converted to about two other languages. The LittleMan has just the x, y and z variables of the finger. It will control position, if tuned, about as well as the standard position control model. When the target is moving in a random way, you can fit it to human behavior.

Many things don’t fit human behavior when the target is moving in a known, predictable pattern. Human subjects will ‘lead’ in front of the target at some points. The model will always be behind the target. Subjects will draw the shape of about the same size. The model will draw a much larger shape (or much smaller, depending on tuning).

This is what happens for the model and for the LittleMan (red is the target, always the same size, green is the model, path size increasing):

This specific model was fitted to subject behavior in normal random target tracking, has delay, etc.

RM For example, I presume your belief that people slow down their hand movement through curves is based on the power law.

Not really. You can see it just by plotting speed over time. This is from a real experiment, showing only the very slow and very fast situation. The target is moving with a constant speed along an elliptical path. When it is slow, the subject more or less matches the speed. When the target is fast, the subject’s speed oscillates from relatively fast to relatively slow.

The slow parts of the subject’s fast trajectory (red) are in the ‘corners’ of the ellipse path, I could add more plots of curvature for the target’s trajectory and the subject’s trajectory if don’t believe.

And yes, the affine velocity in fast movement is relatively constant, but it is not constant in slow movement.

RM:. You should be able to get it to move in exactly the same way as the people do, which would mean that the model movement data should result in the same regression solutions for the power law as for the people. If it doesn’t then try to figure out what has to be changed about the model to get it to fit.

Yeah, it doesn’t. What needs changing are controlled variables (adding on top). I’ve got many variables tested, turned out negative, and some of them are starting to look pretty good.

Lately, I’ve been thinking the situation is rather analogous to driving a car - if you want to drive at 20km/h on average, on a curved road, there is no need to slow down in the curve - you can keep your speed practically always at 20. If you want to keep average speed at 100km/h, you are going to have to slow down in the curves somewhat to maintain ‘accuracy’, or staying in your lane, and this will reduce your average speed, so you will compensate by increasing the speed in flat parts of the road. There are many ways to measure (perceive) accuracy and average velocity (or maybe rhythm). Either way, both situations - driving a car and tracing a figure - can be thought of as path control.

RM: This is nice work.

AM: The physics is not the problem, the controlled variables are the problem.

RM: You may be right. But I’d like to know how you know. I guess the first thing I’d like to know is whether the human subject(s) were in the same situation as the Little Man; that is, were they tracking a moving target with their finger tip? Actually, I’ve now read ahead and I think they were.

AM: I do have the LittleMan converted to about two other languages. The LittleMan has just the x, y and z variables of the finger. It will control position, if tuned, about as well as the standard position control model. When the target is moving in a random way, you can fit it to human behavior.

RM: This is a well known phenomenon and it points to the possibility that one of the main missing controlled variables may be control of a temporal pattern or sequence. Bill demonstrated this in one of his “portable demos” described in the 1960 paper, part 2. Have a person track your finger as it moves about in a fairly random pattern. When you abruptly stop moving your target finger the subject stops almost immediately. Now have the subject track your finger as it moves in a circle. The tracking is as good or better than it was with the random target but when you stop the movement the subject keeps moving in a circular pattern for about 1/4 sec or so. Clearly stopping the circular target was a disturbance to a higher level variable than was stopping the random target.

AM: This is what happens for the model and for the LittleMan (red is the target, always the same size, green is the model, path size increasing):

RM: I presume this is not what happens with the human subjects. Do you have the corresponding positional plots for a person. It would be nice to see them in comparison to the model. I presume the model parameters were the same for all target speeds? If so, one way to get the model to track better as a function of speed might be to adjust those parameters for the different speeds. If that improved the fit of the model to the human it could suggest that another set of controlled variables might be the parameters of the output functions of systems at all levels of the model.

AM: This specific model was fitted to subject behavior in normal random target tracking, has delay, etc.

RM: How did it do? Was it fit separately to each speed or was the fit “one size fits all” for all speeds.

RM For example, I presume your belief that people slow down their hand movement through curves is based on the power law.

AM: Not really. You can see it just by plotting speed over time. This is from a real experiment, showing only the very slow and very fast situation. The target is moving with a constant speed along an elliptical path. When it is slow, the subject more or less matches the speed. When the target is fast, the subject’s speed oscillates from relatively fast to relatively slow.

RM: It would be nice to see the curvature of the target plotted along with the speed of the hand. Also a running measure of deviation of hand from target. If the target is moving at constant speed and the hand is really slowing around curves and speeding up on straight aways then the finger should be behind the target on curves and ahead on straight aways. Which would mean that the person may not be controlling for staying on target but for staying on the path of the target, which suggests control of a higher level sequence kind of perception, as noted above.

AM: The slow parts of the subject’s fast trajectory (red) are in the ‘corners’ of the ellipse path, I could add more plots of curvature for the target’s trajectory and the subject’s trajectory if don’t believe.

RM: Why wouldn’t I believe you? But I would like to see plots of the target curvature and deviation of finger from target over time – overlaid on the above graph of finger (hand) speed over time.

AM: And yes, the affine velocity in fast movement is relatively constant, but it is not constant in slow movement.

RM: Very interesting.

RM:. You should be able to get it to move in exactly the same way as the people do, which would mean that the model movement data should result in the same regression solutions for the power law as for the people. If it doesn’t then try to figure out what has to be changed about the model to get it to fit.

AM: Yeah, it doesn’t. What needs changing are controlled variables (adding on top). I’ve got many variables tested, turned out negative, and some of them are starting to look pretty good.

RM: : Could you tell me what you’ve tried that didn’t work and what did?Maybe I could suggest something. Did you try adding physics?

AM: Lately, I’ve been thinking the situation is rather analogous to driving a car

RM: Maybe. As I understand it race drivers adjust their speed around curves to prevent the car from going past the “limits of adhesion”. So the variable being controlled is probably something like their sense of angular acceleration. That may be true in the hand movement study too but you said physics wasn’t involved. But if they are – if going around a curve applies a force that would move the hand away from the intended path if the movement were not slowed – then that might indeed be why you see a slowing at curves

RM: Again,this is great work; I think you’re on the right track. I think I could help you out a little, too, but you might not want to hear any of my nonsense;-)

Best

Rick

• if you want to drive at 20km/h on average, on a curved road, there is no need to slow down in the curve - you can keep your speed practically always at 20. If you want to keep average speed at 100km/h, you are going to have to slow down in the curves somewhat to maintain ‘accuracy’, or staying in your lane, and this will reduce your average speed, so you will compensate by increasing the speed in flat parts of the road. There are many ways to measure (perceive) accuracy and average velocity (or maybe rhythm). Either way, both situations - driving a car and tracing a figure - can be thought of as path control.
[/quote]

Here is how one person tracked, looking at the target on the screen, and moving a pen on a graphics tablet (Instead of holding a mouse, the person was using a wacom tablet with pen).

Units in mm of movement of the pen. There is a decrease in accuracy with the speed of movement, but the general shape of each cycle is generally elliptical of similar size to target path shape.

Here are speed and curvature for the pen movements for the fast situation

The same for target:

First plot, the speed of the pen and curvature of its path over time shows that in this subject, speed was low when curvature was high, and speed was high when curvature was low. At the same time, the target he was following was not slowing down nor speeding up, and had a completely flat speed profile (green line on second plot).

This relationship between speed and curvature for subject’s movement can be shown in a speed-curvature plot:

The relationship is not linear, but looks much nicer on the log-log plot:

The log-log plot shows the ‘physical’ meaning of the speed-curvature power law - when the curvature is high (‘corners’ of the ellipse) the speed will tend to be lower than when the curvature is low (‘flat parts’ of the ellipse). In this case, the r2 is not particularly high, it it usually higher, I guess because of not super accurate tracking, but the relationship is still pretty obvious.

These sorts of correlations are why everyone gets upset when you say that the power law is a statistical artifact. It can be a statistical artifact due to noise, as Maoz, Portugaly, Flash and Weiss (2005) showed, I can illustrate that if needed, but is generally a real phenomenon that happens at high speeds of hand movement. By “real phenomenon” I mean that people really do slow down their hand movement speed in curved parts of the path, if the overall speed is high. If the overall movement speed is low - a person can pick and choose whatever speed profile they want - constant speed, or high in the corners and slow in the flat parts. At high speed, they cannot choose.

The lead-follow analysis shows somewhat erratic change in angular separation, but generally the separation is around 0 radians. If you take the center of the ellipse as (0,0) of the coordinate system, and then measure the angle between the going from the cursor to the center and the line going from the center to the target, you get this pattern for this subject (same data as above, for the fast situation). Some subjects show a ‘nicer’ pattern, especially after practicing for a while.

This data is from one of the very first experiments we did in the lab, in late 2016, all credit to Alex for coming up with it and leading the analysis. There is much more data from that experiment, different target speed profiles, a whole range on average speeds, several subjects, etc, this one is just for illustration purposes, but everywhere we see the same pattern in the speed-curvature relationship. Whatever the target speed profile, when the average speed is high, subjects have the same beta of 1/3 in the power law.

RM: Now have the subject track your finger as it moves in a circle. The tracking is as good or better than it was with the random target but when you stop the movement the subject keeps moving in a circular pattern for about 1/4 sec or so. Clearly stopping the circular target was a disturbance to a higher level variable than was stopping the random target.

Yep. Very well known phenomenon in PCT, I even had the comparison of tracking a random and a circular target as one experiment for my B.A. in psychology. Control of patterns or paths is not that much explored in modelling in PCT.

On the standard position tracking model and the LittleMan - there is not much to say. Both have position tracking as the only level or top level control system. If you put a fast target elliptical target, they will not follow as accurately as humans do. If you put higher gain, they might be more accurate, but still never go to ‘lead’ in front of the target. Also importantly - they react differently to disturbances when the gain is higher, much faster.

The physics is certainly involved in the whole process, and changes in physics, such as drawing in water, or having bigger arms or stronger muscles, can change the power law exponent, or the point in the average speed where the power law appears, etc. But adding more physical realism to the model will not help in finding controlled variables.

One thing that sort-of worked was control of phase difference (the angular separation) along with radius (distance from center). Those are two controlled variables. Then you have an oscillator that changes frequency of oscillation in proportion to the phase difference, and amplitude in proportion to radius difference. The output of the oscillator is then the reference for the position tracking system. In engineering, this is called a phase-locked loop, there are a few bits in the archives about it. Works ok, maintains size, goes a bit into lead sometimes, slows down in the corners. It is limited to ellipses, though, the delays are not very realistic, and there are some instability issues, etc, had it presented last year as a poster, but it needs more work.

For the car analogy, there is really a whole line of research, some 80 years old, on what “visual cues” and “information” drivers extract from the visual scene and use to “guide” driving. This is a beautiful paper: theory of driving perception and control (2019) pdf

It is not an exact analogy to moving hands, but there are interesting parts.