# Misunderstanding Behavior with Math

I would appreciate comments on the presentation by Richard Kennaway and myself gave at the IAPCT Conference last month in Manchester. A copy of the slides is here:

I may not have been obvious but the aim of the talk was to get researchers to turn away from conventional approaches to studying behavior and start doing research based on an understanding of organisms as living control systems. Do you think we made that point?

Anyway, any comments would be most welcome because we are thinking of turning this into a paper to be published in a high impact journal and we want it to be in as good a shape as possible.

Best Rck

I find it promising that you do not refer to the speed-curvature power law as a statistical error. I quite agree that it is irrelevant in the sense of being present in the movement program, but then so does about half of the researchers who published papers on the power law. Something else, probably on a higher level, is controlled, and a trajectory is “simply” produced.

It is crucial to find the controlled variables in this type of human movement (following predictable paths), but in order to demonstrate that a model explains human behavior, the behavior of the model has to match the behavior of the human. That is why laws of behavior are important. That is why Bill showed how the LittleMan demo also has bell-shaped velocity profiles (so does any PD control system, that was the point).

The 2/3 power law has been found in relatively fast movements (for slow movements, the exponent changes, and so does the r2, so often there is no power law at all). If you take a “standard” position tracking model to follow a fast target along an ellipse, you don’t get the reference trajectory. That model is not a good demonstration of what constitutes a PCT explanation of movement control of predictable paths.

The second thing is I cannot find evidence for your claim on gait planarity: “It is assumed that the system is trying to produce that behavior, or is playing back a representation of it.” Quite the opposite, as stated in Catavitello et al (2018) “The planar covariation law may emerge from the coupling of neural oscillators with limb mechanical oscillators”; Maybe gait planarity is not a good example of what you’re trying to show.

It is crucial to find the controlled variables in this type of human movement

Yes, that’s the main point of the paper. Actually, the point is that it’s crucial to find the controlled variables in all purposeful behavior. Of course, the other point is that you find them using the test for the controlled variable, not by looking for mathematical “laws” of overt behavior.

but in order to demonstrate that a model explains human behavior, the behavior of the model has to match the behavior of the human. That is why laws of behavior are important. That is why Bill showed how the LittleMan demo also has bell-shaped velocity profiles (so does any PD control system, that was the point).

Actually, that wasn’t the point I thought Bill was making. I thought his point was that the bell-shaped velocity profiles are a side-effect of control. That’s the take away I got from Bill’s statement that " the invariances noted by the authors were simply side-effects of the operation of the control systems of the arm interacting with the dynamics of the physical arm…The path which Atkeson, Hollerbach…are treading is a blind alley".

Best
RIck

The second thing is I cannot find evidence for your claim on gait planarity: “It is assumed that the system is trying to produce that behavior, or is playing back a representation of it.”

Credit where credit is due: Richard came up with that way of saying it. I agreed.

Quite the opposite, as stated in Catavitello et al (2018) “The planar covariation law may emerge from the coupling of neural oscillators with limb mechanical oscillators”; Maybe gait planarity is not a good example of what you’re trying to show.

Actually, I think that statement is an even better way of stating why gait planarity an excellent example of what we are trying to show: examples of mathematics used to misunderstand behavior. Thanks!

Best

Rick

Yes, that’s the main point of the paper. Actually, the point is that it’s crucial to find the controlled variables in all purposeful behavior. Of course, the other point is that you find them using the test for the controlled variable, not by looking for mathematical “laws” of overt behavior.

The point can be made much more effectively by finding the controlled variables. Then - demonstrating that as a side effect of controlling those variables, trough the interaction with the physical system (arm and the environment), it is possible to get the same overt behavior.

Arguing about how it is important to find controlled variables without actually finding them is counter-productive for the purpose of “getting researchers away from conventional approaches”.

What I’m saying on gait planarity - it seems that your argument is a straw man. They are suggesting the same thing as you - that this overt behavior might emerge or arises without anything planning it explicitly.

I posted this via email but apparently it didn’t show up here so I’m re-posting it. If we can’t reply by email that seems like a big problem for this way of having discussions.
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I’m responding by email to see how it works.

Yes, that’s the main point of the paper. Actually, the point is that it’s crucial to find the controlled variables in all purposeful behavior. Of course, the other point is that you find them using the test for the controlled variable, not by looking for mathematical “laws” of overt behavior.

The point can be made much more effectively by finding the controlled variables. Then - demonstrating that as a side effect of controlling those variables, trough the interaction with the physical system (arm and the environment), it is possible to get the same overt behavior.

That’s a good point. Actually, that is what I try to do but apparently I don’t do it explicitly enough. In every example we give in the paper – LOT, power law, velocity profiles, gait planarity – we describe a control model that produces those phenomena as a side effect. All control models control variables but I’m afraid we only explicitly describe the controlled variables in the object interception model that produces the observed LOT as a side effect. The controlled variables in that model as vertical and horizontal optical velocity. The controlled variables in the model that accounts for the power law are the x and y coordinates of the position of the moved object. The controlled variables in the little man are joint angles and difference between joint angles. I’m pretty sure Richard would know what they are because he probably uses some similar controlled variables in his model of his six-legged walking arachnid named Archy which accounts for the observed gait planarity found in that gaits of all kinds of creatures.

Arguing about how it is important to find controlled variables without actually finding them is counter-productive for the purpose of “getting researchers away from conventional approaches”.

I will try to be more explicit in this paper (not yet written, by the way) about the controlled variables in the models that account for the mathematical regularities in overt behavior as side-effects of control. But your comments makes me realize that we have to do a better job of explaining why we’re always arguing (haranguing might be a better word;-) for the importance of doing research aimed at finding the controlled variables around which behavior is organized. The reason being that behavior IS control and this controlling is organized around the maintenance of perceptual variables in fixed or varying reference states. So research focused on the study of observed regularities in overt behavior (or in irrelevant aspects of what might be considered input variables, as in LOTs) is, as Bill said, headed down a blind alley.

Another thing your comments make me realize is that we have to make it clear that the observation of possible controlled variables is the first step in doing research on purpose. Your comments suggest that the first step in research on purposeful behavior is the discovery of mathematical regularities in overt behavior, like the power law and gait planarity. But this is precisely what we are saying is the wrong way to go about studying the purposeful behavior of living control systems. The fact that the control models account for the observed mathematical regularities described in the paper doesn’t mean that the way to go about studying purposeful system is to start by observing mathematical regularities in behavior and then trying to figure out how a control model might account for the. The way to go about it is to start with some behavior of interest that clearly involves control; a behavior like catching, pointing or walking. Then guess at the variables that seem to be controlled and then test this using some version of the Test for the Controlled Variable.

What I’m saying on gait planarity - it seems that your argument is a straw man. They are suggesting the same thing as you - that this overt behavior might emerge or arises without anything planning it explicitly.

I don’t think that’s quite the same as what we are saying, which is that the overt behavioral regularity called “planarity” is a side effect of control.

Best

Rick

I strongly disagree with most of what you wrote, but I’ll just comment on the use of “side effect of control”. Without specifying the full mechanism, the expression is meaningless. For example, the bell-shape profile of the tangential velocity of the hand in a reaching movement is (hypothetically) a side effect of a step change in the reference signal or a step change in the disturbance to a position (and velocity?) control system, where the output function is a low-pass filter with such-and-such properties. If the output function were different, you could get a near step-shape in the velocity too, instead of the bell shape.

I strongly disagree with most of what you wrote,

That’s fine. When dealt with properly, disagreements can lead to discovery. Let me know what you think we disagree about and maybe we can figure out a way to subject it to empirical test.

but I’ll just comment on the use of “side effect of control”. Without specifying the full mechanism, the expression is meaningless. For example, the bell-shape profile of the tangential velocity of the hand in a reaching movement is (hypothetically) a side effect of a step change in the reference signal or a step change in the disturbance to a position (and velocity?) control system, where the output function is a low-pass filter with such-and-such properties. If the output function were different, you could get a near step-shape in the velocity too, instead of the bell shape.

I think the expression “side effect of control” is meaningful in the context of understanding that behavior is a control process. We give this context in our presentation at the IAPCT meeting and I will certainly give it in any paper that is based on that presentation. In the case of the tangential velocity profiles, that means that the observed behavior – the movement of the finger from one position to another – is a process of control, as you note in your two alternative explanations of the behavior. The fact that this movement is controlled can be easily demonstrated by the fact that the same behavior is repeated in varying circumstances (a consistent result is produced in the face of varying disturbance).

The “side effect” of this controlling that we discussed in our presentation was the invariance of the profile; the fact that the normalized shape of the profile is the same bell-shaped shape for different distances and speeds of movement. It’s a side-effect of control in the sense that there is nothing in the model (Bill’s “Little Man”) that is designed to produce this invariance. So the observed invariance has nothing to do with how the behavior is produced. Yet this invarance has been taken to show just that – something important about how the movement is produced. That’s the problem with side-effects of control; that are red herrings that are leading researchers down blind alleys.

The bell-shaped shape of the velocity profiles is also a side-effect of controlling. And, as you mention, this shape will probably be influenced by characteristics of the control system itself, such as nature of the output function. But a researcher who knows that they are seeing control will certainly take this into account when modeling the behavior.

The fact that Bill’s “Little Man” fits the actually observed bell-shaped velocity profiles means that Bill picked something very close to the right parameters – control variables, output functions, etc --for the control systems in the Little Man, and he did this without trying to get the model to produce invariant, normalized bell shaped tangential velocity profiles. He did it by trying to make a model that appeared to track a target with its finger the way a real person does.

But ultimately, the most important thing to keep in mind when studying the behavior of living things is that you are studying the behavior of an input control system. So the first thing to think about when trying to understand the behavior of such systems is “what variables might this system be controlling?”, Keep your eye on that ball and you will never be seduced by the scarlet hussy of attractive mathematical regularities that are side effects of controlling these variables.

Best

Rick

Yes, the context is important for saying “the side effect of control”. We agree on that. The context are all the parameters of the hypothetical control system, as the observed invariance could be affected by changes in any of the parameters, so it needs to be clear which parameters are responsible for the invariance.

Controlled variables are also invariances in behavior. The near constant distance between a cursor and a target in humans tracking pseudorandom targets is an invariance; a behavioral law. You can express quantitatively how the amount of error changes with frequency spectrum (difficulty) in the target trajectory. This distance between the cursor and the target happens to also be a controlled variable in this case. Sometimes the invariance is a side effect, sometimes it is the main effect.

Either way, invariances in behavior serve as test to reject models. If a model proposed to explain some behavior does not behave as the human in the same situation, the model is wrong. Maybe Bill didn’t “look at invariances” to create the LittleMan but he sure would modify it it didn’t show the same bell-shaped profile in velocity. The blind alley is the inverse dynamics approach as the explanation of human movement. The invariances found still need to be explained.

The strawman accusation is still here. I mean by this that you are missrepresenting the argument of Catavitello et at. on gait planarity when you say “It assumed that the system is trying to produce that behavior, or is playing back a representation of it”. They say no such thing, as far I can find in that paper.

The way to go about it is to start with some behavior of interest that clearly involves control; a behavior like catching, pointing or walking. Then guess at the variables that seem to be controlled and then test this using some version of the Test for the Controlled Variable.

The interesting behavior for the 2/3 power law was fast drawing of ellipses (not just any voluntary behavior, as you state in the presentation).

You mention empirical tests: to show that the 2/3 power law is not a side effect of control of position, one way is to follow a fast target along an ellipse. Humans do it very well up to about half a second per cycle, but the position control model does not. Another way is to create a slow non-power-law trajectory, say constant speed over elliptical trajectory, and the model will track it, maintainting constant speed, which is a non power law trajectory. So, position is not the controlled variable, and the power law is not a side effect of control of position [with standard gain, delay etc in the loop].

The power law in the humans tracking helicopter trajectories is probably a side effect of the helicopter trajectories already being power law. In other words, the humans slow down in the same place as the helicopters. Other possibility is the noise (as in the paper by Flash that you quote).

I’m responding by email to see how it works.

Yes, that’s the main point of the paper. Actually, the point is that it’s crucial to find the controlled variables in all purposeful behavior. Of course, the other point is that you find them using the test for the controlled variable, not by looking for mathematical “laws” of overt behavior.

The point can be made much more effectively by finding the controlled variables. Then - demonstrating that as a side effect of controlling those variables, trough the interaction with the physical system (arm and the environment), it is possible to get the same overt behavior.

That’s a good point. Actually, that is what I try to do but apparently I don’t do it explicitly enough. In every example we give in the paper – LOT, power law, velocity profiles, gait planarity – we describe a control model that produces those phenomena as a side effect. All control models control variables but I’m afraid we only explicitly describe the controlled variables in the object interception model that produces the observed LOT as a side effect. The controlled variables in that model as vertical and horizontal optical velocity. The controlled variables in the model that accounts for the power law are the x and y coordinates of the position of the moved object. The controlled variables in the little man are joint angles and difference between joint angles. I’m pretty sure Richard would know what they are because he probably uses some similar controlled variables in his model of his six-legged walking arachnid named Archy which accounts for the observed gait planarity found in that gaits of all kinds of creatures.

Arguing about how it is important to find controlled variables without actually finding them is counter-productive for the purpose of “getting researchers away from conventional approaches”.

I will try to be more explicit in this paper (not yet written, by the way) about the controlled variables in the models that account for the mathematical regularities in overt behavior as side-effects of control. But your comments makes me realize that we have to do a better job of explaining why we’re always arguing (haranguing might be a better word;-) for the importance of doing research aimed at finding the controlled variables around which behavior is organized. The reason being that behavior IS control and this controlling is organized around the maintenance of perceptual variables in fixed or varying reference states. So research focused on the study of observed regularities in overt behavior (or in irrelevant aspects of what might be considered input variables, as in LOTs) is, as Bill said, headed down a blind alley.

Another thing your comments make me realize is that we have to make it clear that the observation of possible controlled variables is the first step in doing research on purpose. Your comments suggest that the first step in research on purposeful behavior is the discovery of mathematical regularities in overt behavior, like the power law and gait planarity. But this is precisely what we are saying is the wrong way to go about studying the purposeful behavior of living control systems. The fact that the control models account for the observed mathematical regularities described in the paper doesn’t mean that the way to go about studying purposeful system is to start by observing mathematical regularities in behavior and then trying to figure out how a control model might account for the. The way to go about it is to start with some behavior of interest that clearly involves control; a behavior like catching, pointing or walking. Then guess at the variables that seem to be controlled and then test this using some version of the Test for the Controlled Variable.

What I’m saying on gait planarity - it seems that your argument is a straw man. They are suggesting the same thing as you - that this overt behavior might emerge or arises without anything planning it explicitly.

I don’t think that’s quite the same as what we are saying, which is that the overt behavioral regularity called “planarity” is a side effect of control.

Best

Rick

RM So they tend to get kind of defensive when the explanation of these regularities is shown to be that they are a side effect of controlling.

AM: You’re guessing that is the reason why they’re defensive. It just might be they are defensive because you accused them of ignorance in mathematics, and of not knowing how to calculate basic properties of trajectories, etc.

RM: Well, I hate to go all psychoanalytic on you but it seems to me that this is a clear case of projection;-) I think it was “they” who were accusing me of ignorance of mathematics and of not knowing how to calculate basic properties of trajectories. Far be it from me to accuse anyone of these things since I am not much of a mathematician and know little about calculating basic properties of trajectories. All I know is that I proposed a simple, testable control model of movement trajectory production that accounts for the existence of the “power law” and the reaction was that it was not very encouraging, to say the least.

There is no projection. Look at it this way - people have been calculating the speed-curvature relationship for 30 years. Then you and Shaffer show up claiming that the power law relationship is a statistical artifact, and that the proper way to calculate is to correct for the omitted variable bias. This is very directly saying to all the mathematicians who did speed-curvature analysis: “you are bad at math, look at how you should do this”. Then you got a reply saying “no, you are bad at math, there is no statistical artifact”.

You did propose a testable model that might account for the existence of the power law, that is very good. The problem is that you did not test your model in relevant situations. I have suggested (up in this thread, Misunderstanding Behavior with Math - #11 by amatic ) several ways to test it. If you do the tests, you will see that your model completely fails to account for human behavior.

RM: Well, I hate to go all psychoanalytic on you but it seems to me that this is a clear case of projection;-) I think it was “they” who were accusing me of ignorance of mathematics and of not knowing how to calculate basic properties of trajectories. Far be it from me to accuse anyone of these things since I am not much of a mathematician and know little about calculating basic properties of trajectories. All I know is that I proposed a simple, testable control model of movement trajectory production that accounts for the existence of the “power law” and the reaction was that it was not very encouraging, to say the least.>

AM: There is no projection. Look at it this way - people have been calculating the speed-curvature relationship for 30 years. Then you and Shaffer show up claiming that the power law relationship is a statistical artifact, and that the proper way to calculate is to correct for the omitted variable bias. This is very directly saying to all the mathematicians who did speed-curvature analysis: “you are bad at math, look at how you should do this”. Then you got a reply saying “no, you are bad at math, there is no statistical artifact”.

RM: So it would be more appropriate to say that I was perceived to be accusing rather than intentionally accusing. What you saw as “accusing” was a side effect of me controlling for something else. Again, “they” were taking a side effect of control for an intentionally produced (controlled) result;-)

AM: You did propose a testable model that might account for the existence of the power law, that is very good. The problem is that you did not test your model in relevant situations. I have suggested (up in this thread, Misunderstanding Behavior with Math - #11 by amatic ) several ways to test it. If you do the tests, you will see that your model completely fails to account for human behavior.

RM: Yes, I remember seeing that but I didn’t understand how it rejected my control as showing that the the power law is a side effect of control. I think it would be great if you could give a detailed description of your test of the model. This is precisely what I hoped would happen in this discussion on "PCT Science; tests of the control theory model against actual data. If my model doesn’t account for yor data then it would be nice to see a model that does. And that model would also have to fit the data that my model fits (the data on power law conforming elliptical movement of a cursor produced by non-power law conforming, non-elliptical movement of a mouse). I would replicate your experiment myself but I don’t think I’m equipped to be able to do it.

Best

Rick

So it would be more appropriate to say that I was perceived to be accusing rather than intentionally accusing. What you saw as “accusing” was a side effect of me controlling for something else.

There are not that many different interpretations of your “that thing is a statistical artifact”, though. I did notice you stopped mentioning the omitted bias hypothesis. Looks like you changed your opinion or aren’t so sure anymore.

As for the testing of the model, I’m sure you can do it. Just make a target move along an ellipse in about 1 cycle per second. A human can track this target without much problem. A position tracking model does not follow the target just like a human. So, model rejected.

Position tracking model is great for tracking pseudorandom targets, that works great, and is confirmed many times. The speed-curvature power law is not consistently found in random scribbling, it often has low r2, etc, so random movement is not relevant.

RM: So it would be more appropriate to say that I was perceived to be accusing rather than intentionally accusing. What you saw as “accusing” was a side effect of me controlling for something else.

AM: There are not that many different interpretations of your “that thing is a statistical artifact”, though.

Yes, I can think of only two: 1) I was accusing “them” of not knowing how to do math or 2) I was pointing out an interesting fact about the consequence of doing a regression analysis without taking into account the mathematical relationship between the variables being regressed. Things could have gone so much better if “they” had chosen interpretation 2.

AM: I did notice you stopped mentioning the omitted bias hypothesis. Looks like you changed your opinion or aren’t so sure anymore.

I stopped talking about it because it is apparently a fact that is well known in the field. Both Pollick & Sapiro (1997) and Maoz, Portugaly, Flash & Weiss (2006) had already shown that the “power law” coefficient you get from regressing curvature on velocity depends on the correlation between what they called the affine velocity variable (and what we called the “omitted cross-product variable”) and the dependent (velocity) variable. If this correlation is 0, which it will be when the omitted variable is constant, then the power coefficient will be exactly 2/3 (or 1/3 depending on how velocity and curvature are computed).

RM: Our conclusion, based on this fact, was that the power coefficient depends on characteristics of the movement produced and has nothing to do with how it is produced; Pollick & Sapiro and Maoz, Portugaly, Flash & Weiss (2006) came to a different conclusion. Pollick & Sapiro actually suggested that the power law results from the fact that people control affine velocity. Seems like a hypothesis worth testing.

AM: As for the testing of the model, I’m sure you can do it. Just make a target move along an ellipse in about 1 cycle per second. A human can track this target without much problem. A position tracking model does not follow the target just like a human. So, model rejected.

RM: I really think it would be nice if you could post more details about this experiment. You’ve apparently done some research that shows that a control model can’t account for some controlling that people do. So it sounds like this study has some very important implications not only for the power law but for PCT as a model of behavior as well (to say the least)! So how about some details. In exactly what way does the model deviate from the behavior of the human? Were you unable to adjust the parameters of the model so that it fit the human behavior? Does the power law hold for the human but not the model? Etc.

AM: Position tracking model is great for tracking pseudorandom targets, that works great, and is confirmed many times. The speed-curvature power law is not consistently found in random scribbling, it often has low r2, etc, so random movement is not relevant.

RM: That is consistent with my idea that the finding of a power law depends on the nature of the trajectory of the movement and says nothing about how the movement was produced. Movement trajectories with constant affine velocity (like elliptical trajectories) will follow the “power law” while other movement trajectories will differ from the power law to the extent that the affine component of the velocity is correlated with the total velocity.

RM But I would really like the details on the failure of the PCT model to account for the rapid elliptical movements of humans. That sounds like a very important finding!

Best

Rick

Right, you were pointing out “interesting consequences”…

Do the experiment and you’ll see. Super easy, nothing to it.

Do the experiment and you’ll see. Super easy, nothing to it.

OK, I’ll try when I get a chance. So it’s a pursuit tracking task with cursor (c) tracking an elliptical target (t) that is making an orbit once per second. Is that right? Then I fit the data with a model controlling t-c. And what I see is… what? Poor fit of model to data? A model that doesn’t produce a power law? The end of the world as we know it?

Best

Rick

The setup is correct. You can use parameters (gain, delay, damping) from tracking a random target, and see what happens. Position control does not explain drawing ellipses quickly.

Drawing ellipses quickly is a behavior where you can consistently find very strong (high r2) speed-curvature relationship with exponents near 1/3 (or 2/3).

The setup is correct. You can use parameters (gain, delay, damping) from tracking a random target, and see what happens. Position control does not explain drawing ellipses quickly.

I don’t know when I’ll get to this, but this comment made me realize that you are probably dealing with a higher level control system when the subjects are drawing an ellipse quickly. They are not controlling the position of the cursor relative to the moving target; they are matching their movement pattern to the target movement pattern. This is actually closer to the model I proposed (and the one described in our reply to the criticisms of our paper. In my model, the subject is assumed to be moving the reference for the cursor position in an elliptical pattern. These variations in the reference are assumed to be made by the outputs of a higher level system that is controlling for seeing an elliptical movement trajectory.

My model was rejected because it was considered to be cheating to put an autonomously varying reference for elliptical movement into the model. In other words, giving the model autonomy (the very essence of what we think of as the behavior of a living system) was cheating.

Drawing ellipses quickly is a behavior where you can consistently find very strong (high r2) speed-curvature relationship with exponents near 1/3 (or 2/3).

Yes, to the extent that the secularly varied reference is elliptical and there is good control so that the actual cursor movement is elliptical you will get a 1/3 (or 2/3) power law because the affine velocity component of an elliptical movement is constant.

I will try to eventually get to this; but I’m not in a bog rush because I’m detecting a great deal of hostility to what I think of as the PCT approach to understanding the power law. So I’ve got plenty of other problems in my life so unless you are really interested in continuing this discussion I think maybe we can just drop it now.

Best

Rick