Side Effects of Control

One way to test for controlled variables is to guess at what they are and then put them in a model and see how well the model fits the data. I was able to demonstrate how this is done using object interception data collected by Dennis Shaffer and I described the results of that testing in this paper. The main result was that a model that controlled vertical optical velocity fit the data somewhat better than one that controlled vertical optical acceleration. I used this approach to testing for controlled variables to model movement behavior that follows the power law.

A control model that successfully fits curved movement data – including yours --will show the apparent “slowing down through curves” power relationship between velocity and curvature. I describe a control model that fits power-law-conforming movement data in the behavioral illusions paper. That model perfectly accounts for cursor movements and the mouse movements that produced them. In both cases, the movement appears to slow down through curves, although there was nothing in the model that was designed to produce that result. The apparent slowing down though curves is a side-effect of control that results from the mathematical relationship between velocity, curvature and affine velocity.

Actually, I did all those things. But thanks for the tip.

That was good! Also not a formal TCV, but ok. It did not explain why do people slow down in curves in other types of movement, like handwriting or drawing ellipses, where the power law was initially found.

Apparent slowing down? I had real slowing down in my experiments. Raw speed and curvature plots over time show an obvious correlation, not merely an apparent correlation.

If you had only “apparent” slowing down - that means you had a statistical artifact, and there was no need to make a model to explain the findings - the statistics explain it all.

If the slowing down was real, then the model might explain it as a side effect of controlling the cursor-target distance. But you had some VERY slow movements, didn’t you?

The cursor-target distance control model produces much bigger ellipses than people in the same task, so it does NOT account for behavior or the power law in this task.

The correlation is real; a result of the mathematical relationship between velocity, curvature and affine velocity. What is apparent is that the slowing down occurs in response to the curvature.

I made the cursor movements in the Figure below at a speed of about 8 cm/s. The mouse movement were made over a much smaller distance, so the speed was ~2cm/s.

I don’t known if those movements would be considered very slow but the position control model accounted for both movements with great accuracy. And in both cases there was a negative power relationship between curvature and speed for both participant and model

Yes, but your model with the higher order phase lock system accounted for both the slow and high speed movement, right? If so, you should see about the same power relationship for the model as found for the participant in each condition.

Nope. There is only the correlation between speed and curvature. No affine velocity in the analysis.

You said: “apparent “slowing down through curves” power relationship betweeen velocity and curvature”

So, you made a mistake: the correlation is real, but the interpretation might be a mistake?

Yep, too slow. If you draw whole ellipse in one second, that might be showing a real power law. This is the situation where people CANNOT draw elliptic shaped without slowing down in curves. And this is real slowing down, not just apparent.

Yes, that is a different model.

Yes, of course. But, as Maoz et al showed, it’s the correlation between the variable omitted from the regression, affine velocity, and the variable included , curvature, that determines how much beta (the power coefficient relating curvature to velocity) will deviate from -1/3. The fast movements are closer to an ellipse, where the correlation between curvature and affine velocity will be close to zero. So beta will be close to -1/3.

It’s not clear from this what my mistake was but you correctly state what I meant to say: the correlation between curvature and velocity is real, but the interpretation of that correlation might be a mistake.

Ah, now I think I see why you think that. You get a beta <>-1/3 and a low R^2 with the slow movement and a beta close to -1/3 and a high R^2 with the fast movement. Is that it?

Maoz et al don’t talk about omitted variables.

If you do multiple regression with logC and logVa as predictors, and logV as the criterion variables, you will always get the coefficients from the formula, regardless of the correlation between C and V.

The mistake was saying that the correlation was apparent. This would imply that people do NOT slow down in curves. In fact, in fast movements - they do slow down in curves.

The main thing is the R^2, the fit to the power law. If the trajectory has an R^2 < 0.75 in my analysis - this means it is not a power-law movement.

Empirically, slow elliptic movements did not have a good fit to the power law, and fast movement did. Betas had different values, depending on the speed, but this is irrelevant because of the low R^2.

Yes, it was implied. They showed how the observed beta would be affected by the correlation between curvature and affine velocity, the latter being the variable that is never included in the regression used to evaluate the fit to the power law.

It might have helped if we had called affine velocity something other than the “omitted” variable because calling it that implies that power law researchers have been purposefully omitting that variable from their analysis. And, of course, they haven’t. We should have just called it a “third variable” that influences the observed beta in the regression used to evaluate fit to the power law.


I don’t think I said that the correlation is apparent; if I did, then that would definitely have been a mistake. But I think I said that the correlation is real; what is apparent is the appearance that people slow down in curves.

Maybe our problem is just linguistic. The term “slow down” implies (to me, anyway) that the slowing is done actively, presumably in response to the increase in curvature. But what we see in the correlation between speed and curvature is just an association between these variables. This association is more accurately described as “movement is slower through high curvature than through low curvature segments of a movement”. If this is what you meant when you said that people “slow down through curves” then you were, indeed, describing something that is real (the correlation between speed and curvature) and not apparent (active slowing in response to curvature).

OK, thanks. I’ll try to find out what, if anything, our analysis implies about the relationship between R^2 and observed beta.

It might be interesting to try an elliptical variant on Powers’s “square-circle” demo, in which the subjects were actually tracking one shape while perceiving themselves to be tracking a different shape. It should be possible to do a tracking study in which the physical track follows a shape with curvature variations anti-correlated with the shape they perceive themselves to be tracking.

The current discussions of the Power Law seem to me disembodied, utterly irrelevant, musings divorced from reality. Philosophical discussions about the number of angels that can dance on the head of a pin.

I understand that math is the language of physics, but to me, it is most important to have a sense of the physics to which you apply the math. What does the math represent. The various equations you all squabble about mean nothing to me.

A huge weakness throughout discussions on CSGnet seems to me to be that a background with solid understanding of physics (the world we are part of and live in) is not required for entry into academic psychology. As a result, most psychology deals with appearances, not explanations.

Some participants on CSGnet have solid credentials re physics, others basically none.

I guess the Power Law deals with appearances.

I have ignored Powel Law talk as silly, and the math discussions as a waste of time, but here is my attempt to understand what scientists who take it seriously may be talking about.

There must be more to drawing an ellips or curve than curve and speed change.

Of course: mass and force. Acceleration = Force/Mass.

It seems to me silly to discuss the Power Law without dynamics.

A race car driver slows down as she approaches a curve. Why?
Her soft tires grip the pavement well, creating a coefficient of friction that approaches 1. That means that if the car weighs and presses on the ground with 1,000 lbs, tires can create a friction force of 1,000 pounds (to start, stop, or turn). That allows the car to take the curve at 1G acceleration (1,000 lbs of centripetal force applied to 1,000 lbs mass). Steer too much and you will lose the grip on the pavement and your car will fly straight off the track.

If her car is sophisticated, it will feature wings front and back that press the car down at high wind speed, creating a higher force on the pavement, say 2,000 lbs. That allows the car to take the curve at 2G.

Our moon is subject to a centripetal force from gravity that creates a curve so the moon stays in its orbit.

Earth is subject to varying centripetal force from gravity that creates an elliptical orbit.

Your hand, finger, and pen have mass too, so unless your muscles can generate unlimited force, you will naturally slow down to negotiate the curve.

Clearly, there are physical constraints that govern the speed and curvature of the car as well as the finger.

What emerges here is the possible path, the curve, or ellips.

Does the Power Law apply to all of this?

If the moon slows down, it will drop into a lower orbit. Does that change anything?

Do the race car wings change the power law for negotiating that curve on the race track?

Just wondering what the Power Law is all about.

I think that is a good idea. But I have already done something similar. I had participants do a task that captures something of your suggestion. I had participants draw elliptical cursor movements “free hand” on the screen while a very non-elliptical disturbance was being applied to the cursor. The results for one run were presented as Figure 9 in Marken et al (2022). Here is a graph of the results:

The solid lines are the cursor movements and the dashed lines are the mouse movements made by the participant and model. In both cases, the cursor movements follow a -1/3 power law pretty closely; the mouse movements don’t. I think this shows that the power law is an irrelevant side effect of the actions (non-elliptical mouse movements) that produce the controlled result (fairly elliptical cursor movements).

Hi Dag,

I think you are partially right, and I am sorry I have had to continue the discussion, but I have done it only because Rick cannot admits his coarse conceptual errors in his argumentation. These errors have almost nothing to do with PCT but because these arguments are used to ground a certain PCT view, they are in a danger to discredit whole PCT if continued.

(The core of the error is Rick’s assumption that if there is a mathematical dependence between three variables, like curvature, velocity and affine velocity of a trajectory or width, height and area of a rectangle, it follows that there is also a mathematical dependence between the two variables curvature and velocity or width and height. But that assumption holds only if we know the value of the third variable which is produced from the two other or if we know that it remains constant. In reality the third value will not always remain constant and we can determine its value only bay calculating it from the two other.)

I hope Adam will correct and give a better answer to what follows. I think that power law of curved movement (there also other power law phenomena) means that in (very) many cases or case types – but not all – there is a negative power relationship between curvature and velocity, and very often (but not always) the power coefficient is -1/3. Some how there must of course be a physical background but that is just the complicated question. For example when we just observe elliptical movement we perceive its velocity constant if there is that power relation which means that in reality the velocity slows down in steep curves. Also in Ricks last message the cursor follows power law even if it does not have any determined mass but the mouse and hand do not follow even though they clearly have a mass.

Your car example is nice because there the slowing in the curves is a controlled variable. Often we have to assume that power law phenomenon is not controlled but a side effect.

What a coincidence. I was continuing the discussion because you (and Martin, Adam, Warren, Bruce, and now Dag – sorry if I left anyone out) cannot admit your conceptual errors that have everything to do with PCT (because we’re dealing with a CONTROL process) and reflect a surprisingly deep lack of understanding of PCT, particularly in how the theory applies to actual behavior.

It’s neither an assumption nor an error. Here is the derivation of that “dependence”, per Maoz et al:

The reason velocity, v(t), curvature, kappa(t), and affine velocity, alpha(t), are mathematically related is because they are all functions of the same variables, the first and second derivatives of x and y, which are the coordinates of the changing position of the entity being moved. What this means is that when you are making a planar movement you are simultaneously varying its velocity, curvature and affine velocity.

This fact about the power law is something that I noticed when I was first asked about how PCT explains it – though at that time all I knew was that velocity and curvature were being produced simultaneously --and I noted that near the beginning of the Marken and Shaffer (2017) paper:

Somehow all those physicists who were working on the power law well before I came along failed to notice this fact. But it was this fact that led me to the PCT explanation of the power law: that it was an irrelevant side effect of controlling the x,y position of the entity being moved.

Yes, that is a correct description of the phenomenon.

We already know the physical background of the power law. It’s described in the quote above from Marken & Shaffer (2017): Muscle forces are applied to the entity being moved (a finger or a stylus, for example). Those forces must be enough to compensate for disturbances which include the mass of the entity being moved and any resistive forces (friction, air resistance) affecting it.

In Dag’s example the main controlled variable is probably centrifugal force. This variable is controlled by controlling two other variables, the speed and degree of curvature of the path of the car. Speed and path curvature are controlled as the means of staying ahead of the other cars and, simultaneously, staying on the road. Staying ahead of the other cars means going as fast as possible and hugging the curves as tightly as possible; staying on the road involves controlling the position of the car relative to the shape of the road while varying one’s speed and tightness around curves so centrifugal force doesn’t exceed the limits of adhesion.

This is a situation where you would see some “real” slowing through curves. But in this case the slowing is done in “response” to the disturbance of the curvature of the road, which is independent of the driver’s actions (wheel turning). The curvature is a disturbance to the curvature of the path the driver has to take to stay on the road. But as a side effect it is a disturbance to control of centrifugal force because if, to stay on the road, the driver has to take a tightly curved path, she has to slow down to keep from going at a speed that produces a centrifugal force that exceeds the limit of adhesion to the road.

So in the case of a race driver going around a race course, a negative correlation between speed and the disturbance curvature of the road would reflect a true disturbance-output relationship of a control system – the system controlling the perception of centrifugal force. If, however, you looked at the relationship between the curvature and speed of the path of the car itself, you would find something close to the -1/3 power law, which would be an irrelevant side effect of the driver controlling the position of the car.

I’ll just end by saying that the importance of understanding the PCT explanation of the power law of movement, as an irrelevant mathematical/ statistical side effect of control, is that it serves as an excellent acid test of one’s understanding of PCT. And so far, no one in this discussion (other than me) has passed the test.

To quote one of our great (and hopefully soon to be incarcerated) ex-presidents: Sad.