Side Effects of Control

[rsmarken]

@amatic

@rsmarken

One of the first papers I published on what is now called PCT (when I still hadn’t mastered the “that-which” distinction, I’m afraid) was about distinguishing intentional from accidental behavior. What I called “accidental” behavior is behavior that is an unintended side effect of intentional behavior, which, of course, is the behavior that is recognized in PCT as control.

I mention this because the topic of “side effects” came up in the recent discussion of the power law of movement. Several times in that discussion I described the power law as an irrelevant side effect of control. While there seemed to be general agreement that the power law is, indeed, a side effect of control, there were objections to my calling it “irrelevant”. The main objection seemed to be that, by calling the power law “irrelevant”, I was saying that it was not worth studying it. And that is not what I meant.

You should stick with PCT on this - find controlled variables.

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.

The phenomenon is that people slow down in curves in most “natural” movement (“natural” meaning not too slow or too fast). The phenomenon is VERY RELEVANT because any model that purports to explain movement needs to also slow down in curves just like the people do, or not slow down, just like the people do.

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.

Learn the basics of PCT, do some experiments, build a model, then you can understand what is relevant and what is irrelevant.

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

[amatic]

@rsmarken

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.

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.

@rsmarken

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.

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.

[amatic]

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.

[rsmarken]

@amatic

@rsmarken

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.

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.

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.

[rsmarken]

@amatic

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?

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

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.

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.

[amatic]

@rsmarken

The correlation is real; a result of the mathematical relationship between velocity, curvature and affine velocity.

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

What is apparent is that the slowing down occurs in response to the curvature.

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?

@rsmarken

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.

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.

@rsmarken

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.

Yes, that is a different model.

[rsmarken]

@amatic

@rsmarken

The correlation is real; a result of the mathematical relationship between velocity, curvature and affine velocity.

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

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.

What is apparent is that the slowing down occurs in response to the curvature.

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?

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.

@rsmarken

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.

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.

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?

[amatic]

@rsmarken

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.

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.

@rsmarken

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.

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.

@rsmarken

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?

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.

[rsmarken]

@amatic

@rsmarken

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.

Maoz et al don’t talk about omitted variables.

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.

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.

Correct.

@amatic

@rsmarken

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.

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.

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).

@rsmarken

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?

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.

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

[MartinT]

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.

[Archivist]

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.

[rsmarken]

@MartinT

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.

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).

[EetuP]

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.

[rsmarken]

@EetuP

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.

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.

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 …

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.

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.

Yes, that is a correct description of the phenomenon.

Some how there must of course be a physical background but that is just the complicated question.

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.

Your [Dag’s] 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.

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.

[EetuP]

Who could have believed that after years of insisting that we should always start from data and not from theory, you now manage to invent a previously (and also afterwards) unknown mathematical law which makes all data obsolete. No mean feat! But, please read on.

RM: Here is the derivation of that “dependence”, per Maoz et al:

It doesn’t matter at all how the “derivation” is done, the end result is that we have the original two variables, in this case velocity (V) and curvature (C), whose internecine correlation we are interested in, and the third variable which is a product of these two in the form (V^x * C^y), in this case it happens to be the so called affine velocity (Va or alpha in original text) in the form (Va = V * C^1/3).

RM: 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.

Yes, they are all functions of the same variables, the x and y components of the movement, but they are that in a certain order and with certain internecine relations: V and C are independent variables so that they do no way determine each other: velocity can be same in steep and gentle curve, and the same curve can be moved with different velocities. Instead affine velocity is a function of both velocity and curve and fully determined by them. It cannot “affect back” to them. This is exactly the same situation as with width, height and area of rectangular.

In the musings about causality in the quotation (as a picture in the original message) from Marken and Shaffer (2017) are bad flaws. The curvature and velocity measured at the same point are simultaneous and so cannot causally affect each other according to the normal view of event causality, that is true. But nothing prevents the curvature in the previous measure point from affecting the velocity in the next point. This can work also in another way around: if you drive too fast to the curve you can slide out from the road. But still much worse failure is to think that the missed third variable which could affect both velocity and curvature could be a variable which itself is a function of them both! Instead as far as I can see the power law researchers have just been searching for those third variables, which of course must be independent physical forces, like just the centripetal force mentioned by Dag.

RM: 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.

Yes, they must be enough to that and in addition they sometimes somehow cause the power relationship between the velocity and curvature, and this is the question: why and how and when?

RM: 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 centripetal 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.

If there happens to be the curvature-velocity relationship something close to the -1/3 power law and that relationship suits to keep the car on the road then it is not at all an irrelevant side effect but just the means for the driver’s control.

RM: 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.

It is always easy to pass self-set tests. I personally really hope that no one else will ever pass it, for the sake of the future of PCT. And I am sorry about your downfall.

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

Aah, now I understand! You have moved to the post-truth world of alternative facts. Your inventions of that mystical law of the relationship between two variables because of the relationship between three variables and the OVB analysis fit very well to that world. Now, by setting a suitable form and exponents to the third variable and adding it to the predictor variables of a correlation analysis you can produce a strong and exceptionless correlation between any possible two variables with just the slope and power coefficients you happen to wish for. Whoah, if the news about this spread to our world, no one will any more take seriously the unusually high correlations typical to the PCT research.

It would be sad if it were not so funny at the same time.

[rsmarken]

@EetuP

RM: Here is the derivation of that “dependence”, per Maoz et al:

It doesn’t matter at all how the “derivation” is done, the end result is that we have the original two variables, in this case velocity (V) and curvature (C), whose internecine correlation we are interested in, and the third variable which is a product of these two in the form (V^x * C^y), in this case it happens to be the so called affine velocity (Va or alpha in original text) in the form (Va = V * C^1/3).

True.

RM: 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.

Yes, they are all functions of the same variables, the x and y components of the movement, but they are that in a certain order and with certain internecine relations: V and C are independent variables so that they do no way determine each other: velocity can be same in steep and gentle curve, and the same curve can be moved with different velocities. Instead affine velocity is a function of both velocity and curve and fully determined by them. It cannot “affect back” to them. This is exactly the same situation as with width, height and area of rectangular.

True

In the musings about causality in the quotation (as a picture in the original message) from Marken and Shaffer (2017) are bad flaws. The curvature and velocity measured at the same point are simultaneous and so cannot causally affect each other according to the normal view of event causality, that is true. But nothing prevents the curvature in the previous measure point from affecting the velocity in the next point.

If curvature at time t affects speed at time t+dt then what affects the curvature a time t+dt? If it’s the speed at time t then we seem to have the same situation that would exist if the causal link were between the simultaneous occurrence of speed and curvature: curvature causing the speed that causes the curvature that causes the speed…

RM: 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.

Yes, they must be enough to that and in addition they sometimes somehow cause the power relationship between the velocity and curvature, and this is the question: why and how and when?

But isn’t that answered by the control model of movement behavior? The “why” is the varying reference signal that specifies the desired position of the entity being moved; the “how” is an error signal that continuously causes exactly the right amount of output (force) to keep the perceived state of that entity matching the reference; and the “when” depends on the transport lag in the control loop.

RM: 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 centripetal 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.

If there happens to be the curvature-velocity relationship something close to the -1/3 power law and that relationship suits to keep the car on the road then it is not at all an irrelevant side effect but just the means for the driver’s control.

As I said in the quote above, that would, indeed, be the case – a power law curvature - velocity relationship would not be an irrelevant side effect of control-- if the curvature being measured is that of the road (a variable that is independent of the driver’s actions and, thus, qualifies as a potential disturbance to a controlled variable) and the velocity is that of the car (which is a variable that is dependent on the driver’s actions and could also affect a potential controlled variable).

RM: 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.

It is always easy to pass self-set tests. I personally really hope that no one else will ever pass it, for the sake of the future of PCT. And I am sorry about your downfall.

I agree, and it was rude for me to say that “no one in this discussion (other than me) has passed the test.” It was uncalled for. I apologize. But I don’t understand what you think is my “downfall”.

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

Aah, now I understand! You have moved to the post-truth world of alternative facts…

Oh, I see. You interpreted my (also uncalled for) statement about the “great” ex-president as support for Trump. Don’t worry, it’s not. The only thing about Trump that I consider “great” is that he serves as a great example to my granddaughter of what a human being should not be.

[EetuP]

Rick,

Encouraging that you agree the first two quotations from my previous message!

RM: If curvature at time t affects speed at time t+dt then what what affects the curvature a time t+dt? If it’s speed at time t then we seem to have the same situation that would exist if the causal link were between the simultaneous occurrence of speed and curvature: curvature causing the speed that causes the curvature.

We should be very careful with “affecting” and “causing”. Probably very many things or events can affect the speed and curvature, in some (rare) cases they can even affect each other. (As in my example too much speed can straighten the curve in a slippery surface.) But I can’t se how they could ever cause each other. Rather in our normal example cases it the moving organism which causes both the speed and curvature by using its legs or other means in certain ways.

(I forgot to mention about the quotation from your paper that if v and c cannot be cause and effect because of their simultaneity then of course neither affine velocity can it because it is as well simultaneous.)

RM: But isn’t that answered by the control model of movement behavior? The “why” is the varying reference signal that specifies the desired position of the entity being moved; the “how” is an error signal that continuously causes exactly the right amount of output (force) to keep the perceived state of that entity matching the reference; and the “when” depends on the transport lag in the control loop.

That is an interesting answer to my question: Why and how and when the muscle forces cause the power relationship between the velocity and curvature? Have you, or Adam, tested how the transport lag affects the power law relationship in an elliptical movement?

RM: I agree, and it was rude for me to say that “no one in this discussion (other than me) has passed the test.” It was uncalled for. I apologize. But I don’t understand what you think is my “downfall”.

Thanks, I appreciate your apologizing. With your “downfall” I meant your fate that you passed the test. That you had become to erroneously believe that from the dependence relationship between those three variables follows the dependence between two of them. And that it is adequate to add the third variable as an predictor to the correlation analysis between those two.

[rsmarken]

@EetuP

RM: But isn’t that answered by the control model of movement behavior? The “why” is the varying reference signal that specifies the desired position of the entity being moved; the “how” is an error signal that continuously causes exactly the right amount of output (force) to keep the perceived state of that entity matching the reference; and the “when” depends on the transport lag in the control loop.

That is an interesting answer to my question: Why and how and when the muscle forces cause the power relationship between the velocity and curvature? Have you, or Adam, tested how the transport lag affects the power law relationship in an elliptical movement?

In the PCT model, the observed relationship between velocity and curvature is an irrelevant side-effect of controlling for the varying position of the moved entity. The particular relationship between velocity and curvature you find will depend on the trajectory of that movement and on how the movement is filtered, if filtering is done prior to the regression analysis.

The PCT model predicts that two different aspects of the control process will affect the observed relationship between velocity and curvature: 1) the particular trajectory specified by the secularly varying reference specification for the entity moved, which affects the shape of the trajectory that is produced and 2) characteristics of functions in the control loop – specifically, the output function gain and slowing, transport lag and characteristics of the feedback function – that affect the nature of the temporal filtering done by the control loop.

Exactly how variations in these two factors affect the nature of the observed relationship between the velocity and curvature of a particular movement is TBD (to be determined), if someone thinks that that is worth determining. But I have shown that changes in the feedback function can change the nature of the observed relationship between velocity and curvature when the model is controlling for making an elliptical movement.

The difference in the observed velocity - curvature relationship for an ellipse drawn by a model with a high versus a low gain feedback function mimicked rather well the difference in the velocity - curvature relationship that is observed when a person draws an ellipse in the air versus under water.

[MartinT]

I think that if I were to try to make a PCT model of, say, driving along a winding road, one perception that would come into the model would be the most distant point along the road that the driver could see. If I were the driver on a road with a cliff obscuring the right-hand curve just ahead, I might take the curve a bit slower than if the road were on a flat plane where I could see a long way ahead. And if the road followed a curving ridge line with a steep drop-off both sides, I might take the curve just a bit slower than if it were on the flat. From a PCT viewpoint the view from the point (the moving car) would seem to matter, since it would affect the controlled perception of where the car was in its lane.

[rsmarken]

@MartinT

I think that if I were to try to make a PCT model of, say, driving along a winding road, one perception that would come into the model would be the most distant point along the road that the driver could see.

I think that’s unlikely and unnecessary. Unlikely because the determination of “farthest” would be rather imprecise when that point is beyond the distance where perceptual distance can be determined by binocular disparity (about 2 meters for humans, I believe). And unnecessary because controlling for the farthest point won’t keep you on the road since you might have to go in a direction that takes you off the road in order to keep that point perceived as farthest. Maybe the variable should be optical angle of the direction of the car relative to that farthest point. But even that doesn’t sound promising.

When I drive on curved roads, especially one that runs along cliffs, like the beautiful highway 1 here in California, I think I control for staying in my lane and as far from the cliff edge as possible – without going into the oncoming lane – when my lane is on the cliff side. I adjust my speed in inverse proportion to the steepness of the approaching curve as well as to the “blindness” of the curve. So I’m adjusting my speed to control for a lot of variables, including the loudness of the screams of the passengers;-)