Behavioral Illusions: The Basis of a Scientific Revolution

[rsmarken]

Hi Adam

AM: These sorts of correlations are why everyone gets upset when you say that the power law is a statistical artifact.

RM: They certainly do get upset. I think if they calmed down a bit they would see that I simply showed that the curvature and velocity of curved movements are mathematically confounded variables. If you want to see if either one is controlled you would have to find a way to look at the effect of a disturbance on each variable while factoring out the confounding effect of the other. I think that would be a great research project.

AM: On the standard position tracking model and the LittleMan - there is not much to say. Both have position tracking as the only level or top level control system. If you put a fast target elliptical target, they will not follow as accurately as humans do.

RM: Have you looked to see if there is a power law relationship between curvature and velocity for the Little Man tracking an elliptical target? To the extent that the tracking is reasonably good I predict that there will be.

Best

Rick

[amatic]

RM: Was the instantaneous pen speed (upper graph) measured in the same way as the instantaneous target speed (lower graph)? If both were measured as angular velocity, how did you generate elliptical target movement with constant angular velocity?

Speed was measured as displacement over time, in the same way for the target and for the pen. Not as angular velocity, just normal speed as the magnitude of the velocity vector.

I suppose there are many ways to generate elliptical trajectories with a constant speed, we used formulas from Huh and Sejnowski (2015) for generating pure frequency curves with any given beta. Another way is using a list of points in an ellipse, and then setting the time span between successive points as dt[i] = distance[i] /speed. Then you get a list of points and a list of time spans, which can be interpolated and used for target trajectories.

RM: They certainly do get upset. I think if they calmed down a bit they would see that I simply showed that the curvature and velocity of curved movements are mathematically confounded variables.

From my analysis of human hand trajectories - the slowing down in high curvature is real, not a statistical artifact.

How do you propose to verify your claim? What test would prove that it is correct or not correct, other than the omitted variable bias test? Can you think of any other way?

RM: Have you looked to see if there is a power law relationship between curvature and velocity for the Little Man tracking an elliptical target? To the extent that the tracking is reasonably good I predict that there will be.

We’ve looked at the position tracking model in 2D in detail, and also a 2-segment arm modeled after the LittleMan. If we want good tracking, then the target speed needs to be low. In that case, the model cursor follows whatever speed profile the target has, so you did not predict correctly. If the target speed is constant - no power law in the model. If the target speed is correlated with curvature, so will be the model’s speed.

If the target speed is high, then the model is not following very accurately at all. In many cases it does show speed-curvature correlation. This seems to be the effect of “filtering”, where you can see the target as a near-sinusoid, and then the filtering by the model converts it to a pure sinusoid.

[rsmarken]

Hi Adam

RM: Was the instantaneous pen speed (upper graph) measured in the same way as the instantaneous target speed (lower graph)?

AM: Speed was measured as displacement over time, in the same way for the target and for the pen. Not as angular velocity, just normal speed as the magnitude of the velocity vector.

RM: So you measured speed as
?
Or was it something else? It looks like that’s what you used in the power law plots. I presume those are measures of V plotted against R (curvature) for the participants movements. However, if the values of R are measure of target curvature and those measures of R can be shown to be uncorrelated with the corresponding R’s for the participant’s movement then the power law relationship between R and V would provide some real evidence that the participant is slowing down through curves.

AM: I suppose there are many ways to generate elliptical trajectories with a constant speed, we used formulas from Huh and Sejnowski (2015) for generating pure frequency curves with any given beta. Another way is using a list of points in an ellipse, and then setting the time span between successive points as dt[i] = distance[i] /speed. Then you get a list of points and a list of time spans, which can be interpolated and used for target trajectories.

RM: But doesn’t that affect your measures of curvature?

RM: They certainly do get upset. I think if they calmed down a bit they would see that I simply showed that the curvature and velocity of curved movements are mathematically confounded variables.

AM: From my analysis of human hand trajectories - the slowing down in high curvature is real, not a statistical artifact.

RM: You are not the only one who has concluded that. I believe you are all wrong.

AM: How do you propose to verify your claim? What test would prove that it is correct or not correct, other than the omitted variable bias test? Can you think of any other way?

RM: Nope. I think these two equations will do:


RM: This shows that there is a mathematical correlation between curvature (R or A) and speed (V or A), the size of the correlation dependent on the correlation between curvature and affine velocity (D). So the degree to which any movement trajectory fits a power law depends on the nature of the trajectory itself, not on the way it the trajectory was generated.

RM: Have you looked to see if there is a power law relationship between curvature and velocity for the Little Man tracking an elliptical target? To the extent that the tracking is reasonably good I predict that there will be.

AM: We’ve looked at the position tracking model in 2D in detail, and also a 2-segment arm modeled after the LittleMan. If we want good tracking, then the target speed needs to be low. In that case, the model cursor follows whatever speed profile the target has, so you did not predict correctly. If the target speed is constant - no power law in the model. If the target speed is correlated with curvature, so will be the model’s speed.

RM: I guess you are saying that the model doesn’t behave like humans in the same situation. That’s a surprise to me since my simple tracking model does; it produces a power law when the person does and no power law when the person doesn’t (which is actually most of the time). But I’ll check it out myself with the Little Man when I get a chance.

Best regards

Rick

[amatic]

RM: So you measured speed as
?

Yep. Root of the x speed squared plus the y speed squared, Pythagora’s theorem, pretty old stuff.

RM: It looks like that’s what you used in the power law plots. I presume those are measures of V plotted against R (curvature) for the participants movements.

It was curvature as C = 1/R, where R is the circle inscribed in some three very close points on the curve

RM However, if the values of R are measure of target curvature and those measures of R can be shown to be uncorrelated with the corresponding R’s for the participant’s movement then the power law relationship between R and V would provide some real evidence that the participant is slowing down through curves.

I don’t see the logic there. Participant’s and target’s C are uncorrelated, though.

Or if you prefer radius of curvature:

RM: But doesn’t that affect your measures of curvature?

Not sure what you mean, yes and no. Curvature is always the same in the same spatial points, but if the target is moving at different speeds, then it will appear at points of same curvature at different times, here is a plot, orange and red are from a PL trajectory, and blue and green from a constant speed trajectory.

AM: From my analysis of human hand trajectories - the slowing down in high curvature is real, not a statistical artifact.
RM: You are not the only one who has concluded that. I believe you are all wrong.

I know you do. Why is speed over time not convincing? That is the simplest evidence I can think of. It certainly looks like the speed is going from low to high, and it looks relatively slow exactly in the parts of high curvature, and relatively fast in parts of low curvature. The formula for speed is pretty basic too, no room for statistical tricks.

RM: Nope. I think these two equations will do:


RM: This shows that there is a mathematical correlation between curvature (R or A) and speed (V or A), the size of the correlation dependent on the correlation between curvature and affine velocity (D). So the degree to which any movement trajectory fits a power law depends on the nature of the trajectory itself, not on the way it the trajectory was generated.

I don’t see what that means. The degree to which any trajectory fits a power law depends on the trajectory. It can be generated by a person or generated by a computer program with any degree of correlation.

The power law itself does not tell you how it was generated - did someone claim otherwise?

RM: I guess you are saying that the model doesn’t behave like humans in the same situation. That’s a surprise to me since my simple tracking model does; it produces a power law when the person does and no power law when the person doesn’t (which is actually most of the time). But I’ll check it out myself with the Little Man when I get a chance.

That is not what I’m saying. For random movements of the target - the model will behave just like a person, and produce power law in the same situations. If the helicopter is slowing down in curved parts, then probably the person will slow down too, if he is tracking well. If the helicopter is not slowing down, the person might not either. The model will reproduce that behavior beautifully, great stuff. Not relevant for the power law, or rather for movement along ‘predictable’ paths. In helicopters, the ‘appearance’ of the power law depends largely on helicopter movements and how well the person can track them. In fast elliptical movements, for example, target speed profile does not really matter, as long as the cycle frequency (and average speed) are high, the person will slow down in curved parts, and go fast in other parts.

What I was saying is that for slow movements in an elliptical trajectory - if the speed of the target is constant, the model will also have a constant speed, because it is tracking well. If the target speed is not constant, and is varying with curvature, then the model will vary its speed accordingly.

When the target speed is on average high, as I showed on those plots, the model will not track accurately, neither position nor velocity, and the power law might appear, but in this case it doesn’t matter, because the accuracy is so low.

[amatic]

RM: But I’ll check it out myself with the Little Man when I get a chance.

Please do. The LittleMan has position control as top level systems, so it will behave more-less the same as just a position control system. If you just take x and y position control loops, standard proportional-and-slowing, and then a very fast target, you will see for yourself exactly what happens. The LittleMan will not behave very differently, beacuse it also does not have the top level systems demonstrated in Bill’s portable demonstration of following circular targets.

[rsmarken]

Hi Adam

RM: However, if the values of R are measure of target curvature and those measures of R can be shown to be uncorrelated with the corresponding R’s for the participant’s movement then the power law relationship between R and V would provide some real evidence that the participant is slowing down through curves.

AM: I don’t see the logic there. Participant’s and target’s C are uncorrelated, though.

RM: The logic is that if there is a power law relationship between target curvature and the subject’s pen velocity then it could not be attributed to the mathematical dependency that exists between measures of curvature and velocity in the subjects pen movement. It would show that subjects are, indeed, slowing down at curves, perhaps because they are controlling some constant relationship (ratio?) between target curvature and pen speed.

RM: But this conclusion would only be valid if target curvature is completely uncorrelated with the subject’s pen curvature and according to your graphs that is definitely the case. So if you correlate target curvature with pen speed and find a power law relationship then that would be pretty good evidence that the subject is, indeed, slowing down at curves. But given the data at hand I think that it is highly unlikely that you will find this. You will almost certainly find no clear relationship at all between target curvature and subject pen velocity, which would actually be pretty good evidence that subject’s don’t slow down at curves, at least in this situation.

RM: You are not the only one who has concluded that. I believe you are all wrong.

AM: I know you do. Why is speed over time not convincing?

RM: Because that relationship is a mathematical consequence of the variance in another variable – a confounding variable: affine velocity.

RM: …So the degree to which any movement trajectory fits a power law depends on the nature of the trajectory itself, not on the way it the trajectory was generated.

AM The power law itself does not tell you how it was generated - did someone claim otherwise?

RM: I guess I just assumed that that was why people have been studying the power law for all these years. It looks like researchers are trying to understand why movements follow the power law (when they do) in order to learn something about how movements are produced. I thought that was why some researchers were developing all these models of movement that are aimed at showing why people produce movements that follow a power law. I thought that was why I was asked for the PCT explanation of the power law. I thought that was why you are searching for the controlled variables that explain the power law – variables that that you just can’t seem to find. But if you don’t think that the power law tells anything about how movement is produced (spoiler alert: it’s control of perception) then what is your interest in the power law?

RM: The power law is of interest to me because it’s another great example of a “red herring pulled across the path of progress” in understanding human nature and, thus, a line of research that is preventing the scientific revolution Bill described in the 1978 Psych review paper – a revolution which, by the way, is the topic of this thread.

Best

Rick

[amatic]

RM: The logic is that if there is a power law relationship between target curvature and the subject’s pen velocity then it could not be attributed to the mathematical dependency that exists between measures of curvature and velocity in the subjects pen movement. It would show that subjects are, indeed, slowing down at curves, perhaps because they are controlling some constant relationship (ratio?) between target curvature and pen speed.

You are mixing something here that is not usually mixed. No one is measuring target curvature vs subject velocity for the power law. It is subject pen velocity vs subject pen curvature.

Other variants of this experiment include rhythmical movement, as in drawing the ellipse every 1 second, following a metronome rhythm, and following a template on paper with a pen. Also in there, the measures are pen speed and pen curvature.

AM: Why is speed over time not convincing?
RM: Because that relationship is a mathematical consequence of the variance in another variable – a confounding variable: affine velocity.

Do you agree that the speed is not constant in that plot?

The green line is oscillating between about 150 mm/s and about 300m/s, that is a factor of 2 slowdown at some parts of the curve. Do you agree this is correct?

Is the problem exactly the location of the slowdown, that is, do you think they are slowing down in the less curved parts?

RM: I guess I just assumed that that was why people have been studying the power law for all these years. It looks like researchers are trying to understand why movements follow the power law (when they do) in order to learn something about how movements are produced. I thought that was why some researchers were developing all these models of movement that are aimed at showing why people produce movements that follow a power law. I thought that was why I was asked for the PCT explanation of the power law.

The question usually goes like this: when making fast curved movements, hand speed is correlated with curvature; why does the brain select exactly this trajectory, from the infinite set of possible trajectories?

One answer is - because those trajectories are optimal in some way, they spend least energy, they are the smoothest, etc, etc.

Another answer is - the brain doesn’t really care about trajectories, it is not controlling trajectories, there are other perceptual variables involved, and the slowing down in curved parts is really a side effect of controlling something else.

RM: I thought that was why you are searching for the controlled variables that explain the power law – variables that that you just can’t seem to find. But if you don’t think that the power law tells anything about how movement is produced (spoiler alert: it’s control of perception) then what is your interest in the power law?

The power law is really just an interesting check on the model. Other checks are that the model produces trajectory at the same accuracy level as human subjects, that there is similar lead and follow dynamics, that the model generalized across different curves, etc.

The red herring is simply assuming that the brain selected the trajectory, ‘cares’ about the trajectory, etc. That path potentially includes a lot of complicated computation that might not be possible in the brain in real time, that’s why it is a red herring, because it leads to complicated models.

[amatic]

This is the same data as before, from tracking a target moving with high constant speed along an elliptical path. There is no calculation of the power law, there is no calculation of curvature, there is just speed calculated as distance over time between two very close points. Each point’s color shows the speed of movement.

[amatic]

Do you see that in this case the speed of the subject’s pen was lower in the curved parts and higher in the straight parts? Blue = slow, red=fast.

That is the very regular phenomenon that is described in shorthand form as the speed-curvature power law. The slowing is real, not a statistical artifact, the plot above has absolutely no statistics, just plain old speed. Dead simple. You can try it at home.

So, what does a good scientist do when he realizes he has published mistakes?

[rsmarken]

Hi Adam

RM: The logic is that if there is a power law relationship between target curvature and the subject’s pen velocity then it could not be attributed to the mathematical dependency that exists between measures of curvature and velocity in the subjects pen movement. It would show that subjects are, indeed, slowing down at curves, perhaps because they are controlling some constant relationship (ratio?) between target curvature and pen speed.

AM: You are mixing something here that is not usually mixed. No one is measuring target curvature vs subject velocity for the power law. It is subject pen velocity vs subject pen curvature.

RM: I know. I think they should be looking at the relationship between target curvature and subject pen velocity. That’s the negative relationship that should exist if subjects are slowing down in curves as the means of maintaining accuracy.

RM: Because that relationship is a mathematical consequence of the variance in another variable – a confounding variable: affine velocity.

AM: Do you agree that the speed is not constant in that plot?

RM: Yes

AM: The green line is oscillating between about 150 mm/s and about 300m/s, that is a factor of 2 slowdown at some parts of the curve. Do you agree this is correct?

RM: Yes

AM: Is the problem exactly the location of the slowdown, that is, do you think they are slowing down in the less curved parts?

RM: No.

RM: I guess I just assumed that that was why people have been studying the power law for all these years. It looks like researchers are trying to understand why movements follow the power law (when they do) in order to learn something about how movements are produced. I thought that was why some researchers were developing all these models of movement that are aimed at showing why people produce movements that follow a power law. I thought that was why I was asked for the PCT explanation of the power law.

AM: The question usually goes like this: when making fast curved movements, hand speed is correlated with curvature; why does the brain select exactly this trajectory, from the infinite set of possible trajectories?

RM: I know. That question assumed a causal model of movement. It is not the question you would ask if you could see that behavior is control.

Best

Rick

[rsmarken]

Hi Adam

@amatic

This is the same data as before, from tracking a target moving with high constant speed along an elliptical path. There is no calculation of the power law, there is no calculation of curvature, there is just speed calculated as distance over time between two very close points. Each point’s color shows the speed of movement.

Do you see that in this case the speed of the subject’s pen was lower in the curved parts and higher in the straight parts? Blue = slow, red=fast.

That is the very regular phenomenon that is described in shorthand form as the speed-curvature power law. The slowing is real, not a statistical artifact, the plot above has absolutely no statistics, just plain old speed. Dead simple. You can try it at home.

So, what does a good scientist do when he realizes he has published mistakes?

RM: I imagine he would try to learn from it. Am I the scientist you are talking about? If so, what mistake do you think I made?

Best

Rick

[amatic]

Thanks for staying in the discussion.

RM: I know. I think they should be looking at the relationship between target curvature and subject pen velocity. That’s the negative relationship that should exist if subjects are slowing down in curves as the means of maintaining accuracy.

Maybe. Do you mean target curvature at the time instant where the pen is slowing down, or do you mean curvature of the target path at the spatial point where the pen is slowing down? What if the pen is not exactly on the same path that the target passed?

You don’t really need a target to confirm what the pen is doing. Slowing down in curved parts was noticed a long time ago in writing with an Edison pen, the mechanism that later became the tattoo pen.
history of the tattoo machine

That was a machine intended for copying documents, it made constant 50Hz needle oscillations, and someone noticed that in the more curved parts, the dots were closer together than in the straight parts, like in the letter a above.

RM: Yes. Yes. No.

Great!

AM: The question usually goes like this: when making fast curved movements, hand speed is correlated with curvature; why does the brain select exactly this trajectory, from the infinite set of possible trajectories?
RM: I know. That question assumed a causal model of movement. It is not the question you would ask if you could see that behavior is control.

The question assumes that the person has the ability to produce desired trajectories. The solution can be open-loop command of trajectory. I guess this is what you call “causal model of movement”. But it can also be a closed loop force-speed-position control. I think both are wrong solutions as models of human movement, but for different reasons. First solution, because it is not really control, it works good only for industrial robots in stable environments.

Second solution - because it requires very fast loops and high power actuators, and humans don’t really have those. Trajectory control is quite possible at low speeds, I think, people can follow slow, predictable targets fairly well. The problems appear at high speeds, and I think people somehow gradually transition to a different mode of control when the target is fast and predictable.

RM: what mistake do you think I made?

AM: From my analysis of human hand trajectories - the slowing down in high curvature is real, not a statistical artifact.
RM: You are not the only one who has concluded that. I believe you are all wrong.

This is the mistake. Not believing that speed is lower in curved parts, and higher in flatter parts.*

There could be questions about the statistics of the power law, and weather the movements can be summarized in a single measure, and how general is the law, and at what speeds it appears, and what is ‘unconstrained movement’, and when it can appear without actual slowing down in curves as a statistical artifact, and the difference between ‘spin angular velocity’ and just speed as used in the measurements, etc, etc, etc.

In my opinion, based on analyzing human movement data, movements at high speed along known paths are well summarized with the speed-curvature power law, but with different exponents depending on the shapes of the paths. Slow movements are not at all fitting power laws.

*However, instantaneous speed is not a statistical measure. It is just a derivative of position, nothing fancy going on. All experiments I’ve conducted so far, and there was a ton of them, show that people simply cannot maintain constant high speed when moving along curved paths, even when explicitly asked to do so. They always slow down in curved parts.

[rsmarken]

Hi Adam

AM: Thanks for staying in the discussion.

RM And thanks to you too!

RM: I know. I think they should be looking at the relationship between target curvature and subject pen velocity. That’s the negative relationship that should exist if subjects are slowing down in curves as the means of maintaining accuracy.

AM: Maybe. Do you mean target curvature at the time instant where the pen is slowing down, or do you mean curvature of the target path at the spatial point where the pen is slowing down?

RM: I would start by looking at the correlation between between target curvature and pen speed at different leads/lags between target and cursor. If there is not a strong correlation at any lead/lag (as I think there won’t be) then I’d move on to something else.

AM: What if the pen is not exactly on the same path that the target passed?

RM: I don’t think that matters though you could definitely keep that data.It might be a useful covariate.

RM: what mistake do you think I made?

AM: From my analysis of human hand trajectories - the slowing down in high curvature is real, not a statistical artifact.

RM: You are not the only one who has concluded that. I believe you are all wrong.

AM: This is the mistake. Not believing that speed is lower in curved parts, and higher in flatter parts.

RM: To paraphrase the Cowardly Lion ( I do believe that speed is lower in curved parts, and higher in flatter parts. I do I do I do. I do I do.I do believe that speed is lower in curved parts, and higher in flatter parts.

RM: What I don’t believe is that hand trajectories slow down in high curvature areas. This implies that the slowing was done in response to the high curvature. This can’t be the case, as we noted in the “Cause and Correlation” section at the beginning of Marken and Shaffer, 2017).

RM: All that the power law shows is that speed is slower in curved parts and faster in flatter parts of a movement. The Marken & Shaffer papers show that this association is a result of the mathematical relationship that exists between speed and curvature. The “statistical artifact” we refer to is that the relationship between speed and curvature often follows a “power law” – where the “law” is that the exponent of that relationship is close to 1/3 or 2/3, depending on how speed and curvature are measured.

RM: This is an artifact because the power “law” is found using simple regression analysis that omits one of the variables that influences the shape of the mathematical relationship between speed and curvature. That variable – which we call the “cross product variable” in our paper – is actually a measure of affine velocity.

RM: If the affine velocity of a movement is constant then the relationship between speed and curvature will be a power function with exponent 1/3 or 2/3, depending on how speed and curvature are measured. The power law is a statistical artifact in the sense that the power law relationship between speed and curvature would always be seen if the omitted variable – affine velocity – were always included in the regression analysis.

AM: However, instantaneous speed is not a statistical measure. It is just a derivative of position, nothing fancy going on. All experiments I’ve conducted so far, and there was a ton of them, show that people simply cannot maintain constant high speed when moving along curved paths, even when explicitly asked to do so. They always slow down in curved parts.

RM: My guess is that this might be because there is higher correlation between affine velocity and curvature when movements are slow than when they are fast. And it’s this correlation that affects the degree of deviation of the observed power relationship between speed and curvature from the 1/3 or 2/3 "law.
The equation for affine velocity is:

.
The equation for curvature is:

RM: I suspect that that the correlation between the measure of affine velocity and curvature, R, could be affected by the speed of movement. It may just be a computational thing – with speed affecting the accuracy of the difference computations that are estimates of the first and second derivatives of movement in X and Y. And perhaps because of this the filtering that is regularly done to smooth other the data might affect the correlation between affine velocity and R more than for slow than for fast movements.

RM: Given that there is a mathematical relationship between measures of affine velocity and curvature I think power law researchers have given too little attention to the effects that filtering might have on the results and why filtering might have such effects. I strongly suspect that the filtering is has a lot to do with whether or not you find the power law and that this is because the filtering affects the correlation between affine velocity and curvature.

Best

Rick

[amatic]

RM: To paraphrase the Cowardly Lion ( I do believe that speed is lower in curved parts, and higher in flatter parts. I do I do I do. I do I do.I do believe that speed is lower in curved parts, and higher in flatter parts.

You do? Well, that is great. I though you didn’t. Let’s leave measures of curvature and affine velocity out of this discussion for a while, and stay with the plot of the path with just speed marked in rainbow color.

You agree that there lower speed in the “visually measured” curved parts than in the ‘flat’ parts. You also agree that the target is moving at a constant speed, and that the subject is not correctly tracking neither the position nor the speed of the target?

RM: What I don’t believe is that hand trajectories slow down in high curvature areas. This implies that the slowing was done in response to the high curvature. This can’t be the case, as we noted in the “Cause and Correlation” section at the beginning of Marken and Shaffer, 2017).

Why would “slowing down” imply a response to high curvature? To me, slowing down in high curvature means that high curvature and relatively lower speed happen at the same time in fast hand movement. People generally explicitly state that correlation does not imply causation in statistics classes. That is common knowledge. Did someone explicitly state that the speed-curvature correlation implies causation?

[rsmarken]

Hi Adam

RM: To paraphrase the Cowardly Lion ( I do believe that speed is lower in curved parts, and higher in flatter parts. I do I do I do. I do I do.I do believe that speed is lower in curved parts, and higher in flatter parts.

AM: You do? Well, that is great. I though you didn’t. Let’s leave measures of curvature and affine velocity out of this discussion for a while, and stay with the plot of the path with just speed marked in rainbow color.

AM: You agree that there lower speed in the “visually measured” curved parts than in the ‘flat’ parts. You also agree that the target is moving at a constant speed, and that the subject is not correctly tracking neither the position nor the speed of the target?

RM: I agree that speed tracking doesn’t happen but I don’t remember the report of position tracking quality. How bad was it? Compared to what? If you are still getting the power law then perhaps position tracking was good enough. As long as they traced out something close to an ellipse they will get the power law. I base that on this little quote from Gribble and Ostry (1996) :

G&O: In this paper we demonstrate that simulated elliptical tracing
movements obey the power law, even when the modeled
control signals have constant tangential velocity – in other
words, the power law relation is observed in kinematics even
when no relation between curvature and velocity exists in
the modeled control signals.

RM: Your constant tangential velocity target is like their constant tangential velocity control signal. And like you, they found that ellipses generated with constant velocity control signals obeyed the power law which shows reduced tangential velocity of movement with increased curvature. THe power law was found because what was drawn was an ellipse, even though the signal specifying the time to move through the curves of the ellipse were constant! I presume the same would be found if those signals varied over time.

RM: What I don’t believe is that hand trajectories slow down in high curvature areas. This implies that the slowing was done in response to the high curvature. This can’t be the case, as we noted in the “Cause and Correlation” section at the beginning of Marken and Shaffer, 2017).

AM: Why would “slowing down” imply a response to high curvature?

RM: I think that is the typical way “slowing down” is interpreted in ordinary conversation. And I got the impression from discussions on CSGNet that that is what people meant by “slowing down” through curves. But if it’s not the way you interpret it then mazel tov.

AM: To me, slowing down in high curvature means that high curvature and relatively lower speed happen at the same time in fast hand movement.

RM: I believe that happens in slow movement as well; it just that such movements may not as regularly have a power exponents of 1/3 or 2/3. At least that’s what I have found.

AM: People generally explicitly state that correlation does not imply causation in statistics classes. That is common knowledge. Did someone explicitly state that the speed-curvature correlation implies causation?

RM: No, I can’t recall anyone explicitly stating it. But, then, I don’t recall anyone explicitly stating that it didn’t.

Best regards

Rick

[amatic]

RM: I agree that speed tracking doesn’t happen but I don’t remember the report of position tracking quality. How bad was it? Compared to what? If you are still getting the power law then perhaps position tracking was good enough. As long as they traced out something close to an ellipse they will get the power law.

Position tracking of constant high speed target was worse compared to the tracking of constant low speed target. I’ve posted a plot of for the three cases of tracking, and also the speeds over time, but here is a speed-colored plot.

Bold - not correct. Elliptical path in no way guarantees the power law or any speed-curvature correlation. For a small range of slow speeds, subjects can maintain constant speed. Anything constant is uncorrelated with curvature, which is varying. No speed-curvature power law here, even though the path is elliptical. Same data as above in the “slow” condition.

As for Gribble and Ostry, that is a wonderful paper. Their model with a constant reference signal behaves very much how the LittleMan behaves, and how the position tracking model behaves.

RM: The power law was found because what was drawn was an ellipse, even though the signal specifying the time to move through the curves of the ellipse were constant! I presume the same would be found if those signals varied over time.

No. The power law was found because the speed of the target was higher than the model could follow. The model was acting as a low pass filter because of the slowing in the output. In their case muscles are causing the ‘slowing’. In the case of the our tracking model, the slowing is caused by the leaky integrator and the transport delay.

Why do you presume when you could check for yourself what happens when a model tracks a fast target? Judging by your predictions, you would be surprised.

RM: I think that is the typical way “slowing down” is interpreted in ordinary conversation. And I got the impression from discussions on CSGNet that that is what people meant by “slowing down” through curves. But if it’s not the way you interpret it then mazel tov.

No, definitely not, and no one else in published papers on the power law interprets speed at time t to be the response to the stimulus of curvature at time t. Slowing down in curves is a property of the response - speed and curvature are simultaneous aspects of the same hand trajectory, measured at the same time instant.

Stimuli are either entire shapes plus rhythm markers, or moving targets, and the response is the entire hand trajectory.

RM: I believe that happens in slow movement as well; it just that such movements may not as regularly have a power exponents of 1/3 or 2/3. At least that’s what I have found.

The slowing down of hand speed in high curvature in tracking tasks does not happen if the target is relatively slow. No correlation between speed and curvature, so no exponent.

[rsmarken]

Hi Adam

AM: As for Gribble and Ostry, that is a wonderful paper. Their model with a constant reference signal behaves very much how the LittleMan behaves, and how the position tracking model behaves.

RM: My impression is that their model is open loop: the “control signal” was not a reference for input (a perceived aspect movement) but a command for output (force). See their Figure 1. But I see that their “control signals” go through a bunch of transforms (via the muscle models) so that may be how they get their result.

RM: But this model can be rejected in a New York minute by simply comparing the behavior of the model to that of a subject in the same situation when continuously varying force disturbances are applied to the hand (or whatever is being moved by the muscle forces). Then you get something like this:

RM: The cursor movement follows the power law; the mouse movement that produced it doesn’t. This seems to me to be a nice, simple demonstration of the futility of trying to understand behavior based on interesting mathematical properties of the side effects of control.

AM: The slowing down of hand speed in high curvature in tracking tasks does not happen if the target is relatively slow. No correlation between speed and curvature, so no exponent.

RM: I tracked a very slow moving elliptical target (40 seconds to track the complete ellipse of 1000 samples, so the rate was 25 samples/sec). The V-R power coefficient was .34, r^2=.44, and the A-C power coefficient was .66, r^2 = .74. These values depend to a staggering extent on the filtering.

RM: I really have to get to other work now so, which it has been fun, I am going to have to end this conversation now. I wish I could convince you to stop playing with the power law and start studying movement as a control process. But, as Mr. Zimmerman says, “You do what you must do and you do it well”.

Best regards

Rick

[amatic]

Sigh. So many things I disagree with in that one little post of yours. And we did not even come to affine velocity and the abuse of the omitted variable bias…

At least we did come to an agreement of what the behavioral illusion is, after struggling with causal functions and feedback quantities.

We also agreed that taking side effects to reflect something about the organism is not a behavioral illusion, and is not even a mistake, side effects do reflect properties of the control loop. Speed profiles certainly do depend on properties of the system.

Side effects of measurement (noise) do not reflect system properties, so not filtering or filtering too much is a problem. That is a whole different matter, but also not a behavioral illusion.

And we also agreed that humans move at lower speed in areas of high curvature when tracking constant high speed targets along an ellipse! Not a statistical artifact.

Still, it’s been pretty difficult talking to you.

[rsmarken]

Hi Adam

@amatic

AM: Sigh. So many things I disagree with in that one little post of yours. And we did not even come to affine velocity and the abuse of the omitted variable bias…

At least we did come to an agreement of what the behavioral illusion is, after struggling with causal functions and feedback quantities. …

Still, it’s been pretty difficult talking to you.

RM: I think the problem is just that we are interested in different things. I’m interested in understanding the controlling done by living systems, which to me means understanding their behavior in terms of the perceptual variables they control. You are interested in something else (what is ain’t exactly clear), though you sometimes talk as though we are interested in the same thing; for example, you use the term controlled variable a lot, which kind of confused me. Anyway, whatever it is you are interested in, your research apparently fits your interests just fine, but it is irrelevant to mine, except perhaps as an example of how not to do the kind of research I’m interested in doing.

RM: So we disagree because we have been making the mistake of thinking that we are both interested in the same thing. Now maybe we can avoid disagreements by just minding our own business. It’s just like you go your way and I’ll go mine. You obviously do a great job of doing research relevant to your interests and I do what I think is a reasonable job of doing research relevant to mine. So keep up the good work but I don’t really need to hear about it just as you don’t need to hear about my work. I think we’ll both get a lot more done that way.

Best

Rick

[amatic]

It took us a good part of a month to see that your papers on the power law contain mistakes from a PCT perspective - your definitions of a behavioral illusion were not correct, neither in the first nor the second paper. Mistake right there in the title of your two papers.

You agreed they were mistakes, so why this tone? You resent being proven incorrect? Are you afraid more mistakes will come up? Because there are many of them. I’m just starting, and all this “you are interested in something else, not controlled variables” nonsense just pushes me to continue.

Of course, feel free to ignore my posting on this forum if you are not getting things done because of them, or for any other reason.