Adam and Rick,
Thank you and please continue. This discussion has been very instructive for me – and I believe for anyone who has followed – about PCT, power law research, science general and especially about dialogue and human nature.
There really seems to be many issues left.
Hi Eetu
EP: Adam and Rick,
EP: Thank you and please continue. This discussion has been very instructive for me – and I believe for anyone who has followed – about PCT, power law research, science general and especially about dialogue and human nature.
RM: Well, by popular demand;-)
EP: There really seems to be many issues left.
RM: For me there is only one issue: What kind of research do we do to understand the behavior of what we know to be living control systems. That is, what kind of research do we do to test the PCT model of purposeful behavior. Adam claims that power law research is just the kind of research we should be doing. Indeed, he says it’s the kind of research Powers was advocating in his 1978 Psych Review paper. I claim that power law research is just the kind of research Powers was advocating against in the 1978 Psych Review paper. I wrote the two papers about the power law (M&S 2017, M & S 2018) being an example of a behavioral illusion to make that point – that studying the power law is just the kind of blind alley Powers said researchers were going down by studying invariant tangential velocity profiles (a mathematical characteristic of overt behavior that is the same kind of “law” as the power law).
RM: Adam has been arguing that I made all kinds of mistakes in those papers. In his last couple of posts he seems to believe that I have acknowledged those mistakes. I don’t. It is not a mistake to call the power law a behavioral illusion because it is; It is not a mistake to say that side-effects of control tell you nothing about the properties of the control system because they don’t; and it is not a mistake to say that the apparently slower velocity of movement through curves is a statistical artifact because it is.
RM: I doubt that Adam and I will ever come to agreement about these things. But this conflict did lead me to have an “up a level” experience, which is that what I want to do is the kind of research Powers encouraged in the 1978 Psych Review paper and elsewhere; research aimed at determining the variables organisms control, how they control them and why. Bill didn’t go into a lot of detail about how to do this kind of research; he mainly produced demonstrations of principle. So I know the principles. But what I would like is some help from like-minded people about how to go about doing this kind of research.
RM: Once I went “up a level” and realized that my higher level goal was getting help with doing PCT-based research the conflict over the power law evaporated. I don’t care what the power law people think of my evaluation of their research; that research is basically irrelevant to me. I wrote the critiques of the power law in the hopes that it would convince researchers to drop that research and start doing research from a PCT perspective. Not only was that a fantasy but many of the power law people thought of themselves as doing PCT-based research – the kind of research Powers described in the Psych Review paper. But once I went up a level I realized that what the power law people think or say about their own research was of no interest to me. They could call it PCT research or whatever they wanted. I realized that whatever they called it – even if they said it was exactly the kind of research Powers begged them to do – it was not the kind of research I wanted to do; so there was no need to argue anymore. No more conflict.
RM: So I’m doing the PCT Research seminar in hopes of getting some discussion going on how to do PCT research (as I understand it). And the power law will NOT be any part of that discussion; there will be no discussion of what’s wrong with conventional research; just a discussion of how to do PCT research in what I (and what I’m quite sure Bill Powers) would consider to be the right way.
Best
Rick
Well, I’ll let you rest a few days, do your seminar, and then I’ll be back to discussing your last two papers as good examples of how not to do PCT-based research. That would actually be a good title for the topic, as a little revenge for this nonsense you’re saying I said about “power law research”.
And also revenge for the taksies backsies! To think I called a mazeltov on our agreement… tctctc. What a schmuck.
AM: Well, I’ll let you rest a few days, do your seminar, and then I’ll be back to discussing your last two papers as good examples of how not to do PCT-based research.
RM: I hope you’ll also explain how I should have done it. That would, indeed, be very interesting.
AM: That would actually be a good title for the topic, as a little revenge for this nonsense you’re saying I said about “power law research”.
RM: Always with the “nonsense”
AM: And also revenge for the taksies backsies! To think I called a mazeltov on our agreement… tctctc. What a schmuck.
RM: What agreement?
Best
Rick
What agreement? This agreement!
AM:you do see that, after finding the controlled variable, taking side effects of control to tell you something about the behaving mechanism is NOT a behavioral illusion?
RM: Yes, I do see that it should not longer be called a behavioral illusion. A better term for it would be a “waste of time”.
AM: We agree! It is not a behavioral illusion! Mazel tov!
And you talked like the Cowardly Lion too,
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.
But now you did a takesy-backsy:
RM: It is not a mistake to say that side-effects of control tell you nothing about the properties of the control system because they don’t; and it is not a mistake to say that the apparently slower velocity of movement through curves is a statistical artifact because it is.
Not good for the image of the teacher, Master Marken, so many changes of opinion. A good teacher has solid knowledge of the subject and if he made a mistake, he acknowledges it and continues with the new knowledge. Not you. First claiming something wrong, then being corrected, then evading the correction, then saying yeah ok, maybe that was a mistake (mazeltov), and then again back to claiming the wrong thing. Tomorrow you may be at any of those points, spinning like a compas needle to wherever it looks like “I was not wrong”. Yeah, you were.
Hi Adam
AM: What agreement? This agreement!
AM:you do see that, after finding the controlled variable, taking side effects of control to tell you something about the behaving mechanism is NOT a behavioral illusion?
RM: Yes, I do see that it should not longer be called a behavioral illusion. A better term for it would be a “waste of time”.
AM: We agree! It is not a behavioral illusion! Mazel tov!
RM: I guess the attempt at humor was lost on you. I was tired of explaining the same thing over and over to you and so I just called it “a waste of time” rather than a “behavioral illusion” since the name doesn’t matter. But the fact is that taking side effects of control to tell you something about the mechanism that produces the behavior is both a behavioral illusion and a waste of time.
AM: And you talked like the Cowardly Lion too,
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: But now you did a takesy-backsy:
RM: It is not a mistake to say that side-effects of control tell you nothing about the properties of the control system because they don’t; and it is not a mistake to say that the apparently slower velocity of movement through curves is a statistical artifact because it is.
RM: As I noted above, I believe I have consistently said that taking the side-effects of control to tell you something about the behaving mechanism is a waste of time. Obviously, that’s because these side-effects tell you nothing about the properties of the system.
RM: And with regard to going slower in tight curves and faster in flatter ones, power “law” studies (law in quotes because the exponent of the power function varies considerably around the “legal” values – 1/3 and 2/3) generally find a positive power exponent – indicating slower velocity in tighter curves. But this relationship is forced by the mathematical relationship that exists between measures of curvature and velocity. Mathematically, curvature and velocity are related by a 1/3 or 2/3 power function, depending on how these two variables are measured. But the velocity measure is simultaneously dependent on a third variable, affine velocity, by a 1/3 power function.
RM: So if you did the appropriate regression analysis to determine the function relating curvature to velocity – the appropriate regression analysis being one that includes both curvature and affine velocity as predictors of velocity – you would always find exactly a 1/3 or 2/3 power function (R^2 of 1.0) relationship between curvature and velocity (and a 1/3 power relationship between affine velocity and velocity). If, however, you leave the affine velocity measure out of the regression analysis – as is the custom – then how close the exponent of the power function comes to 1/3 or 2/3 (and how close the regression coefficient comes to 1.0) depends on the degree to which curvature covaries with affine velocity.
RM: Since for most trajectories the covariation between affine velocity and curvature is very low, you will typically find a power function close to 1/3 or 2/3. So you will generally see that velocity is lower when curves are tighter. But this relationship is a statistical artifact that results from the mathematical 1/3 or 2/3 relationship between curvature and velocity.
RM: What is artifactual about the speed-curvature relationship is that the closeness of the power coefficient to 1/3 or 2/3 depends on the covariation of the omitted affine velocity variable with the included curvature predictor variable for that trajectory. Because this covariance is slightly different from one trajectory to another it looks like the power law is not a result of a mathematical relationship between curvature and velocity because the calculated power exponent varies around 1/3 or 2/3 a bit (or a lot). So the observed power law looks like some kind of physical or psychological law. But it is really just an artifact of the math.
AM: Not good for the image of the teacher, Master Marken, so many changes of opinion.
RM: I don’t believe I have changed my opinion that much. But even if I did that would not make me a schmuck, at least not the way they used the term in my family. A schmuck wasn’t the person who had what some considered a wrong opinion; it was the person who brow beat that person into admitting that she was wrong. It wasn’t the Hollywood ten who were schmucks for not admitting they were wrong by confessing to having been a member of the communist party; it was Senator McCarthy who was a schmuck for forcing them to publicly admit it. It wasn’t Galileo who was the schmuck for not admitting he was wrong about the sun being at the center of the solar system; it was the people of the church who forced him to admit it.
RM: The schmuck is the person who needs another person to confess their wrongness so that they themselves can feel right; the mensch is the person who simply shows that the other person is wrong without ever demanding a confession. I leave it as an exercise for you to determine who have been the schmucks in this power law discussion, from the start (yes, there is more than one).
Best
Rick
I think only schmucks do takesy-backsies.
It is a bad strategy. That is how you guarantee you’ll be wrong somewhere along the spin. Now you gotta go with “oh I was just joking”, and “you’re the inquisition, forcing me, Galileo, to tell a lie”. Such drama. I imagine you quietly saying, as you exit the courtroom, “Eppur si don’t reflect properties…”.
Changing your opinion or seeing you were not correct does not make you schmuck, it is the going back and forth with your opinions (or spinning like a dreidel)
I’ve shown some examples of the research we did in the lab on the topic of drawing fast ellipses and similar shapes, and you were all “oh, that is great Adam, wonderful work”. And I was thinking “Pft. As if you had any criteria”. It was just basic tracking.
What I see in your comments and evaluations of papers I’ve posted links to, comments about other people’s work, and your own work, is a lack of solid criteria for what makes good research. What aspects of it are good, and what are not so good.
Bill took his criteria mostly from physics, I think, and rejected most criteria from psychology and life sciences. Criteria might be a good topic for discussion. First design an input function for what is good research, then the how will solve itself.
Hi Adam
AM: I think only schmucks do takesy-backsies.
RM: Well. I may eventually have to take this back but it’s my impression that both schmucks and mensches do “takesy-backsies” but only schmucks demand that there are no take backs. I agree that keeping agreements is a good idea, in principle, especially when they are financial contracts. But in scientific discussions it is often the case that people think (and say) that they agree with the point being made but later realize (based on data and/or other considerations) that they no longer agree. Sometimes doing a takesy-backsy is a good thing, even though it involves going back on an agreement (going back on the agreement in the US constitution that people can be property was, I think, a very good takesy-backsy, for example; I don’t know if Kelvin ever took back his claim, based on physics calculations, that the world wasn’t old enough for evolution to have occurred, but he should have).
RM: I don’t believe I have done any particularly important takesy-backsies anyway. Since my first encounter with the power law (of movement; I was already very familiar with Stevens’ power law of psychophysics-- also a behavioral illusion, by the way) I understood it to be an example of a behavioral illusion; that it is a mathematically constrained side-effect of controlling the position of a limb and understood that there is nothing you can learn about the controlling involved in producing smooth, directed limb movements from these side effects. Clearly you don’t agree with these points. So I think we have irreconcilable differences and it’s time for a divorce. All I ask for is thanks for getting you the job in Alex’s lab; you can have custody of the kids (Martin T. and Bruce A.)
AM: Changing your opinion or seeing you were not correct does not make you schmuck,
it is the going back and forth with your opinions (or spinning like a dreidel)
RM: So is there a particular number of opinion changes that I’m allowed before I am deemed a schmuck? And why do you care whether or not I agree with you anyway? Why can’t you just assume that I completely disagree with you and content yourself with showing, to your satisfaction, that I am completely wrong?
AM: I’ve shown some examples of the research we did in the lab on the topic of drawing fast ellipses and similar shapes, and you were all “oh, that is great Adam, wonderful work”. And I was thinking “Pft. As if you had any criteria”. It was just basic tracking.
RM: I got excited because the measure of curvature in the target track is mathematically independent of the measure of the speed of the pen. So the curvature of the target track could be a disturbance to a controlled variable that is also affected by the speed of movement of the pen; that is, the controlled variable could be something like the ratio of (or difference between) target curvature and pen speed. But my enthusiasm was tempered after I saw some of your data and thought about it some more. But there still might be a way to get some information about what the subject is controlling in such such an experiment. The problem is the fact that there is a mathematical relationship between measures of velocity and curvature in both target and pen movement. I don’t know how you overcome that confounding bu there might be a way.
AM: What I see in your comments and evaluations of papers I’ve posted links to, comments about other people’s work, and your own work, is a lack of solid criteria for what makes good research. What aspects of it are good, and what are not so good.
RM: Well, I do my best. I’m sorry I don’t live up to your standards.
AM: Bill took his criteria mostly from physics, I think, and rejected most criteria from psychology and life sciences. Criteria might be a good topic for discussion. First design an input function for what is good research, then the how will solve itself.
RM: If you think the power law research is good research then your criteria for evaluating research are clearly quite different than mine. But I would be interested in knowing what your criteria for evaluating research are.
Best
Rick
Rick,
I understand that what you most want to do is that kind of research which you think is the right and best way. But I think it does not entitle you to withdraw this discussion.
Adam has given some clear and interesting arguments (data based!) in support of them really being mistakes. If you just say that this kind of discussion does not interest you, it casts a long shadow of doubt over the premises of the kind of research you exclaim the only right way to do scientific research.
Best
Eetu
Hi Eetu
EP: Adam has given some clear and interesting arguments (data based!) in support of them really being mistakes. If you just say that this kind of discussion does not interest you, it casts a long shadow of doubt over the premises of the kind of research you exclaim the only right way to do scientific research.
RM: I agree that Adam makes very convincing (and data based) arguments. But so do I;-) (See M & S 2017, M & S 2018. Nevertheless, I lost this argument in the eyes of much of the PCT research community, such as it is, long before Adam entered the fray.
RM: Those who made or were convinced by arguments like Adam’s weren’t going to do the kind of research that I (and Bill) claim to be the only right way to study the behavior of living control systems anyway. And that’s because they all have a vested interest of some kind in doing research the conventional way. So it’s really useless to keep arguing about this. As they say, you can’t get a person to understand something when their [paycheck, prestige, reputation] depends on their not understanding it.
RM: Those who are willing to try or are already doing research the way Bill argued that it should be done are signed up for my seminar and/or will get my forthcoming book on The Study of Living Control Systems. I myself have a lot to learn about how to do that new kind of research. So I want to try to focus on that rather than on what’s wrong with the conventional approach.
Best regards
Rick
Rick,
Yes you made some interesting and surprising arguments in those papers and they received strong and to my eyes quite reasonable criticisms. I did not found out that you ever did respond argument by argument to them. Now when Adam again
states that criticism in an outstandingly clear and simple way you still refuse to answer. Why? Is there something more important than argumentative scientific discussion? This discussion is the only way how science can progress. It does not progress if everybody
just keep doing their own way of research.
I think the most interesting and important issue in this debate started by Adam is not so much about “on what’s wrong with the conventional approach” or what’s right with it or any other approach but rather just what is wrong with your
original arguments in those mentioned papers. Adam for example gave strong arguments that the empirical dependence between curvature and velocity cannot generally (and logically) be a mathematical, statistical or measurement created artefact as you keep claiming.
If your attitude leans on the fact the that on the one hand you have followers who already believe you and on the other hand those who criticize you do it only because of their vested interests then you are driving PCT to sectarianism.
But that is the spirit of our time, the time of private and alternative truths.
Hi Eetu
EP:I think the most interesting and important issue in this debate started by Adam is not so much about “on what’s wrong with the conventional approach” or what’s right with it or any other approach but rather just what is wrong with your original arguments in those mentioned papers. Adam for example gave strong arguments that the empirical dependence between curvature and velocity cannot generally (and logically) be a mathematical, statistical or measurement created artefact as you keep claiming.
RM: OK, Great.Tell me what you found to be Adam’s "strong arguments that the empirical dependence between curvature and velocity cannot generally (and logically) be a mathematical, statistical or measurement created artefact " (as I keep claiming) and I’ll answer them for you.
Best
Rick
Hi Eetu
RM: I said:
RM: OK, Great.Tell me what you found to be Adam’s "strong arguments that the empirical dependence between curvature and velocity cannot generally (and logically) be a mathematical, statistical or measurement created artefact " (as I keep claiming) and I’ll answer them for you.
RM: While you’re thinking about that I’d also be interested in what you think of Adam’s ideas about the relevance of the power law to PCT.
RM: Oh, and I should also comment on this from your earlier post:
EP: If your attitude leans on the fact the that on the one hand you have followers who already believe you and on the other hand those who criticize you do it only because of their vested interests then you are driving PCT to sectarianism.
RM: I hope that I don’t have followers (in the religious sense); it would creep me out. I think the people who have signed up for the seminar have some interest in PCT research and think I might know something about it. I count some of the people enrolled in the seminar as students with little or no experience with PCT research and others as peers who have been engaged in PCT research for some time. I’m actually giving the seminar in the hopes of getting some ideas – from experts and as well as those looking at it with fresh eyes – about how to do PCT-based research.
RM: And I don’t think all criticism of me comes from vested interests – I make mistakes and when they are correctly pointed out I learn from that criticism. But some criticism clearly does come from vested interest, particularly criticism of my analysis of the power law, which, if it were correct (and I’m pretty sure it is) would be a big practical problem for people whose research careers are based on the idea that the power law is an important phenomenon.
Anyway, I look forward to learning what you think about all this.
Best regards
Rick
Sorry, I am busy with teaching at the moment, so I must try to write very concisely.
You are still not answering but just throw the question back to me. Perhaps you have no input functions for criticism and counter arguments? That would explain why you ask me to repeat them again. But OK will repeat the most important argument
with my own words: Even a child knows that if you have a simple normal coordinate space and a moving point in it then the velocity of the moving point can no way depend on the place of the point or on the direction of its movement. They are independent phenomena.
Adam just showed this wit data and diagrams. If there is a (real) change in the velocity of the point it is not caused by mathematics or statistics. You have a real phenomenon which requires a real explanation.
As for your question about the relevance of the power law to PCT, I must say I don’t know. Power law phenomenon is an interesting phenomenon for research by its own right. I think it is great the Adam tries to apply PCT to that research.
PCT is relevant for it like for many other questions but it is just a dream that PCT could just annihilate those questions away from science.
Hi Eetu
EP: You are still not answering but just throw the question back to me.
RM: Sorry, Eetu. I was just asking what you thought were Adam’s “strong arguments that the empirical dependence between curvature and velocity cannot generally (and logically) be a mathematical, statistical or measurement created artefact”. I can’t explain why I think those arguments are wrong unless you tell me what they are.
EP: Perhaps you have no input functions for criticism and counter arguments? That would explain why you ask me to repeat them again.
RM: Why all the anger? I didn’t ask you to repeat your question. You never told me what Adam’s arguments were that convinced you that the empirical dependence between curvature and velocity cannot be a mathematical, statistical or measurement created artefact. I just asked you what those arguments were.
EP: But OK will repeat the most important argument with my own words: Even a child knows that if you have a simple normal coordinate space and a moving point in it then the velocity of the moving point can no way depend on the place of the point or on the direction of its movement. They are independent phenomena. Adam just showed this with data and diagrams. If there is a (real) change in the velocity of the point it is not caused by mathematics or statistics. You have a real phenomenon which requires a real explanation.
RM: I know that the power law is a real phenomenon. And what I have shown is that velocity is mathematically dependent on curvature, not direction. This is just a mathematical fact, acknowledged even by power law researcher themselves (1,2) The relationship is described by this equation:
(1)
where A is angular velocity, C is curvature and D is something called affine velocity.
RM: The power law is the empirical finding of a 2/3 power relationship between C (curvature) and A (velocity) in the movements produced by living organisms. That is, the power law is the finding that the empirical relationship between measurements of C and A is:
(2)
where beta is typically close to 2/3.
RM: But note that there is a 2/3 power relationship between C and A in the mathematical relationship between C and A (equation 1)! The only difference between equation 1 – the mathematical relationship between C and A – and equation 2 – with beta = 2/3 – is the variable D in equation 1. Apparently this remarkable “coincidence” – the similarity between equations 1 and 2 – didn’t make a big impression on power law researchers but it made a big impression on me.
RM: I knew that the empirical power law – equation 2 – was found using simple linear regression, where the log of the observed values of C were regressed on the logs of the observed values of A to determine the best fitting power law relationship between curvature and velocity. That is they do a regression analysis using the equation:
(3)
to find the value of beta that gives the best fit to a power relationship between observed values of C and A. The regression analysis typically results in a value of beta nearly equal to 2/3 and an R^2 value greater than .9 (indicating a very good fit to the power equation.
RM: But it is clear from equation 1 that the results of this regression are artifactual because the regression analysis leaves out one of the variables on which the value of A depends; it leaves out the variable D, the measure of affine velocity. The appropriate regression analysis for predicting log (A) from log (C) would be a multiple regression analysis that include two predictors of log(A): log (C) and log (D). When you do this analysis what you always find is that the beta for predicting log (A) from log (C) is exactly 2/3, because that’s what equation 1 says it is mathematically. That is, you always find that the regression equation for predicting log(A) from log(C) is exactly what you would expect from equation 1:
log (A) = 1∕3 ⋅ log (D) + 2∕3 ⋅ log (C ) (4)
This is true for all movement trajectories, even those of the planets in their orbits.
RM: What this means is that the empirical power “law” – the apparent decrease in velocity as the 2/3 power of curvature – depends on variations in a variable – affine velocity – that is not included in the typical regression analysis that is used to find the power “law”. If affine velocity is nearly constant – as it apparently is in most trajectories (2) – then the exponent of the power relationship found by regressing C on A (omitting D) will be close to the true mathematical value of this exponent (per equation 1).
RM: Another way of saying it is like this: The 2/3 power law decrease in movement velocity (A) as a function of movement curvature (C) is a mathematical fact (per equation 1). The degree to which this mathematical relationship is observed in actual movement, by regressing measures of C on measures of A, depends on the degree to which the affine velocity of the movement (the variable D) is constant and, therefore, uncorrelated with variation in curvature, C.
RM: I’m not aware of anyone – let alone Adam – showing that there is anything wrong with this analysis. But maybe I missed it so feel free to show me where I am mistaken.
EP: As for your question about the relevance of the power law to PCT, I must say I don’t know. Power law phenomenon is an interesting phenomenon for research by its own right. I think it is great the Adam tries to apply PCT to that research.
PCT is relevant for it like for many other questions but it is just a dream that PCT could just annihilate those questions away from science.
RM: PCT doesn’t annihilate questions away from science (whatever that means). PCT explains the power law as an irrelevant side effect of movement which can’t tell you anything about how that movement was produced. So I guess PCT does kind of annihilate the power law as a phenomenon that can tell you anything about how movements are produced; but PCT does show that the power law is an interesting example of a behavioral illusion that has kept researchers going down a blind alley for many years.
References
1.Pollick F, Sapiro G (1997) Constant affine velocity predicts the 1/3 power law of planar motion perception generation. Vision Res, 37:347–353
2. Maoz U, Portugaly E, Flash T, Weiss Y (2006) Noise and the 2/3 power
law. Adv Neural Inf Proc Syst 18:851–858
Rick, not really angry, just wondering because I think that others – at least Martin – has stated the same criticism already earlier.
What does your first equation tell us? Isn’t it a definition of that something called affine velocity? If you measure A and C then you can calculate D with this equation. As they say in your first reference: “This means that the 1/3 power
law, experimentally found in the studies of hand-drawing and planar motion perception, implies motion at constant affine velocity." So what have we found? That according to the definition of your first equation, we can call 1/3 power law also “constant affine
velocity law”. Nothing is solved, just a new name given to that empirical finding which should be explained. One explanation could be that humans typically control for constant affine velocity in their movements – as heavenly bodies and larvae do not. Other
possible explanations are for example physical properties of our nerves or muscles or something else.
btw. If you say that power law is an artificial mathematical construct created by the researchers then it cannot be a side effect of the control of the studied subjects (but rather the side effect of the researchers). I can’t even see what
the claim that power law is a mathematical artefact has to do with PCT.
One simple thing forgot. You said: “velocity is mathematically dependent on
curvature, not direction”. What is curvature? Isn’t a change of direction? How can velocity be independent of direction but dependent on the change of direction?
Hi Eetu
EP: Rick, not really angry, just wondering because I think that others – at least Martin – has stated the same criticism already earlier.
RM: And I believe I have answered those criticisms. I’ve published the answers as two articles in a peer-reviewed journal. I knew the answers in those papers wouldn’t convince people, like Martin, who were arguing against my PCT interpretation of the power law. But the fact that those papers were almost instantly accepted for publication gave me a brief moment of happiness in the midst of the otherwise depressing experience of dealing with the self-anointed PCT experts on CSGNet after Bill passed away (and was no longer there to argue with them himself).
EP: What does your first equation tell us? Isn’t it a definition of that something called affine velocity?
RM: No,that equation
is the mathematical relationship between the measures of curvature (C ) and velocity (A) that are used by power law researchers to test to determine whether a “power law” relationship exists between those variables in the movement produced by a living system. The definition of affine velocity is:
D = 
RM: It’s the cross product of the first and second derivative of movement in the X and Y dimensions. Since affine velocity means something like parallel movement in both dimensionsI think affine velocity is maximum when D = 0.
EP: As they say in your first reference: “This means that the 1/3 power law, experimentally found in the studies of hand-drawing and planar motion perception, implies motion at constant affine velocity." So what have we found? That according to the definition of your first equation, we can call 1/3 power law also “constant affine velocity law”.
RM: Actually, that would not be correct. My simulations show that, for arbitrary trajectories, affine velocity is almost never a constant. You find the 1/3 (or 2/3) power law relationship between curvature and velocity when affine velocity is not correlated with curvature. That is, you find it when the predictor variable (affine velocity) that is omitted from the regression analysis used to determine whether there is a power law relationship between curvature and velocity is uncorrelated with the predictor variable (curvature) that is included in that regression.
EP: Nothing is solved, just a new name given to that empirical finding which should be explained. One explanation could be that humans typically control for constant affine velocity in their movements – as heavenly bodies and larvae do not.
RM: That would be a possible explanation that could be tested using methods based on PCT (the TCV). But there is no evidence that affine velocity is constant in movements produced by living systems. All we know is that affine velocity is typically not strongly correlated with curvature in the movements made by living systems. What I found when I first started working on this (and what Maoz et al had also found before me but I only became aware of later) is that this is true of almost all randomly generated two dimensional movement trajectories. This fact about movement trajectories actually explains why the power law is so often observed in the trajectories produced by living systems – it’s almost impossible to produce a trajectory that isn’t close to a “power law” because in almost all trajectories there is a low correlation between affine velocity and curvature.
EP: Other possible explanations are for example physical properties of our nerves or muscles or something else.
RM: Yes, but fortunately the PCT explanation of the power law (as an irrelevant mathematical side effect of movement) would save researches from going down that very costly blind alley.
EP: btw. If you say that power law is an artificial mathematical construct created by the researchers then it cannot be a side effect of the control of the studied subjects (but rather the side effect of the researchers). I can’t even see what the claim that power law is a mathematical artefact has to do with PCT.
RM: The power law is NOT an artificial mathematical construct created by the researchers; it’s an actual mathematical relationship between the variables (measures of curvature and velocity) researchers use in the regression analysis to determine whether the there is a “power law” relationship between those variables in the particular movement under study.
RM: As far as the relationship of the power law to PCT, that relationship is the same as that of almost any finding of conventional psychology to PCT; it is a finding that results from failure to see that behavior IS control. This fact is the fundamental observation on which PCT is based; it is prior to theory.
RM: In the case of the power law, all that research is based on the assumption that movement is a generated output. If that were actually the case then characteristics of observed movements – such as the power law relationship between curvature and velocity-- would, indeed, tell us something about how the movement was generated. But, in fact, movement is controlled input, which is evident from the fact that consistent movement trajectories are produced in the face of variable disturbances.
RM: So in order to understand how the movement is produced you have to know what inputs are being controlled. Regularities in the movement can hint at what variables are under control. But in order to find out what variables are actually being controlled you have to do some version of the test for the controlled variable.
RM: Focusing on trying to figure out why you often observe a power law relationship between curvature and velocity of movement is not going to help you find the variables being controlled in movement. It’s just a blind alley – that is its relationship to PCT – and my paper showing that the power law is just a statistical artifact is my attempt to plant a sign at the entrance to that blind alley saying “do not enter”. I knew that this sign was likely to be ignored. But I think it’s worth it to keep trying. Bill certainly thought so.
Best
Rick
Hi Eetu
EP: One simple thing forgot. You said: “velocity is mathematically dependent on
curvature, not direction”. What is curvature?
RM: In power law research curvature is measured as either radius of curvature:
or degree of curvature:
And velocity is measured as tangential velocity:
or angular velocity
RM: All the variables used in power law analysis – R, C, V and A – are calculated from the data, which are the X,Y coordinates of the movement at each instant in the movement’s trajectory.
EP: Isn’t a change of direction? How can velocity be independent of direction but dependent on the change of direction?
RM: You can call it what you like. If you want to call the variables R and C measures of “change in direction” rather than “curvature” that’s fine with me. The mathematical dependence is between the variables V and R (and A and C), regardless of what you call them.
Best
Rick
Oops, sorry, I found out that I have misused the English terms. Instead of velocity I should have been speaking about speed. Velocity is a vector and is thus of course dependent on direction of the movement. Speed means just the moved length
in a time independently of the direction of the movement. An object or point can move with a stable speed absolutely independently of its direction of movement AND of the changes of its direction of the movement. A quick change of the direction of the movement
is a turn or angle, a slower change of direction is a curvature. This is a basic mathematical and logical truth which should not be muddled up by such complicated subtleties as affine velocity etc.
As Adam wrote: “the ‘physical’ meaning of the speed-curvature power law - when the curvature is high (‘corners’ of the ellipse) the speed will tend to be lower than when the curvature is low (‘flat parts’ of the ellipse). (Behavioral Illusions: The Basis of a Scientific Revolution)”
And this does not happen always, so it is not any mathematical dependence but a real physical and explanation requiring phenomenon.