I think I finally understand what Martin et al are getting at and I can see why they think I am being evasive. So I will start working on an explanation that, I hope, they will find to be responsive. I’ll try to post something by this evening. If not, it will have to wait until Friday.
Best, Rick
Now that I’ve re-read your and Eetu’s posts I think this is good advice. I’m not going to reply to your post until I hear whether Martin and Eetu agree with it.
But while we’re waiting, could you please tell me if you agree with Eetu that my post to which he replied was a good example of dysfunctional discussion?
Best, Rick
I am not the judge. It may appear that I am because I am talking about how to make such a judgement. This is important, because the quality of discussion is a function of collective control that any of us can participate in and, in my opinion, we are each responsible for this.
I think Eetu is irked that <Eetu-perception>your responses have been non sequiturs </Eetu-perception>. Eetu, please correct me if this guess is wrong about your CV and the disturbance.
I believe (subject to correction by Eetu and Martin) that the following illustrates why your replies appear to them to be non sequiturs.
You evidently believe conversely that you have ‘answered their objections’.
This has the hallmarks of people talking past each other, i.e. thinking that they are talking about the same thing when they are talking about different things (equivocation à deux?). No dispute can be resolved in which the parties are talking about different things.
To be sure of talking about the same thing, go back a step and ask a different question: what is the core of the problem as they see it? The way to ask that question is to study their statements of the core issue and then paraphrase them, in your own words, and then to ask them if your restatement is a good paraphrase with the same meaning. Note that if you let any whisp of refutation intrude it does not have the same meaning. You must see things from their point of view. (That should sound familiar.) Repeat as necessary until they agree that you understand what they perceive to be the problem.
They have given you their paraphrases of your understanding of the relation between equations (4) and (5), i.e. that (ẋ - ẏ) in (4) and (ẋ - ẏ) in (5) are interchangeable. You have not objected to these paraphrases.
There follow some quotations and links to help you have a clear focus on their formulation of the core issue as they see it, so you can paraphrase it, without at this time adding any “but that’s wrong because”. If you have a good refutation later, it will be more effective for being expressed in their terms. To limit confusion, use the notation and equation numbers that they’re using.
Here’s Martin’s restatement in post #4, with a reference to “the quote you offer from [his] original critique” in post #3.
In your image (screenshot?) of that quotation in post #3, the equations were distorted. Here’s a clear screenshot of the entire relevant passage as I see it in my PDF. I have taken the liberty of healing a page break in this screenshot, and the search term “critical mistake” was highlighted by my PDF reader.
My paraphrase of this, to which no one so far has objected, is that (4) refers to actual experimental measurements (the values of ẋ and ẏ are “determined by the velocity observed in an experiment”, emphasis added), a finite set, whereas (5) refers to the infinite set of all possible velocities. In Martin’s words, “In (4) ẋ and ẏ are determined by the velocity observed in an experiment, whereas in (5) they are arbitrary parameters, valid for any velocity whatever (including the observed velocity), which is not the same at all.”
In post #9 Eetu has rooted this in the observation that (4) concerns experimental values of two variables V and R for a specific trajectory but (5) concerns the general relationship of V, R, and D for any possible trajectory, that it says nothing about the relationship of V and R unless a stable value is stipulated for D, in which case by tautology the relationship of V and R conforms to the 1/3 power law.
I don’t understand the need to wait until Eetu and I say whether we agree with Bruce, but if it matters to you, Rick, I say I do.
The reasons for waiting were not Rick’s but mine, my acute awareness of my limitations—particularly limitations of my mathematical competence and limitations of my competence to impute motivations.
[I tried to reply from email but discourse cut the message from beginning. It is now corrected here.]
First I want to say that we are now discussing only the first (“preparatory”) part of Rick’s argument. I think it is no use to waste too much energy to this stage because the substantial grave error happens in second part. But the error begins already here.
I’ll attempt a paraphrase of what Eetu, Martin, et al. are saying. If it’s not actually a paraphrase, they’ll tell us, and you can ignore this, Rick, without responding to it.
Thank’s Bruce. for me it seems that you are at least nearly paraphrasing what I tried to say, but there are perhaps some unnecessary complications. Let’s see.
Richard Fineman famously advised us that “The first principle is that you must not fool yourself and you are the easiest person to fool.” I think these folks are saying that you are fooling yourself. The form of the self-deception is equivocation, taking two different things to be the same.
This is at least part of the diagnosis.
Everyone agrees that the V and R variables in (4) are instantiated with values of “the movement data” (Rick), “the measurement values” (Eetu) of particular movement paths. For each path, the relationship between V and R is given in the data, and a particular value of D can be calculated.
Yes.
According to Martin and Eetu and others the variables in (5) are not instantiated with any particular values because (5) is a generalization over all possible values of all three variables, V, R, and D. No ratio or proportion between V and R can be calculated from (5) unless D is held constant. (This is true generally of an equation with three variables, no?)
Yes, this is generally true of all this kind of three variable equations. We can know or infer something about the behavior of one or two variables if we know something about the behavior of the other variables. From this kind of strictly defined relationship between three variables we can infer nothing about the relationship between two of them if we do not know how the third behaves.
Holding D constant is kinda like instantiating D with data, except that values of D are not observational data. D is not observed as such, it is calculated from observed values of V and R. In (4), particular values of D are dependent upon and derived from particular observed values of V and R. In the different specific instantiations of V and R in (4), substituting specific measurements of velocity and curvature, D can have a range of values, and the proportional relation of V and R is variable.
D need not be necessarily held constant because it remains constant quite spontaneously IF the trajectory happens to obey the 1/3 power law (about the relationship between V and R). If the trajectory does not obey (or manifest) just that variant of power relationships between V (velocity) and R (curvature) then D is not constant but varies somehow.
When D is made a constant rather than being dependent upon the variable values of V and R, the proportional relation of V and R is constant. Making D a constant with some arbitrary value in (5) kinda looks like presuming the conclusion (petitio principii, ‘begging the question’).
I cannot say whether or that Rick thinks that D is always constant, because he admits that not all possible trajectories obey 1/3 power law. Rather he seems to think that there is some kind of “mathematical pressure” for all trajectories to come near that power law. But there is no (mathematical) mechanism which could create this kind of pressure. D has no privilege over other possible third elements in those kinds of three-part equations between V, R, and some third variable. Neither there is any metaphysical power or controller which would try to stabilize that D or any other third variable. D stabilizes if there happens to prevail that special power law relationship between V and R. And the question remains, why they behave so sometimes but not always?
rsmarken:
In both cases [i.e. in both equation 4 and equation 5] they are computed from the X and Y values of the movement data
In (4), “the movement data” are data for a particular trajectory and D is derived from V and R; in (5), “the movement data” are all possible values of all three variables V, R, D and no constant proportion between V and R (or any pair of them) can be derived unless one of the three values is given a constant value. “All possible values” is a generalization.
Yes.
A finite set of observational data can be organized in a table Tn. Holding any one variable to a constant value excerpts a subset of that tabulation and in that subset the ratio of the other two values is constant. The values tabulated in Tn are not all possible values. The variables V, R, D in (5) ‘refer to’ imagined values in an imagined tabulation T∞ of all possible values for them. In imagined table T∞ the only way to get a constant ratio of any two values is by excerpting a subset of T∞ in which the third value has a fixed value.
For me this idea of observational table here feels like a complication, but it can be helpful for someone else.
The variables R, V and D in both equations (the denominator on the right in equation 4 is equivalent to D) refer to measures of variable aspects of any curved movement (such as variations in the instantaneous velocity of the movement, V, over time).
“Refer to” has two meanings. In (4), the variables R and V ‘refer to’ the data for each trajectory, one at a time, and D ‘refers to’ these data indirectly, by solving equation (4) for those particular values. In (5), the variables R, V, and D ‘refer to’ ranges of observed values for R and V and calculated values for D. These may be imagined to be in a tabulation of all possible values, though in practice only a subset of all possible values are the observed data and the generalization of (5) enables unobserved values for two variables to be extrapolated from any stipulated value of the third.
Yes, they refer in different ways in different stages – there happens quite of a metamorphosis. First the instantaneous values of R and V are determined in (2) and (3) based on the dotted (time derivative) values of x and y:
But then Rick thinks that contents of the brackets in in both equations are equivalent and replaces the bracketed part of (3) with D and gets the equations (4) and (5). However, as Martin said, these bracketed parts are not equivalent. It is perhaps not very easy to see their difference just by looking at equations. But think that (2) is a measurement of that current velocity and (3) is the measurement of that current curvature. Then note that the curvature is not dependent of the velocity of the object that is moving through it. The object can move through just the same curve with different velocities: you can drive the same bend of the road sometimes slower and sometimes faster. So the bracketed part – which is the velocity – is needed in (3) but it can be any velocity, not just that measured velocity of (2). Why seeing the difference between the bracketed parts of these equations is especially difficult to Rick, is that he thinks already beforehand that there must be some mathematical dependence between velocity and curvature – he needs it to show that power law research is pointless tinkering.
So the eq. (5) has lost its connection to the measurements of the original trajectory and it is valid for any velocity and any curvature – and as such tells nothing about the relationship between velocity and curvature, except if we happen to know that D remains constant.
Eetu
Thanks Bruce, good analysis and good advices!
Rick discusses like politicians. If a politician is asked a question which she cannot understand or cannot answer or she knows that the answer would be disadvantageous to her then she does not even try to answer that question but answers some other unasked question which she can answer with a story she thinks is advantageous to her career.
Again, Discourse cut my message, but I corrected it to: http://discourse.iapct.org/t/discussion-of-the-speed-curvature-power-law/15990/16?u=eetup
Of course we all attribute motivations to others, a propensity that probably goes quite far back in evolutionary time. It can be important in collective control (though not the accidental or coincidental collective control that results from confluence or conflict of effects of control outputs in a shared environment).
I think we attribute motives by associating observed behavior with memory of (apparently) controlled consequences of such actions in the past. If they are memories of our own actions (of course we do perceive our own behavioral outputs), then we also remember what CVs we were controlling. Either way, we can imagine what we would control if we were doing what that individual is doing. So there is something of confession in every attribution (the quip “the hand with the pointing finger has three more pointing back”.)
PCT admonishes us to verify these imagined perceptions. Do we know how to do that?
Please note that I did not claim that Rick would have similar motivation as that ideal typical politician. I meant rather that for me it causes a similar frustration when he does not answer the criticism but says that he does not agree because … and then states just the same claim again which was criticized.
My attempted paraphrase got a passing grade from both Martin and Eetu, with some additional aspects contributed by Eetu.
Can you lay out a summary of their position in your own words, for purposes of mutual verification?
It is customary for a critical article in a journal to begin by summarizing the views that it will criticize. I’m wondering now how much churning of controversy might be avoided if the authors first sent their summary to the other side for verification that it is a fair and accurate representation. Equivocation can still elude detection, but the straw man form of equivocation would be harder to erect. (That’s a generalization, not a pointing finger!)
Here is my summary of what I think is their position. I’m basing it on Martin’s comment in Exp. Brain Res. I believe he is saying that our mistake was to consider the term in parentheses in equation 1 below to be the same as the term in parentheses in equation 2. The difference that we supposedly missed is that the derivatives of x and y in equation 1 are with respect to time while those in equation 2 are with respect to space.
If this were the case it would invalidate our simple algebraic derivation of equation 3, above, because we get the numerator of equation 3 – V^3 – by noting that the numerator in 2 is equivalent to equation 1 raised to the 3rd power.
Our derivation of the relationship between V and R, shown below, depends on the correctness of the derivation of equation 3.
If our derivation of equation 3 is incorrect then equation 4 would also be incorrect, not only because the numerator – V^3 – was incorrectly derived, but also because the first and second derivatives of x and y that make up the denominator, D, shown below, would be incommensurate with the derivatives that were used to calculate V on the left side of the equation.
So if we made the mistake Martin claims we made – failing to note that the derivatives of x and y in equations 1 and 2 are taken with respect to different dimensions, time and space, respectively – then our derivation of equation 7 relating V to R and D, would be incorrect.
My reply to this is that, to my knowledge, it is not true that the derivatives used in the calculation of V (equation 1) are taken with respect to time while those used in the calculation of R (equation 2) are taken with respect to space. The piece of evidence that supports that conclusion is the fact that the derivatives in equations 1 and 2 are all notated the same way, suggesting that they are all taken with respect to the same dimension and that dimension is time.
Equations 1 and 2 are used by power law researchers to calculate the velocity, V, and curvature, R, variables that are used in their regression analysis to test for conformity of their data to a power law. If these calculations really involved calculating derivatives with respect to different dimensions I can’t help but think that that very important fact would be noted in their papers.
Another reason for thinking that the derivatives of x and y are not different in equations 1 and 2 is because other power law researchers – specifically Pollick & Sapiro G (1997) Constant affine velocity predicts the 1/3 power law of planar motion perception generation. Vision Res, 37:347–353 and Maoz U, Portugaly E, Flash T, Weiss Y (2006) Noise and the 2/3 power law. Adv Neural Inf Proc Syst 18:851–858 – have derived the same relationship between V and R as did Marken & Shaffer (2017), the one shown as equation 7.
If this doesn’t answer the claim that my math is wrong then feel free to claim victory.
Best, Rick
A quick fact check. Again, I claim no mathematical expertise, but in [Martin Taylor 2016.09.15.13.41], it appears to me that Martin’s objection is that (1) is with respect to a particular trajectory while (2) is an abstraction over an infinite set of possible trajectories. Specifically, he says that the V in one equation (velocity during a particular trajectory) is not necessarily the same as the V in the other (velocities during possible trajectories), and velocity refers to both time and space in both instances. Yes, (1) refers to V and R during a particular time interval while (2) is timeless, but time vs. space is not the issue that he stated.
The conversion of [Martin Taylor 2016.09.15.13.41] to a Discourse post is difficult to read and omits some data, so I have created a PDF from my csgnet/gmail copy. The ‘Bruce’ that Martin mentions is Bruce Abbott.
That was in 2016, eight years ago. Is there someplace else that Martin abandons this objection and replaces it with an objection that the derivatives of one equation are in respect to time and the derivatives of the other are in respect to space?
First I was going to describe another current political discussion maneuver: mis-re-interpretation of what others have said, but perhaps it is enough to say just that my interpretation about what Martin said is quite different. For to support my interpretation I attach the Martin’s article. Nowhere there reads that “the derivatives of x and y in equation 1 are with respect to time while those in equation 2 are with respect to space**.”** (The numbered equations in Rick’s message I am replying to.) Instead there reads that Marken & Schaffer “noted the visual similarity between the expression under the square root sign in (4) [inside the bracket in eq. 1 mentioned above] and the expression inside the bracket of (5) [eq. 2 mentioned above] and treated them as being the same thing. In (4), however, ẋ (dx/dt) and ẏ (dy/dt) are values observed in an experiment and are used to compute the corresponding velocity, whereas in (5) they are arbitrary parameters, corresponding to any velocity whatever (including the observed velocity).”
I do not even understand what would mean the derivatives of x and y were with respect to space. Of course (or at least naturally) they all are time derivatives: the first derivatives (one dot above) mean x and y velocities and second derivatives (two dots) mean x and y accelerations. If the curve of the trajectory is not drawn or otherwise determined beforehand, then it must be calculated from the movement of the studied moving object. The object moves with some velocity which is calculated in every check point from the first derivatives of its x and y transitions. The curve is calculated from the changes of x and y velocities by the x and y accelerations. So the derivatives in both equations are in this relation similar. This is not your error – if I interpret Martin’s article right.
Instead, your error is to think that by replacing the contents of the brackets in eq. 2 with V you had unveiled a mathematical dependence relation or a function between R and V, curvature and velocity. In the first part (“Mathematical background”) of his article Martin proves with dimensional analysis and by testing different velocity values the case – which I think should be self-evident to everyone – that the curve and velocity are mathematically independent and the eq. 2 will give the right and same curvature independently from what is the velocity of the studied object.
So I would say (once again) that your error is not that you replace the contents of the brackets in eq. 2 with V (the measured velocity). You can of course do it. You can replace it with any possible velocity. But the error is to think that it somehow shows or creates a mathematical relationship between R and V. This is an astonishing error, but still bigger error waits in the OVB.
Taylor2018_Article_CommentsOnMarkenAndShafferTheP.pdf (841 KB)
Yes, and his justification for saying that is at the end of that post [[Martin Taylor 2016.09.15.13.41]] where he says:
MT: Actually one does not have to use mathematics to see the falsity of Rick’s point 4*. As has been pointed out many times, curvature has no relation to time, whereas velocity does. At the risk of inducing boredom by repetition, curvature has the dimension (1/length), while velocity has a dimension (length/time). Although they may be related in experimental observations, they are not, and could not be, mathematically related.
Martin made this point (that curvature is related to space while velocity is related to time) in his comment on Marken & Shaffer (2017) in Exp. Brain Res. That paper was the basis for my description of what I believed Martin thought was our mistake.
Actually it was six, not eight, years ago and the objections to my analysis of the power law have been stated in many different ways during that time and I’ve tried to answer them substantively but, apparently, my answers have never satisfied my opponents. And vice versa.
So after six years of debate about the power law, some of which has taken place on CSGNet, some as published papers in journals (Exp. Brain Res.), and many now in IAPCT Discourse, we are clearly no closer to agreement than we were when this all started.
However, I think the fact that this discussion has been so persistent and so heated suggests that there is something very basic at stack here. I was getting at what it is in my earlier post where I quoted Bill on PCT as a paradigm shift. I think the power law is an excellent example of what Bill was talking about – a problem that no longer needs to be solved due to the paradigm shift that is PCT. I think the power law debate shows that some people think of PCT as a paradigm shift, and some don’t.
I think you may have a point there. I’ll re-read Martin’s Mathematical Analysis section more carefully and see where I might have made my mistake.
While I am trying to better understand what you think is my math mistake in the derivation of the equation relating velocity to curvature of movement, I would appreciate it if you could tell me whether Maoz U, Portugaly E, Flash T, Weiss Y (2006) made the same mistake. After all, they derived the same equation we did (equation (4) below) for the relationship between velocity and curvature from the same equations for computing velocity, V, (what they call v(t)) and curvature, R, (what they call kappa(t), which is curvature measured as 1/R) as follows:
If they did make the same mistake just say “yes” and no further explanation is needed. But if they didn’t could you please explain why. I think that would go a long way toward helping me understand my own mistake.
==========
I have now read and re-read the Mathematical Background section of your comment in Exp Brain Res and I still can’t understand how you got to your conclusion:
Any way the equation is examined, R and V are mathematically completely independent of each other, even if experiments suggest that in many situations they are not factually independent. The research question is why mathematical independence does not imply measured independence in those experimental and observational situations.
I understand this conclusion and, if it were correct, it would certainly mean that our mathematical/statistical explanation of the approximately 1/3 power relationship that is typically found between R on V is wrong.
I think your proof of the mathematical independence of V and R has something to do with the dimensional analysis of equation 3 (which is the crucial step in our derivation of the mathematical relationship between V and R). So if you could go over that I’d appreciate it. I’d also appreciate it if you could explain why this analysis does or doesn’t apply to the Moaz et al derivation of the mathematical relationship between R and V. They used the equivalent of your equation 3 to do it, as you can see here:
The right hand side of their equation 3 is the equivalent of your equation 3 except that V^3 is in the denominator because they are measuring curvature as 1/R rather than R. But that shouldn’t make a difference, should it? According to your comment they are still inappropriately deriving a mathematical dependence between V and R (actually, 1/R) in equation 4. So they must have made the same mistake we did, right?
Anyway, looking forward to your clarifications.
Best, Rick
The only point I have ever tried to make is that your original equation 4 is based on experimental data, but equation 5 applies to any curve and any velocity profile at all. That your data fit equation 5 is no surprise.
It may be relevant that Moaz et al. preface their derivation of their (3) and (4) from the Frenet-Serret formulas with the phrase “for any regular planar curve parameterized with t …”. If the distinction is between the specific and the general, both their formulae are universal generalizations.
My equation 4 is based on algebra, not experimental data. And equation 5 certainly does apply to any curve (with no straight segments). The fact that the the data from all curved movements is fit by equation 5 is not supposed to be surprising; it is a devastating proof that, for any curved movement, when log(R) is regressed on log(V), the data will be fit well by something close to a 1/3 power function when the correlation between the included predictor, log (R), and the omitted covariate, log (D), is small. And Moaz et al showed, via simulation, that the correlation between log(R) and log(D) for most randomly generated movements is, indeed, small.
I read your Exp. Brain Res paper criticizing Marken & Shaffer (2017) several times now and I find it completely unconvincing. Your conclusion that R and V are mathematically independent does not follow from the dimensional analysis in equation 3a that presumably shows this to be the case. And both Maoz et al and I have demonstrated via simulation that it is empirically false; R and V are demonstrably mathematically dependent on each other per equation 4 in Maoz et al (2006) and equation 5 in Marken & Shaffer (2017).
I predict with high confidence that you will reject my evaluation of your criticism of Marken & Shaffer (2017). But I am done with this. You have been able to successfully convince most everyone involved in this discussion, including yourself, that I have made a grave mathematical error in my analysis of the power law. You did it with some pretty snazzy mathematical razzle dazzle, but I am not fooled.
rsmarken:
