Affine velocity (was Re: Behav. illusions)

[Martin Taylor 2018.03.16.17.36]

Thanks. I think I understand. Affine velocity has, so far as I can

see, no relation to affine geometry as described in the various
Wikipedia articles with “affine” in their titles. It has to do with
how velocity is perceived after an affine transformation. In the
end, affine velocity is simply VR-1/3 . Pollack and Sapiro
show that if VR-1/3 is constant, then V is proportional
to R1/3 . That’s a pretty trivial finding, isn’t it?. It
does, however, explain in more convoluted language why your paper
simply states a tautology.

But what does it have to do with the velocity with which people and

other organisms who control some perception or other in tracing
curves do their tracing? And how does it explain the wide range of
exponents actually observed in experiments? If people did control
“affine velocity”, what causes these deviations in their
performances? That is, and has been, the experimental question –
under what conditions do the subjects, human or non-human, produce
what relationship between curvature and velocity, and why.

Martin

image427.png

image428.png

···

[Rick Marken 2018-03-16_12:38:01]

[Martin Taylor 2018.03.15.11.20]

            MT: I must confess to ignorance, here, so would you mind

explaining “affine velocity”. I realize that you don’t
want to respond to the criticisms made in my comment on
your curvature paper, but I’m not criticizing here.

          RM: According to Moaz et al  affine velocity is defined

as

          which, of course, is our "cross-product" variable D (as

mentioned in our rebuttal to you rebuttal). A more
detailed explanation of affine velocity is given in
Pollack and Sapiro (1997), which is available here:

          [https://www.sciencedirect.com/science/article/pii/S0042698996001162](https://www.sciencedirect.com/science/article/pii/S0042698996001162)

                        RM: The velocity of the movement was

surely also being controlled and as we noted
in our rebuttal to the rebuttal it is most
likely that the velocity being controlled is
affine rather than tangential or angular.

image428.png

image427.png

···

[Rick Marken 2018-03-17_09:45:00]

[Martin Taylor 2018.03.16.17.36]

MT: Thanks. I think I understand. Affine velocity has, so far as I can

see, no relation to affine geometry as described in the various
Wikipedia articles with “affine” in their titles. It has to do with
how velocity is perceived after an affine transformation. In the
end, affine velocity is simply VR-1/3 . Pollack and Sapiro
show that if VR-1/3 is constant, then V is proportional
to R1/3. That’s a pretty trivial finding, isn’t it?.

RM: Yes, it is mathematically somewhat trivial but scientifically quite significant. Power law researchers find that when organisms make curved movements V is found to be nearly proportional to R1/3.
 Thus, we have the power “law”,Â
V= R1/3, an observed relationship between the curvature ® and velocity (V) of curved movements that presumably tells us something about the mechanisms that produced these movements. Â

RM: However, both Pollack and Sapiro (1997) and Moaz et al (2006) have shown that there is a mathematical relationship between V and R of the form:Â V=Â R1/3Â *Â alpha1/3
, where alpha =Â
VR-1/3, the affine velocity. So it’s true that when alpha is constant, V will be proportional to R1/3
. But this just proves the point we made in our paper: the empirical power law is a mathematical artifact. The “law” is found to hold to the extent that the affine velocity throughout a curved movement trajectory is constant. If the trajectory is one where affine velocity is constant throughout the movement then V will be found to be proportional to
R1/3
.  If the trajectory is one where affine velocity varies along with curvature  then V will be found to be proportional to R raised to an exponent other than 1/3.
So the extent to which the observed relationship between V and R conforms to the power law depends on what movement trajectory was produced, not on how it was produced.Â

Â

MT: But what does it have to do with the velocity with which people and

other organisms who control some perception or other in tracing
curves do their tracing?

RM: Affine velocity may be one of the perceptual variables people control when they produce curved movements.Â

Â

MT: And how does it explain the wide range of

exponents actually observed in experiments?

 RM: It is explained by the wide range of correlations between affine velocity and curvature for different movement trajectories. That’s what our OVB analysis shows; it is also shown by the equivalent analyses of Pollack and Sapiro (1997) and Moaz et al (2006).Â

MT: If people did control

“affine velocity”, what causes these deviations in their
performances?

RM: It may be because people can control affine velocity better for some kinds of trajectories than for others. This difference in ability to control affine velocity would result in different degrees of correlation between affine velocity and curvature. And it is these differences in the correlation between affine velocity and curvature that results in different power exponents being found for the relationship between R and V using regression analysis that leaves affine velocity out of the analysis.

Â

MT: That is, and has been, the experimental question --

under what conditions do the subjects, human or non-human, produce
what relationship between curvature and velocity, and why.

RM: That’s the wrong question because we now know that what relationship between curvature and velocity is observed-- as determined by regression analysis that leaves affine velocity out of the analysis – depends on what trajectory was produced, not on how it was produced or the conditions under which it was produced. The right question is “what variables are being controlled when people produce curved movement trajectories.”

Best

          RM: According to Moaz et al  affine velocity is defined

as

          which, of course, is our "cross-product" variable D (as

mentioned in our rebuttal to you rebuttal). A more
detailed explanation of affine velocity is given in
Pollack and Sapiro (1997), which is available here:Â

          [https://www.sciencedirect.com/science/article/pii/S0042698996001162](https://www.sciencedirect.com/science/article/pii/S0042698996001162)

Rick

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[Martin Taylor 2018.03.17.13,45]

      You missed the word

“sometimes” or perhaps “often”.

      I think we all agree that

when we find what variables are controlled when fly larvae
approach food, and when people and other animals make movements
that approximate power-law relationships between curvature and
velocity, we will understand why the power is often near 1/3 (or
2/3, depending on which relation you look at), and under what
conditions it has some other value.

      I'm not sure who invented

the term, but it is a misleading one, since it has nothing to do
with the mathematics of affine transformations other than being
the projection of a velocity through an affine transformation.
What remains unclear to me is why this might be interesting when
we are looking for reasons why the person, animal, or larva might
under some conditions appear to control for constant “affine
velocity” and under other conditions not control for velocity,
affine or otherwise.

      I do wish you would

respond to at least one of the criticisms I made in my published
comment, or for that matter, in over a year of CSGnet interaction,
that show your second sentence to be a non-sequitur.

As you said, "affine velocity" is the D in your paper. You cite

Pollack and Shapiro, but not my simpler way of doing the maths they
do to show that D is easily decomposable into V time a pure function
of geometric variables, and is therefore of no value in an
expression that purports to explain or describe the value of V. It
is not clear why you cite Pollack and Shapiro approvingly, but fail
to approve my criticism, instead refusing to mention it in your
rebuttal, instead substituting an entirely different criticism that
you pulled out of thin air. If you believe Pollack and Shapiro, you
have to believe me, too. You can’t have one without the other.

Suppose "affine velocity" varies as radius of curvature minus 6 cm,

what power will be found when the radius of curvature varies from 10
cm to 2 cm in different parts of, say, an ellipse?

      There's no "So" there.

It’s a simple fact. If the curve tracer chooses to stop a while or
reverse direction during the track, the observed relation between
V and R will not conform well to the power law, whatever exponent
you choose.

Maybe it is, but what evidence can you adduce to suggest that it is

other than a simple possibility? After all, other experimenters have
found that minimum jerk (rate of change of acceleration) is usually
close to fitting the data, which makes perception of jerk an equally
plausible possibility for a variable people control, with a
reference value of zero. Personally, i find the concept of “zero” to
be less complex than that of “affine velocity”, so the Ockham’s
Razor vote would go to that, if they were the only two variables in
question.

Again, it would be really nice if you would address the published

criticism of that statement, instead of refuting a criticism never
made. Even if your “refutation” had addressed the actual criticism,
you would still have to answer why and under what conditions the
correlations change so consistently.

To the CSGnet readership more generally, I apologise for this

message, since I said I was going to avoid putting my hands on the
curvature-velocity tar-baby again. But now I suppose I have got
stuck again. I will try to wash my hands of it, but with what
success, I do not know.

Martin

image427.png

image428.png

···

[Rick Marken 2018-03-17_09:45:00]

[Martin Taylor 2018.03.16.17.36]

            MT: Thanks. I think I understand. Affine velocity has,

so far as I can see, no relation to affine geometry as
described in the various Wikipedia articles with
“affine” in their titles. It has to do with how velocity
is perceived after an affine transformation. In the end,
affine velocity is simply VR-1/3 . Pollack and
Sapiro show that if VR-1/3 is constant, then
V is proportional to R1/3 . That’s a pretty
trivial finding, isn’t it?.

            RM: Yes, it

is mathematically somewhat trivial but scientifically
quite significant. Power law researchers find that when
organisms make curved movements V is found to be nearly
proportional to R1/3.

                      RM: According to Moaz et al  affine

velocity is defined as

                      which, of course, is our "cross-product"

variable D (as mentioned in our rebuttal to
you rebuttal). A more detailed explanation of
affine velocity is given in Pollack and Sapiro
(1997), which is available here:

                      [https://www.sciencedirect.com/science/article/pii/S0042698996001162](https://www.sciencedirect.com/science/article/pii/S0042698996001162)

            Thus, we have

the power “law”,
V= R1/3 ,
an observed relationship between the curvature
® and velocity (V) of curved movements that presumably
tells us something about the mechanisms that produced
these movements.

            RM: However,

both Pollack and Sapiro (1997) and Moaz et al (2006)
have shown that there is a mathematical relationship
between V and R of the form: V= R1/3 * alpha1/3 ,
where alpha =
VR-1/3 ,
the affine velocity.

            So it's true

that when alpha is constant, V will be proportional to R1/3 .
But this just proves the point we made in our paper: the
empirical power law is a mathematical artifact.

The “law” is
found to hold to the extent that the affine velocity
throughout a curved movement trajectory is constant. If
the trajectory is one where affine velocity is constant
throughout the movement then V will be found to be
proportional to
R1/3 .

                               If

the trajectory is one where affine velocity varies
along with curvature then V will be found to be
proportional to R
raised to an exponent other than 1/3.

            So the extent

to which the observed relationship between V and R
conforms to the power law depends on what movement
trajectory was produced, not on how it was
produced.

            MT: But what does it

have to do with the velocity with which people and other
organisms who control some perception or other in
tracing curves do their tracing?

          RM: Affine velocity may be one of the perceptual

variables people control when they produce curved
movements.

            MT: And how does it

explain the wide range of exponents actually observed in
experiments?

          RM: It is explained by the wide range of correlations

between affine velocity and curvature for different
movement trajectories. That’s what our OVB analysis shows;

[Rick Marken 2018-03-18_17:15:07]

[Martin Taylor 2018.03.17.13,45]

RM: Thus, we have the power "law", V= R1/3, an observed relationship between the curvature (R) and velocity (V) of curved movements that presumably tells us something about the mechanisms that produced these movements. Â

MT: I think we all agree that when we find what variables are controlled when fly larvae approach food, and when people and other animals make movements that approximate power-law relationships between curvature and velocity, we will understand why the power is often near 1/3 (or 2/3, depending on which relation you look at), and under what conditions it has some other value.

RM: Great. But you won't discover those variables without testing to see what they are.Â

RM: However, both Pollack and Sapiro (1997) and Moaz et al (2006) have shown that there is a mathematical relationship between V and R of the form:Â V=Â R1/3Â *Â alpha1/3, where alpha =Â VR-1/3, the affine velocity.

MT: ... What remains unclear to me is why this might be interesting when we are looking for reasons why the person, animal, or larva might under some conditions appear to control for constant "affine velocity" and under other conditions not control for velocity, affine or otherwise.

RM: You can't look for the conditions under which organisms control for constant affine velocity until you find at least one condition under which they do. This involves testing for the controlled variable.

RM: So it's true that when alpha is constant, V will be proportional to R1/3. But this just proves the point we made in our paper: the empirical power law is a mathematical artifact.

MT: I do wish you would respond to at least one of the criticisms I made in my published comment, or for that matter, in over a year of CSGnet interaction, that show your second sentence to be a non-sequitur.

RM: I did respond to your criticisms in my rebuttal to your paper. If you think they are not correct, feel free to respond to them and explain why they are not.

RM: The "law" is found to hold to the extent that the affine velocity throughout a curved movement trajectory is constant. If the trajectory is one where affine velocity is constant throughout the movement then V will be found to be proportional to R1/3.

MT: As you said, "affine velocity" is the D in your paper. You cite Pollack and Shapiro, but not my simpler way of doing the maths they do to show that D is easily decomposable into V time a pure function of geometric variables, and is therefore of no value in an expression that purports to explain or describe the value of V.

RM: But that's not what they showed. They showed the V is a function of both C and D, as we did. And that when D is constant, the relationship between V and C will follow a 1/3 power relationship. They proved it mathematically and tested it using regression of C on V and we and Maoz et al proved it statistically and tested it using OVB analysis.Â

MT: It is not clear why you cite Pollack and Shapiro approvingly, but fail to approve my criticism, instead refusing to mention it in your rebuttal, instead substituting an entirely different criticism that you pulled out of thin air. If you believe Pollack and Shapiro, you have to believe me, too. You can't have one without the other.

RM: Actually, I can. I cite Pollack and Shapiro because they and Maoz et al discovered the same thing we discovered in our paper; that when you regress C on V omitting the variable D from the analysis you will find that the relationship between C and V is fit by a power law with a coefficient that approximates 1/3 to an extent determined by the correlation between C and the omitted variable D. I don't believe this is what you showed in your criticism of our paper.

RM: If the trajectory is one where affine velocity varies along with curvature  then V will be found to be proportional to R raised to an exponent other than 1/3.

MT: Suppose "affine velocity" varies as radius of curvature minus 6 cm, what power will be found when the radius of curvature varies from 10 cm to 2 cm in different parts of, say, an ellipse?

 RM: I have no idea. To do it exactly you would have to determine, either mathematically or computationally, how these variations affect the correlation between D (affine velocity) and C. I could work it out with my spreadsheet models if you are really interested.

RM: So the extent to which the observed relationship between V and R conforms to the power law depends on what movement trajectory was produced, not on how it was produced.Â

MT: There's no "So" there. It's a simple fact. If the curve tracer chooses to stop a while or reverse direction during the track, the observed relation between V and R will not conform well to the power law, whatever exponent you choose.

RM: Reversals and stops result in different trajectories that will follow the 1/3 or 2/3 power law to the extent that affine velocity is not correlated with curvature throughout the trajectory. The "hows" I'm referring to are the neural and muscular mechanisms that result in the trajectory. That is what the power law is supposed to tell us about. Our paper shows that it's those "hows" that the power law can't tell us about.

RM: Affine velocity may be one of the perceptual variables people control when they produce curved movements.Â

MT: Maybe it is, but what evidence can you adduce to suggest that it is other than a simple possibility?

RM: I explained why affine velocity was a possible controlled variable if a person is trying to control for making a curved movement with constant speed. But the test for the controlled variable always starts with a hypothesis about the controlled variable that is a "simple possibility". Then you iteratively work toward getting a better definition of the controlled variable(s). That's what PCT based research is about.Â
Â

MT: After all, other experimenters have found that minimum jerk (rate of change of acceleration) is usually close to fitting the data, which makes perception of jerk an equally plausible possibility for a variable people control, with a reference value of zero. Personally, i find the concept of "zero" to be less complex than that of "affine velocity", so the Ockham's Razor vote would go to that, if they were the only two variables in question.

RM: Those are output-generation models; minimum jerk is used as a parameter of the output waveform that generates the trajectory open loop. Such models would fail if the trajectory were produced against known disturbances. The model of trajectory production must be a control model (as Figure 1 in our rebuttal paper shows) and if there is a way to make minimum jerk a perceptual variable, it could be a candidate for a controlled variable.Â

MT: And how does it explain the wide range of exponents actually observed in experiments?

 RM: It is explained by the wide range of correlations between affine velocity and curvature for different movement trajectories. That's what our OVB analysis shows;

MT: Again, it would be really nice if you would address the published criticism of that statement, instead of refuting a criticism never made.

RM: I believe we did refute it in the section entitled "Tangled up in statistics" (at least you ya gotta love the section titles).Â
Â

MT: Even if your "refutation" had addressed the actual criticism, you would still have to answer why and under what conditions the correlations change so consistently.

RM: I don't know if we "had to" do that. But I agree that it would be interesting to do. I think the paper on movement in water versus air is what you're talking about. Maybe I will do what you suggest with that data.Â
Â

MT: To the CSGnet readership more generally, I apologise for this message, since I said I was going to avoid putting my hands on the curvature-velocity tar-baby again. But now I suppose I have got stuck again. I will try to wash my hands of it, but with what success, I do not know.

RM: Well I think it's useful. But do what makes you feel best. I hope you can come to the IAPCT meeting so we can discuss it there. Should make good theater;-)
Best
Rick

···

--
Richard S. MarkenÂ
"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

[Martin Taylor 2018.03.19.15.59]

If you remember, that is exactly what your detractors have been

saying for over a year, over your objections that there was no need,
since you had solved the problem mathematically.

.

See [Martin Taylor 2018.03.08.23.07] for just a few examples, and

[Martin Taylor 2018.03.08.23.07] for an even more outrageous example
which I had not seen because my download attempts always cut off at
the same place before it. I saw it because I was sent a PDF of the
whole thing. I have to presume you read neither of those messages.

Bottom line in case you don't still have the two messages (though

you could get them from Dag’s archives): for just about every point
that I made in my published comment, you substituted an invented
criticism drawn from thin air that you could rebut, and falsely
attributed your personal inventions to me. I didn’t analyze Alex’s
criticisms, but I would be surprised to find much difference.

No. I promised myself I would not set foot in the USA while that

dangerous megalomaniac is titular President. It’s bad enough living
next door, where some people have been infected, but I pity you who
live in the middle of the disease and are essentially unprotected. I
hope the vaccination provided by your constitution proves strong
enough to withstand the onslaught.

Martin
···

[Rick Marken 2018-03-18_17:15:07]

[Martin Taylor 2018.03.17.13,45]

              MT: I

think we all agree that when we find what variables
are controlled when fly larvae approach food, and when
people and other animals make movements that
approximate power-law relationships between curvature
and velocity, we will understand why the power is
often near 1/3 (or 2/3, depending on which relation
you look at), and under what conditions it has some
other value.

          RM: Great. But you won't discover those variables

without testing to see what they are.

                          RM:

Thus, we have the power “law”, V= R1/3 ,
an observed relationship between
the curvature ® and velocity (V) of
curved movements that presumably tells us
something about the mechanisms that
produced these movements.


MT: I do wish you
would respond to at least one of the criticisms I made
in my published comment, or for that matter, in over a
year of CSGnet interaction, that show your second
sentence to be a non-sequitur.

          RM: I did respond to your criticisms in my rebuttal to

your paper. If you think they are not correct, feel free
to respond to them and explain why they are not.

      ...

        RM: Well I think it's useful. But do

what makes you feel best. I hope you can come to the IAPCT
meeting so we can discuss it there. Should make good
theater;-)

image434.png

···

[Rick Marken 2018-03-20_18:16:50]

[Martin Taylor 2018.03.19.15.59]

MT: If you remember, that is exactly what your detractors have been

saying for over a year, over your objections that there was no need,
since you had solved the problem mathematically.

RM: Actually, I don’t remember anything like that. Â

MT: See [Martin Taylor 2018.03.08.23.07] for just a few examples, and

[Martin Taylor 2018.03.08.23.07] for an even more outrageous example
which I had not seen because my download attempts always cut off at
the same place before it. I saw it because I was sent a PDF of the
whole thing. I have to presume you read neither of those messages.

MT: Bottom line in case you don't still have the two messages (though

you could get them from Dag’s archives):

RM: I probably have them but it’s difficult for me to search for them.Â

Â

MT: for just about every point

that I made in my published comment, you substituted an invented
criticism drawn from thin air that you could rebut, and falsely
attributed your personal inventions to me.Â

RM: OK, here is what you say is our “critical mistake”:

RM: I answered this by pointing out that it is incorrect to say that dx/dt and dy/dt are arbitrary parameters in the calculation of curvature. In doing so you imply that we were incorrect to calculate curvature from the data, as we did for velocity. In fact,power law researchers calculate both velocity (your equation 4) and curvature (your equation 5) using the values dx/dt and dy/dt (and d2x/dt and d2y/dt) that are derived from the trajectory data. Contrary to your claimÂ
dx/dt and dy/dt are not parameters in the equation for curvature.

MT:Â
I didn’t analyze Alex’s criticisms, but I would be surprised to find much difference.

RM: They seem to differ considerably from you on this “critical mistake”. Indeed, just yesterday Adam confirmed this in a post to CSGNet when he said "
Both C&V [curvature and velocity] are calculated from the response trajectory…". So why don’t you start by you explaining why we (and Adam and I presume the rest of the folks in Alex’s lab)Â are wrong and you are right about our “critical mistake”.Â

Best

Rick


Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

          RM: Great. But you won't discover those variables

without testing to see what they are.Â

          RM: I did respond to your criticisms in my rebuttal to

your paper. If you think they are not correct, feel free
to respond to them and explain why they are not.

[Martin Taylor 2018.03.22.10.08]

I wasn't going to reply to this rather breathtakingly cavalierly

dismissive message, but I broke down.

No, I wouldn't expect you to remember, or to go back over the

myriads of messages that repeated it, if messages from a couple of
weeks ago are too old for you to remember or find. And even if you
did, I imagine you would read them now as carefully as you
apparently did then.

Here's just one example [Martin Taylor 2017.07.27.17.17 Note: The

time stamp wrongly shows 2017, but it was 2016] “* What we need is
an application of the Test for the Controlled Variable… Maybe
there are other hypothetical perceived and controlled variables
that could be disturbed, but at the moment only viscosity has been
used as a disturbance, and so far as I know, it hasn’t been
proposed as a perceived and controlled variable. It does affect
the observed performance, though. What does the observer see as
stabilized against this kind of disturbance, and why? That’s the
kind of question I would be asking, rather than endlessly
rearranging the equations that describe curvature.”*

Oh, dear. It simply doesn't matter to you that I pointed out six

false statements in your rebuttal that described criticisms that you
said I made, substituting in each case a criticism I didn’t make
that you could rebut.

I should have thought that, as a serious scientist, you would have

been properly concerned to set the record straight by showing that
the actual criticisms were misguided, but since you don’t care…

No, I did not. Read the last sentence of what you quoted. Or read

the section called “Mathematical Background” in my comment (or just
equations 1, 2, and 3, if the text is too much.)

Exactly so.

Let me put the question directly. Do you or do you not believe "*      that

only the velocity observed in the experiment can correspond to the
x-dot and y-dot in the expression for R."*

Because Alex and Adam are correct. The velocity and curvature *** in
a particular trial of an experiment*** are indeed
calculated from the response trajectory (which includes the timing
that was observed). I have never, contrary to your claim, said that
this was not the case.

Since you can't or won't go back and look for two easily found

messages from this month, I guess there’s no point in reminding you
that all this was told to you a year and a half ago in [Martin
Taylor 2016.08.16.13.13], which summarized several earlier messages.

I am working under the best-case assumption that you invented your

own criticisms in place of the ones that were actually made because,
as you demonstrate in your message, you could not understand the
actual criticisms in my comment. Since I had made real criticisms,
you had to rebut something rather than let them stand. Not knowing
what actually was being criticised, you only way to find something
to rebut was to invent something you could actually understand.

It would be less than generous to imply the alternative, that you

did understand the various criticisms.

Martin

image434.png

···

[Rick Marken 2018-03-20_18:16:50]

[Martin Taylor 2018.03.19.15.59]

            MT: If you remember, that is exactly what your

detractors have been saying for over a year, over your
objections that there was no need, since you had solved
the problem mathematically.

RM: Actually, I don’t remember anything like that.

                        RM: Great. But you won't discover those

variables without testing to see what they
are.

            MT: See [Martin Taylor 2018.03.08.23.07] for just a few

examples, and [Martin Taylor 2018.03.08.23.07] for an
even more outrageous example which I had not seen
because my download attempts always cut off at the same
place before it. I saw it because I was sent a PDF of
the whole thing. I have to presume you read neither of
those messages.

            MT: Bottom line in

case you don’t still have the two messages (though you
could get them from Dag’s archives):

          RM: I probably have them but it's difficult for me to

search for them.

                        RM: I did respond to your criticisms in

my rebuttal to your paper. If you think they
are not correct, feel free to respond to
them and explain why they are not.

            MT: for just about

every point that I made in my published comment, you
substituted an invented criticism drawn from thin air
that you could rebut, and falsely attributed your
personal inventions to me.

RM: OK, here is what you say is our “critical mistake”:

          RM: I answered this by pointing out that it is

incorrect to say that dx/dt and dy/dt are arbitrary
parameters in the calculation of curvature. In doing so
you imply that we were incorrect to calculate curvature
from the data, as we did for velocity.

          In fact,power law researchers calculate both velocity

(your equation 4) and curvature (your equation 5) using
the values dx/dt and dy/dt (and d2x/dt and d2y/dt) that
are derived from the trajectory data.

          Contrary to your claim 
            dx/dt

and dy/dt are not parameters in the equation for
curvature.

        MT: 
          I

didn’t analyze Alex’s criticisms, but I would be surprised
to find much difference.

        RM: They seem to differ considerably from you on this

“critical mistake”. Indeed, just yesterday Adam confirmed
this in a post to CSGNet when he said "
Both
C&V [curvature and velocity] are calculated from the
response trajectory…". So why don’t you start by you
explaining why we ( and
Adam and I presume the rest of the folks in Alex’s lab) are
wrong and you are right about our “critical mistake”.

image434.png

···

[Rick Marken 2018-03-22_11:07:06]

[Martin Taylor 2018.03.22.10.08]Â

MT: No, I did not. Read the last sentence of what you quoted. Or read

the section called “Mathematical Background” in my comment (or just
equations 1, 2, and 3, if the text is too much.)

RM: What we actually said was that X.dot and Y.dot are variables calculated from the movement trajectory data and that they are the variables used in the calculation of both R and V.Â

This is quite different than asserting that “only the velocity observed in the experiment can correspond to the X.dot and Y.dot used in the expression for R”. As you demonstrated in your “Mathematical Background” section you let anything you want correspond to the X.dot and Y.dot used in the expression for R. But imagining that X.dot and Y.dot correspond to something other than the velocities observed in the experiment is irrelevant to the fact thatÂ
X.dot and Y.dot DO correspond to the velocities observed in the experiment; and these measures of X.dot and Y.dot are used in the calculation of the variables, V and R that are used in the regression analysis that is used to test whether the movement trajectory follows the power law.Â

MT: Exactly so.

RM: OK, so you agree with us.Â

Â

MT: Let me put the question directly. Do you or do you not believe “* that
only the velocity observed in the experiment can correspond to the
x-dot and y-dot in the expression for R.”*

RM: I answered it above but I’ll answered it again. No, I do not believe “that only the velocity observed in the experiment can correspond to the x-dot and y-dot in the expression for R.” What I believe is that only the velocity observed in the experiment corresponds to the x-dot and y-dot that are used in the calculation R (and V).Â

MT: Because Alex and Adam are correct. The velocity and curvature *** in
a particular trial of an experiment*** are indeed
calculated from the response trajectory (which includes the timing
that was observed). I have never, contrary to your claim, said that
this was not the case.

RM: Great. But why the emphasis on “in a particular trial of an experiment”. Since both V and R are based on calculations of the same velocity variables, the values of X.dot and Y.dot (and X.double.dot and Y.double.dot) are the same in the Gribble/Ostry equations for both V and R. So our derivation of the mathematical relationship between V and R (and between A and C) is correct.Â

Â

MT:Â  Since you can't or won't go back and look for two easily found

messages from this month, I guess there’s no point in reminding you
that all this was told to you a year and a half ago in [Martin
Taylor 2016.08.16.13.13], which summarized several earlier messages.

RM: And it was as wrong then as it is now.Â

MT: I am working under the best-case assumption that you invented your

own criticisms in place of the ones that were actually made because,
as you demonstrate in your message, you could not understand the
actual criticisms in my comment. Since I had made real criticisms,
you had to rebut something rather than let them stand. Not knowing
what actually was being criticised, you only way to find something
to rebut was to invent something you could actually understand.

MT: It would be less than generous to imply the alternative, that you

did understand the various criticisms.

RM: Think what you like.Â

BestÂ

Rick


Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery

            MT: for just about

every point that I made in my published comment, you
substituted an invented criticism drawn from thin air
that you could rebut, and falsely attributed your
personal inventions to me.Â

RM: OK, here is what you say is our “critical mistake”:

          RM: I answered this by pointing out that it is

incorrect to say that dx/dt and dy/dt are arbitrary
parameters in the calculation of curvature. In doing so
you imply that we were incorrect to calculate curvature
from the data, as we did for velocity.

          RM: In fact,power law researchers calculate both velocity

(your equation 4) and curvature (your equation 5) using
the values dx/dt and dy/dt (and d2x/dt and d2y/dt) that
are derived from the trajectory data.Â

          RM: Contrary to your claim 
            dx/dt

and dy/dt are not parameters in the equation for
curvature.

        MT:Â 
          I

didn’t analyze Alex’s criticisms, but I would be surprised
to find much difference.

        RM: They seem to differ considerably from you on this

“critical mistake”. Indeed, just yesterday Adam confirmed
this in a post to CSGNet when he said "
Both
C&V [curvature and velocity] are calculated from the
response trajectory…". So why don’t you start by you
explaining why we ( and
Adam and I presume the rest of the folks in Alex’s lab)Â are
wrong and you are right about our “critical mistake”.Â

martin, at this point i do not understand what you are trying to achieve wasting digital ink with rick – forget about it. contemplate a tree thru the window; it is gonna make you more good than this.

image434.png

···

[Rick Marken 2018-03-22_11:07:06]

[Martin Taylor 2018.03.22.10.08]Â

            MT: for just about

every point that I made in my published comment, you
substituted an invented criticism drawn from thin air
that you could rebut, and falsely attributed your
personal inventions to me.Â

RM: OK, here is what you say is our “critical mistake”:

          RM: I answered this by pointing out that it is

incorrect to say that dx/dt and dy/dt are arbitrary
parameters in the calculation of curvature. In doing so
you imply that we were incorrect to calculate curvature
from the data, as we did for velocity.

MT: No, I did not. Read the last sentence of what you quoted. Or read

the section called “Mathematical Background” in my comment (or just
equations 1, 2, and 3, if the text is too much.)

RM: What we actually said was that X.dot and Y.dot are variables calculated from the movement trajectory data and that they are the variables used in the calculation of both R and V.Â

This is quite different than asserting that “only the velocity observed in the experiment can correspond to the X.dot and Y.dot used in the expression for R”. As you demonstrated in your “Mathematical Background” section you let anything you want correspond to the X.dot and Y.dot used in the expression for R. But imagining that X.dot and Y.dot correspond to something other than the velocities observed in the experiment is irrelevant to the fact thatÂ
X.dot and Y.dot DO correspond to the velocities observed in the experiment; and these measures of X.dot and Y.dot are used in the calculation of the variables, V and R that are used in the regression analysis that is used to test whether the movement trajectory follows the power law.Â

MT: Exactly so.

RM: OK, so you agree with us.Â

Â

MT: Let me put the question directly. Do you or do you not believe “* that
only the velocity observed in the experiment can correspond to the
x-dot and y-dot in the expression for R.”*

RM: I answered it above but I’ll answered it again. No, I do not believe “that only the velocity observed in the experiment can correspond to the x-dot and y-dot in the expression for R.” What I believe is that only the velocity observed in the experiment corresponds to the x-dot and y-dot that are used in the calculation R (and V).Â

          RM: In fact,power law researchers calculate both velocity

(your equation 4) and curvature (your equation 5) using
the values dx/dt and dy/dt (and d2x/dt and d2y/dt) that
are derived from the trajectory data.Â

          RM: Contrary to your claim 
            dx/dt

and dy/dt are not parameters in the equation for
curvature.

        MT:Â 
          I

didn’t analyze Alex’s criticisms, but I would be surprised
to find much difference.

        RM: They seem to differ considerably from you on this

“critical mistake”. Indeed, just yesterday Adam confirmed
this in a post to CSGNet when he said "
Both
C&V [curvature and velocity] are calculated from the
response trajectory…". So why don’t you start by you
explaining why we ( and
Adam and I presume the rest of the folks in Alex’s lab)Â are
wrong and you are right about our “critical mistake”.Â

MT: Because Alex and Adam are correct. The velocity and curvature *** in
a particular trial of an experiment*** are indeed
calculated from the response trajectory (which includes the timing
that was observed). I have never, contrary to your claim, said that
this was not the case.

RM: Great. But why the emphasis on “in a particular trial of an experiment”. Since both V and R are based on calculations of the same velocity variables, the values of X.dot and Y.dot (and X.double.dot and Y.double.dot) are the same in the Gribble/Ostry equations for both V and R. So our derivation of the mathematical relationship between V and R (and between A and C) is correct.Â

Â

MT:Â  Since you can't or won't go back and look for two easily found

messages from this month, I guess there’s no point in reminding you
that all this was told to you a year and a half ago in [Martin
Taylor 2016.08.16.13.13], which summarized several earlier messages.

RM: And it was as wrong then as it is now.Â

MT: I am working under the best-case assumption that you invented your

own criticisms in place of the ones that were actually made because,
as you demonstrate in your message, you could not understand the
actual criticisms in my comment. Since I had made real criticisms,
you had to rebut something rather than let them stand. Not knowing
what actually was being criticised, you only way to find something
to rebut was to invent something you could actually understand.

MT: It would be less than generous to imply the alternative, that you

did understand the various criticisms.

RM: Think what you like.Â

BestÂ

Rick


Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.�
                --Antoine de Saint-Exupery


Alex Gomez-Marin, PhD

Research Group Leader

Instituto de Neurociencias

behavior-of-organisms.org

[Martin Taylor 2018.03.22.14.23]

You are probably right, but I hope there are more readers than just

Rick. In case there are, I hope to at least partially insulate them
against Rick’s repeated assertions that his reliance on magical
non-physical illogic is correct and can be used in further
developments and uses of PCT. I feel that not to try vaccination against the infection would be
improper. Having said that, I acknowledge that I said I wasn’t going
to get dragged into the pit again, but have done so. I’m hoping I
can make this my last message on the matter, but this time I do not
think I can promise. Facts do matter.
True, it is indeed “quite different”, but the rest of your paper
depends so strongly on it, that it is important to note that you do
in fact rely on that assumption. I should ask, though, if this “quite different” assertion is what
you think I wrongly claimed, why did you in your rebuttal (and in
the message segment in the nested quote) not rebut it, instead of
rebutting the falsehood that I claimed that you were wrong to
calculate both curvature and velocity from the experimental data, That experimenters correctly computed the velocities and curvatures
using the sampled data from their experiments, yes.
Yes.
No, no, Noooo, NOOOO. Your calculations (and theirs) are correct FOR
THAT TRIAL OF THAT EXPERIMENT. When you use them as a general formula instead of as the results of
a set of sample measurements, you do the same as if you were to say
“Today I observed that the temperature is two degrees Celsius less
than the proportion of cloud cover in the sky” and generalize to say
“When I look at a thermometer I know how cloudy it is without
looking at the sky”.
But of course, you have told us often enough that your logic is
correct, so I guess you really can tell how cloudy it is by looking
at a thermometer, or at least would be able to if you measured one
temperature and looked at the sky once.
Maybe you could explain exactly what is and was wrong. Do you find
fault with my equations 1, 2, and 3? If so, where is the mistake? Do
you find fault with my analysis of “D” that made no reference to
curvature, but that showed “D” to be velocity times a function of
spatial variables?
I’m afraid at this moment I can’t think of other possibilities for
where I was (and am) wrong, so either you thought of something else,
or you found a mistake in one or other of these two derivations.
It’s something you have never told me, in all the times you have
said I was wrong. Maybe you can be a bit more explicit now, and tell
me where I have been wrong (along with so many others who have
commented on your analysis) all these many months?
Martin

image434.png

···

On 2018/03/22 2:15 PM, Alex Gomez-Marin
wrote:

      martin, at this point i do not understand what

you are trying to achieve wasting digital ink with rick –
forget about it. contemplate a tree thru the window; it is
gonna make you more good than this.

              [Rick Marken

2018-03-22_11:07:06]

[Martin Taylor 2018.03.22.10.08]

                                MT:

for just about every point that I
made in my published comment, you
substituted an invented criticism
drawn from thin air that you could
rebut, and falsely attributed your
personal inventions to me.

                              RM: OK, here is what you say is our

“critical mistake”:

                              RM: I answered this by pointing out

that it is incorrect to say that dx/dt
and dy/dt are arbitrary parameters in
the calculation of curvature. In doing
so you imply that we were incorrect to
calculate curvature from the data, as
we did for velocity.

                  MT: No, I did

not. Read the last sentence of what you quoted. Or
read the section called “Mathematical Background”
in my comment (or just equations 1, 2, and 3, if
the text is too much.)

                RM: What we actually said was that X.dot and

Y.dot are variables calculated from the movement
trajectory data and that they are the variables used
in the calculation of both R and V.
This
is quite different than asserting that “only the
velocity observed in the experiment can correspond
to the X.dot
and Y.dot used in the expression for R”.

                  As

you demonstrated in your “Mathematical Background”
section you let anything you want correspond to
the X.dot and Y.dot used in the expression for R.
But imagining that X.dot and Y.dot correspond to
something other than the velocities observed in
the experiment is irrelevant to the fact that
X.dot
and Y.dot DO correspond to the velocities
observed in the experiment; and these measures
of X.dot and Y.dot are used in the calculation
of the variables, V and R that are used in the
regression analysis that is used to test
whether the movement trajectory follows the
power law.

                              RM: In fact,power law researchers

calculate both velocity (your equation
4) and curvature (your equation 5)
using the values dx/dt and dy/dt (and
d2x/dt and d2y/dt) that are derived
from the trajectory data.

MT: Exactly so.

RM: OK, so you agree with us.

RM: Contrary to your claim dx/dt
and dy/dt are not parameters in the
equation for curvature.

                  MT: Let me put the question directly. Do

you or do you not believe “* that only the
velocity observed in the experiment can
correspond to the x-dot and y-dot in the
expression for R.”*

                  RM: I

answered it above but I’ll answered it again. No,
I do not believe " that
only the velocity observed in the experiment can
correspond to the x-dot and y-dot in the
expression for R." What I believe is that only
the velocity observed in the experiment
corresponds to the x-dot and y-dot that are
used in the calculation R (and V).

  Yes, and so do I and I

think all your other critics. How about dealing with the actual
criticism instead of reiterating things everyone can agree on as
though they had some relevance?

                            MT: 
                              I

didn’t analyze Alex’s criticisms, but
I would be surprised to find much
difference.

                            RM: They seem to differ considerably

from you on this “critical mistake”.
Indeed, just yesterday Adam confirmed
this in a post to CSGNet when he said "
Both
C&V [curvature and velocity] are
calculated from the response
trajectory…". So why don’t you start
by you explaining why we ( and
Adam and I presume the rest of the
folks in Alex’s lab) are
wrong and you are right about our
“critical mistake”.

                  MT: Because

Alex and Adam are correct. The velocity and
curvature *** in a particular trial of an
experiment*** are indeed calculated from
the response trajectory (which includes the timing
that was observed). I have never, contrary to your
claim, said that this was not the case.

                RM: Great. But why the emphasis on "in a

particular trial of an experiment". Since both V
and R are based on calculations of the same velocity
variables, the values of X.dot and Y.dot (and
X.double.dot and Y.double.dot) are the same in the
Gribble/Ostry equations for both V and R.

                So our derivation of the mathematical

relationship between V and R (and between A and C)
is correct.

      MT:  Since you can't or won't go back and

look for two easily found messages from this month, I guess
there’s no point in reminding you that all this was told to
you a year and a half ago in [Martin Taylor 2016.08.16.13.13],
which summarized several earlier messages.

RM: And it was as wrong then as it is now.