# Power Law re-boot

[Roger Moore 2016.09.15.22.12 BST]

1. THE MATHEMATICS

Equation 8 states that the instantaneous velocity V is a function of the first-derivative of any movement decomposed into X and Y coordinate space. This means that V can be calculated from the velocity in X (Xdot) and the velocity in Y (Ydot). It also means that V is independent of any other variables - i.e. V is independent of any acceleration parameters and hence, by definition, INDEPENDENT OF CURVATURE.

Equation 9 states that the radius (inverse curvature) R of a curve is a function of the first and second derivatives of any movement decomposed into X and Y coordinate space. This means that R can be calculated from the velocity in X (Xdot), the velocity in Y (Ydot), the acceleration in X (Xdoubledot) and the acceleration in Y (Ydoubledot).

Rick’s equation relating V to R and D is correct in that the derivation contains no errors. However, it is NOT CORRECT to interpret V as a function of R only. In Rick’s equation, V is a function of R and D. This means that D CANNOT BE IGNORED. For all values of X and Y, D has a value which cancels R exactly, and leads to a true result - V=V (which has already been pointed out).

Hence, the mathematics tells us that V and R vary independently, exactly as one would expect. There is NO hidden “law” relating V and R.

1. THE PROBLEM

If the observed effect - slowing during curved trajectories - is the result of a closed-loop negative-feedback control system, then it is possible to hypothesise an explanation as follows:

a) Let Vr be the intended (reference) velocity.

b) Let Vp be the perceived velocity.

c) Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e) If Vp=Vr, then e=0 and Vm will remain unchanged.

f) If Vp>Vr, then Vm will be reduced until e=0.

So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature. I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature. In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly. (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

Of course, this only explains the slowing, it doesn’t explain the so-called ‘Power Law’. In order to derive the necessary relations, it would be necessary to analyse the particular perceptual system(s) involved in estimating speed. E.g., for a visual perceptual system, I would look to an estimate of velocity based on something like ‘optical flow’. In other cases one might have to consider the characteristics of proprioceptive feedback. In each case, the hypothesis is that the perceived speed is OVERESTIMATED and hence gives rise to a compensatory reduction in velocity.

This is a scientific hypothesis that can be tested by modelling/simulation of perceptual systems with the appropriate characteristics, and then comparing those characteristics with those found in living systems.

Roger

···

Prof ROGER K MOORE* BA(Hons) MSc PhD FIOA FISCA MIET

Chair of Spoken Language Processing
Vocal Interactivity Lab (VILab), Sheffield Robotics
Speech & Hearing Research Group (SPandH)
Department of Computer Science, UNIVERSITY OF SHEFFIELD
Regent Court, 211 Portobello, Sheffield, S1 4DP, UK

• Winner of the 2016 Antonio Zampolli Prize for "*Outstanding Contributions *

*to the Advancement of Language Resources & Language Technology *

Evaluation within Human Language Technologies"

e-mail: r.k.moore@sheffield.ac.uk
web: http://www.dcs.shef.ac.uk/~roger/
twitter: @rogerkmoore
Tel: +44 (0) 11422 21807
Fax: +44 (0) 11422 21810
Mob: +44 (0) 7910 073631

Editor-in-Chief: COMPUTER SPEECH AND LANGUAGE
(http://www.journals.elsevier.com/computer-speech-and-language/)

Hi Roger, thank goodness for your arrival on the scene! It was getting very messy, and your analysis, incorporating both the mathematics and the organism’s perspective on its velocity, is very neat indeed. And scientific!

Warren & Vyv

···

Prof ROGER K MOORE* BA(Hons) MSc PhD FIOA FISCA MIET

Chair of Spoken Language Processing
Vocal Interactivity Lab (VILab), Sheffield Robotics
Speech & Hearing Research Group (SPandH)
Department of Computer Science, UNIVERSITY OF SHEFFIELD
Regent Court, 211 Portobello, Sheffield, S1 4DP, UK

• Winner of the 2016 Antonio Zampolli Prize for "*Outstanding Contributions *

*to the Advancement of Language Resources & Language Technology *

Evaluation within Human Language Technologies"

e-mail: r.k.moore@sheffield.ac.uk
web: http://www.dcs.shef.ac.uk/~roger/
twitter: @rogerkmoore
Tel: +44 (0) 11422 21807
Fax: +44 (0) 11422 21810
Mob: +44 (0) 7910 073631

Editor-in-Chief: COMPUTER SPEECH AND LANGUAGE
(http://www.journals.elsevier.com/computer-speech-and-language/)

[Martin Taylor 2016.09.15.23.09]

``````      [Roger
``````

Moore 2016.09.15.22.12 BST]

``````      2.
``````

THE PROBLEM

``````      If
``````

the observed effect - slowing during curved trajectories - is
the result of a closed-loop negative-feedback control system,
then it is possible to hypothesise an explanation as follows:

``````      a)
``````

Let Vr be the intended (reference) velocity.

``````      b)
``````

Let Vp be the perceived velocity.

``````      c)
``````

Let e=Vr-Vp

``````      d).
``````

Let Vm be the output velocity, where Vm is some function of e.

``````      e)
``````

If Vp=Vr, then e=0 and Vm will remain unchanged.

``````      f)
``````

If Vp>Vr, then Vm will be reduced until e=0.

``````      So,
``````

given that an increase in ‘curvature’ is observed to elicit a
reduction in ‘speed’, it is possible to deduce that Vp>Vm
in the presence of curvature. I.e. the ‘perception’ of
velocity is affected by the ‘disturbance’ of curvature. In
other words, when going around a curve, an organism perceives
itself to be going faster than it really is, and it slows
accordingly. (This certainly concurs with my own experiences
when driving with the cruise control switched on - a speed
that feels comfortable on the straight can feel very
uncomfortable when you encounter even the slightest bend.)

``````      Of
``````

course, this only explains the slowing, it doesn’t explain the
so-called ‘Power Law’. In order to derive the necessary
relations, it would be necessary to analyse the particular
perceptual system(s) involved in estimating speed. E.g., for
a visual perceptual system, I would look to an estimate of
velocity based on something like ‘optical flow’. In other
cases one might have to consider the characteristics of
proprioceptive feedback. In each case, the hypothesis is that
the perceived speed is OVERESTIMATED and hence gives rise to a
compensatory reduction in velocity.

``````      This
``````

is a scientific hypothesis that can be tested by
modelling/simulation of perceptual systems with the
appropriate characteristics, and then comparing those
characteristics with those found in living systems.

Roger

``````That's a different possibility from any I had considered, and a very
``````

interesting one. It raises the whole question of the perception of
time as a function of the information rate of control. Time drags
when you are bored, but seems to fly by when you are productively
“doing” something. Maybe the time component of velocity changes its
perceptual rate on curves.

``````But would this account for the fly larva seeking the source of food
``````

odour? And would you see there being a difference in principle
between cases in which the perception is OF the moving object (e.g.
finger drawing a pattern) rather than FROM the moving object (e.g.
car driver)?

``````After reading your message, I went to Google Scholar to see if
``````

anyone had studied the perception of velocity on curves, and found
two interesting papers by Viviani and Stucchi (1989 and 1992; and --the PDF download link of the latter works but the resulting URL is
too long to quote). In the first, they showed that if subject
adjusted the velocity of a point moving around an ellipse or in a
squiggle pattern until it was perceived to move over the whole
trajectory at a uniform pace, the true velocity conformed to a power
law relation with curvature (usually rather less than the 1/3
power). The second showed that the velocity profile affected the
perceived shape of a trajectory. If the point trajectory was
circular, but the velocity varied as it would when someone drew an
ellipse, the circle appeared flattened. So there does seem to be a
relation between perceived velocity and perceived curvature, even
when looking at rather than from the moving point.
For the record, apart from the possible experiments I suggested to
Alex a while back, one of which was to use the Powers
“Circle-Square” illusion to tease apart perceptual from motor
effects in the power-law, the approach I’ve been working on has been
related to the effective transport lag of the control loop, thinking
of how far ahead along the trajectory one might need to look at a
target direction or lateral acceleration in order to be on course
when one got there. Changes of curvature would be what matters, not
curvature as such, because at least in a non-viscous medium a
constant lateral acceleration force keeps you on a constant
curvature section of the track – setting the car wheels at a
particular angle will do that. It’s a quite different way to look at
the problem than yours is – or is it, if you also consider the
“time drags” perception?
Thanks a lot for chiming in with a new idea.
Martin

···

[From Bruce Abbott (2016.09.16.0900 EDT)]

Thanks for this, Roger.Â Perhaps now the debate with Rick over his mistaken analysis can be laid to rest.Â I, for one, have no interest in continuing it.

Thanks also for your interesting proposal – I nnever would have thought in terms of speed-perception being influenced by curvature.Â I do recall one incident in my own life that relates to the issue.Â A car turned in front of me from the left-hand lane, cutting across my lane to make a right turn.Â I tried to make the turn with him but we ended up side-swiping, after which my car went over the curb into a vacant lot, just missing a tree.Â During this whole incident, everything seemed to go into slow motion, and today, over 50 years later, I can still remember every moment.Â This slow-motion effect is well known but I am not aware of an explanation for it.Â It is opposite the effect you are proposing for entering a turn – there it would seem that one perceives the speed as increasing, as though time were going by more quickly.

As I noted in a previous post, a power-law relation may emerge under a variety of circumstances, owing to different factors being at play.Â Thus there may be no one answer to the question of what enforces such a relation between speed and curvature.Â A different PCT model may be required to account for each case in which a power-law (or other) relation is consistently observed.

Bruce

[Roger Moore 2016.09.15.22.12 BST] –

1. THE MATHEMATICS

Equation 8 states that the instantaneous velocity V is a function of the first-derivative of any movement decomposed into X and Y coordinate space. This means that V can be calculated from the velocity in X (Xdot) and the velocity in Y (Ydot). It also means that V is independent of any other variables - i.e. V is independent of any acceleration parameters and hence, by definition, INDEPENDENT OF CURVATURE.

Equation 9 states that the radius (inverse curvature) R of a curve is a function of the first and second derivatives of any movement decomposed into X and Y coordinate space. This means that R can be calculated from the velocity in X (Xdot), the velocity in Y (Ydot), the acceleration in X (Xdoubledot) and the acceleration in Y (Ydoubledot).

Rick’s equation relating V to R and D is correct in that the derivation contains no errors. However, it is NOT CORRECT to interpret V as a function of R only. In Rick’s equation, V is a function of R and D. This means that D CANNOT BE IGNORED. For all values of X and Y, D has a value which cancels R exactly, and leads to a true result - V=V (which has already been pointed out).

Hence, the mathematics tells us that V and R vary independently, exactly as one would expect. There is NO hidden “law” relating V and R.

1. THE PROBLEM

If the observed effect - slowing during curved trajectories - is the result of a closed-loop negative-feedback control system, then it is possible to hypothesise an explanation as follows:

a) Let Vr be the intended (reference) velocity.

b) Let Vp be the perceived velocity.

c) Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e) If Vp=Vr, then e=0 and Vm will remain unchanged.

f) If Vp>Vr, then Vm will be reduced until e=0.

So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature. I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature. In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly. (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

Of course, this only explains the slowing, it doesn’t explain the so-called ‘Power Law’. In order to derive the necessary relations, it would be necessary to analyse the particular perceptual system(s) involved in estimating speed. E.g., for a visual perceptual system, I would look to an estimate of velocity based on something like ‘optical flow’. In other cases one might have to consider the characteristics of proprioceptive feedback. In each case, the hypothesis is that the perceived speed is OVERESTIMATED and hence gives rise to a compensatory reduction in velocity.

This is a scientific hypothesis that can be tested by modelling/simulation of perceptual systems with the appropriate characteristics, and then comparing those characteristics with those found in living systems.

Roger

[From Rick Marken (2016.09.16.1000)]

···

Roger Moore (2016.09.15.22.12 BST)–

RM: Wow, a post from James Bond;-) Since your initials are the same as mine I’ll note your comments with JB.

JB: Rick’s equation relating V to R and D is correct in that the derivation contains no errors. However, it is NOT CORRECT to interpret V as a function of R only.

RM: Actually, I don’t. V is a function of R and V^3/R (which I call D) as follows:

V = D^1/3R1/3 (1)

JB: In Rick’s equation, V is a function of R and D. This means that D CANNOT BE IGNORED.

RM: Right!!

JB: For all values of X and Y, D has a value which cancels R exactly, and leads to a true result - V=V (which has already been pointed out).

RM: You lost me there, Mr. Bond. Let’s look at the linear version of equation (1):

log(V) = 1/3log(D) + 1/3log® (2)

RM: There is no canceling of R going on that I can see; log (V) is a function of log(D) and log® with the coefficients shown. As far as this formula implying that V = V, that is true of every mathematical equation. For example, if x = y +k*z we can replace the right side of the equation with x so that x = x. Try it. You’ll see that it works as smoothly as your Aston Martin.

JB: Hence, the mathematics tells us that V and R vary independently, exactly as one would expect. There is NO hidden “law” relating V and R.

RM: Actually, there is a mathematical law relating V and R; it’s given by equation (2).

JB: 2. THE PROBLEM

If the observed effect - slowing during curved trajectories - is the result of a closed-loop negative-feedback control system, then it is possible to hypothesise an explanation as follows:

RM: This is an excellent description of the illusion to which power law researchers are enthusiastically succumbing; it appears that curved trajectories have the effect of slowing the movement through them. I think your model is an excellent way to expose the illusion.

JB: a) Let Vr be the intended (reference) velocity.

b) Let Vp be the perceived velocity.

c) Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e) If Vp=Vr, then e=0 and Vm will remain unchanged.

f) If Vp>Vr, then Vm will be reduced until e=0.

JB: So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature. I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature. In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly.

RM: I think the best way to see the problem with this model is to set it up as a working model and see what happens. But I’ll just make some qualitative comments on it. First, the controlled variable in this model is velocity, Vp. The output that affects this variable is also velocity, Vm. You will find that that’s a big problem if you try to build a working version of your model. The output of a control system must be a variable that affects the controlled variable, Vp. That means the output should be something like variations in the X, Y coordinates of whatever it is that is producing the movement, Vp.

RM: Another problem is that the presumed “cause” of the variation in velocity – curvature – doesn’t appear in the model. Your verbal description suggests that your are assuming that curvature, C, affects the state of the controlled variable, Vp. So you should have a term like Vp = C + Vm somewhere in the model since the state of a controlled variable is a function of disturbances (curvature, C) and output (Vm)–though you are still going to have to work on that output variable)

RM: You go on to say that an increase in curvature results in an increase in Vp relative to Vm suggesting that Vm is the reference to which the perception of velocity is being compared. I think you must mean that an increase in curvature results in an increase in Vp relative to Vr, which is what your equations say. Assuming that this is the case, then you have a model that (if it works) will keep Vp = Vr and if Vr is constant then the model will move at a constant velocity through all curves, which is not what is observed.

RM: The reason you are having trouble developing a proper control model of the behavior you see (the apparent slowing of movement through curves) is that there is no evidence, in the trajectory itself, of any variable being controlled. Power law researchers are not interested in studying control; they don’t even know what it is. So they see the power law as evidence of environmental constraints on movement (an S-R view). Control theorists who don’t understand the nature of the phenomenon that control theory explains (control) see the power law as a disturbance-output relationship that is evidence of the existence of controlled variable to which to which they are unable to point but just must be there.

RM: This is why the first thing people who are interested in applying control theory to behavior should learn is the nature of the phenomenon that control theory explains: CONTROL.

Best regards

Rick

Of course, this only explains the slowing, it doesn’t explain the so-called ‘Power Law’. In order to derive the necessary relations, it would be necessary to analyse the particular perceptual system(s) involved in estimating speed. E.g., for a visual perceptual system, I would look to an estimate of velocity based on something like ‘optical flow’. In other cases one might have to consider the characteristics of proprioceptive feedback. In each case, the hypothesis is that the perceived speed is OVERESTIMATED and hence gives rise to a compensatory reduction in velocity.

This is a scientific hypothesis that can be tested by modelling/simulation of perceptual systems with the appropriate characteristics, and then comparing those characteristics with those found in living systems.

Roger

Prof ROGER K MOORE* BA(Hons) MSc PhD FIOA FISCA MIET

Chair of Spoken Language Processing
Vocal Interactivity Lab (VILab), Sheffield Robotics
Speech & Hearing Research Group (SPandH)
Department of Computer Science, UNIVERSITY OF SHEFFIELD
Regent Court, 211 Portobello, Sheffield, S1 4DP, UK

• Winner of the 2016 Antonio Zampolli Prize for "*Outstanding Contributions *

*to the Advancement of Language Resources & Language Technology *

Evaluation within Human Language Technologies"

e-mail: r.k.moore@sheffield.ac.uk
web: http://www.dcs.shef.ac.uk/~roger/
twitter: @rogerkmoore
Tel: +44 (0) 11422 21807
Fax: +44 (0) 11422 21810
Mob: +44 (0) 7910 073631

Editor-in-Chief: COMPUTER SPEECH AND LANGUAGE
(http://www.journals.elsevier.com/computer-speech-and-language/)

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Martin Taylor 2016.09.16.13.10]

``````Aye, and there's the rub. There's none so blind as they who will not
``````

see.
Just try entering the value of log (D) into that equation, and you
would be able to see it. (Hint: log(D) = 3*log(V) - log® ).
Martin

···

On 2016/09/16 12:58 PM, Richard Marken
wrote:

[From Rick Marken (2016.09.16.1000)]

``````              Roger
``````

Moore (2016.09.15.22.12 BST)–

``````          RM: Wow, a post from James Bond;-) Since your initials
``````

are the same as mine I’ll note your comments with JB.

``````              JB:
``````

Rick’s equation relating V to R and D is correct in
that the derivation contains no errors. However, it
is NOT CORRECT to interpret V as a function of R
only.

``````          RM: Actually, I don't. V is a function of R and V^3/R
``````

(which I call D) as follows:

V = D^1/3R1/3 (1)

``````              JB:
``````

In Rick’s equation, V is a function of R and D. This
means that D CANNOT BE IGNORED.

RM: Right!!

``````              JB:
``````

For all values of X and Y, D has a value which cancels
R exactly, and leads to a true result - V=V (which has
already been pointed out).

``````          RM: You lost me there, Mr. Bond. Let's look at the
``````

linear version of equation (1):

log(V) = 1/3log(D) + 1/3log® (2)

RM: There is no canceling of R going on that I can see;

[From Rick Marken (2016.09.16.1050)]

···

Martin Taylor (2016.09.16.13.10)–

``````MT: Aye, and there's the rub. There's none so blind as they who will not
``````

see.

``````MT: Just try entering the value of log (D) into that equation, and you
``````

would be able to see it. (Hint: log(D) = 3*log(V) - log® ).

RM: Thanks. I see. So log (V) = log (V), just as it should. Equation (2) shows that log(V) can be decomposed into two sources of variance, log(D) and log®. And, indeed, when these two sources of variance are taken into account in a regression analysis, all of the variance in log(V) is accounted for (R^2 = 1.0). When only log® is used in the regression, as in power law research, only a portion of the variance in log(V) is accounted for and the estimate of the coefficient of log® is biased relative to it’s true vale (1/3), unless the covariance of log(D) and log® is 0 (as it is in perfect ellipses.

Best

Rick

log(V) = 1/3log(D) + 1/3log® (2)

RM: There is no canceling of R going on that I can see;

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

https://en.m.wikipedia.org/wiki/File:Groundhog_Day_(movie_poster).jpg

···

Martin Taylor (2016.09.16.13.10)–

``````MT: Aye, and there's the rub. There's none so blind as they who will not
``````

see.

``````MT: Just try entering the value of log (D) into that equation, and you
``````

would be able to see it. (Hint: log(D) = 3*log(V) - log® ).

RM: Thanks. I see. So log (V) = log (V), just as it should. Equation (2) shows that log(V) can be decomposed into two sources of variance, log(D) and log®. And, indeed, when these two sources of variance are taken into account in a regression analysis, all of the variance in log(V) is accounted for (R^2 = 1.0). When only log® is used in the regression, as in power law research, only a portion of the variance in log(V) is accounted for and the estimate of the coefficient of log® is biased relative to it’s true vale (1/3), unless the covariance of log(D) and log® is 0 (as it is in perfect ellipses.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

log(V) = 1/3log(D) + 1/3log® (2)

RM: There is no canceling of R going on that I can see;

Hi Roger, thank goodness for your arrival on the scene! It was getting very messy, and your analysis, incorporating both the mathematics and the organism's perspective on its velocity, is very neat indeed. And scientific!

Warren & Vyv

RKM: Thanks both. As you can guess, it was an attempt to get the discussion back on the rails (using my last breath of fresh air before teaching starts next week) - one can tolerate only so much noisy traffic in one's in-box

[Roger Moore 2016.09.15.22.12 BST]

1. THE MATHEMATICS

Equation 8 states that the instantaneous velocity V is a function of the first-derivative of any movement decomposed into X and Y coordinate space. This means that V can be calculated from the velocity in X (Xdot) and the velocity in Y (Ydot). It also means that V is independent of any other variables - i.e. V is independent of any acceleration parameters and hence, by definition, INDEPENDENT OF CURVATURE.

Equation 9 states that the radius (inverse curvature) R of a curve is a function of the first and second derivatives of any movement decomposed into X and Y coordinate space. This means that R can be calculated from the velocity in X (Xdot), the velocity in Y (Ydot), the acceleration in X (Xdoubledot) and the acceleration in Y (Ydoubledot).

Rick's equation relating V to R and D is correct in that the derivation contains no errors. However, it is NOT CORRECT to interpret V as a function of R only. In Rick's equation, V is a function of R and D. This means that D CANNOT BE IGNORED. For all values of X and Y, D has a value which cancels R exactly, and leads to a true result - V=V (which has already been pointed out).

Hence, the mathematics tells us that V and R vary independently, exactly as one would expect. There is NO hidden "law" relating V and R.

2. THE PROBLEM

If the observed effect - slowing during curved trajectories - is the result of a closed-loop negative-feedback control system, then it is possible to hypothesise an explanation as follows:

a) Let Vr be the intended (reference) velocity.

b) Let Vp be the perceived velocity.

c) Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e) If Vp=Vr, then e=0 and Vm will remain unchanged.

f) If Vp>Vr, then Vm will be reduced until e=0.

So, given that an increase in 'curvature' is observed to elicit a reduction in 'speed', it is possible to deduce that Vp>Vm in the presence of curvature. I.e. the 'perception' of velocity is affected by the 'disturbance' of curvature. In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly. (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

Of course, this only explains the slowing, it doesn't explain the so-called 'Power Law'. In order to derive the necessary relations, it would be necessary to analyse the particular perceptual system(s) involved in estimating speed. E.g., for a visual perceptual system, I would look to an estimate of velocity based on something like 'optical flow'. In other cases one might have to consider the characteristics of proprioceptive feedback. In each case, the hypothesis is that the perceived speed is OVERESTIMATED and hence gives rise to a compensatory reduction in velocity.

This is a scientific hypothesis that can be tested by modelling/simulation of perceptual systems with the appropriate characteristics, and then comparing those characteristics with those found in living systems.

Roger

--------------------------------------------------------------------------------------------
Prof ROGER K MOORE* BA(Hons) MSc PhD FIOA FISCA MIET

Chair of Spoken Language Processing
Vocal Interactivity Lab (VILab), Sheffield Robotics
Speech & Hearing Research Group (SPandH)
Department of Computer Science, UNIVERSITY OF SHEFFIELD
Regent Court, 211 Portobello, Sheffield, S1 4DP, UK
* Winner of the 2016 Antonio Zampolli Prize for "Outstanding Contributions

to the Advancement of Language Resources & Language Technology
Evaluation within Human Language Technologies"

···

On 16 September 2016 at 00:21, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

On 15 Sep 2016, at 22:12, Prof. Roger K. Moore <<mailto:r.k.moore@sheffield.ac.uk>r.k.moore@sheffield.ac.uk> wrote:

e-mail: <mailto:r.k.moore@dcs.shef.ac.uk>r.k.moore@sheffield.ac.uk
web: <https://urldefense.proofpoint.com/v2/url?u=http-3A__www.dcs.shef.ac.uk_-7Eroger_&d=CwMFaQ&c=8hUWFZcy2Z-Za5rBPlktOQ&r=-dJBNItYEMOLt6aj_KjGi2LMO_Q8QB-ZzxIZIF8DGyQ&m=TtsOi2zHq871hux2nXRvDpPTt1Ew16hcLq0eJEhot90&s=20XIZ_UnesKjxiV370aQnw9h40xzi-j-ncCfuPjPHVA&e=>http://www.dcs.shef.ac.uk/~roger/
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Fax: +44 (0) 11422 21810
Mob: <tel:%2B44%20%280%29%207910%20073631>+44 (0) 7910 073631

[Fred Nickols (2016.09.17.1402 ET)]

Professor Moore (aka RKM):

I have a question about the section of your post labeled âThe Problem.â?

RKM:

a) Let Vr be the intended (reference) velocity.

b) Let Vp be the perceived velocity.

c) Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e) If Vp=Vr, then e=0 and Vm will remain unchanged.

f) If Vp>Vr, then Vm will be reduced until e=0.

RKM: So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature. I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature. In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly. (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

FN: Woudnât it also be possible to account for the reduction in speed as a result of a lowering of Vr?Â If I understand what you say above, you say the slowing occurs because the organism perceives itself as going faster than it really is.Â I would have said the organism slows because its reference signal for velocity has been reduced.Â If it is actually a control system, it is keeping perceived speed aligned with intended speed.Â In order for it to slow, there would have to be a decrease in Vr.Â Vp is quite simply the perceived velocity.Â The organism has no knowledge of ârealâ? velocity.Â It simply keeps Vp aligned with Vr and in order for any slowing to take place it would have to be the result of a reduction in Vr.

FN: Do I have that correct?

FN: If so, then given the hierarchical nature of PCT, a reduced Vr would have to come from the next higher level in the hierarchy.Â My uneducated guess would be that Vr comes from some higher level systems concerned with matters such as staying on the road, safety, etc.Â As an experienced driver of all kinds of paved and unpaved roads, when I find myself coming upon a curve I tend to slow down and the sharper I perceive the curve to be the more I slow down.Â I think what Iâm doing is reducing the Vr in your description above.

FN: Am I muddying the waters?

Regards,

Fred Nickols, Consultant

My Objective is to Help You Achieve Yours

DISTANCE CONSULTING LLC

âAssistance at a Distanceâ?SM

···

From: Prof. Roger K. Moore [mailto:r.k.moore@sheffield.ac.uk]
Sent: Saturday, September 17, 2016 1:37 PM
To: csgnet@lists.illinois.edu
Subject: Re: Power Law re-boot

On 16 September 2016 at 00:21, Warren Mansell wmansell@gmail.com wrote:

Hi Roger, thank goodness for your arrival on the scene! It was getting very messy, and your analysis, incorporating both the mathematics and the organism’s perspective on its velocity, is very neat indeed. And scientific!

Warren & Vyv

RKM: Thanks both. As you can guess, it was an attempt to get the discussion back on the rails (using my last breath of fresh air before teaching starts next week) - one can tolerate only so much noisy traffic in one’s in-box

On 15 Sep 2016, at 22:12, Prof. Roger K. Moore r.k.moore@sheffield.ac.uk wrote:

[Roger Moore 2016.09.15.22.12 BST]

1. THE MATHEMATICS

Equation 8 states that the instantaneous velocity V is a function of the first-derivative of any movement decomposed into X and Y coordinate space. This means that V can be calculated from the velocity in X (Xdot) and the velocity in Y (Ydot). It also means that V is independent of any other variables - i.e. V is independent of any acceleration parameters and hence, by definition, INDEPENDENT OF CURVATURE.

Equation 9 states that the radius (inverse curvature) R of a curve is a function of the first and second derivatives of any movement decomposed into X and Y coordinate space. This means that R can be calculated from the velocity in X (Xdot), the velocity in Y (Ydot), the acceleration in X (Xdoubledot) and the acceleration in Y (Ydoubledot).

Rick’s equation relating V to R and D is correct in that the derivation contains no errors. However, it is NOT CORRECT to interpret V as a function of R only. In Rick’s equation, V is a function of R and D. This means that D CANNOT BE IGNORED. For all values of X and Y, D has a value which cancels R exactly, and leads to a true result - V=V (which has already been pointed out).

Hence, the mathematics tells us that V and R vary independently, exactly as one would expect. There is NO hidden “law” relating V and R.

1. THE PROBLEM

If the observed effect - slowing during curved trajectories - is the result of a closed-loop negative-feedback control system, then it is possible to hypothesise an explanation as follows:

a) Let Vr be the intended (reference) velocity.

b) Let Vp be the perceived velocity.

c) Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e) If Vp=Vr, then e=0 and Vm will remain unchanged.

f) If Vp>Vr, then Vm will be reduced until e=0.

So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature. I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature. In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly. (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

Of course, this only explains the slowing, it doesn’t explain the so-called ‘Power Law’. In order to derive the necessary relations, it would be necessary to analyse the particular perceptual system(s) involved in estimating speed. E.g., for a visual perceptual system, I would look to an estimate of velocity based on something like ‘optical flow’. In other cases one might have to consider the characteristics of proprioceptive feedback. In each case, the hypothesis is that the perceived speed is OVERESTIMATED and hence gives rise to a compensatory reduction in velocity.

This is a scientific hypothesis that can be tested by modelling/simulation of perceptual systems with the appropriate characteristics, and then comparing those characteristics with those found in living systems.

Roger

Prof ROGER K MOORE* BA(Hons) MSc PhD FIOA FISCA MIET

Chair of Spoken Language Processing
Vocal Interactivity Lab (VILab), Sheffield Robotics
Speech & Hearing Research Group (SPandH)
Department of Computer Science, UNIVERSITY OF SHEFFIELD
Regent Court, 211 Portobello, Sheffield, S1 4DP, UK

• Winner of the 2016 Antonio Zampolli Prize for "*Outstanding Contributions *

*to the Advancement of Language Resources & Language Technology *

Evaluation within Human Language Technologies"

e-mail: r.k.moore@sheffield.ac.uk
web: http://www.dcs.shef.ac.uk/~roger/
twitter: @rogerkmoore
Tel: +44 (0) 11422 21807
Fax: +44 (0) 11422 21810
Mob: +44 (0) 7910 073631

Editor-in-Chief: COMPUTER SPEECH AND LANGUAGE
(http://www.journals.elsevier.com/computer-speech-and-language/)

[From Rick Marken (2016.09.17.1140)]

···

Fred Nickols (2016.09.17.1402 ET)–

Â

FN: Professor Moore (aka RKM):

Â

FNI have a question about the section of your post labeled âThe Problem.â?

Â

FN: Am I muddying the waters?

RM: Not at all. I think it would be impossible to muddy them any more than they are already muddied.

Best

Rick

Â

Â

Regards,

Â

Fred Nickols, Consultant

My Objective is to Help You Achieve Yours

DISTANCE CONSULTING LLC

âAssistance at a Distanceâ?SM

Â

Â

Â

Â

Â

From: Prof. Roger K. Moore [mailto:r.k.moore@sheffield.ac.uk]
Sent: Saturday, September 17, 2016 1:37 PM
To: csgnet@lists.illinois.edu
Subject: Re: Power Law re-boot

Â

On 16 September 2016 at 00:21, Warren Mansell wmansell@gmail.com wrote:

Hi Roger, thank goodness for your arrival on the scene! It was getting very messy, and your analysis, incorporating both the mathematics and the organism’s perspective on its velocity, is very neat indeed. And scientific!

Warren & Vyv

Â

RKM: Â Thanks both.Â As you can guess, it was an attempt to get the discussion back on the rails (using my last breath of fresh air before teaching starts next week) - one can tolerate only so much noisy traffic in one’s in-box

Â

On 15 Sep 2016, at 22:12, Prof. Roger K. Moore r.k.moore@sheffield.ac.uk wrote:

[Roger Moore 2016.09.15.22.12 BST]

Â

Â

1.Â THE MATHEMATICS

Â

Equation 8 states that the instantaneous velocity V is a function of the first-derivative of any movement decomposed into X and Y coordinate space.Â This means that V can be calculated from the velocity in X (Xdot) and the velocity in Y (Ydot).Â It also means that V is independent of any other variables - i.e. V is independent of any acceleration parameters and hence, by definition, INDEPENDENT OF CURVATURE.

Â

Equation 9 states that the radius (inverse curvature) R of a curve is a function of the first and second derivatives of any movement decomposed into X and Y coordinate space.Â This means that R can be calculated from the velocity in X (Xdot), the velocity in Y (Ydot), the acceleration in X (Xdoubledot) and the acceleration in Y (Ydoubledot).

Â

Rick’s equation relating V to R and D is correct in that the derivation contains no errors.Â However, it is NOT CORRECT to interpret V as a function of R only.Â In Rick’s equation, V is a function of R and D.Â This means that D CANNOT BE IGNORED.Â For all values of X and Y, D has a value which cancels R exactly, and leads to a true result - V=V (which has already been pointed out).

Â

Hence, the mathematics tells us that V and R vary independently, exactly as one would expect.Â There is NO hidden “law” relating V and R.

Â

Â

2.Â THE PROBLEM

Â

If the observed effect - slowing during curved trajectories - is the result of a closed-loop negative-feedback control system, then it is possible to hypothesise an explanation as follows:

Â

a)Â Let Vr be the intended (reference) velocity.

b)Â Let Vp be the perceived velocity.

c)Â Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e)Â If Vp=Vr, then e=0 and Vm will remain unchanged.

f)Â If Vp>Vr, then Vm will be reduced until e=0.

Â

So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature.Â I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature.Â In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly.Â (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

Â

Of course, this only explains the slowing, it doesn’t explain the so-called ‘Power Law’.Â In order to derive the necessary relations, it would be necessary to analyse the particular perceptual system(s) involved in estimating speed.Â E.g., for a visual perceptual system, I would look to an estimate of velocity based on something like ‘optical flow’.Â In other cases one might have to consider the characteristics of proprioceptive feedback.Â In each case, the hypothesis is that the perceived speed is OVERESTIMATED and hence gives rise to a compensatory reduction in velocity.

Â

This is a scientific hypothesis that can be tested by modelling/simulation of perceptual systems with the appropriate characteristics, and then comparing those characteristics with those found in living systems.

Â

Roger

Â

Prof ROGER K MOORE* BA(Hons) MSc PhD FIOA FISCA MIET

Chair of Spoken Language Processing
Vocal Interactivity Lab (VILab), Sheffield Robotics
Speech & Hearing Research Group (SPandH)
Department of Computer Science, UNIVERSITY OF SHEFFIELD
Regent Court, 211 Portobello, Sheffield, S1 4DP, UK

Â

• Winner ofÂ the 2016 Antonio Zampolli Prize for "*Outstanding ContributionsÂ *

*to the Advancement of Language Resources & Language TechnologyÂ *

Evaluation within Human Language Technologies"

e-mail:Â r.k.moore@sheffield.ac.uk
web:Â http://www.dcs.shef.ac.uk/~roger/
twitter: @rogerkmoore
Tel: +44 (0) 11422 21807
Fax: +44 (0) 11422 21810
Mob: +44 (0) 7910 073631

Editor-in-Chief: COMPUTER SPEECH AND LANGUAGE
(http://www.journals.elsevier.com/computer-speech-and-language/)

Â

Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Fred
Nickols (2016.09.17.1402 ET)]

Â

``````        Professor
``````

Moore (aka RKM):

Â

``````        I
``````

have a question about the section of your post labeled â€œThe
Problem.â€?

[From Fred Nickols (2016.09.17.1540 ET)]

Thanks, Martin.Â A few more comments.Â Letâ€™s take a road I travel a lot.Â Itâ€™s a two-lane, paved highway, US-36.Â There is a stretch where a straight road enters a series of curves.Â The highway speed limit is 55.Â Iâ€™m doing 55 and when I approach the first curve itâ€™s posted as a 35 mph curve.Â I usually slow to 40 and experience no problems, no excessive drift and no sense of going too fast.Â If I were to enter that curve at 50 mph (which I did on one occasion) I would experience way too much drift and have the sense of going way too fast.Â The one time I went too fast I stepped on the brake just as I entered the curve because I could tell I was going too fast.Â In no case was I concerned with measured speed; I was concerned with my own sense of how fast I was traveling and I didnâ€™t have to look at the speedometer or have someone handy with a radar gun to tell me my â€œrealâ€? speed.Â It was my perception of speed that mattered and in relation to my notion (i.e., reference condition) of just how fast was safe and how fast was unsafe. Â As a boy in Iowa, traveling on gravel roads, I learned early on how fast was safe and how fast was unsafe for what kinds of curves.Â Naturally, that safe speed or reference condition was different for paved roads – but not by a whole loot.

Anyway, thanks for responding.Â I will continue to think that whatâ€™s driving reduced speed when entering a curve owes to a lowering of the reference signal for â€œsafe speed.â€?

Fred

···

From: Martin Taylor [mailto:mmt-csg@mmtaylor.net]
Sent: Saturday, September 17, 2016 2:50 PM
To: csgnet@lists.illinois.edu
Subject: Re: Power Law re-boot

[Martin Taylor 2016.09.17.14.37]

[Fred Nickols (2016.09.17.1402 ET)]

Professor Moore (aka RKM):

I have a question about the section of your post labeled â€œThe Problem.â€?

I realize this is addressed to Roger, but since he has mentioned that he will be soon starting his teaching load, I hope you won’t mind if I offer a partial answer. If I get it wrong, I hope Roger will correct me.

RKM:

a) Let Vr be the intended (reference) velocity.

b) Let Vp be the perceived velocity.

c) Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e) If Vp=Vr, then e=0 and Vm will remain unchanged.

f) If Vp>Vr, then Vm will be reduced until e=0.

RKM: So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature. I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature. In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly. (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

FN: Woudnâ€™t it also be possible to account for the reduction in speed as a result of a lowering of Vr?

Yes. That’s a distinct alternative possibility.

If I understand what you say above, you say the slowing occurs because the organism perceives itself as going faster than it really is. I would have said the organism slows because its reference signal for velocity has been reduced. If it is actually a control system, it is keeping perceived speed aligned with intended speed.

Yes, but notice that it is PERCEIVED speed, not radar-measured road speed.

In order for it to slow, there would have to be a decrease in Vr. Vp is quite simply the perceived velocity.

Roger’s suggestion, backed up by personal experience, is that the relation between real-world and perceived speed changes on a curve.

The organism has no knowledge of â€œrealâ€? velocity. It simply keeps Vp aligned with Vr and in order for any slowing to take place it would have to be the result of a reduction in Vr.

Or an increase in Vp. Roger suggests that for him Vp increases despite a constant measured road speed when he enters a curve, at least when driving.

FN: Do I have that correct?

In that it could be a reduction in Vr, Yes; in that it must be a reduction in Vr, No.

FN: If so, then given the hierarchical nature of PCT, a reduced Vr would have to come from the next higher level in the hierarchy. My uneducated guess would be that Vr comes from some higher level systems concerned with matters such as staying on the road, safety, etc. As an experienced driver of all kinds of paved and unpaved roads, when I find myself coming upon a curve I tend to slow down and the sharper I perceive the curve to be the more I slow down. I think what Iâ€™m doing is reducing the Vr in your description above.

All very reasonable. Indeed, it is not unlikely that both effects occur. The real issue is how either or both of these effects would lead to a 1/3 or 1/4 power law, when one would happen rather than the other unless there is a continuum of exponents applicable over a range of conditions, and when no power law at all would be observed.

FN: Am I muddying the waters?

Not at all, at least as I perceive the situation.

Martin

[From
Fred Nickols (2016.09.17.1540 ET)]

Â

``````        Thanks,
``````

Martin.Â A few more comments.Â Letâ€™s take a road I travel a
lot.Â Itâ€™s a two-lane, paved highway, US-36.Â There is a
stretch where a straight road enters a series of curves.Â
The highway speed limit is 55.Â Iâ€™m doing 55 and when I
approach the first curve itâ€™s posted as a 35 mph curve.Â I
usually slow to 40 and experience no problems, no excessive
drift and no sense of going too fast.Â If I were to enter
that curve at 50 mph (which I did on one occasion) I would
experience way too much drift and have the sense of going
way too fast.Â The one time I went too fast I stepped on the
brake just as I entered the curve because I could tell I was
going too fast.Â In no case was I concerned with measured
speed; I was concerned with my own sense of how fast I was
traveling and I didnâ€™t have to look at the speedometer or
have someone handy with a radar gun to tell me my â€œrealâ€?
speed.Â It was my perception of speed that mattered and in
relation to my notion (i.e., reference condition) of just
how fast was safe and how fast was unsafe. Â As a boy in
Iowa, traveling on gravel roads, I learned early on how fast
was safe and how fast was unsafe for what kinds of curves.Â
Naturally, that safe speed or reference condition was
different for paved roads – but not by a whole lot.

Â

``````        Anyway,
``````

thanks for responding.Â I will continue to think that whatâ€™s
driving reduced speed when entering a curve owes to a
lowering of the reference signal for â€œsafe speed.â€?

[From Rick Marken (2016.09.17.1615]

···

Martin Taylor (2016.09.17.14.37)–

``````MT: I realize this is addressed to Roger, but since he has mentioned
``````

that he will be soon starting his teaching load, I hope you won’t
mind if I offer a partial answer. If I get it wrong, I hope Roger
will correct me.

RKM:

Â

``````        a)Â
``````

Let Vr be the intended (reference) velocity.

``````        b)Â
``````

Let Vp be the perceived velocity.

``````        c)Â
``````

Let e=Vr-Vp

``````        d).
``````

Let Vm be the output velocity, where Vm is some function of
e.

``````        e)Â
``````

If Vp=Vr, then e=0 and Vm will remain unchanged.

``````        f)Â
``````

If Vp>Vr, then Vm will be reduced until e=0.

Â

``````        RKM:
``````

So, given that an increase in ‘curvature’ is observed to
elicit a reduction in ‘speed’, it is possible to deduce that
Vp>Vm in the presence of curvature.Â I.e. the
‘perception’ of velocity is affected by the ‘disturbance’ of
curvature.Â In other words, when going around a curve, an
organism perceives itself to be going faster than it really
is, and it slows accordingly.Â (This certainly concurs with
my own experiences when driving with the cruise control
switched on - a speed that feels comfortable on the straight
can feel very uncomfortable when you encounter even the
slightest bend.)

Â

``````        FN:
``````

Woudnâ€™t it also be possible to account for the reduction in
speed as a result of a lowering of Vr?Â

MT: Yes. That’s a distinct alternative possibility.

RM: I guess my comments on Roger’s model didn’t make much of an impression in this new, high integrity world of PCT.Â But I wonder why you (and Bruce A.) – the expert PCT modelers – don’t seem to have any problem with Roger’s model above. Does it really look like a correct control of velocity model to you? So you not see any problems with it? Anything missing? Are you confident that you could implement this as a working model?Â

Best

Rick

``````        If I understand what you say above, you say the slowing
``````

occurs because the organism perceives itself as going faster
than it really is.Â I would have said the organism slows
because its reference signal for velocity has been reduced.Â
If it is actually a control system, it is keeping perceived
speed aligned with intended speed.

``````Yes, but notice that it is PERCEIVED speed, not radar-measured road
``````

speed.

``````        In order for it to slow, there would have to be a decrease
``````

in Vr.Â Vp is quite simply the perceived velocity.

``````Roger's suggestion, backed up by personal experience, is that the
``````

relation between real-world and perceived speed changes on a curve.

``````        Â
``````

The organism has no knowledge of â€œrealâ€? velocity.Â It simply
keeps Vp aligned with Vr and in order for any slowing to
take place it would have to be the result of a reduction in
Vr.

``````Or an increase in V<sub>p</sub>. Roger suggests that for him V<sub>p</sub>
increases despite a constant measured road speed when he enters a
``````

curve, at least when driving.

Â

``````        FN:
``````

Do I have that correct?

``````In that it could be a reduction in V<sub>r</sub>    , Yes; in that it
``````

must be a reduction in Vr, No.

Â

``````        FN:
``````

If so, then given the hierarchical nature of PCT, a reduced
Vr would have to come from the next higher level in the
hierarchy.Â My uneducated guess would be that Vr comes from
some higher level systems concerned with matters such as
staying on the road, safety, etc.Â As an experienced driver
of all kinds of paved and unpaved roads, when I find myself
coming upon a curve I tend to slow down and the sharper I
perceive the curve to be the more I slow down.Â I think what
Iâ€™m doing is reducing the Vr in your description above.

``````All very reasonable. Indeed, it is not unlikely that both effects
``````

occur. The real issue is how either or both of these effects would
lead to a 1/3 or 1/4 power law, when one would happen rather than
the other unless there is a continuum of exponents applicable over a
range of conditions, and when no power law at all would be observed.

Â

``````        FN:
``````

Am I muddying the waters?

``````Not at all, at least as I perceive the situation.

Martin
``````

Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Martin Taylor 2016.09.17.22.56]

[From Rick Marken (2016.09.17.1615]

``````True. Apart from misrepresenting what Roger wrote, they referred to
``````

Roger’s words as if he had proposed a model, and finished with a
generic propaganda blast. Therefore I for one saw them as irrelevant
and uninteresting.

``````Of course not. Roger never proposed a model. Why do you suggest he
``````

tried to do so? Why do you talk about a “control of velocity model”
when none was suggested or implied? Would you like to use what he
did suggest as a component of a model that you might perhaps design
and simulate? If you did, then we might have a model to talk about.

``````Oh well, I guess nothing changes and you will read my words as
``````

carefully as you read his and as you usually do mine, but here’s a
paraphrase of his suggestion. It’s short.

``````[Paraphrase of RKM] If velocity perception is affected by curvature
``````

so that a given speed is perceived as faster when you are going
round a curve than on the straight, then you will slow down if the
velocity reference value does not change.

``````Does that sound like a proposal for a model to you? I suppose it
``````

does. To me it doesn’t.

``````Here's a longer paraphrase, incorporating what I perceive to be
``````

Roger’s background intention. I could be wrong about that, but I
think not.

``````[Paraphrase 2] As part of whatever is going on, velocity is
``````

presumably controlled, and a velocity control unit would presumably
be part of whatever model is eventually created to account for the
various power-function variants. It’s possible that in this future
model we might need to incorporate a perception of curvature in the
velocity perceptual function instead of, or in addition to whatever
variations in the velocity reference value are provided from higher
levels.

``````That's what I get from Roger's words. That's obviously different
``````

from the impression you wanted to promulgate in the “comments”
message you wanted us to be impressed by: that Roger is a PCT
incompetent who produces unimplementable models and proposes them as
problem solutions, so therefore no CSGnet reader should take any
notice of him. It would not have suited your purpose to quote
Roger’s immediately following paragraph in your “comments” message,
would it? That’s the paragraph that starts: "Of course, this only
explains the slowing. It doesn’t explain the so-called “Power Law”.

``````Martin
``````
···

Martin Taylor (2016.09.17.14.37)–

``````            MT: I realize this is addressed to Roger, but since he
``````

has mentioned that he will be soon starting his teaching
load, I hope you won’t mind if I offer a partial answer.
If I get it wrong, I hope Roger will correct me.

RKM:

``````                      a)
``````

Let Vr be the intended (reference) velocity.

``````                      b)
``````

Let Vp be the perceived velocity.

``````                      c)
``````

Let e=Vr-Vp

``````                      d).
``````

Let Vm be the output velocity, where Vm is
some function of e.

``````                      e)
``````

If Vp=Vr, then e=0 and Vm will remain
unchanged.

``````                      f)
``````

If Vp>Vr, then Vm will be reduced until
e=0.

``````                      RKM:
``````

So, given that an increase in ‘curvature’ is
observed to elicit a reduction in ‘speed’, it
is possible to deduce that Vp>Vm in the
presence of curvature. I.e. the ‘perception’
of velocity is affected by the ‘disturbance’
of curvature. In other words, when going
around a curve, an organism perceives itself
to be going faster than it really is, and it
slows accordingly. (This certainly concurs
with my own experiences when driving with the
cruise control switched on - a speed that
feels comfortable on the straight can feel
very uncomfortable when you encounter even the
slightest bend.)

``````                      FN:
``````

Woudn’t it also be possible to account for the
reduction in speed as a result of a lowering
of Vr?

``````            MT: Yes. That's a distinct alternative
``````

possibility.

``````          RM: I guess my comments on Roger's model didn't make
``````

much of an impression in this new, high integrity world of
PCT.

``````          But I wonder why you (and Bruce A.) -- the expert PCT
``````

modelers – don’t seem to have any problem with Roger’s
model above. Does it really look like a correct control of
velocity model to you? So you not see any problems with
it? Anything missing? Are you confident that you could
implement this as a working model?

[Roger K. Moore 2016.09.17.18.40 BST]

···

Hi Martin,

``````That's a different possibility from any I had considered, and a very
``````

interesting one. It raises the whole question of the perception of
time as a function of the information rate of control. Time drags
when you are bored, but seems to fly by when you are productively
“doing” something. Maybe the time component of velocity changes its
perceptual rate on curves.

RKM: The variable perception of time is an interesting angle, but I’m not sure that it applies here. As Bruce points out, time appears to slow during high cognitive load - the exact opposite of the effect we’re trying to account for. FWIW, I like to think of time perception as as a process of ‘sampling’ - with the ‘sampling rate’ proportional to the ‘information rate’. This has interesting consequences for behaviour but, I think, it can be ruled out as a hypothesis supporting the ‘Power Law’.

``````But would this account for the fly larva seeking the source of food
``````

odour? And would you see there being a difference in principle
between cases in which the perception is OF the moving object (e.g.
finger drawing a pattern) rather than FROM the moving object (e.g.
car driver)?

RKM: From what I see from the literature (e.g. the reference you cite below), adherence to the ‘power law’ is observed in both behaviour and perception. This is nice result since a PCT-based explanation would place the emphasis on perception.

``````After reading your message, I went to Google Scholar to see if
``````

anyone had studied the perception of velocity on curves, and found
two interesting papers by Viviani and Stucchi (1989 and 1992; <
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.294.3871&rep=rep1&type=pdf>

and http://link.springer.com/article/10.3758/BF03208089
–the PDF download link of the latter works but the resulting URL is
too long to quote). In the first, they showed that if subject
adjusted the velocity of a point moving around an ellipse or in a
squiggle pattern until it was perceived to move over the whole
trajectory at a uniform pace, the true velocity conformed to a power
law relation with curvature (usually rather less than the 1/3
power). The second showed that the velocity profile affected the
perceived shape of a trajectory. If the point trajectory was
circular, but the velocity varied as it would when someone drew an
ellipse, the circle appeared flattened. So there does seem to be a
relation between perceived velocity and perceived curvature, even
when looking at rather than from the moving point.

RKM: Indeed, a good find and possible evidence for the effect I hypothesised.

``````For the record, apart from the possible experiments I suggested to
``````

Alex a while back, one of which was to use the Powers
“Circle-Square” illusion to tease apart perceptual from motor
effects in the power-law, the approach I’ve been working on has been
related to the effective transport lag of the control loop, thinking
of how far ahead along the trajectory one might need to look at a
target direction or lateral acceleration in order to be on course
when one got there. Changes of curvature would be what matters, not
curvature as such, because at least in a non-viscous medium a
constant lateral acceleration force keeps you on a constant
curvature section of the track – setting the car wheels at a
particular angle will do that. It’s a quite different way to look at
the problem than yours is – or is it, if you also consider the
“time drags” perception?

RKM: My original idea related to ‘optical flow’, and I suspect that angular acceleration might be key.

Thanks a lot for chiming in with a new idea.

RKM: Cheers

``````Martin
``````

[Roger K. Moore 2016.09.18.10.30 BST]

···

On 16 September 2016 at 13:59, Bruce Abbott bbabbott@frontier.com wrote:

[From Bruce Abbott (2016.09.16.0900 EDT)]

Â

RKM: Â Hi Bruce,

Â

Thanks for this, Roger.Â Perhaps now the debate with Rick over his mistaken analysis can be laid to rest.Â I, for one, have no interest in continuing it.

RKM: Â We can always hope, but looking ahead in my timeline, I see that’s unlikely ;-(Â

Â

Thanks also for your interesting proposal – I never would have thought in terms of sppeed-perception being influenced by curvature.Â I do recall one incident in my own life that relates to the issue.Â A car turned in front of me from the left-hand lane, cutting across my lane to make a right turn.Â I tried to make the turn with him but we ended up side-swiping, after which my car went over the curb into a vacant lot, just missing a tree.Â During this whole incident, everything seemed to go into slow motion, and today, over 50 years later, I can still remember every moment.Â This slow-motion effect is well known but I am not aware of an explanation for it.Â It is opposite the effect you are proposing for entering a turn – there it would seem that one perceives the speed as increasing, aas though time were going by more quickly.

Â

RKM: Â I’ve also experienced this effect and, as you say, it would have the opposite effect to the one we’re trying to explain.Â So, as I said in my reply to Martin, I think time dilation is a hypothesis that we can rule out with respect to velocity control.

Â

As I noted in a previous post, a power-law relation may emerge under a variety of circumstances, owing to different factors being at play.Â Thus there may be no one answer to the question of what enforces such a relation between speed and curvature.Â A different PCT model may be required to account for each case in which a power-law (or other) relation is consistently observed.

Â

RKM: Â I agree - the so-called ‘Power Law’ could be a compilation of many different effects.Â But, of course, in Science we constantly strive to find simple parsimonious theories (like PCT) that account for a wide number of observations.

Â

Bruce

Â

Â

[Roger Moore 2016.09.15.22.12 BST] –

Â

Â

1.Â THE MATHEMATICS

Â

Equation 8 states that the instantaneous velocity V is a function of the first-derivative of any movement decomposed into X and Y coordinate space.Â This means that V can be calculated from the velocity in X (Xdot) and the velocity in Y (Ydot).Â It also means that V is independent of any other variables - i.e. V is independent of any acceleration parameters and hence, by definition, INDEPENDENT OF CURVATURE.

Â

Equation 9 states that the radius (inverse curvature) R of a curve is a function of the first and second derivatives of any movement decomposed into X and Y coordinate space.Â This means that R can be calculated from the velocity in X (Xdot), the velocity in Y (Ydot), the acceleration in X (Xdoubledot) and the acceleration in Y (Ydoubledot).

Â

Rick’s equation relating V to R and D is correct in that the derivation contains no errors.Â However, it is NOT CORRECT to interpret V as a function of R only.Â In Rick’s equation, V is a function of R and D.Â This means that D CANNOT BE IGNORED.Â For all values of X and Y, D has a value which cancels R exactly, and leads to a true result - V=V (which has already been pointed out).

Â

Hence, the mathematics tells us that V and R vary independently, exactly as one would expect.Â There is NO hidden “law” relating V and R.

Â

Â

2.Â THE PROBLEM

Â

If the observed effect - slowing during curved trajectories - is the result of a closed-loop negative-feedback control system, then it is possible to hypothesise an explanation as follows:

Â

a)Â Let Vr be the intended (reference) velocity.

b)Â Let Vp be the perceived velocity.

c)Â Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e)Â If Vp=Vr, then e=0 and Vm will remain unchanged.

f)Â If Vp>Vr, then Vm will be reduced until e=0.

Â

So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature.Â I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature.Â In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly.Â (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

Â

Of course, this only explains the slowing, it doesn’t explain the so-called ‘Power Law’.Â In order to derive the necessary relations, it would be necessary to analyse the particular perceptual system(s) involved in estimating speed.Â E.g., for a visual perceptual system, I would look to an estimate of velocity based on something like ‘optical flow’.Â In other cases one might have to consider the characteristics of proprioceptive feedback.Â In each case, the hypothesis is that the perceived speed is OVERESTIMATED and hence gives rise to a compensatory reduction in velocity.

Â

This is a scientific hypothesis that can be tested by modelling/simulation of perceptual systems with the appropriate characteristics, and then comparing those characteristics with those found in living systems.

Â

Roger

Â

[Roger K. Moore 2016.09.18.10.45 BST]

···

Hi Fred,

Thanks for your question.Â I see that Martin already responded in exactly the way I would have.Â The only thing that I would add is that your solution - varying Vr - effectively pushes the problem up one level.Â As you say, “the sharper I perceive the curve to be the more I slow dow”.Â So, in your interpretation there is the perception of sharpness (i.e. curvature), and that would be compared to a reference curvature (presumably, no curvature?) with the difference leading to a change in Vr.Â This is a plausible hypothesis which could also be tested empirically.

Notice that our two hypotheses are positing different perceptual signals/controlled variables.Â Mine is a one-level system based on an assumption of systematic velocity measurement error, and yours is a two-level system based on the direct perception of curvature.

Cheers

Roger

On 17 September 2016 at 19:15, Fred Nickols fred@nickols.us wrote:

[Fred Nickols (2016.09.17.1402 ET)]

Â

Professor Moore (aka RKM):

Â

I have a question about the section of your post labeled â€œThe Problem.â€?

Â

RKM:

Â

a)Â Let Vr be the intended (reference) velocity.

b)Â Let Vp be the perceived velocity.

c)Â Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e)Â If Vp=Vr, then e=0 and Vm will remain unchanged.

f)Â If Vp>Vr, then Vm will be reduced until e=0.

Â

RKM: So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature.Â I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature.Â In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly.Â (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

Â

FN: Woudnâ€™t it also be possible to account for the reduction in speed as a result of a lowering of Vr?Â If I understand what you say above, you say the slowing occurs because the organism perceives itself as going faster than it really is.Â I would have said the organism slows because its reference signal for velocity has been reduced.Â If it is actually a control system, it is keeping perceived speed aligned with intended speed.Â In order for it to slow, there would have to be a decrease in Vr.Â Vp is quite simply the perceived velocity.Â The organism has no knowledge of â€œrealâ€? velocity.Â It simply keeps Vp aligned with Vr and in order for any slowing to take place it would have to be the result of a reduction in Vr.

Â

FN: Do I have that correct?

Â

FN: If so, then given the hierarchical nature of PCT, a reduced Vr would have to come from the next higher level in the hierarchy.Â My uneducated guess would be that Vr comes from some higher level systems concerned with matters such as staying on the road, safety, etc.Â As an experienced driver of all kinds of paved and unpaved roads, when I find myself coming upon a curve I tend to slow down and the sharper I perceive the curve to be the more I slow down.Â I think what Iâ€™m doing is reducing the Vr in your description above.

Â

FN: Am I muddying the waters?

Â

Regards,

Â

Fred Nickols, Consultant

My Objective is to Help You Achieve Yours

DISTANCE CONSULTING LLC

â€œAssistance at a Distanceâ€?SM

Â

Â

Â

Â

Â

From: Prof. Roger K. Moore [mailto:r.k.moore@sheffield.ac.uk]
Sent: Saturday, September 17, 2016 1:37 PM
To: csgnet@lists.illinois.edu
Subject: Re: Power Law re-boot

Â

On 16 September 2016 at 00:21, Warren Mansell wmansell@gmail.com wrote:

Hi Roger, thank goodness for your arrival on the scene! It was getting very messy, and your analysis, incorporating both the mathematics and the organism’s perspective on its velocity, is very neat indeed. And scientific!

Warren & Vyv

Â

RKM: Â Thanks both.Â As you can guess, it was an attempt to get the discussion back on the rails (using my last breath of fresh air before teaching starts next week) - one can tolerate only so much noisy traffic in one’s in-box

Â

On 15 Sep 2016, at 22:12, Prof. Roger K. Moore r.k.moore@sheffield.ac.uk wrote:

[Roger Moore 2016.09.15.22.12 BST]

Â

Â

1.Â THE MATHEMATICS

Â

Equation 8 states that the instantaneous velocity V is a function of the first-derivative of any movement decomposed into X and Y coordinate space.Â This means that V can be calculated from the velocity in X (Xdot) and the velocity in Y (Ydot).Â It also means that V is independent of any other variables - i.e. V is independent of any acceleration parameters and hence, by definition, INDEPENDENT OF CURVATURE.

Â

Equation 9 states that the radius (inverse curvature) R of a curve is a function of the first and second derivatives of any movement decomposed into X and Y coordinate space.Â This means that R can be calculated from the velocity in X (Xdot), the velocity in Y (Ydot), the acceleration in X (Xdoubledot) and the acceleration in Y (Ydoubledot).

Â

Rick’s equation relating V to R and D is correct in that the derivation contains no errors.Â However, it is NOT CORRECT to interpret V as a function of R only.Â In Rick’s equation, V is a function of R and D.Â This means that D CANNOT BE IGNORED.Â For all values of X and Y, D has a value which cancels R exactly, and leads to a true result - V=V (which has already been pointed out).

Â

Hence, the mathematics tells us that V and R vary independently, exactly as one would expect.Â There is NO hidden “law” relating V and R.

Â

Â

2.Â THE PROBLEM

Â

If the observed effect - slowing during curved trajectories - is the result of a closed-loop negative-feedback control system, then it is possible to hypothesise an explanation as follows:

Â

a)Â Let Vr be the intended (reference) velocity.

b)Â Let Vp be the perceived velocity.

c)Â Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e)Â If Vp=Vr, then e=0 and Vm will remain unchanged.

f)Â If Vp>Vr, then Vm will be reduced until e=0.

Â

So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature.Â I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature.Â In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly.Â (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

Â

Of course, this only explains the slowing, it doesn’t explain the so-called ‘Power Law’.Â In order to derive the necessary relations, it would be necessary to analyse the particular perceptual system(s) involved in estimating speed.Â E.g., for a visual perceptual system, I would look to an estimate of velocity based on something like ‘optical flow’.Â In other cases one might have to consider the characteristics of proprioceptive feedback.Â In each case, the hypothesis is that the perceived speed is OVERESTIMATED and hence gives rise to a compensatory reduction in velocity.

Â

This is a scientific hypothesis that can be tested by modelling/simulation of perceptual systems with the appropriate characteristics, and then comparing those characteristics with those found in living systems.

Â

Roger

Â

Prof ROGER K MOORE* BA(Hons) MSc PhD FIOA FISCA MIET

Chair of Spoken Language Processing
Vocal Interactivity Lab (VILab), Sheffield Robotics
Speech & Hearing Research Group (SPandH)
Department of Computer Science, UNIVERSITY OF SHEFFIELD
Regent Court, 211 Portobello, Sheffield, S1 4DP, UK

Â

• Winner ofÂ the 2016 Antonio Zampolli Prize for "*Outstanding ContributionsÂ *

*to the Advancement of Language Resources & Language TechnologyÂ *

Evaluation within Human Language Technologies"

e-mail:Â r.k.moore@sheffield.ac.uk
web:Â http://www.dcs.shef.ac.uk/~roger/
twitter: @rogerkmoore
Tel: +44 (0) 11422 21807
Fax: +44 (0) 11422 21810
Mob: +44 (0) 7910 073631

Editor-in-Chief: COMPUTER SPEECH AND LANGUAGE
(http://www.journals.elsevier.com/computer-speech-and-language/)

Â

[Roger K. Moore 2016.09.18.11.15 BST]

···

Hi again Fred,

I see that Martin has also responded to this - excellent.

I just want to pick up on one line: “In no case was I concerned with measured speed; I was concerned with my own sense of how fast I was traveling and I didnât have to look at the speedometer or have someone handy with a radar gun to tell me my ârealâ? speed.” Â That is exactly compatible with my suggestion.Â I was not claiming that Vr was fixed, merely that Vm varies as a function of the effect of curvature on Vp.Â Vr can be anything you like which, as I understand the ‘Power Law’, is precisely what is observed - Vr can follow any trajectory, but the corresponding Vm at any instant is influenced by the curvature.

Cheers

Roger

On 17 September 2016 at 20:57, Fred Nickols fred@nickols.us wrote:

[From Fred Nickols (2016.09.17.1540 ET)]

Â

Thanks, Martin.Â A few more comments.Â Letâs take a road I travel a lot.Â Itâs a two-lane, paved highway, US-36.Â There is a stretch where a straight road enters a series of curves.Â The highway speed limit is 55.Â Iâm doing 55 and when I approach the first curve itâs posted as a 35 mph curve.Â I usually slow to 40 and experience no problems, no excessive drift and no sense of going too fast.Â If I were to enter that curve at 50 mph (which I did on one occasion) I would experience way too much drift and have the sense of going way too fast.Â The one time I went too fast I stepped on the brake just as I entered the curve because I could tell I was going too fast.Â In no case was I concerned with measured speed; I was concerned with my own sense of how fast I was traveling and I didnât have to look at the speedometer or have someone handy with a radar gun to tell me my ârealâ? speed.Â It was my perception of speed that mattered and in relation to my notion (i.e., reference condition) of just how fast was safe and how fast was unsafe.Â As a boy in Iowa, traveling on gravel roads, I learned early on how fast was safe and how fast was unsafe for what kinds of curves.Â Naturally, that safe speed or reference condition was different for paved roads  but not by a whole lot.

Â

Anyway, thanks for responding.Â I will continue to think that whatâs driving reduced speed when entering a curve owes to a lowering of the reference signal for âsafe speed.â?

Â

Fred

Â

From: Martin Taylor [mailto:mmt-csg@mmtaylor.net]
Sent: Saturday, September 17, 2016 2:50 PM
To: csgnet@lists.illinois.edu
Subject: Re: Power Law re-boot

Â

[Martin Taylor 2016.09.17.14.37]

[Fred Nickols (2016.09.17.1402 ET)]

Â

Professor Moore (aka RKM):

Â

I have a question about the section of your post labeled âThe Problem.â?

I realize this is addressed to Roger, but since he has mentioned that he will be soon starting his teaching load, I hope you won’t mind if I offer a partial answer. If I get it wrong, I hope Roger will correct me.

Â

RKM:

Â

a)Â Let Vr be the intended (reference) velocity.

b)Â Let Vp be the perceived velocity.

c)Â Let e=Vr-Vp

d). Let Vm be the output velocity, where Vm is some function of e.

e)Â If Vp=Vr, then e=0 and Vm will remain unchanged.

f)Â If Vp>Vr, then Vm will be reduced until e=0.

Â

RKM: So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature.Â I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature.Â In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly.Â (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)

Â

FN: Woudnât it also be possible to account for the reduction in speed as a result of a lowering of Vr?

Yes. That’s a distinct alternative possibility.

If I understand what you say above, you say the slowing occurs because the organism perceives itself as going faster than it really is.Â I would have said the organism slows because its reference signal for velocity has been reduced.Â If it is actually a control system, it is keeping perceived speed aligned with intended speed.

Yes, but notice that it is PERCEIVED speed, not radar-measured road speed.

In order for it to slow, there would have to be a decrease in Vr.Â Vp is quite simply the perceived velocity.

Roger’s suggestion, backed up by personal experience, is that the relation between real-world and perceived speed changes on a curve.

Â The organism has no knowledge of ârealâ? velocity.Â It simply keeps Vp aligned with Vr and in order for any slowing to take place it would have to be the result of a reduction in Vr.

Or an increase in Vp. Roger suggests that for him Vp increases despite a constant measured road speed when he enters a curve, at least when driving.

Â

FN: Do I have that correct?

In that it could be a reduction in Vr, Yes; in that it must be a reduction in Vr, No.

Â

FN: If so, then given the hierarchical nature of PCT, a reduced Vr would have to come from the next higher level in the hierarchy.Â My uneducated guess would be that Vr comes from some higher level systems concerned with matters such as staying on the road, safety, etc.Â As an experienced driver of all kinds of paved and unpaved roads, when I find myself coming upon a curve I tend to slow down and the sharper I perceive the curve to be the more I slow down.Â I think what Iâm doing is reducing the Vr in your description above.

All very reasonable. Indeed, it is not unlikely that both effects occur. The real issue is how either or both of these effects would lead to a 1/3 or 1/4 power law, when one would happen rather than the other unless there is a continuum of exponents applicable over a range of conditions, and when no power law at all would be observed.

Â

FN: Am I muddying the waters?

Not at all, at least as I perceive the situation.

Martin