RM: The [something] is the position of the finger. That is clearly a controlled variable.
In a tracking task, yes. If you have a target moving in front of you, then controlling the position of the finger relative to the target explains the movement trajectory quite nicely. But a tracking task is not a curve tracing task. If you have a curve in front of you, or if you are instructed to draw a shape, there is no target to follow.Â
Bill is saying that his control architecture is far simpler that the one proposed by Hollerbach and Atkeson, and it still produces the bell shaped velocity profiles when moving from point to point. He is not saying that the velocity profiles are a statistical illusion, as you claim with OVB. He is not claiming it is an example of a behavioral illusion, as you are. That is all - he is saying is that in the same task, a very simple control architecture, coupled with arm dynamics and environmental forces, can produce the same velocity profiles.Â
I’m perfectly in agreement with that. If we want to explain power law trajectories in humans, we need to create control systems that produce power law trajectories in those same tasks. That is really what a lot of people are doing, just with more complicated (or in other ways different) control systems.Â
RM: It is consistently observed in research on curved movements. That’s why they call it a law. But the exponent is not always observed to be exactly 1/3 or 2/3. Our OVB analysis shows exactly why this is the case. It predicts exactly how much the observed power exponent will deviate from 1/3 or 2/3. The deviation is proportional to the covariance between log D and the measures of curvature and velocity.
RM: And we never said it was. Our analysis predicts that there are trajectories where the fit to a power law (using only curvature as the predictor) will be poor or non-existent, depending on the above mentioned covariance between log D and the curvature and velocity variables.
One of many issues with using the OVB here, is that that analysis gives you a “deviation from the true exponent” regardless of r2, the coefficient of determination of regression analysis between logC and logA. Run the OVB on any sort of non-power law trajectory, and you’ll still get a deviation from 2/3, as if it means something.Â
You’re trying to explain how to calculate relatively trivial things to mathematicians and physicists, and using some obscure statistical trickery. I mean, sure, if that rocks your boat, if you don’t have better things to do, but don’t call that PCT.Â
···
On Sat, Nov 11, 2017 at 8:18 AM, Richard Marken rsmarken@gmail.com wrote:
[From Rick Marken (2017.11.10.2310)]
 Erling Jorgensen (2017.11.10 1445 EST)]
EP:Â If different muscle forces can produce the SAME movement trajectory on different occasions then why could they not produce also the similar correlation on those occasions?
RM:Â Because it’s not the muscle forces alone that are producing the similar correlation (by which I presume you mean the power law relationship between curvature and velocity). This is why the existence of the power law tells nothing about how the movement is produced. The existence of the power law is a side effect of controlling the position of the finger (or pen or whatever) when tracing out a curved trajectory of movement.
EJ:Â This certainly sounds like a contradiction:Â "The existence of the power law is a side effect of controlling [something]…"Â
RM: The [something] is the position of the finger. That is clearly a controlled variable.
Â
EJ: and yet "the existence of the power law tells nothing about how the movement is produced."Â
RM: I don’t see the contradiction. The power law doesn’t tell us anything about how the movement is produced; control theory does. The movement is the output of a control loop that varies to counter varying disturbances so as to keep the controlled variable (finger position) in the varying reference state.Â
RM: Since the varying position of the finger is (or is closely related to) a controlled variable, the power law is a measure of a feature of the controlled variable itself. It’s like measuring the curvature and velocity of the movement of the cursor in a one-dimensional pursuit tracking task. In that case there will be no power law because curvature is constant (at 0). The lack of a power law in this case obviously doesn’t tell us that the movements in this tracking task are produced differently than those in the two-dimensional case. Indeed, we know that the the lack of a power law relationship in this case tells us nothing about how movements are produced. That’s why we don’t do a power law analysis of tracking tasks. Control theory tells us how the tracking movements are produced in this one-dimensional task; and it tells us how they are produced in the two-dimensional movement task as well.Â
RM: The finding of a consistent power law in the case of two dimensional movements seems like it is telling us something about how movement is produced because it seems to say that people slow down through curves, which is consistent with our intuitions about how we drive through curves. But the curvature measured in power law studies is not independent of the velocity measure, as it is when we are driving on a curvy road. In power law studies both curvature and velocity are dependent variables – both being simultaneously produced by the combined effect of muscle and gravitational forces. So the existence of the power law relationship between curvature and velocity is a very seductive illusion – an illusion in the the sense that it looks like there is a cause-effect relationship between curvature and output (or between disturbance and output in PCT terms) that tells us something about how movement is produced in terms of how the velocity of movement is varied in response to variations in curvature. But it doesn’t.Â
EJ: If it is a reliable effect, then we should explore WHETHER it can tell us something about the movement.Â
RM: You can see very reliable side effects of controlling that tell you nothing about how that controlling is done. Here’s a relevant post (which I posted before; maybe it won’t get ignored this time) where Bill Powers makes the same point about invariant (reliably produced) trajectory profiles that I am making regarding the invariant power law.Â
[From Bill
Powers (950527.0950 MDT)]
Â
RE:
trajectories vs. system organization
Â
BP: In a great
deal of modern behavioral research, trajectories of movement
are
examined in the hope of finding invariants that will reveal secrets
of
behavior. This approach ties in with system models that compute
inverse
kinematics and dynamics and use motor programs to produce
actions
open-loop. These models assume that the path followed by a limb
or the
whole body is specified in advance in terms of end-positions and
derivatives
during the transition, so the path that is followed reflects
the computations
that are going on inside the system.
Â
BP: It is this
orientation that explains papers like
Â
Atkeson,
C. G. and Hollerback, J.M.(1985); Kinematic features of
unrestrained
vertical arm movements. The Journal of Neuroscience 5,
#9,
2318-2330.
Â
BP: In the
described experiments, subjects move a hand in the vertical plane
at various
prescribed speeds from a starting point to variously located
targets,
and the positions are recorded as videos of the positions of
illuminated
targets fastened to various parts of the arm and hand.
Â
BP: The
authors constructed a tangential-velocity vs time profile of the
wrist
movement for various speeds, directions, and distances of
movement.
They normalized the profiles to a fixed magnitude, then to a
fixed
duration, and found that the curves then had very nearly the same
shape.
Using a “similarity” calculation, they quantified the measures of
similarity.
Â
BP: They were
then able to compare these normalized tangential velocity
profiles
across various directions and amounts of movement and show that
the
treated profiles were very close to the same. They conclude:
Â
   Â
Taken together, shape invariance for path and tangential velocity
   Â
profile indicates that subjects execute only one form of trajectory
   Â
between any two targets when not instructed to do otherwise. The
   Â
only changes in trajectory are simple scaling operations to
   Â
accomodate different speeds. Furthermore, subjects use the same
   Â
tangential velocity profile shape to make radically different
   Â
movements, even when the shapes of the paths are not the same in
   Â
extrinsic coordinates. Different subjects use the same tangential
   Â
velocity profile shape.
Â
   Â
… this would be consistent with a simplifying strategy for joint
   Â
torque formation by separation of gravity torques from dynamic
   Â
torques and a uniform scaling of the tangential velocity profile
   Â
…Â (p. 2325)
Â
   Â
… if the motor controller has the ability to fashion correct
   Â
torques for one movement, why does it not use this same ability for
   Â
all subsequent movements rather than utilize the dynamic scaling
   Â
properties? Among the possibilities we are considering, the first
   Â
is a generalized motor tape where only one movement between points
   Â
must be known if the dynanmic components in equation 6 are stored
   Â
separately…A second possibility is a modification of tabular
   Â
approaches [ref] where the dimensionality and parameter adjustment
   Â
problem could be reduced by separate tables for the four components
   Â
in equation 6. (p. 2326)
Â
BP: This paper
was sent to me by Greg Williams as a source of data about
actual
hand movements, for comparison with the hand movements generated
by Little
Man v. 2, the version using actual arm dynamics for the
external
part of the model. The model’s hand movements were, as Greg
will
attest, quite close to those shown in this paper, being slightly
curved
lines connecting the end-points. Forward and reverse movements
followed
somewhat different paths, and by adjustment of model parameters
this
difference, too, could be reproduced.
Â
BP: What is interesting is that the fit between the Little Man
and the realdata was found without considering tangential velocity
profiles or doingany scaling or normalization. In
other words, the invariances noted bythe authors were simply side-effects of the operation of the
controlsystems of the arm interacting with the dynamics of the
physical arm. In
the Little
Man there is no trajectory planning, no storage of movement
parameters,
no table-lookup facility, no computation of invariant
velocity
profiles. The observed behavior is simply a reflection of the
organization
of the control system and the physical plant.
Â
BP: The path which Atkeson, Hollerbach (and many others at MIT
andelsewhere) are treading is a blind alley,
because no matter howcarefully the observations are made and the invariances are
calculated,there will be no hint of the control-system organization,
the SIMPLEcontrol-system organization, that (I claim) is actually
creating the**observed trajectories. [Italics mine – RM]
RM: Both the invariant tangential velocity profiles that Bill talks about here and the power law of movement that Shaffer and I talk about in our paper are reliable side effects of control. And Bill demonstrates that the invariant tangential velocity profiles found by Atkeson & Hollerback are, indeed, side effects of control in the same way that the Marken/Shaffer paper demonstrates that the the power law is a side effect of control: by showing that both of these “invariants” are produced by a control model of movement, a model that is produced without considering tangential velocity profiles or power laws, respectively. If Atkeson & Hollerback were on CSGNet when Bill wrote this post I am quite certain that they would have responded to Bill’s analysis of invariant tangential velocity profiles as aggressively as Alex, Martin and Bruce have responded to my analysis of the invariant power law of movement. As a friend of mine said recently (when I was discussing this power law thing with him) “Emperors don’t take it well when told that they are naked”.
EJ: Now there may be some ambiguity about that word "movement." I grant you, if it simply means “muscle forces,” we have good reason in PCT to ask how a changing force leads to a consistent effect.  But the correlation is not between degree of force and degree of curvature. It is with angular speed. So that moves us up the hierarchy of potential controlled perceptions.Â
RM: This is completely incomprehensible to me. The only controlled variable we know about is the position of the pointer (finger) tracing out the movement trajectory. We know that this variable is controlled because a consistent result is being produce in the face of varying disturbance. The power law is simply a measure of the relationship between two measures – velocity and curvature – of this variable. That relationship says nothing about a higher level variable being controlled; or about any variable being controlled for that matter. The relationship is found for all kinds of curved movements, whether the movements are controlled results (as they when the movement is intentionally produced) or not (as they are for the movement of inanimate objects, like Frisbees.Â
Â
EJ: A rate-of-change variable such as angular speed (or linear speed) could certainly be a controlled transition.  And indeed, I think PCT teaches us that ambiguous words like “movement” are often constituted perceptually as defined relationships, that bring about temporally circumscribed events, involving controlled transitions, of particular body configurations. There are lots of ways to see movement-related regularities emerge, because from PCT we know those get enacted via controlled perceptual results.Â
RM: It’s certainly true that people control the speed with which they move and that they control the degree of curvature through which they move their limbs. But the power law tells us nothing about whether these variables are being controlled when a person produces curved trajectories. You have to set up experiments where you can test to see whether either or both of these variables are being controlled. These experiments will require that you be able to introduce disturbances that have relatively independent effects on these variables. This will require some ingenuity; but such experiments will not get done if researchers continue to go down the blind alley (or follow the red herring) of investigating the side effects rather than the central feature of control: controlled variables.
EJ: The power law shows there is often a relative consistency where movements along a sharper curve are slower than movements along a more open curve.Â
RM:" Yes, and, as I said above, that’s what makes the illusion so compelling.Â
Â
“Slower” to me suggests higher up in a PCT hierarchy, because higher level perceptions have to operate with a slower time constant. That’s the beauty of your “Hierarchical Behavior of Perception” demo (at MindReadings.com), where a configuration of square vs. circle can be controlled at a faster rate than a transition of clockwise vs. counter-clockwise, which can be controlled at a faster rate than a sequence of small-medium-large or vice versa. Those relative timing issues [can] tell us something about relative placement within a hierarchy of controlled perceptions.Â
RM: I think you misunderstand what the demo shows. Slower means that the controlled variable – like speed --Â not a state of that variable – like “slower” --Â Â is higher in the hierarchy;
EJ: The implication for me is to consider that where some kind of power law between speed and curvature shows up, there are likely two different levels of perceptions being controlled. Although as Alex has acknowledged, it is dogged work to track that down.Â
RM: I think whatever work power law researchers are doing is “dogged” because they are doing it down a blind alley. The power law says nothing about what variable is controlled, though I don’t think power law researchers have any idea that behavior is organized around controlled variables anyway. The red herring they are chasing is the idea that the power law reflects some kind of biological and/or kinematic constraint on how movement is produced. There is no mention of controlled variables in any paper on the power law (except Marken & Shaffer, of course).
RM: Our statistical analysis simply shows why this side effect (a power law with a coefficient close to 1/3 or 2/3) is consistently observed. Â
EJ: Well, then your statistical analysis has to be flawed. Because it is not consistently observed.  Â
RM: It is consistently observed in research on curved movements. That’s why they call it a law. But the exponent is not always observed to be exactly 1/3 or 2/3. Our OVB analysis shows exactly why this is the case. It predicts exactly how much the observed power exponent will deviate from 1/3 or 2/3. The deviation is proportional to the covariance between log D and the measures of curvature and velocity.
RM: One of the things the Zago, et al. (2017) paper shows is that “The power law is not obligatory mathematically,”
RM: And we never said it was. Our analysis predicts that there are trajectories where the fit to a power law (using only curvature as the predictor) will be poor or non-existent, depending on the above mentioned covariance between log D and the curvature and velocity variables.
Â
EJ: in the section of their paper with that same heading. As demonstrated in their Figure 4, ellipses can be traced at various speed profiles. Slowing down with increasing curvature is just one possibility, leading to the customary power law relationship. But progressive acceleration over one cycle is another possibility. And deliberately slowing down as the curvature opens out and lessens is yet another.Â
Yes, and it is predicted perfectly by the OVB analysis.
EJ: Moreover, the Zago, et al. (2017) paper shows several physical systems where there is no necessary relationship between speed and curvature. They are laid out in their Figure 5, where accelerations related to gravity (e.g., ideal binary stars, a projectile with and without drag, a pendulum, a weight on orthogonal springs) may or may not lead to a power-law approximation, but usually don’t. So, contrary to what you assert, the power law cannot just be "a statistical consequence of how curvature and velocity are measured."Â
RM: It’s all accounted for by our analysis. The fit to a power law depends on the nature of the movement trajectory itself, not on how it was produced.Â
EJ: Figure 3 of the Zago, et al. (2017) paper shows a further demonstration that the power law effect, whatever it is attributable to, is not just a statistical artifact of how curvature is calculated. They use three different methods of calculating curvature, and as Fig. 3B shows, their plotted time profiles are virtually identical, as were their estimates of the power law exponent as 0.78 or 0.76 for that set of experiments.Â
RM: Yes, that was interesting. We will deal with this in our rebuttal. I’ve already figured out why it happens. Â
EJ: This doesn’t even get into the question that Martin, and Alex, and Bruce A. have repeatedly raised, as well as myself on at least one occasion, that the additional predictor variable for Velocity of your Omitted Variable Bias proposal includes the Velocity term as one of its arguments. You can’t have Velocity predicting itself. That is simply a tautology. Standard mathematical practice (and I am far from a mathematician) is to get all references to one variable on the same side of the equation.Â
RM: The answer to that question is that their analysis is wrong.We will explain why in our paper. I’ll just say that the proof that their analysis is wrong is given in a paper Zago et al refer to as a reference supposedly proving that our analysis is wrong. I would never have found that paper were it not for their kind efforts to set me straight.
EJ:Â I keep going back to what I take to be a PCT dictum:Â When living systems produce a regularity that is not spurious, start by suspecting perceptual control in one form or another.
RM: I have never heard that dictum and it sure doesn’t sound like PCT to me. Maybe you were thinking of this: Systems that produce a consistent result in the face of variable disturbances may be controlling that result (or some result related to it).Â
EJ: I don’t think it has yet been determined which controlled results are in play here. But I’d like to see what Adam M. and his colleagues come up with.Â
RM: As I said, the only variable that is likely being controlled in studies of intentionally produced curved movement is the trajectory of the movement itself. There are surely other aspects of this movement that are also being controlled but what they are will never be determined by looking intensively at irrelevant side effects of control.Â
BestÂ
Rick
Â
All the best,
Erling
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–
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
