[Martin Taylor 2016.09.15.13.41]
[From Rick Marken (2016.09.15.1030)]
The fact that you see no sign, despite the flaw having been pointed
out (to quote myself) “six ways from Sunday” and two more ways since
then, is the problem. I don’t know how Bruce will respond, but I’ll
just state it without further explanation. (I see you asked me to do
the same, so I no longer feel I am pre-empting Bruce).
You've had the explanation may times, starting less than an hour
after you first proposed your analysis. But here goes again.
So far, so good.
So there is, but that V isn't necessarily the same V. It's any
variable AT ALL. And that possibility includes that it could be the
same. If x/0 = infinity and 3/0 = infinity, x COULD BE 3, but you
can’t use x = 3 in any further computations. Likewise, since V could
be just about anything at all (the limits of that statement have
been stated previously) you can’t use that V as being the observed
values of V as a function of x and y in any further analysis.
Nope. They are using linear regression to find whether there is a
linear relationship between variables that are NOT mathematically
linearly related.
Nope.
Nope.
Nope.
Nope.
No you haven't, not in what you have presented on CSGnet.
OK,
-----Restatement-------
1.1 Researchers observe the velocity of something that moves in
space. They quantify this movement in Cartesian coordinates.
1.2. Researchers determine the trajectory of the moving object and
quantify it as a curve in the same Cartesian space.
1.3. Researchers use the measurements in Cartesian space to
determine the curvature of the trajectory at each point along it.
For convenience they use the measures they already took in 1 and 2.
2. Researchers note that their observed velocities have a
relationship with the curvature they measure.
3. Rick notices that the curvature equation looks as though it can
be transposed so as to show a variable “V” as if it could be
determined from the right-hand side of the equation. He does this by
hiding the identical “V” in a newly defined variable on the right
side, though he seems to be unaware that this is what he is doing.
Then he asserts that this “V” MUST BE the velocity measured in 1.1.
The grounds for this assertion are unclear, but perhaps it is made
because researchers took the convenient approach of using it to make
their curvature measurement. However that may be, the assertion is
false, because the equation actually is V = V*(R/R)1/3 , a
tautology that is true for any “V” whatever, and that allows for no
inferences at all with respect to V.
4. Rick uses his assertion in 3 to state that V and curvature are
mathematically related, which they are not. The rest of the analysis
therefore fails and need not be considered further.
-------end restatement--------
Actually one does not have to use mathematics to see the falsity of
Rick’s point 4. As has been pointed out many times, curvature has no
relation to time, whereas velocity does. At the risk of inducing
boredom by repetition, curvature has the dimension (1/length), while
velocity has a dimension (length/time). Although they may be related
in experimental observations, they are not, and could not be,
mathematically related.
That fact alone should have led Rick to question his quaint notion
that equation 2 (presumably Gribble and Ostry’s equation 9) shows a
relationship between V and R. He could have seen immediately that V3
occurs in the numerator and denominator of that equation, and
cancels out. Having seen that, he would never have posted his
analysis in the first place, and would have saved everyone two
months of uncomprehending nonsense.
Martin
···
BruceAbbott (2016.09.15.0800 EDT)–
RM: So your complaint isthat my analysis assumes that V and R must be
in the same fixed relation from one moment to
another. But that’s precisely what my analysis
doesn’t assume. My analysis is completely
blind to the moment to moment relation between
V and R values.Â
Â
BA:ÂNo, that’s not my complaint! But
before we get into this, how about complying
with my request? I asked you to restate my
explanation why your analysis is based on a
conceptual error. We do seem to be making
progress – you say that you read it. That’s a
good start. All that’s left is for you to
restate my explanation.
RM: Actually, I think it's about time for you to start
restating my explanation of the power law. All you have
done throughout this discussion is repeat that I have made
a “conceptual error” and that if I would just understand
that error then I would finally see the light and know
that my analysis is invalid. This doesn’t cut it with me
anymore.Â
RM: If, indeed, I am making a conceptual error that
invalidates my analysis then show me how it does that * in
terms of my analysis* . In order to do this you have
to know what my analysis is. And I have seen no sign that
you do.
So I suggest that you prove to me that you understand
my analysis before you start telling me what’s wrong with
it. Â
RM: A good way to prove that you understand my analysis
is to explain exactly how my “conceptual error”
invalidates it. That is, describe what you think if my
analysis and then show me exactly how my “conceptual
error” invalidates it. Here’s a quick summary of my
analysis, in case you’ve forgotten (most important steps
are bolded):
1. Power law researchers measure the instantaneous
velocity (V) and curvature (R) that occur throughout a
movement using the following formulas:
2. The power law is determined using linear regression
analysis with the logarithms of these measures of curvature
and velocity as the predictor and criterion variables,
respectively. The regression equation is:Â
log (V) = a + b* log (R) Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (1)
3. There is a linear relationship between the logs of V
and R when V and R are computed using equations 8 and 9, as
follows:
log (V) = 1/3 * log(V^3/R) + 1/3*log(R) Â Â (2)
4. ** So power law researchers are using linear
regression to determine whether there is a linear
relationship between variables that are mathematically
linearly related, per equation 2**.
** 5. However, they are using equation 1 to determine the
relationship that is actually defined by equation 2. So
there is a “missing variable” (V^3/R, also known as D) in
their prediction equation**.
** 6. Because of this missing variable, a linear
regression analysis using equation 1 will find a value of
b (the coefficient of log(R), known as the power
coefficient) that is “biased” relative to its true value
of 1/3.**
6. The degree to which b is a biased estimate of the true
coefficient of log(R) is proportional to the ratio of the
covariance of R and D to the variance of R: cov(R,D)/var(R).
7. This means that, for any movement trajectory, it is
possible to predict exactly what power coefficient will be
found relating R to V from the folllowing equation:Â
b = b* + a  * cov(R,D)/var(R).Â
where b = observed power coefficient, b* = true power
coefficient, a = true coefficient of D
That is, you can predict exactly what the power
coefficient will be without analyzing (using log-log
regression) the relationship between R and V. I have
confirmed this fact using many movement trajectories, some
made by humans, some by helicopters.
8. The conclusion of my analysis is that the power law
coefficient that is found for different movements depends
completely on characteristics of the movement trajectory
itself and has nothing to do with how it was produced.Â
RM: Now please restate my analysis and tell me how my
“conceptual error” invalidates this analysis. Â