[From Bruce Abbott (2016.08.23.1810 EDT)]
Bruce Abbott (2016.08.21. 2310 EDT) –
BA: To maintain your assertion that V must depend on R, you will have to disprove my mathematical proof of its falsity with respect to the empirical relationships, AND explain how V and R in my spreadsheet can be linearly related (as they are!) despite R being computed by the ratio of V-cubed to D. Your move.
As I have received no such reply after two days and you have during this same period posted several times on CSGnet, I assume that you have decided just to ignore my challenge. Perhaps I asked too much of you.
Let’s make this simple. You have asserted as mathematical fact the following, based on an algebraic manipulation of the formula for the radius of curvature, R, in which you solve the equation for V:
Rick Marken (2016.08.21.1620) –
RM: This implies that you are thinking of curvature as a disturbance and velocity as the output that compensates for this disturbance. So you are, indeed, looking at the power law as an S-R (actually, a disturbance-output) relationship, with curvature ® as the disturbance and velocity (V) as the output. The controlled variable would then be some function of curvature and velocity (CV = f( R, V).The problem is that, for this to be true, R and V must have independent effects on the CV. And they can’t because the value of R depends completely on the value of V and vice versa.
I take this statement to constitute a prediction based on your belief that the equation for R implies that V and R, as observed in the data, cannot have independent effects; rather, V must always be a function of R to the one-third power (more or less; you leave yourself a bit of wiggle room by hypothesizing that leaving D out of the regression permits some of the relationship to be observed into the error term). Therefore, if a data set can be found in which V and R do not vary as you predict, this would constitute a firm refutation of your statement.
In contrast, Martin Taylor and I have shown both mathematically and empirically that V and R as observed in the data can have almost any relationship. As we’ve noted several times, this is because R depends on the ratio or V-cubed to D, so that R is a pure length. The length of the radius of curvature does not force the tangential velocity with a point moves along the curve take on any particular value. This position leads to an equally firm prediction that one can easily find relationships between V and R that do not conform to your prediction.
But I’ve already demonstrated one such data set. My spreadsheet demonstrates that V can be an inverse linear function of R, even though in the formula for computing R, R = V-cubed/D. Furthermore, I could have made the relationship between R and V in the data almost anything, even though V is a function of the cube-root of R in the formula.
In case you’ve forgotten the damning evidence, here is the table from my spreadsheet:
V
R
D
5.0
10.0
12.5000
5.1
9.9
13.3991
5.2
9.8
14.3478
5.3
9.7
15.3481
5.4
9.6
16.4025
5.5
9.5
17.5132
5.6
9.4
18.6826
5.7
9.3
19.9132
5.8
9.2
21.2078
5.9
9.1
22.5691
6.0
9.0
24.0000
If you plug the values of V and D into the formula for computing the radius of curvature, R, you will get the R-values shown on each row of the table. That is,
R = V-cubed/D. Yet if you regress log V onto log R, you do not get values of V that are proportional to the third root of R. And if you include D in the regression, you do get back the formula for R (in log-log form), just as I stated would happen.
The data can lead to only one conclusion: your analysis of how V and R must be related in the data must be rejected. As a scientist, you have no other choice.
I’ve shown you all this before but thus far you have refused to grapple with its implications for your analysis. Can I expect you to do so anytime soon?
Bruce