RM: Yes, I think it might be worth it to figure out why this particular side effect occurs. Given the level of misunderstanding of PCT on this site I think it might be worth spending some time on it.
Fine! But you should also figure out why it sometimes occurs and sometimes not. And when it occurs it can have also other power coefficient than 1/3. Your OVB does not answer this. I claims that it occurs always with 1/3 even when it empirically does not occur.
RM: While x and y are 2 df of planar movement, velocity (V) and curvature (C) are not. That’s because V and C (or R) are functions of both x and y see equations 2 and 3 in Marken & Shaffer (2017).
But think: A turn (as a minimal part of curvature) happens when the relationship between the velocities of x and y components of the movement changes. For example x accelerates and y either accelerates less, or keeps constant velocity, or accelerates more, or the same in another way around. Now because x and y are 2 df of planar movement, you cannot predict the change of the velocity of one component from the change of the velocity of the other component. That is why in just the same curvature the total velocity can freely decelerate, remain constant, or accelerate. From the independence of x and y velocities follows the independence of the total velocity and the curvature, no matter what you equations seem to say.
About the equations [V = R1∕3 * D1∕3]; [Width * Height = Area]; [Perimeter = Width + Height]; and also that of the angular speed could be added here [A = C * V], you said: “All these equations are equivalent. They would all be linear if you take the log of both sides in the equations for V and Area.” The first sentence is simply true depending on what you mean here with equivalence of different equations, but they are all similar so that the left side and right side are equivalent: There is always the same value both side when the variables are instantiated and calculations made. So the relationships between the sides are linear even without taking the logs, aren’t they?..
RM: No. Only the equation for P = H + W is linear.
Here is something strange. The value of the Area of an rectangle is always exactly the same as Width times Height. What you mean by saying that it is not linear. (This probably not an important question in relation to our discussion topic, but I just don’t understand.)
And if the equations above are equivalent then there must also prevail a necessary mathematical relationship between the height and with of all possible rectangles. Do you think so? If you think so, then what is the power coefficient of that relationship?
RM: In A = H * W the power coefficient of H is 1 and the power coefficient of W is 1. The equation can be written A = H^1*W^1. The same is true for H and W in P = H + W and for C and V in A = C * V.
Aah! You think that the A = H * W sets or unveils a mathematical relationship between A and W so that there is a linear negative correlation between them, don’t you? But that is true only if the area is constant. If the areas of the rectangles may freely change then that correlation will vanish. Similarly IF affine velocity remains constant THEN there is 1/3 power relationship between velocity and curvature. But if affine velocity may freely change then there is no power relationship or it has a different coefficient.
RM: …There is no law against using regression analysis to confirm a mathematical relationship between variables. And in our case, that use of regression had enormous scientific value because it showed that when you do a regression on only a subset of the variables (V and R) involved in a mathematical relationship that involves those variables and one other (D) your result will deviate from the true relationship between the included two by an amount that is proportional to the covariance between the included and omitted predictor. This is a scientifically important finding because it confirms the PCT explanation of the observed behavior.
A regression analysis which gives exactly the same result from any random data is as valuable as a stopped watch! The result may describe the data by change like the stopped watch shows right time twice a day.
In Marken & Schaffer 2018 you say that:
- “[T]he movement trajectories of some physical systems, such as the orbits of the planets, do not follow the 1/3 or 2/3 power law, a fact that we confirmed for ourselves.”
- “We also found that the power law is no more obligatory for biological systems than it is for physical systems”
- “whether a trajectory is produced by a biological or physical system, a regression analysis based on Eqs. 3 or 4 always finds that the “true” power coefficient for the relationship between the speed and curvature of the movement trajectory is 1/3 (for R versus V) or 2/3 (for C versus A), with an R2 value of 1.0 in both cases”.
So you have admitted yourself that your OVB method gives fabricated results that do not fit – have nothing to do – with the empirical data from which they are calculated!
BTW. How would you proceed if you had a large data of different rectangles and you wanted to know how their widths correlate with their heights?
Eetu