[From Erling Jorgensen (2017.11.17 1626 EST)]
Richard Kennaway (2017.11.14 15:33 GMT)
Rick Marken (2017.11.17.1240)
RK: …[Huh and Sejnowsky (2015)] get into a mixture of power laws depending on frequency, which looks perilously close to the tautology that log V = 1/3 log D + 1/3 log R. …
RM: I have done multiple regression analyses on many different mathematically defined trajectories (including pendulum trajectories, circular trajectories and random squiggle trajectories) as well as for the 41 different actual and modeled pursuit trajectories of the pursuers of toy helicopters, and the toy helicopter trajectories themselves; I’ve done it for the trajectories of the mouse doing two dimensional tracking or making arbitrary two dimensional movements; I’ve done it for filtered versions of these trajectories (using varying filter bandwidths) and I’ve done it for the raw trajectory data. In all cases, the multiple regression coefficients of both D and R are exactly 1/3 and the coefficients of D and C are exactly 1/3 and 2/3. And the multiple R^2 value in all cases is always exactly 1.0.
EJ: The fact that all these different situations come out to exactly those predicted coefficients is evidence to me that it probably is dealing with a tautology.
RM: … the regression used to find the power law doesn’t care whether the variables included are derivatives with respect to time or space; it doesn’t care about the temporal relationship between the values of the variables that are entered into the regression.
EJ: If spatial variables are being confounded with time derivatives, this suggests to me that regression analysis is the wrong tool for exploring these distinctions, whether its linear or multiple regression.
All the best,
Erling