[From Rick Marken (2006.12.18.0850)]
Bill Powers (2006.12.17.1125 MST)
Martin Taylor 2006.12.17.22.22 --
CR is the control ratio -- (fluctuations in the disturbance)/(fluctuations in the perceptual signal).
But suppose this amplitude ratio CR is 10 -- that is, fluctuations of the disturbance correspond to fluctuations in the perceptual signal one tenth as large. If there is no noise in the system, the correlation between the perceptual signal and the disturbance will be 1.0, not 0.1 as you imply (1/CR = 0.1).
I think Martin means that CR is a _variance_ ratio: CR = var(d)/ var(p).
Ar you OK with the result that "the absolute value of the minimum correlation between d and o must be sqrt(1 - 1/CR^2)?" if there's no loop transport delay, no noise, and the output function is a pure integrator?
No, I'm not. Without any noise or delay, the correlation between d and o will be 1.000, the maximum possible, regardless of their relative amplitudes.
Yes. But I think Martin is talking about a variance ratio, not an amplitude ratio. In which case his formula looks pretty good. Well, actually, I think it probably should just be just 1/CR^2. In that case, when there is no control, CR = 1 (the variance of d = the variance of p) and the minimum absolute correlation between d and o is 1.0. When the control is nearly perfect, the variance in d is >> than the variance in p, CR ~infinity and the minimum correlation between d and o is ~ 0.
Best
Rick
ยทยทยท
----
Richard S. Marken Consulting
marken@mindreadings.com
Home 310 474-0313
Cell 310 729-1400