[Martin Taylor 2006.12.18.17.05]
[From Bill Powers (2006.12.17.1125 MST)]
Martin Taylor 2006.12.17.22.22 --
Without any noise or delay, the correlation between d and o will be 1.000, the maximum possible, regardless of their relative amplitudes.
OK. That's a direct contradiction of my analysis. I'd appreciate seeing your derivation. If you can't find a problem in my derivation, and I can't find a problem in yours, there must be a subtle problem somewhere. If your claim is true for all values of the gain, then my derivation must be wrong. I'd really, really, appreciate it if you could show me where.
To be precise, we are talking about a loop in which there's no transport lag, the output function is C*integral (error) dt, and you are saying that for all values of C, the correlation between d and o is 1.000.
I presume, although you did not mention it, that your claim does not include C<=0, but that for any non-zero and non-negative value of C. even .00001, the perfect correlation must hold.
If what you say is true, then of course my derivation is simply wrong, and I'd like to find the flaw.
I must be missing something here. The phase shift has nothing to do with the correlation, either -- only with amplitude ratios. Correlation describes the degree of randomness in a relationship, not a ratio of amplitudes.
I don't know why you would assert that phase shift has nothing to do with correlation. Just imagine a trivial situation, and ask what is the correlation between cos(x) and cos(x + phi). Phi is the phase shift, and when it's zero the correlation is unity, and when it's 90 degrees the correlation is zero. In the more complex situation we are talking about, the phase shift across the integrator has everything to do with the correlations between d and o and between d and p. At least it does in my analysis. If you want to demonstrate the error in my analysis, you have to do more than simply assert that "phase shift has nothing to do with correlation".
Of course correlation describes the precision with which one variable tracks variations in another, as you say. Correlation is equal to the cosine of the angle between the vectors representing two waveforms, isn't it? Isn't that a representation of the degree of randomness in a relationship? That the ratio of amplitudes between p and d turns out to be equal to their maximum correlation is an interesting result, not a definition.
I do wish you would _read_ the derivation you so severely criticise. All that the derivation produces as a result is that the MAXIMUM correlation between disturbance and perception signals is equal to the ratio of their fluctuation amplitudes, which will be between zero and 1.0. It's just what falls out of the derivation.
I've said over and over that the derivation seems too simple, and I'm hoping someone will demonstrate the flaw that may well exist, but all I get is commentary on my explanatory e-mails, which seem after all not to be too explanatory. Please, please, refer to the original <http://www.mmtaylor.net/PCT/Info.theory.in.control/Control+correl.html> and tell me what's wrong with it.
(For Rick's benefit, I should say that I used the term Control Ratio differently from him, taking amplitude rather than variance).
Using the seemingly straightforward derivation referenced, I then used the expression sin^2(x) + cos^2(x) = 1 to derive the mimimum correlation between d and o. Again, that might be wrong, but I can't see why it should be, and if it is, I'd like to be shown wherein it is wrong.
The maximum correlation between d and p and the minimum correlation between d and o both should be the actual correlation in the absence of noise, transport lag, and reference variation, and that ought to be testable in a simulation that uses a very fine value of dt.
Is it that you're using "correlation" not in the statistical sense but in an informal sense?
I must be suffering some serious misapprehension here.
Apparently. I think it may be a bit like my misapprehension in the first few days after I heard of PCT. I couldn't believe that we were talking about actual, analyzable control systems in the engineering sense. Once I got over that misapprehansion, analyses such as the one on correlation we are talking about became possible. Why don't you just assume that I'm trying to be moderately mathematically rigorous, as you have recently urged? I may well be wrong, but I'm not being excessively informal (even though I drank 2/3 of a botle of nice wine a few hours ago
[From Bill Powers (2006.12.18.0020 MST)]
I found an old "track-analyze program" and set it so the model uses no delay and a perfect output integrator instead of the optimum delay and damping ("leakage") settings.
Unfortunately, you included only a trace of a person's track and a model handle track with a record of the deviation between them, rather than a disturbance track along with the model handle track and a record of the perceptual signal. We haven't been discussing how well the model fits what a person does. We are dealing with the correlations in the signals within a model. We need to see the correlations among d, o and p signals in the model. You probably could get them from the same run of the model as the one you graphed with the person's track.
I know you can't create a computer-based model with zero transport lag, but you ought to be able to provide a reasonable simulation test of the analysis by using dt very small compared to 1/W (where W is the bandwidth of the disturbance signal). We should be able to test the correctness of the analysis, at least to a first approximation, by using a very small dt, a low value of c, and a band-limited noise as a disturbance.
The analysis says that the correlation between perceptual and disturbance signals should be (close to) p/d the ratio of perceptual signal fluctuation amplitude (p) to disturbance signal fluctuation amplitude (d), and the correlation between disturbacne and output signals should be (close to) sqrt(1-(p/d)^2). I say "close to" because of the inevitable artefacts of digital simulation.
Just putting an example number on it, if control reduces the perceptual RMS fluctuation to 1/10 of the disturbance RMS, correlation between the perceptual and disturbance signal should be no greater than 0.1, and that between disturbance and output should be no less than approximately 0.98 (sqrt 0.99).
I'd be quite happy to be shown to be wrong, and even happier to be shown analytically where the derivation is wrong.