<Martin Taylor 940205 12:30>
<Martin Taylor 940204 18:30>
Rick Marken (940202.1330)
The model that generated stability factors closest to the one's
observed sampled the equivalent of [ ] ahead. [ Martin: If you are
reading this, can you intuit, based on IT, how far ahead the model had to
predict the since disturbance in order to match the subject's
performance?].Very roughly, 180 msec according to my calculations. My intuition would
have said about double that. I don't know which to trust. But I'll have
to go with 180 msec unless I can find a mistake in the calculation.
Yes, I think I found a mistake. Here's the rationale, the calculation,
and the mistake.
Rick found a stability factor of 6.67 with a random target, and 15.85 with
a sine wave target when the model was augmented by applying a phase-advanced
sinusoid as the reference signal. The stability factor is the ratio between
the variance in the CEV that would have been expected if the control-system
output and the disturbance wave were uncorrelated to the variance actually
observed. That variance is related to the uncertainty of the perceptual
signal. To make the computation, I assumed that the predictor reference
signal brought the residual uncertainty of the disturbance to the same
level as the random one (actually, there's a second mistake. I should have
taken 6.26, the unpredicted pursuit stability factor for the sine wave.
Serves me right for trying to do this while rushing to go home after a
long day's work.)
So, the calculation is based on the notion that there was a reduction in
the amplitude of the effective disturbance wave sufficient to bring the
stability factor up by the ratio 15.85/6.26, a ratio of 1.59 (the square
root of the stability ratio). The reduction is assumed to be caused by
the subtraction of the prediction waveform from the disturbance waveform,
resulting in a sine wave having an amplitude 1/1.59 (0.63) times the
amplitude of the disturbance waveform.
Calling the disturbance wave sin(theta) and the prediction wave sin(theta+phi),
where phi represent the phase advance of the predictor waveform,
the peak amplitude of the difference wave (sin(theta) - sin(theta+phi))
occurs when theta= -phi/2. The amplitude of that peak is
sin(phi/2)-sin(-phi/2) = 2sin(phi/2)
which we want to be 0.63. So sin(phi/2) = 0.32, giving phi/2=18.5 degrees.
My second mistake (the one I spotted first) was to take this value as
the phase advance instead of half the phase advance. The real phase
advance should be 37 degrees.
Rick says that the frequency of the sine wave was 0.3 Hz, giving a period
of 3.3 seconds. A phase advance of 37 degrees give 340 msec, which is
my corrected estimate of the lead time for Rick's predictor reference
signal, in place of the 180 msec I gave in my previous response.
Maybe this is all wrong, but I'll go with it, at least for now.
Martin