<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
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