# Update (was many things but is now Model-Based Testing for Controlled Variables)

[From Rick Marken (2013.04.04.1050)]

MT: I had earlier suggested using a tolerance zone for the perceptual signal,
using the expression

p = sign(qi)*(max(abs(qi)-tol, 0)

RM: I could easily change the model so that p is computed in this way.
But first could you explain the expression "max(N,0)", where N is
abs(qi)-tol.

RM: Never mind, I figured it out. What a maroon I am. But I still need

RM: When I calculate p using your "tolerance zone" equation what
should I use for qi? I think it should be arctan(v/s) for the angle control
model and v for the difference control model. Do you agree or should I do
something different?

RM: I've implemented the "tolerance zone" model using the vertical
distance between cursor and target (v) as qi and it works beautifully
in the sense that the greater the width of the tolerance zone (the
greater the value of tol in the tolerance zone equation above) the
poorer the tracking performance (measured in bits). This means that
the "tolerance zone" model can account for the decrease in performance
with increased horizontal separation between cursor and target (s) by
assuming that the increase in tol is proportional to the increase in
s. And this tolerance zone model accounts for the decrease in
performance with increase in cursor-target separation just as well as
does the angle control model.

So the good news is that Martin's tolerance zone model works exactly
as he predicted, bringing down tracking performance as the the
news is that the tolerance zone model does not fit the time variations
in output (mouse movements) as well as the angle control model; the
difference in the fit of the models to the human data gets more
pronounced as the horizontal separation between cursor and target (s)
increases.

So while both models fit the decrease in performance with increased s
equally well, the angle control model accounts for the detailed time
variations in the output much better than the tolerance zone model.
Since the perception controlled by the tolerance zone model is the
vertical distance between cursor and target (v) the results seem to
favor the idea that the perception controlled in this tracking task is
the angle between target and cursor, arctan(v/s).

Another reason for preferring the angle control over the tolerance
zone model is that no mechanism is proposed to account for the
increase in the width of the tolerance zone with increased horizontal
separation between cursor and target.

Best

Rick

ยทยทยท

--
Richard S. Marken PhD
rsmarken@gmail.com