[From Bill Powers (950222.1200 MST)]
Bruce Abbott (direct post 950221)
I've taken the liberty of including my data using the sine-wave
disturbance below. It would seem that by the second cycle I was
doing very well, but then began to get progressively more out of
phase.
Here are my results:
PERCEPTION: x - y ln(x) - ln(y) x/y
k = 0.0504 18.51 25.38
r = -62 -0.336 0.717
RMS = 26.2 10.3 10.3
Corr = 0.9830 0.9850 0.9851
The middle model says that the perception of lengths is the natural log
of the actual lengths, and that the controlled perception is the
difference in the logs. The reference log perception is -0.336, which
when used as the exponent of e gives an illusion ratio of 0.715,
agreeing with the model based on direct perception of the ratio with an
error of 2 parts in 717, or 0.28%.
Note that the correlation is very insensitive to model fit (only a 0.2%
difference across models), while RMS error is very sensitive (ratio and
log model values are 40% of the linear model values)
It would be interesting to plug in Stevens' Power Law for perception of
lengths to see how it works.
I predict that the advantage of the ratio or log hypothesis over the
linear-difference hypothesis will increase as the range of sizes of the
figure increases. The log version is probably more physiological, since
we know that basic perceptions are logarithm-like. Also computing a
ratio with neurons would be quite difficult if the perceptions were
linear, but simple if they were logarithmic.
Incidentally, the log and ratio models are a direct refutation of one of
our favorite objections to PCT: "Everybody knows that control systems
have to be linear."
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Best,
Bill P.