🚀 go-pugleaf

[From Bill Powers (950213.1055 MST)]

Bruce Abbott (950213.0925 EST)--

     I like your proposal better as it makes explicit the fact that both
     line lengths are perceptions.  However, it seems to me that
     functionally the two models are isomorphic.

Yes, but for one thing I thought of only later. If the perceptual signal
is derived from horizontal minus vertical, with no anisotropy factor,
then a constant non-zero reference signal would make the model maintain
a constant _difference in length_ between the two legs of the T. With
the perceptual signal representing the ratio, horizontal/vertical, a
constant reference signal different from 1 would make the model maintain
a constant _ratio_ of the two sides different from equality.

In the first case, the difference in lengths would be maintained for any
size of the whole figure, while the ratio would vary as the figure
changed sizes. In the second case, the ratio would be maintained for any
size, while the absolute difference in lengths would vary with size. So
one hypothesis should work distinctly better than the other, and you
need only fit the reference signal, without an added anisotropy in the
perceptual function.

In your discussion of the tilted figure, you get an additional
condition, so you might be able to distinguish between a perceptual
anisotropy and a reference signal that isn't exactly 0 (or 1 for ratio
control) by comparing with the vertically-oriented results. You might be
able to combine the two effects; some perceptual anisotropy, some
adjustment of reference signal to account for the remainder.

     In pursuit tracking, do you normally treat the moving spot as part
     of the perceptual input rather than as reference?  I ask because
     the inverted-t demo seems to me to fall into the same class.

Yes, and yes. If the instructions are "keep the cursor on the moving
target" the models are equivalent, but if they are "keep the cursor 1 cm
left of the moving target" they aren't. We tend to take the most common
case as the general case, which is a mistake. If the reference signal is
the target position, then how do you set up the model to keep the error
at 1 cm? If you say that the perceptual signal represents the distance
between target and cursor, then you can account for tracking both on and
beside the target with the same model, the only difference being whether
the reference signal is zero or nonzero.

     An interesting question is how to derive the "anisotropy" for a
     given run from the data.  It would show up as a constant error from
     reference in either the subtractive or ratio model.  This error
     could be fed back into the perceptual function as the anisotropy
     factor during a second fit of the model to data.

That's the ticket, except you don't need the anisotropy factor and can
do it all by adjusting reference signals.
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Bruce Nevin (direct post) --

Bruce, with his usual delicacy, has asked me privately about this
phenomenon of counterclockwise sun-dials. Nobody has ever been so
tactful while managing to point out that I was dead wrong.

Of course the shadows on sun-dials rotate clockwise. William through the
looking-glass, I'm afraid. There went another neuron.
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Martin Taylor (950212.1100) --
RE: E. coli investigations

     I don't think that the difference is interesting, in that the
     problem is set up to deal with ONE step.  If you make two steps in
     a given direction, you can use the data to see what would have been
     the probability you would have of improving the situation after one
     step of double the size.

Actually if the step is taken to be a step in direction rather than
distance, only one added calculation needs to be made: to project the
direction until it intersects a radius to the target at right angles,
and use that as the origin for the next step.

               o  start
                 \
                   \                    o  target
                     \               *
                       \          *
                         \     *
                           \* (next move starts here)

With E. coli moving at a constant velocity, error-squared will decrease
until the path is at right angles to the direction to the target, and
the next tumble will occur almost immediately to create a new direction
(again not, in general, aimed directly at the target).
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Best to all,

Bill P.