T-illusion; info in perception

[From Bill Powers (950221.2210 MST)]

Bruce Abbott [950221.2000 EST]--

     Ah-HA!! So THAT's the problem! In our previous discussions of the
     3CV1 model you indicated that k would be a number between 0 and 1,
     so I have been thinking of k as reflecting the _proportion_ of
     error corrected in one iteration.

The basic problem here is loop gain, and the units of measurement. When
the perceptual signal is

(YLength + c)/(horiz),

the gain of the input function is being multiplied by 1/horiz. The total
loop gain is the gain in the input function TIMES the gain in the
comparator (-1) TIMES the gain in the output function TIMES the gain in
the external link from handle to "cursor." I confess that I haven't
actually sat down and figured out the effects of all these scalings on
the _apparent_ versus _actual_ value of k in the output function, but
this is where the explanation lies. I'll try to work this out more
coherently, but what we're doing now is worth it just as a replication.

I got my analysis program to run, finally, and took some data on me. It
was an interesting experience. I kept thinking things like "This is
ridiculous, I have to keep bouncing my gaze back and forth between the
two lines, and how can anyone say when the lines are "equal", and I'm
really doing terribly at this, I'll be lucky to get a correlation of
0.5." So here are the results"

        x-y x/y
k 0.0291 8.672
r -62 0.75
RMS 13.7 15.3
Corr 0.9545 0.9529

As you can see, my results don't prefer one model over the other.
Perhaps if I practiced more a difference would develop (I really didn't
feel very skillful -- and oh, yeah, I just remembered that I changed the
slowing factor to 0.01 to make it a little more active. I'll try it with
the lower difficulty tomorrow).

There is one problem with my simple program for finding the best k, and
you may have it, too. If I change the starting values I get somewhat
different numbers. This suggests that the RMS error as a function of k
is a somewhat bumpy curve with local minima. I don't think this is a
serious problem because the values differ by what is probably less than
any amount that is meaningful, but it's something to keep in mind. I
suppose that the best way to find out would be to actually plot RMS vs k
over some range around the final value and see what the score is.

     I have been deriving the participant's reference value by computing
     observed V/H or V-H from the data. Is this how you are doing it?
     These values do seem to give the best fits according to the spot
     checks I've made.

That's how I ended up doing it, too. It seems to be accurate enough.

     Is there any utility to looking at the rates of change of the two
      lines? I have a procedure that does this and, as might be
     expected, these are also correlated, although not as well as the
     absolute lengths are.

It probably won't gain much -- it will mostly amplify the high-frequency
noise which doesn't correlate with much.

I'd like to send you (both) my raw data to see if you get the same
numbers I do for the same run. OK? This way we can make sure we're doing
the analysis the same way, or in equivalent ways..

When we've gone as far as we care to with the T-illusion, I'd like to
try a different one, the "plumber's illusion" as I know it.

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This would be compensatory tracking; the slanted line on the left stays
still and the line on the right is disturbed straight up and down. The
object, of course, is to move the right-hand line up and down to make it
line up with the left-hand line. Mabe the central "wall" should be a
filled rectangle.

···

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Martin Taylor (950221 20:30)--

Brief comment on information in perception. In order to detect the
information in the perceptual signal that corresponds to variations in
the disturbing variable, it seems to me that it would be necessary to
know the behavior of the disturbing variable by some means other than
through the perceptual signal. Something like synchronous detection
would be needed -- otherwise there would be no way to detect a component
correlated with variations in the disturbing variable.

This means that while the _analyst_ might be able to calculate what
proportion of the perceptual signal's variations are due to the
variations in the disturbing variable, the control system itself could
not do that because it has no separate channel that tells it what the
disturbance variations actually are. So even if the information exists,
somehow, in the perceptual signal, there is no way the control system -
can find out what it is.
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Best,

Bill P.