[From Bill Powers (950220.2330 MST)]
Bruce Abbott (950219.1445) --
RE my suggested model:
p = vert/horiz;
e = p - ref;
h = h - k * e;
c = h;
O.K., so let's plug in some numbers. Say horiz = 200, vert = 150,
and
r = 1.0. Then:
p = vert/horiz = 0.75
c = 150 - 159 = -9.
e = p - r = 0.75 - 1.00 = -0.25
If k = 1.0 then all error should be eliminated in one clock tick.
Is it?
Let's see:
h = h - k*e = -9 - (1.0 * -0.25) = -9 + .75 = -8.75
vert = 159 + -8.75 = 150.25
p = vert/horiz = 150.25/200 = 0.75125
e = p - r = 0.75125 - 1.00 = -.25875
That doesn't seem to work. Let's start over at the point where we
compute h but using my formula:
h = h - k*(e * horiz) = -9 - (1.0 * (-0.25 * 200)) = -9 +50 = 41
vert = 159 + 41 = 200
p = vert/horiz = 200/200 = 1.0
e = p - r = 1.0 - 1.0 = 0
What you have done by including horiz as a multiplier is effectively to
raise the value of k by 200, which does lead to zero error in one step
when you assume that k = 1.0 and h = c = -9 to begin with.
Try just running my model as stated, but using a value of k of 200
instead of 1. We initialize h and c to -9, from the first data point.
c = -9
p = (159 + -9)/200 = 0.75
e = p - ref = 0.75 - 1.0 = -0.25
h = -9 + 0.25*200 = 41
c = 41
p = (159 + 41)/200 = 1.00
e = 1.0 - 1.0 = -0.0
h = 41 + 0.0*200 = 41
So all the error is corrected in one iteration.
Try it with a different starting value of h and c:
c = h = 100
p = (159 + 100)/200 = 1.295
e = p - ref = 1.295 - 1.0 = 0.295
h = 100 - 0.295*200 = 59
c = 59
p = (159 + 59)/200 = 1.09
e = 1.09 - 1.0 = 0.09
h = 59 - 0.09*200 = 41
c = 41
p = (159 + 41)/200 = 1.00
e = 1.0 - 1.0 = 0.0
h = 41 - 0.0*160 = 41
Now there is a single extra iteration before the final result occurs.
You got the the model to give the "right" answer in one iteration by
writing k*e*horiz, but in doing so you introduced a role for the
horizontal size of the bar that makes no sense model-wise. Why should
that perceived size multiply the error signal? Unless you can come up
with a justification, all that can be said is that by doing this, you
can make convergence happen in a single iteration with k set arbitrarily
to 1.0 and the initial value of h and c set to less than the reference
value.
However, this behavior of h and c would not fit the real data. The real
person takes a number of iterations to achieve correction of a sudden
error, so the value used for k should be less than the 200 used above.
The actual number used for a best fit to the data would thus never be
the value of "1.0*horiz" but some considerably smaller number.
I hope to have my results tomorrow -- I programmed some bugs into my
analysis program which took me most of the day to find.
···
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Martin Taylor (950220 18:50)--
RE: E. coli analysis
The new results are even more interesting, particularly the "probability
of an improvement of at least x%."
Let's interpret this probability as the frequency with which
improvements of at least x% occur. The question is, if the distance
travelled in a good direction is k times the distance travelled in a bad
direction at constant speed, what is the average improvement per
reorganization, and per unit time?
It isn't necessary to have very much improvement per reorganization or
per unit time, if we recall that reorganization can be going on 86400
seconds per day, 365 days per year. Suppose that the error is reduced,
on the average, to 0.999 on each iteration. In 1000 iterations, the
error is reduced to 0.37 of its initial value, and in 3650 to 2.6%. In a
human control system, if the initial error of 100% corresponds to no
control, a remaining error of 2.6% would signify quite good control, on
the average, in all the one-dimensional control systems involved (a loop
gain of 40). That would be 10 improvements per day for one year, which
sounds pretty conservative for a child.
I hope that your mathematical colleague doesn't think that retirement is
going to switch off his mind. I hope he suffers the same fate I did --
his workday will double. You two seem to be approaching a complete
analysis at a pretty good clip; it would be a shame to let it drop
before some definitive conclusion is reached.
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Best to all,
Bill P