Relationships

[From Bill Powers (930918.0700 MDT)]

Rick Marken (930917.1100) --

I just did this with my little Hypercard control
simulator. A scatter plot of temporal variations in (r-p)
against temporal variations in o looks like this:

      > x
      > x x
   o | x x
      > x x x
      >________
         r-p

Try plotting o against integral(r-p). You ought to get a pretty
clear relationship, like a straight line. The only reason there
doesn't seem to be a relationship between error and output in the
kind of experiment you used in showing the lack of relationship
is that in real behavior there is a background of noise. In an
easy tracking task, the person keeps the systematic error
vanishingly small, leaving only the unsystematic error visible.
It's that uncorrected and unsystematic error that causes the
output to be different when you play back the input.

It's not correct to say there is "no relatioship" when you can
easily find one by putting in the right mathematical
transformation of one axis. In our models, where we inject no
noise, all relationships are exact. The correlations we calculate
are naive and only for comparison with the standard way of
calculating them (integrals don't normally come into the
picture).

Even WITH integrals taken into account, the error in easy
tracking tasks is largely noise -- but in hard ones, there is a
visible relationship between error and output, and especially
between the integral of tracking error and output.

Off to take the train to Silverton (for the first time) with
brother- and sister-in-law.

···

------------------------------------------------------------
Best,

Bill P.