Allen's model

[From Rick Marken (930330.2100)]

Bill Powers (930330.2000 MST) --

Allan is using information about o in his method,
as I vaguely understand it now, with your help. It shouldn't be
surprising if he can also reconstruct d.

Still nothing from Allan on this: but I think I must not have
been clear in MY description of what I THOUGHT Allan was doing.
I don't think Allan is going to use information about o (the
o generated in the real, closed loop run) at all. In fact, the
way I conceive of Allan's study, you don't need to save o at
all; all you need is d and p from the closed loop run. Again,
here's what I think Allan is proposing:

To get H(D):

Put p's of varying length into the open loop model:

p --> model --> o

Correlate each resulting o (a vector of 1000 values) with d
(also a vector of 1000 values). As you pointed out in your
first post on this topic, this will involve going through a
whole lot of different p's. At some point you get to a p
which, when put through the model, yields a correlation between
the resulting o and d that is equal to the corerlation between
o and d obtained in the closed loop experiment. The length of
the successful p vector is H(D). I suspect that the successful
p would be the same length as the disturbance vector. So, from
my point of view, Allan is proposing a kind of inverse regression
analysis , where the regression equation is d = model(p). We are
given d and the model (the "regression coefficients") and we are
trying to find values for p that satisfy this equation (with the
outputs of the open loop model being the predictors of d).

To get H(D|P):

We again put p's of varying length into the open loop model,
put this time these p's are added to the p vector that was
observed during the closed loop run. Call the p vector from
the closed loop run p'. So

(p'+ p) --> model -->o

Again we would correlate the resulting o's with d. But this
time, Allan imagines that the correlation between o and
d will be maximum when the length of p is 0 -- that is, when
the p vector is all 0's.

I think you should be able to see, now, why Allan will not
get the results he expects. But I want Allan to explain whether
this is a correct description of his plan. If I explain the
problem I'm sure he'll say that he meant something different and
we'll have to go around and around again. I think we have to
get Allan to commit to a precise method of measuring the
information about d in p (p' in my notation here). I would
like to have the method agreed on in the form of a computer
program (I'll write it if he likes). I think we're pretty close
to the point where we can do this.

So let's get this precisely nailed down in terms of program
steps (I'm all set to run Allan's model as soon as I get his
concurrance about what it is) -- are you with us, Allan?