[From Rick Marken (2009.05.25.0915)]
Martin Taylor (2009.05.24.14.52)--
I said 1/CR is the maximum correlation you should get, if the output
function is a pure integrator.
Are we talking about the same thing? I'm assuming your "i" is
perception. My calculations were for the maximum correlation between
disturbance and perception.
OK, so 1/CR is the maximum correlation between d and i (i is the
perceptual variable in my model). Here are the results of a non-leaky
integrator model:
1/CR .86 .16 .02 .004 .0002
r.di -.42 -.36 -.34 -.31 -.24
I guess you could say that r.di never gets above 1/CR, but I don't
think that really tells you much about the relationship between r.di
and quality of control. And, besides, I thought it was -1/CR that is
the maximum correlation; if that's true and "maximum correlation"
means most positive then the first values (which would be -.86 and
-.42) would violate the rule since -42 is higher than -.86. If,
however, maximum correlation means the absolute size of the
correlation, then all the values above violate the rule, whether your
using 1/CR or -1/CR.
It seems very odd to me that if "i" is indeed perception, that it should
have such a high correlation with output. If the output function is an
integrator, then o is uncorrelated with e (error). If the reference value
is fixed, e is (-p + constant), so p is uncorrelated with o. If i is p, then
r.io should also be 0.0. Something is fishy here, and I'm thinking that
maybe "i" is not perception (or its modelling equivalent, sensory input).
Yes, i is definitely the perceptual variable; it's what is compared to
the reference to produce the error that is integrated into the
output.But the fact is the correlation between i and o that I get from
the model is very different depending on the type of model I use. The
results for a pure, non-leaky integrator model controlling at a level
such that CR = 247 (1/CR =.004) lookx like this:
r.io 0.39
r.do -0.999
r.di -0.36
These are the results with no noise. The moderate positive correlation
between i and o is quite a surprise. This i - o correlation is not
changed much at all when you look at lagged correlation (output
following perception). However, when I put a leak into the integration
I get results that look like this:
r.io -0.1071
r.do -0.9998
r.di 0.1285
This is for a system controlling quite well: CR = 2134. Now I get a
weak negative correlation between i and o. By the way, when I do a
partial correlation on the first results above r.io|d = .728. When I
do it on the second results above, I get r.io|d = .22: position i - o
correlation in both cases. Of course, the actual, true relationship
between i and o is o = -1.0* i. So if partial correlation worked,
r.io|d should be close to -1.0.
What all this is telling me is that using correlational methods to
analyze the causal links in a closed loop system is just useless. The
reason is, I think, very simple. The observed relationship between i
and o is part of a closed loop: i is having an effect on o while o is
having an effect on i. When we look at the correlation between i and o
we tend to look at it through S-R theory glasses and think of it as
reflecting something about the effect of i on o (because, after all,
perception must be guiding action in a control task). But this is just
a perceptual bias. The correlation between i and o also reflects the
simultaneous effect of o on i. So the i - o correlation you observe in
a closed loop task confounds the forward effect of i on o and the
backward effect of o on i. The actual i-o correlation that is
observed, whether positive, negative , large or small, depends on the
nature of the i-->o _and_ o-->i connection of the closed loop system
involved.
I guess my basic conclusion from this long and somewhat frustrating
analysis is kind of what I knew already but didn't really understand
quite as clearly as I do now: you simply cannot use open-loop models
to correctly understand the behavior of a closed loop system; and
regression is an open loop model and people are closed loop systems.
Now I want to get back to using multiple regression (MR) to analyze
the results of my Mind Reading demo;-) (What I'm doing on that is an
appropriate use of MR because I am not using it to study causal
relationship in the system; I'm just using it to see which of several
somewhat inter-correlated disturbances is being resisted).
Best
Rick
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