i.kurtzer (980622.0200)
this goes out to those math whizes out there. If A is correlated with B and B
is correlated with C, must A be correlated with C? Why?
thankyou
i.
[From Richard Kennaway (980622.1412 BST)]
i.kurtzer (980622.0200)
this goes out to those math whizes out there. If A is correlated with B and B
is correlated with C, must A be correlated with C? Why?
The short answer is: no.
For example, suppose A and C are independent random variables (hence
uncorrelated) and B = A+C. Then B will have a positive correlation with
both A and C. A and C could even have some negative correlation and B
would still be positively correlated with A and C.
Another example: suppose that A, B, and C are identically normally
distributed with mean 0, subject to the constraint that A+B+C is constant.
Then each of the variables will have a negative correlation with each of
the others. A case where two negatives do not make a positive.
If the correlations of A and C with B are sufficiently high, it should be
possible to say something about the minimum possible correlation between A
and C, since in the limit where A=B=C, we must have A=C. However, for
correlations around the 0.5 mark I doubt the minimum correlation would be
positive.
-- Richard Kennaway, jrk@sys.uea.ac.uk, http://www.sys.uea.ac.uk/~jrk/
School of Information Systems, Univ. of East Anglia, Norwich, U.K.
[From Bill Powers (980622.0928 MDT)]
i.kurtzer (980622.0200)--
this goes out to those math whizes out there. If A is correlated with B and B
is correlated with C, must A be correlated with C? Why?
There are special cases in which no correlation would be expected. However,
the case of interest in PCT involves a model of the following form:
[organism]
A -----> B ------------> C
B is known to be proportionally affected by A, and at low frequencies, C is
proportionally affected by B (the organism acts like a leaky integrator).
Under these conditions, and ignoring feedback effects, the correlation of C
with A must be less than the correlation of B with either A or C,
disregarding sign.
In a tracking experiment, A is the disturbance, B the controlled variable,
and C the output action. The model is
+ [organism]
A -----> B ------------> C
-| |
<-----x1-------
We observe that the correlation of A with C is nearly (-)1.0, while the
correlations of A with B and B with C are around 0.1 to 0.2, disregarding
sign. If C were proportional to B and B were proportional to A, this would
be impossible -- the correlation from A to C could not be greater in
magnitude than either of the intermediate correlations. The correlation of A
with C should be considerably less than 0.1, given the observed intermediate
correlations.
One explanation could be that between A and B there is a perfect integrator
and between B and C there is another one, so for all frequencies there is a
180-degree phase shift between A and C. This would result in zero
correlation between B and either A or C, and a unity correlation from A to C.
This explanation is ruled out on three accounts. First, the relationship
between A and B in the experiment is known to be one of pure
proportionality; it is not a pure integration. Thus the correlation from A
to C cannot be greater than that from A to B -- but it is.
Second, B is a known proportional (feedback) function of C, and the best-fit
model of the organism from B to C is a leaky integrator, not a pure
integrator. Thus if the variations in A are slow enough, the relationship
from B to C can be made arbitrarily close to a proportional one. Under these
circumstances, the observed correlation between B and C is the lowest. It is
not low because the organism-function is a pure integrator, but in spite of
the fact that it is not.
Finally, third, if there were a 180-degree phase shift around the loop with
a high loop gain, the system would be explosively unstable, which it is not.
The only remaining explanation that I know of is that there is some degree
of noise in the system that is of a high enough frequency that the control
system can't counteract its effects on B. This means that B has two
components, one proportional to A and C, and one that is random. The
proportional component keeps the low-frequency value of B very small
relative to A and C. It is the high-frequency random component that reduces
the correlation of B with A and C to a very low value. This is the only
explanation that is consistent with the conditions as observed and deduced.
When we observe the error in the actual control system during the
experiment, we find that the largest component is indeed a random
flunctuation at a frequency higher than the maximum frequency of disturbance
that the participant can effectively oppose.
The main point of this exercise was to rule out any simple open-loop model
to explain behavior in which actions oppose the effects of disturbances.
Best,
Bill P.