Re: How to create conditional correlated random values given exogenous data
- From: "Reef Fish" <large_nassua_grouper@xxxxxxxxx>
- Date: 7 Nov 2006 10:25:00 -0800
Bob wrote:
I just want to comment on the single word "structure" in "correlationI'm not sure whether I get to the point of your answer. With
structure". Just exactly what IS a correlation structure?
If I tell you that X and Y have ZERO correlation, and I give you 100
points (any 100 you choose) insid a unit square from the origin,
to sprinkle on the X-Y plane to characterize the ZERO -correlation
structure, can you do it? What do you envision to be that correlation
"striuture"?
correlation structure I mean the covariance matrix. And if your
variables X and Y are uniformly distributed the X-Y-pane should be
evenly filled.
I also mean the covariance matrix. What you said is a one-way
implication. When you ask about the structure, you are trying to
draw an inference from the OPPOSITE direction.
What I am saying is that if I tell you X and Y have zero correlation,
and you draw the conclusion that the points resemble evenly
filled points on the unit square, then you are a "damned fool" :-)
The REVERSE inference does NOT hold.
I had a ready made example that you can take X and Y to have
ANY distribution, any correlation (ANY STRUCTURE) on the
X-Y plane. When you fit a linear regression of Y on X, the
FITTED values and the RESIDUALS of that regression will
ALWAYS have zero correlation.
The scatter of residuals versus fitted is anything BUT points
scattered evenly on the fitted-residual plane!
Try this for size. :-) I had used the same for several different
subjects in this group already:
Y FITTED RESIDU
*** ************ ******
1 -11 12
4 3.5527E-15 4
9 11 -2
16 22 -6
25 33 -8
36 44 -8
49 55 -6
64 66 -2
81 77 4
100 88 12
The correlation between FITTED and RESIDU is zero of course.
You can see the definite pattern in the residuals because of the
poor fit. ++------++ nothing random about them.
By the same reasoning, the STRUCTORE of the points could
be "virtually" anything for the correlation to be zero.
Random scatter (shot gun blast) ===> zero correlation
zero orrelation === does NOT imply ==> random scatter.
That was what I was trying to convey to you about your use
of "structure". I think you're trying to dfaw the inference
about the scatter that you are not entitled to, based on the
value of correlation coefficients (any given value).
-- Reef Fish Bob.
This is a very serious question relating to what you are asking.
Perhaps
if you realize how miserably you will FAIL in characterizing a zero
correlation structure, that you will not find ANY structure to
characterize
points that will have a structure of any value, and perhaps then think
of
some other ways of asking your questioni to chieve what you really
wat to do.
-- Reef Fish Bob,
.
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