Karhunen-Loeve Directions and Regression
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Students sometimes ask about the difference between the regression line and the Karhunen–Loeve direction. The obvious answer a professor might give is that they are different animals! The object of this Demonstration is to give a more interesting answer.
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Contributed by: Ian McLeod (March 2011)
Open content licensed under CC BY-NC-SA
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Consider the multivariate random variable with mean
and covariance matrix
. Then the Karhunen–Loeve directions are determined by columns of
in the eigendecomposition
, while the regression of
on
is
,
where is the matrix
with the first row and column removed. In the present example, this simplifies to
and similarly for the regression of
on
.
The related question based on data may be asked. In this case we just replace expectations by their sample estimates.
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