Karhunen-Loeve Directions and Regression

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.
The case of variables, and , is shown. It is assumed that and are bivariate normal with means zero, variances one, and covariance . For , the expected values for the regression lines on and on are shown together with the theoretical Karhunen–Loeve directions. For the estimated regressions and directions are shown along with a scatter plot of the data.


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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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