# Karhunen-Loeve Directions and Regression

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

[more]
Contributed by: Ian McLeod (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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.

## Permanent Citation