Techniques for studying correlation and covariance structure Principle Components Analysis (PCA) Factor Analysis

Principle Component Analysis

Let Assume

Let have a p-variate Normal distribution with mean vector Definition: The linear combination is called the first principle component if is chosen to maximize subject to

Consider maximizing subject to Using the Lagrange multiplier technique Let

Now and

Summary is the first principle component if is the eigenvector (length 1)of S associated with the largest eigenvalue l1 of S.