Techniques for studying correlation and covariance structure

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

2. Principle Component Analysis

3. Let Assume

4. 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

5. Consider maximizing subject to Using the Lagrange multiplier technique Let

6. Now and

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