5.3 Principal Component Analysis Based on Correlation Matrices

There are many situations where conducting principal component analysis based on covariance matrices would not be appropriate, especially for the following two cases:

1. The original variables are in different units. For example, if a variable is expressed in ounces, its variance will be 16 16 = 256 times of that expressed in pounds. Then this variable will exert substantially more influence on the types of principal components we would get because PCA is concerned with explaining total variances.

2. The original variables have different meanings and have vastly different numerical magnitudes. For example, one variable could be clearance in inches. It could range in the units of 0.001 in. The other variable might be pressure; it could range between 35 and 45 psi. Clearly, the second variable will exert much more influence on the PCA analysis.

In these cases, we standardize multivariate variables and use correlation matrix to perform principal component analysis.

Specifically, we would first standardize the random vector X = ( X ₁, X ₂, ..., X _p) by

(5.17)

for i = 1,..., p, where is the sample mean for X _i and s _ii the sample variance for X _i. Let

be the sample correlation matrix for X, and l = ( l ₁, l ₂,..., l _p) be the vector of eigenvalues for R. In this case, principal component...