Some Optimal Properties and Interpretations of Principal Components.
Abstract
Optimal properties are derived and some new geometrical interpretations given for principal components. Typically, our main results concern the simultaneous minimization of eigenvalues of certain covariance matrices which measure the goodness of an approximation. Many popular criteria like total variance and generalized variance, which are increasing functions of the eigenvalues, are then minimized by the best approximator. In other situations, the criterion may not be a monotone function of the eigenvalues. A general optimal class is derived based on the non-negative definite ordering of covariance matrices. Also a result is given for the sequential selection of principal components. In the final section gives a new geometrical interpretation of the sample principal components.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1978
- Accession Number
- ADA054555
Entities
People
- Raul Hudlet
- Richard A. Johnson
Organizations
- University of Wisconsin–Madison