Discriminant Analysis with Singular Covariance Matrices: Methods and Applications to Spectroscopic Data

• Published in: Applied Statistics 1995-01, Vol.44 (1), p.101-115
• Main Authors: Krzanowski, W. J., Jonathan, P., McCarthy, W. V., Thomas, M. R.
• Format: Article
• Language: English
• Subjects: Antedependence modelling; Applications; Biology, psychology, social sciences; Canonical variates; Covariance matrices; Cross‐validation; Datasets; Discriminant analysis; Discriminants; Eigenvalues; Eigenvalues and eigenvectors; Eigenvectors; Error rate estimation; Exact sciences and technology; General Interest Section; Information classification; Insurance, economics, finance; Least squares; Mathematics; Medical sciences; Partial least squares; Probability and statistics; Reliability, life testing, quality control; Ridge techniques; Sciences and techniques of general use; Spectroscopy; Statistics; Wavelengths
• ISSN: 0035-9254; 1467-9876
• DOI: 10.2307/2986198

Currently popular techniques such as experimental spectroscopy and computer-aided molecular modelling lead to data having very many variables observed on each of relatively few individuals. A common objective is discrimination between two or more groups, but the direct application of standard discriminant methodology fails because of singularity of covariance matrices. The problem has been circumvented in the past by prior selection of a few transformed variables, using either principal component analysis or partial least squares. Although such selection ensures non-singularity of matrices, the decision process is arbitrary and valuable information on group structure may be lost. We therefore consider some ways of estimating linear discriminant functions without such prior selection. Several spectroscopic data sets are analysed with each method, and questions of bias of assessment procedures are investigated. All proposed methods seem worthy of consideration in practice.