Maximum likelihood principal component analysis with correlated measurement errors: theoretical and practical considerations

Procedures to compensate for correlated measurement errors in multivariate data analysis are described. These procedures are based on the method of ma ximum likelihood principal component analysis (MLPCA), previously described in the literature. MLPCA is a decomposition method similar to conventional PCA, but it takes into account measurement uncertainty in the decompositio n process, placing less emphasis on measurements with large variance. Altho ugh the original MLPCA algorithm can accommodate correlated measurement err ors, two drawbacks have limited its practical utility in these cases: (1) a n inability to handle rank deficient error covariance matrices, and (2) dem anding memory and computational requirements. This paper describes two simp lifications to the original algorithm that apply when errors are con-elated only within the rows of a data matrix and when all of these row covariance matrices are equal. Simulated and experimental data for three-component mi xtures are used to test the new methods. It was found that inclusion of err or covariance information via MLPCA always gave results which were at least as good and normally better than PCA when the true error covariance matrix was available. However, when the error covariance matrix is estimated from replicates, the relative performance depends on the quality of the estimat e and the degree of correlation. For experimental data consisting of mixtur es of cobalt, chromium and nickel ions, maximum likelihood principal compon ents regression showed an improvement of up to 50% in the cross-validation error when error covariance information was included. (C) 1999 Elsevier Sci ence B.V. All rights reserved.