Сравнительный анализ качества робастных модификаций метода главных компонент при сжатии коррелированных данных

Abstract

Principal component analysis is one of the methods traditionally used to solve the problem of reducing the dimensionality of a multidimensional vector with correlated components. We constructed the principal components using a special representation of the covariance or correlation matrix of the indicators observed. The classical principal component analysis uses Pearson sample correlation coefficients as estimates of the correlation matrix elements. These estimates are extremely sensitive to sample contamination and anomalous observations. To robustify the principal component analysis, we propose to replace the sample estimates of correlation matrices with well-known robust analogues, which include Spearman's rank correlation coefficient, Minimum Covariance Determinant estimates, orthogonalized Gnanadesikan --- Kettenring estimates, and Olive --- Hawkins estimates. The study aims to carry out a comparative numerical analysis of the classical principal component analysis and its robust modifications. For this purpose, we simulated nine-dimensional vectors with known correlation matrix structures and introduced a special metric that allows us to evaluate the quality of data compression. Our extensive numerical experiment has shown that the classical principal component analysis boasts the best compression quality for a Gaussian distribution of observations. When observations are characterised by a Student's t-distribution with three degrees of freedom, as well as when a cluster of outliers, individual anomalous observations, or symmetric contaminations described by the Tukey distribution are present in the data, it is the Gnanadesikan --- Kettenring and Olive --- Hawkins estimates modifying the principal component analysis that show the best compression quality. The quality of the classical principal component analysis and Spearman’s rank modification decreases in these cases