Robustness of Principal Component Analysis with Spearman’s Rank Matrix

This paper is concerned with robust principal component analysis (PCA) based on spatial sign and spatial rank vectors. The most common PC approach is based on the eigenvectors of the sample covariance matrix; however, this approach is known to be sensitive to outliers. Several robust alternatives based on spatial sign and spatial rank vectors have been discussed including Kendall’s tau or Marden’s rank and Spearman’s rank matrices. Our aims in this paper are to investigate properties of PCA based on Spearman’s rank matrix from theoretical and practical view points and to compare the performance of PCA based on these robust alternatives. A concentration inequality for an estimator of Spearman’s rank matrix is derived, which reveals consistency of the estimator. The influence functions for eigenvalues and eigenvectors of Spearman’s rank matrix are examined which establish the asymptotics for PCA based on Spearman’s rank matrix. Monte Carlo simulations and application to a real dataset show that the performance of PCA using Spearman’s rank matrix is comparable to or sometime better than that of Kendall’s tau and Marden’s rank matrices.