High breakdown estimators for principal components: the projection-pursuit approach revisited

Abstract

Li and Chen (J. Amer. Statist. Assoc. 80 (1985) 759) proposed a method for principal components using projection-pursuit techniques. In classical principal components one searches for directions with maximal variance, and their approach consists of replacing this variance by a robust scale measure. Li and Chen showed that this estimator is consistent, qualitative robust and inherits the breakdown point of the robust scale estimator. We complete their study by deriving the influence function of the estimators for the eigenvectors, eigenvalues and the associated dispersion matrix. Corresponding Gaussian efficiencies are presented as well. Asymptotic normality of the estimators has been treated in a paper of Cui et al. (Biometrika 90 (2003) 953), complementing the results of this paper. Furthermore, a simple explicit version of the projection-pursuit based estimator is proposed and shown to be fast to compute, orthogonally equivariant, and having the maximal finite-sample breakdown point property. We will illustrate the method with a real data example.

Properties of design-based functional principal components analysis

“…Let us remark that the influence functions of the eigenelements are similar to those found in the multivariate framework for classical PCA (Croux and Ruiz-Gazen, 2005). We are now able to state that the remainder term R T defined in equation (13) Proposition 3.3 Suppose the hypotheses (A1), (A2) and (A3) are true.…”

Outlier-Robust PCA: The High-Dimensional Case

“…In the classical setting, we note that the situation is different. In [30], the authors propose a fast approximate Projection-Pursuit algorithm, avoiding the non-convex optimization problem of finding the optimal direction, by only examining the directions defined by sample. In the classical regime, in most samples the signal component is larger than the noise component, and hence many samples make an acute angle with the principal components to be recovered.…”

“…NUMERICAL ILLUSTRATIONS In this section we illustrate the performance of HR-PCA via numerical results on synthetic data. The main purpose is twofold: to show that the performance of HR-PCA is as claimed in the theorems and corollaries above, and to compare its performance with standard PCA, and several popular robust PCA algorithms, namely, Multi-Variate iterative Trimming (MVT), ROBPCA proposed in [18], and the (approximate) Project-Pursuit (PP) algorithm proposed in [30]. Our numerical examples illustrate, in particular, how the properties of the high-dimensional regime discussed in Section II can degrade, or even completely destroy, the performance of available robust PCA algorithms.…”

Identification of Seawater Quality by Multivariate Statistical Analysis in Xisha Islands, South China Sea

“…Since the ratio of samples to the variables is 2:1, the classical principal component analysis might fail. Robust principal component analysis (RPCA) is still effective even if there are a few anomalous observations and even observation samples are less than number of variables [7][8][9]. Thus, RPCA is employed to understand the spatial pattern of water quality.…”