Robust sparse principal component analysis.

A method for principal component analysis is proposed that is sparse and robust at the same time. The sparsity delivers principal components that have loadings on a small number of variables, making them easier to interpret. The robustness makes the analysis resistant to outlying observations. The principal components correspond to directions that maximize a robust measure of the variance, with an additional penalty term to take sparseness into account. We propose an algorithm to compute the sparse and robust principal components. The method is applied on several real data examples, and diagnostic plots for detecting outliers and for selecting the degree of sparsity are provided. A simulation experiment studies the loss in statistical efficiency by requiring both robustness and sparsity.Dispersion measure; Projection-pursuit; Outliers; Variable selection;

Robust functional principal components: A projection-pursuit approach

In many situations, data are recorded over a period of time and may be regarded as realizations of a stochastic process. In this paper, robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting. Our approach combines robust projection-pursuit with different smoothing methods. Consistency of the estimators are shown under mild assumptions. The performance of the classical and robust procedures are compared in a simulation study under different contamination schemes.Comment: Published in at http://dx.doi.org/10.1214/11-AOS923 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Using Robust PCA to estimate regional characteristics of language use from geo-tagged Twitter messages

Principal component analysis (PCA) and related techniques have been successfully employed in natural language processing. Text mining applications in the age of the online social media (OSM) face new challenges due to properties specific to these use cases (e.g. spelling issues specific to texts posted by users, the presence of spammers and bots, service announcements, etc.). In this paper, we employ a Robust PCA technique to separate typical outliers and highly localized topics from the low-dimensional structure present in language use in online social networks. Our focus is on identifying geospatial features among the messages posted by the users of the Twitter microblogging service. Using a dataset which consists of over 200 million geolocated tweets collected over the course of a year, we investigate whether the information present in word usage frequencies can be used to identify regional features of language use and topics of interest. Using the PCA pursuit method, we are able to identify important low-dimensional features, which constitute smoothly varying functions of the geographic location

Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery

PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust subspace learning or robust PCA (RPCA). For long data sequences, if one tries to use a single lower dimensional subspace to represent the data, the required subspace dimension may end up being quite large. For such data, a better model is to assume that it lies in a low-dimensional subspace that can change over time, albeit gradually. The problem of tracking such data (and the subspaces) while being robust to outliers is called robust subspace tracking (RST). This article provides a magazine-style overview of the entire field of robust subspace learning and tracking. In particular solutions for three problems are discussed in detail: RPCA via sparse+low-rank matrix decomposition (S+LR), RST via S+LR, and "robust subspace recovery (RSR)". RSR assumes that an entire data vector is either an outlier or an inlier. The S+LR formulation instead assumes that outliers occur on only a few data vector indices and hence are well modeled as sparse corruptions.Comment: To appear, IEEE Signal Processing Magazine, July 201

Structural Analysis of Network Traffic Matrix via Relaxed Principal Component Pursuit

The network traffic matrix is widely used in network operation and management. It is therefore of crucial importance to analyze the components and the structure of the network traffic matrix, for which several mathematical approaches such as Principal Component Analysis (PCA) were proposed. In this paper, we first argue that PCA performs poorly for analyzing traffic matrix that is polluted by large volume anomalies, and then propose a new decomposition model for the network traffic matrix. According to this model, we carry out the structural analysis by decomposing the network traffic matrix into three sub-matrices, namely, the deterministic traffic, the anomaly traffic and the noise traffic matrix, which is similar to the Robust Principal Component Analysis (RPCA) problem previously studied in [13]. Based on the Relaxed Principal Component Pursuit (Relaxed PCP) method and the Accelerated Proximal Gradient (APG) algorithm, we present an iterative approach for decomposing a traffic matrix, and demonstrate its efficiency and flexibility by experimental results. Finally, we further discuss several features of the deterministic and noise traffic. Our study develops a novel method for the problem of structural analysis of the traffic matrix, which is robust against pollution of large volume anomalies.Comment: Accepted to Elsevier Computer Network

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

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. (c) 2004 Elsevier Inc. All rights reserved.breakdown point; dispersion matrix; influence function; principal components analysis; projection-pursuit; robustness; dispersion matrices; s-estimators; robust; covariance; location; scale;