Mia Hubert

A fast method for robust principal components with applications to chemometrics

When faced with high-dimensional data, one often uses principal component analysis (PCA) for dimension reduction. Classical PCA constructs a set of uncorrelated variables, which correspond to eigenvectors of the sample covariance matrix. However, it is well-known that this covariance matrix is strongly affected by anomalous observations. It is therefore necessary to apply robust methods that are resistant to possible outliers. n nLi and Chen [J. Am. Stat. Assoc. 80 (1985) 759] proposed a solution based on projection pursuit (PP). The idea is to search for the direction in which the projected observations have the largest robust scale. In subsequent steps, each new direction is constrained to be orthogonal to all previous directions. This method is very well suited for high-dimensional data, even when the number of variables p is higher than the number of observations n. However, the algorithm of Li and Chen has a high computational cost. In the references [C. Croux, A. Ruiz-Gazen, in COMPSTAT: Proceedings in Computational Statistics 1996, Physica-Verlag, Heidelberg, 1996, pp. 211–217; C. Croux and A. Ruiz-Gazen, High Breakdown Estimators for Principal Components: the Projection-Pursuit Approach Revisited, 2000, submitted for publication.], a computationally much more attractive method is presented, but in high dimensions (large p) it has a numerical accuracy problem and still consumes much computation time. n nIn this paper, we construct a faster two-step algorithm that is more stable numerically. The new algorithm is illustrated on a data set with four dimensions and on two chemometrical data sets with 1200 and 600 dimensions.

A robust measure of skewness

The asymmetry of a univariate continuous distribution is commonly measured by the classical skewness coefficient. Because this estimator is based on the first three moments of the dataset, it is strongly affected by the presence of one or more outliers. This article investigates the medcouple, a robust alternative to the classical skewness coefficient. We show that it has a 25% breakdown value and a bounded influence function. We present a fast algorithm for its computation, and investigate its finite-sample behavior through simulated and real datasets.

The breakdown value of the L1 estimator in contingency tables

First we derive the maximal breakdown value of regression equivariant estimators in two-way contingency tables under the loglinear independence model. We then prove that the L1 estimator achieves this maximal breakdown value. Finally, we illustrate how these results can be generalized towards the uniform association model for contingency tables with ordered categories.

Similarities Between Location Depth and Regression Depth

ℝ In this paper we first explore the analogies between location depth and regression depth. The location depth of [Tukey (1975)] is a multivariate generalization of rank, and leads to a multivariate median known as the Tukey median or the deepest location. Regression depth was introduced in [Rousseeuw and Hubert (1999b)], and yields the deepest regression which is a new robust regression estimator. Based on the recent literature on depth, we compare several theoretical and computational aspects of depth and of the deepest fits in location and regression.

An improved algorithm for robust PCA

In Croux and Ruiz (1996) a robust principal component algorithm is presented. It is based on projection pursuit to ensure that it can be applied to high-dimensional data. We note that this algorithm has a problem of numerical stability and we develop an improved version. To reduce the computation time we then propose a two-step algorithm. The new algorithm is illustrated on a real data set from chemometrics