Robust high-dimensional precision matrix estimation

The dependency structure of multivariate data can be analyzed using the covariance matrix $\Sigma$. In many fields the precision matrix $\Sigma^{-1}$ is even more informative. As the sample covariance estimator is singular in high-dimensions, it cannot be used to obtain a precision matrix estimator. A popular high-dimensional estimator is the graphical lasso, but it lacks robustness. We consider the high-dimensional independent contamination model. Here, even a small percentage of contaminated cells in the data matrix may lead to a high percentage of contaminated rows. Downweighting entire observations, which is done by traditional robust procedures, would then results in a loss of information. In this paper, we formally prove that replacing the sample covariance matrix in the graphical lasso with an elementwise robust covariance matrix leads to an elementwise robust, sparse precision matrix estimator computable in high-dimensions. Examples of such elementwise robust covariance estimators are given. The final precision matrix estimator is positive definite, has a high breakdown point under elementwise contamination and can be computed fast

Robust Henderson III estimators of variance components in the nested error model

Common methods for estimating variance components in Linear Mixed Models include Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML). These methods are based on the strong assumption of multivariate normal distribution and it is well know that they are very sensitive to outlying observations with respect to any of the random components. Several robust altematives of these methods have been proposed (e.g. Fellner 1986, Richardson and Welsh 1995). In this work we present several robust alternatives based on the Henderson method III which do not rely on the normality assumption and provide explicit solutions for the variance components estimators. These estimators can later be used to derive robust estimators of regression coefficients. Finally, we describe an application of this procedure to small area estimation, in which the main target is the estimation of the means of areas or domains when the within-area sample sizes are small

Robust principal components for hyperspectral data analysis

Remote sensing data present peculiar features and characteristics that may make their statistical processing and analysis a difficult task. Among them, it can be mentioned the volume of data involved, the redundancy, the presence of unexpected values that arise mainly due to noisy pixels and background objects whose responses to the sensor are very different from those of their neighbours. Sometimes, the volume of data and number of variables involved is so large that any statistical analysis becomes unmanageable if data are not condensed in some way. A commonly used method to deal with this situation is Principal Component Analysis (PCA) based on classical statistics: sample mean and covariance matrices. The drawback in using sample covariance or correlation matrices as measures of variability is their high sensitivity to spurious values. In this work we analyse and evaluate the use of some Robust Principal Component techniques and make a comparison of Robust and Classical PCs performances when applied to satellite data provided by the hyperspectral sensor AVIRIS (Airborne Visible/Infrared Imaging Spectrometer). We conclude that some robust approaches are the most reliable and precise when applied as a data reduction technique before performing supervised image classification. © 2009 Springer Berlin Heidelberg.Fil: Lucini, María Magdalena. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Nordeste; Argentina. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura; ArgentinaFil: Frery, Alejandro César. Universidade Federal de Alagoas; Brasi

The 50% breakdown point in simultaneous M-estimation of location and scale

Breakdown point, Simultaneous M-estimation, Robustness,