Robust covariance matrix estimation and multivariate outlier detection

A severe limitation for the application of robust position and scale estimators having a high breakdown point is a consequence of their high computational cost. In this paper we present and analyze several inexpensive robust estimators for the co variance matrix, based on information obtained from projections onto certain sets of directions. The properties of these estimators (breakdown point, computational cost, bias) are analyzed and compared with those of the Stahel-Donoho estimator, through simulation studies. These studies show a clear improvement both on the computational requirements and the bias properties of the Stahel-Donoho estimator. The same ideas are also applied to the construction of procedures to detect outliers in multivariate samples. Their performance is analyzed by applying them to a set of test cases

Contributions to Robust Methods: Modified Rank Covariance Matrix and Spatial-EM Algorithm

Classical multivariate statistical inference methods including multivariate analysis of variance, principal component analysis, factor analysis, canonical correlation analysis are based on sample covariance matrix. Those moment-based techniques are optimal (most efficient) under the normality distributional assumption. They are, however, extremely sensitive to outlying observations, susceptible to small perturbation in data and poor in the efficiency for heavy-tailed distributions. A straightforward treatment is to replace the sample covariance matrix with a robust one. Visuri et al. (2000) proposed a technique for robust covariance matrix estimation based on different notions of multivariate sign and rank. Among them, the spatial rank based covariance matrix estimator that utilizes a robust scale estimator (MRCM) is especially appealing due to its high robustness, computational ease and good efficiency. In this dissertation, properties of the estimator on orthogonal equivariance under any distribution and affine equivariance under elliptically symmetric distributions have been established. The major robustness properties of the estimator are studied by the breakdown point and influence function analysis. More specifically, the finite sample breakdown point is obtained and the upper bound of the finite sample breakdown point can be achieved by a proper choice of univariate robust scale estimator. The influence functions for eigenvalues and eigenvectors of the estimator are derived. They are found to be bounded under some mild assumptions. Moreover, empirical comparisons to popular robust MCD, M and S estimators show that MRCM has a competitive performance on efficiency as well as robustness. With rapid advances in information technology, data have been becoming huge in size and complex in structure. A single elliptical distribution is no longer sufficient to model such data. This motivates a generalization of our notion of MRCM to mixture models. In this dissertation, we propose a robust Spatial-EM algorithm for estimating parameters in the mixture model. Rather than using sample covariance matrix in each M-step, Spatial-EM ingeniously implements MRCM to enhance stability and robustness of the estimation procedure. Analyzing the log-likelihood function, the proposed one is found to be closely related to the maximum likelihood estimator (MLE) of Kotz type mixture model. Comparing with the direct MLE, Spatial-EM has advantages in computation ease as well as stability. Applications of Spatial-EM to data mining become natural. We illustrate procedures how to use Spatial-EM for supervised and unsupervised learning problems. More specifically, robust clustering and outlier detection methods based on Spatial-EM have been proposed. We adopt the outlier detection to taxonomic research on fish species novelty discovery. UCI Wisconsin diagnostic breast cancer data and Yeast cell cycle data are used for clustering analysis. Comparing with the regular EM and many other existing methods such as X-EM and SVM, Spatial-EM demonstrates its competitive classification power and high robustness. Classical multivariate statistical inference methods including multivariate analysis of variance, principal component analysis, factor analysis, canonical correlation analysis are based on sample covariance matrix. Those moment-based techniques are optimal (most efficient) under the normality distributional assumption. They are, however, extremely sensitive to outlying observations, susceptible to small perturbation in data and poor in the efficiency for heavy-tailed distributions. A straightforward treatment is to replace the sample covariance matrix with a robust one. Visuri et al. (2000) proposed a technique for robust covariance matrix estimation based on different notions of multivariate sign and rank. Among them, the spatial rank based covariance matrix estimator that utilizes a robust scale estimator (MRCM) is especially appealing due to its high robustness, computational ease and good efficiency. In this dissertation, properties of the estimator on orthogonal equivariance under any distribution and affine equivariance under elliptically symmetric distributions have been established. The major robustness properties of the estimator are studied by the breakdown point and influence function analysis. More specifically, the finite sample breakdown point is obtained and the upper bound of the finite sample breakdown point can be achieved by a proper choice of univariate robust scale estimator. The influence functions for eigenvalues and eigenvectors of the estimator are derived. They are found to be bounded under some mild assumptions. Moreover, empirical comparisons to popular robust MCD, M and S estimators show that MRCM has a competitive performance on efficiency as well as robustness. With rapid advances in information technology, data have been becoming huge in size and complex in structure. A single elliptical distribution is no longer sufficient to model such data. This motivates a generalization of our notion of MRCM to mixture models. In this dissertation, we propose a robust Spatial-EM algorithm for estimating parameters in the mixture model. Rather than using sample covariance matrix in each M-step, Spatial-EM ingeniously implements MRCM to enhance stability and robustness of the estimation procedure. Analyzing the log-likelihood function, the proposed one is found to be closely related to the maximum likelihood estimator (MLE) of Kotz type mixture model. Comparing with the direct MLE, Spatial-EM has advantages in computation ease as well as stability. Applications of Spatial-EM to data mining become natural. We illustrate procedures how to use Spatial-EM for supervised and unsupervised learning problems. More specifically, robust clustering and outlier detection methods based on Spatial-EM have been proposed. We adopt the outlier detection to taxonomic research on fish species novelty discovery. UCI Wisconsin diagnostic breast cancer data and Yeast cell cycle data are used for clustering analysis. Comparing with the regular EM and many other existing methods such as X-EM and SVM, Spatial-EM demonstrates its competitive classification power and high robustness

MAINT.Data: modelling and analysing interval data in R

We present the CRAN R package MAINT.Data for the modelling and analysis of multivariate interval data, i.e., where units are described by variables whose values are intervals of IR, representing intrinsic variability. Parametric inference methodologies based on probabilistic models for interval variables have been developed, where each interval is represented by its midpoint and log-range, for which multivariate Normal and Skew-Normal distributions are assumed. The intrinsic nature of the interval variables leads to special structures of the variance-covariance matrix, which are represented by four different possible configurations. MAINT.Data implements the proposed methodologies in the S4 object system, introducing a specific data class for representing interval data. It includes functions and methods for modelling and analysing interval data, in particular maximum likelihood estimation, statistical tests for the different configurations, (M)ANOVA and Discriminant Analysis. For the Gaussian model, Model-based Clustering, robust estimation, outlier detection and Robust Discriminant Analysis are also availableinfo:eu-repo/semantics/publishedVersio