Stéphanie Aerts

Multivariate coefficients of variation

In the univariate setting, coefficients of variation are well-known and used to compare the variability of populations characterized by variables expressed in different units or having really different means. When dealing with more than one variable, the use of such a relative dispersion measure is much less common even though several generalizations of the coefficient of variation to the multivariate setting have been introduced in the literature. In this paper, the lack of robustness of the sample versions of the multivariate coefficients of variation (MCV) is illustrated by means of influence functions and some robust counterparts based either on the Minimum Covariance Determinant (MCD) estimator or on the S estimator are advocated. Then, focusing on two of the considered MCVs, a diagnostic tool is derived and its efficiency in detecting observations having an unduly large effect on variability is illustrated on a real-life data set. The influence functions are also used to compute asymptotic variances under elliptical distributions, yielding approximate confidence intervals. Finally, simulations are conducted in order to compare, in a finite sample setting, the performance of the classical and robust MCVs in terms of variability and in terms of coverage probability of the corresponding asymptotic confidence intervals.

Cellwise robust regularized discriminant analysis

Quadratic and Linear Discriminant Analysis (QDA/LDA) are the most often applied classification rules under normality. In QDA, a separate covariance matrix is estimated for each group. If there are more variables than observations in the groups, the usual estimates are singular and cannot be used anymore. Assuming homoscedasticity, as in LDA, reduces the number of parameters to estimate. This rather strong assumption is however rarely verified in practice. Regularized discriminant techniques that are computable in high-dimension and cover the path between the two extremes QDA and LDA have been proposed in the literature. However, these procedures rely on sample covariance matrices. As such, they become inappropriate in presence of cellwise outliers, a type of outliers that is very likely to occur in high-dimensional datasets. In this paper, we propose cellwise robust counterparts of these regularized discriminant techniques by inserting cellwise robust covariance matrices. Our methodology results in a family of discriminant methods that (i) are robust against outlying cells, (ii) cover the gap between LDA and QDA and (iii) are computable in high-dimension. The good performance of the new methods is illustrated through simulated and real data examples. As a by-product, visual tools are provided for the detection of outliers.