Valentin Todorov

Review of sparse methods in regression and classification with application to chemometrics

High‐dimensional data often contain many variables that are irrelevant for predicting a response or for an accurate group assignment. The inclusion of such variables in a regression or classification model leads to a loss in performance, even if the contribution of the variables to the model is small. Sparse methods for regression and classification are able to suppress these variables. This is possible by adding an appropriate penalty term to the objective function of the method.

Review of robust multivariate statistical methods in high dimension.

General ideas of robust statistics, and specifically robust statistical methods for calibration and dimension reduction are discussed. The emphasis is on analyzing high-dimensional data. The discussed methods are applied using the packages chemometrics and rrcov of the statistical software environment R. It is demonstrated how the functions can be applied to real high-dimensional data from chemometrics, and how the results can be interpreted.

Robust tools for the imperfect world

Data outliers or other data inhomogeneities lead to a violation of the assumptions of traditional statistical estimators and methods. Robust statistics offers tools that can reliably work with contaminated data. Here, outlier detection methods in low and high dimension, as well as important robust estimators and methods for multivariate data are reviewed, and the most important references to the corresponding literature are provided. Algorithms are discussed, and routines in R are provided, allowing for a straightforward application of the robust methods to real data.

Detection of multivariate outliers in business survey data with incomplete information

Many different methods for statistical data editing can be found in the literature but only few of them are based on robust estimates (for example such as BACON-EEM, epidemic algorithms (EA) and transformed rank correlation (TRC) methods of Béguin and Hulliger). However, we can show that outlier detection is only reasonable if robust methods are applied, because the classical estimates are themselves influenced by the outliers. Nevertheless, data editing is essential to check the multivariate data for possible data problems and it is not deterministic like the traditional micro editing where all records are extensively edited manually using certain rules/constraints. The presence of missing values is more a rule than an exception in business surveys and poses additional severe challenges to the outlier detection. First we review the available multivariate outlier detection methods which can cope with incomplete data. In a simulation study, where a subset of the Austrian Structural Business Statistics is simulated, we compare several approaches. Robust methods based on the Minimum Covariance Determinant (MCD) estimator, S-estimators and OGK-estimator as well as BACON-BEM provide the best results in finding the outliers and in providing a low false discovery rate. Many of the discussed methods are implemented in the R package

Comparing Classical and Robust Sparse PCA

{\tt{rrcovNA}}

Logratio approach to statistical analysis of 2×2 compositional tables

which is available from the Comprehensive R Archive Network (CRAN) at http://www.CRAN.R-project.org under the GNU General Public License.

Discussion of “The power of monitoring: how to make the most of a contaminated multivariate sample” by Andrea Cerioli, Marco Riani, Anthony C. Atkinson and Aldo Corbellini

A commonly used procedure for reduction of the number of variables in linear discriminant analysis is the stepwise method for variable selection. Although often criticized, when used carefully, this method can be a useful prelude to a further analysis. The contribution of a variable to the discriminatory power of the model is usually measured by the maximum likelihood ratio criterion, referred to as Wilks’ lambda. It is well known that the Wilks’ lambda statistic is extremely sensitive to the influence of outliers. In this work a robust version of the Wilks’ lambda statistic will be constructed based on the Minimum Covariance Discriminant (MCD) estimator and its reweighed version which has a higher efficiency. Taking advantage of the availability of a fast algorithm for computing the MCD a simulation study will be done to evaluate the performance of this statistic.

General approach to coordinate representation of compositional tables: CoDa tables coordinates

The main drawback of principal component analysis (PCA) especially for applications in high dimensions is that the extracted components are linear combinations of all input variables. To facilitate the interpretability of PCA various sparse methods have been proposed recently. However all these methods might suffer from the influence of outliers present in the data. An algorithm to compute sparse and robust PCA was recently proposed by Croux et al. We compare this method to standard (non-sparse) classical and robust PCA and several other sparse methods. The considered methods are illustrated on a real data example and compared in a simulation experiment. It is shown that the robust sparse method preserves the sparsity and at the same time provides protection against contamination.