The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies.

We consider the affine equivariant sign covariance matrix (SCM) introduced by Visuri et al. (J. Statist. Plann. Inference 91 (2000) 557). The population SCM is shown to be proportional to the inverse of the regular covariance matrix. The eigenvectors and standardized eigenvalues of the covariance, matrix can thus be derived from the SCM. We also construct an estimate of the covariance and correlation matrix based on the SCM. The influence functions and limiting distributions of the SCM and its eigenvectors and eigenvalues are found. Limiting efficiencies are given in multivariate normal and t-distribution cases. The estimates are highly efficient in the multivariate normal case and perform better than estimates based on the sample covariance matrix for heavy-tailed distributions. Simulations confirmed these findings for finite-sample efficiencies. (C) 2003 Elsevier Science (USA). All rights reserved.affine equivariance; covariance and correlation matrices; efficiency; eigenvectors and eigenvalues; influence function; multivariate median; multivariate sign; robustness; multivariate location; discriminant-analysis; principal components; dispersion matrices; tests; estimators;

Robust canonical correlations: a comparative study.

Several approaches for robust canonical correlation analysis will be presented and discussed. A first method is based on the definition of canonical correlation analysis as looking for linear combinations of two sets of variables having maximal (robust) correlation. A second method is based on alternating robust regressions. These methods are discussed in detail and compared with the more traditional approach to robust canonical correlation via covariance matrix estimates. A simulation study compares the performance of the different estimators under several kinds of sampling schemes. Robustness is studied as well by breakdown plots.Alternating regression; Canonical correlations; Correlation measures; Projection-pursuit; Robust covariance estimation; Robust regression; Robustness;

Robust canonical correlations: A comparative study.

Several approaches for robust canonical correlation analysis will be presented and discussed. A first method is based on the definition of canonical correlation analysis as looking for linear combinations of two sets of variables having maximal (robust) correlation. A second method is based on alternating robust regressions. These methods axe discussed in detail and compared with the more traditional approach to robust canonical correlation via covariance matrix estimates. A simulation study compares the performance of the different estimators under several kinds of sampling schemes. Robustness is studied as well by breakdown plots.

Combining information in statistical modelling

How to combine information from different sources is becoming an important statistical area of research under the name of Meta Analysis. This paper shows that the estimation of a parameter or the forecast of a random variable can also be seen as a process of combining information. It is shown that this approach can provide sorne useful insights on the robustness properties of sorne statistical procedures, and it also allows the comparison of statistical models within a common framework. Sorne general combining rules are illustrated using examples from ANOVA analysis, diagnostics in regression, time series forecasting, missing value estimation and recursive estimation using the Kalman Filter

Spatial Sign Correlation

A new robust correlation estimator based on the spatial sign covariance matrix (SSCM) is proposed. We derive its asymptotic distribution and influence function at elliptical distributions. Finite sample and robustness properties are studied and compared to other robust correlation estimators by means of numerical simulations.Comment: 20 pages, 7 figures, 2 table

Distributed Robust Learning

We propose a framework for distributed robust statistical learning on {\em big contaminated data}. The Distributed Robust Learning (DRL) framework can reduce the computational time of traditional robust learning methods by several orders of magnitude. We analyze the robustness property of DRL, showing that DRL not only preserves the robustness of the base robust learning method, but also tolerates contaminations on a constant fraction of results from computing nodes (node failures). More precisely, even in presence of the most adversarial outlier distribution over computing nodes, DRL still achieves a breakdown point of at least $\lambda^*/2$, where $\lambda^*$ is the break down point of corresponding centralized algorithm. This is in stark contrast with naive division-and-averaging implementation, which may reduce the breakdown point by a factor of $k$ when $k$ computing nodes are used. We then specialize the DRL framework for two concrete cases: distributed robust principal component analysis and distributed robust regression. We demonstrate the efficiency and the robustness advantages of DRL through comprehensive simulations and predicting image tags on a large-scale image set.Comment: 18 pages, 2 figure