Robust Estimation with Missing Values for Elliptical Distributions

In this chapter, we tackle the problem of robust estimation of the mean and the covariance matrix when the data contains missing values. Classical estimation methods either assume a multivariate Gaussian distributionGaussian distribution, or suppose an unstructured covariance matrix. However, in many applications, the signal is not well described by a Gaussian model, and very often the data can be efficiently approximated by a low-rank model, inducing a low-rank structure on the covariance matrix which naturally accounts for the underlying signal subspace. By making the most of both (i) robustness to non-Gaussianity and (ii) low-rank structure, this chapter reviews various robust estimation procedures based mainly on the expectation–maximization algorithmExpectation-maximization (EM) algorithm which leverages the observed-data likelihoodLikelihood function, where the signal heterogeneity is accounted for through a deterministic scaleScale parameter. Furthermore, the proposed algorithms are designed to handle various patterns of missing values. At the end of the chapter, the performances of the proposed procedures are illustrated on simulated datasets with missing values. We share a link to a code repository for fully reproducible experiments.