  This paper tackles the problem of missing data imputation for noisy and
non-Gaussian data. A classical imputation method, the Expectation Maximization
(EM) algorithm for Gaussian mixture models, has shown interesting properties
when compared to other popular approaches such as those based on k-nearest
neighbors or on multiple imputations by chained equations. However, Gaussian
mixture models are known to be non-robust to heterogeneous data, which can lead
to poor estimation performance when the data is contaminated by outliers or
follows non-Gaussian distributions. To overcome this issue, a new EM algorithm
is investigated for mixtures of elliptical distributions with the property of
handling potential missing data. This paper shows that this problem reduces to
the estimation of a mixture of Angular Gaussian distributions under generic
assumptions (i.e., each sample is drawn from a mixture of elliptical
distributions, which is possibly different for one sample to another). In that
case, the complete-data likelihood associated with mixtures of elliptical
distributions is well adapted to the EM framework with missing data thanks to
its conditional distribution, which is shown to be a multivariate
$t$-distribution. Experimental results on synthetic data demonstrate that the
proposed algorithm is robust to outliers and can be used with non-Gaussian
data. Furthermore, experiments conducted on real-world datasets show that this
algorithm is very competitive when compared to other classical imputation
methods.
