Anna Bartkowiak, Krystyna Zietak, "Correcting possible
non-positiveness of a covariance matrix estimated elementwise
in a robust way"

When estimating the elements of a covariance or cross-product matrix
elementwise we may obtain a negative definite matrix S which 
will not be suitable for some statistical analyses. To amend this deficiency
we propose an approach based on recent results of Cheng and Higham (1998).
Their result permits to construct a correction yielding positive definite
and reasonably well-conditioned matrix.

The procedure is illustrated by a benchmark data set containing outliers:
namely the bushfire data, known to contain masked outliers. We estimate
the covariance matrix for the data elementwise by a simple robust procedure
described by Gnanadesikan and Ketttenring (1972), which yields in that case
negative definite covariance matrix. The Cheng-Higham correction applied 
to the obtained matrix  permits to obtain positive definite covariance
and correlation matrices, suitable for evaluating Mahalanobis distances,
which in turn permits to reveal also the masked outliers.

Keywords: knowledge discovery, identification of outliers,
masking effect, Mahalanobis distance