A Deterministic Algorithm for the Mcd

The minimum covariance determinant (MCD) method is a robust estimator of multivariate location and scatter (Rousseeuw, 1984). The MCD is highly resistant to outliers, and it is often applied by itself and as a building block for other robust multivariate methods. Computing the exact MCD is very hard, so in practice one resorts to approximate algorithms. Most often the FASTMCD algorithm of Rousseeuw and Van Driessen (1999) is used. This algorithm starts by drawing many random subsets, followed by so-called concentration steps. The FASTMCD algorithm is affine equivariant but not permutation invariant. In this article we present a deterministic algorithm, denoted as DetMCD, which does not use random subsets and is even faster. It is permutation invariant and very close to affine equivariant. We illustrate DetMCD on real and simulated data sets, with applications involving principal component analysis, multivariate regression, and classification. Supplemental material (Matlab code of the DetMCD algorithm and the data sets) are available online.