A Fast Algorithm for the Minimum Covariance Determinant Estimator

The minimum covariance determinant (MCD) method of Rousseeuw (1984) is a highly robust estimator of multivariate location and scatter. Its objective is to nd h observations (out of n) whose covariance matrix has the lowest determinant. Until now applications of the MCD were hampered by the computation time of existing algorithms, which were limited to a few hundred objects in a few dimensions. We discuss two important applications of larger size: one about a production process at Philips with n = 677 objects and p = 9 variables, and a data set from astronomy with n =137,256 objects and p = 27 variables. To deal with such problems we have developed a new algorithm for the MCD, called FAST-MCD. The basic ideas are an inequality involving order statistics and determinants, and techniques which we calìselective iteration' and`nested extensions'. For small data sets FAST-MCD typically nds the exact MCD, whereas for larger data sets it gives more accurate results than existing algorithms and is faster by orders of magnitude. Moreover, FAST-MCD is able to detect an exact t, i.e. a hyperplane containing h or more observations. The new algorithm makes the MCD method available as a routine tool for analyzing multivariate data. We also propose the distance-distance plot (or`D-D plot') which displays MCD-based robust distances versus Mahalanobis distances, and illustrate it with some examples. We wish to thank Doug Hawkins and Jos e Agulll o for making their programs available to us. We also want to dedicate special thanks to Gertjan Otten, Frans Van Dommelen en Herman Veraa for giving us access to the Philips data, and to S.C. Odewahn and his research group at the California Institute of Technology for allowing us to analyze their Digitized Palomar data.