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See detailRelaxMCD: smooth optimisation for the Minimum Covariance Determinant estimator
Schyns, Michael ULg; Haesbroeck, Gentiane ULg; Critchley, Frank

in Computational Statistics & Data Analysis (2010), 54(4), 843-857

The Minimum Covariance Determinant (MCD) estimator is a highly robust procedure for estimating the center and shape of a high dimensional data set. It consists of determining a subsample of h points out ... [more ▼]

The Minimum Covariance Determinant (MCD) estimator is a highly robust procedure for estimating the center and shape of a high dimensional data set. It consists of determining a subsample of h points out of n which minimizes the generalized variance. By definition, the computation of this estimator gives rise to a combinatorial optimization problem, for which several approximative algorithms have been developed. Some of these approximations are quite powerful, but they do not take advantage of any smoothness in the objective function. In this paper, focus is on the approach outlined in a general framework in Critchley et al. (2009) and which transforms any discrete and high dimensional combinatorial problem of this type into a continuous and low-dimensional one. The idea is to build on the general algorithm proposed by Critchley et al. (2009) in order to take into account the particular features of the MCD methodology. More specifically, both the adaptation of the algorithm to the specific MCD target function as well as the comparison of this “specialized” algorithm with the usual competitors for computing MCD are the main goals of this paper. The adaptation focuses on the design of “clever” starting points in order to systematically investigate the search domain. Accordingly, a new and surprisingly efficient procedure based on the well known k-means algorithm is constructed. The adapted algorithm, called RelaxMCD, is then compared by means of simulations and examples with FASTMCD and the Feasible Subset Algorithm, both benchmark algorithms for computing MCD. As a by-product, it is shown that RelaxMCD is a general technique encompassing the two others, yielding insight about their overall good performance. [less ▲]

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See detailA relaxed approach to combinatorial problems in robustness and diagnostics
Critchley, Frank; Schyns, Michael ULg; Haesbroeck, Gentiane ULg et al

in Statistics and Computing (2010), 20(1), 99-115

A range of procedures in both robustness and diagnostics require optimisation of a target functional over all subsamples of given size. Whereas such combinatorial problems are extremely difficult to solve ... [more ▼]

A range of procedures in both robustness and diagnostics require optimisation of a target functional over all subsamples of given size. Whereas such combinatorial problems are extremely difficult to solve exactly, something less than the global optimum can be ‘good enough’ for many practical purposes, as shown by example. Again, a relaxation strategy embeds these discrete, high-dimensional problems in continuous, low-dimensional ones. Overall, nonlinear optimisation methods can be exploited to provide a single, reasonably fast algorithm to handle a wide variety of problems of this kind, thereby providing a certain unity. Four running examples illustrate the approach. On the robustness side, algorithmic approximations to minimum covariance determinant (MCD) and least trimmed squares (LTS) estimation. And, on the diagnostic side, detection of multiple multivariate outliers and global diagnostic use of the likelihood displacement function. This last is developed here as a global complement to Cook’s (in J. R. Stat. Soc. 48:133–169, 1986) local analysis. Appropriate convergence of each branch of the algorithm is guaranteed for any target functional whose relaxed form is—in a natural generalisation of concavity, introduced here—‘gravitational’. Again, its descent strategy can downweight to zero contaminating cases in the starting position. A simulation study shows that, although not optimised for the LTS problem, our general algorithm holds its own with algorithms that are so optimised. An adapted algorithm relaxes the gravitational condition itself. [less ▲]

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See detailThe case sensitivity function approach to diagnostic and robust computation: a relaxation strategy
Critchley, Frank; Schyns, Michael ULg; Haesbroeck, Gentiane ULg et al

in Antoch, Jaromir (Ed.) COMPSTAT 2004: Proceedings in Computational Statistics (2004)

Detailed reference viewed: 7 (1 ULg)