The optimisation of multivariate robust design criteria

Traditional univariate robust design criterion are based on means, variances, mean squared error, signal to noise ratios and the like. Multivariate extensions of these criteria are first discussed. The starting point is multivariate mean squared error and its extensions to weighted combinations of multivariate dispersion and distance from target. For both the dispersion and distance the Euclidian metric can be changed, in the usual way, to favour particular directions or orientations in d dimensions. Notions of multivariate dispersion orderings are also introduced, based on special definitions of stochastic ordering. Definitions of Pareto boundaries that include both multivariate mean and dispersion are introduced and the implications for multivariate optimisation are discussed. Finally, an example highlights the problems faced in choosing between competing design solutions, and how the methods described can be applied to aid in the selection of optimal designs