Space-filling designs with a Dirichlet distribution for mixture experiments

Uniform designs are widely used for experiments with mixtures. The uniformity of the design points is usually evaluated with a discrepancy criterion. In this paper, we propose a new criterion to measure the deviation between the design point distribution and a Dirichlet distribution. The support of the Dirichlet distribution, is defined by the set of d-dimensional vectors whose entries are real numbers in the interval [0,1] such that the sum of the coordinates is equal to 1. This support is suitable for mixture experiments. Depending on its parameters, the Dirichlet distribution allows symmetric or asymmetric, uniform or more concentrated point distribution. The difference between the empirical and the target distributions is evaluated with the Kullback-Leibler divergence. We use two methods to estimate the divergence: the plug-in estimate and the nearest-neighbor estimate. The resulting two criteria are used to build space-filling designs for mixture experiments. In the particular case of the flat Dirichlet distribution, both criteria lead to uniform designs. They are compared to existing uniformity criteria. The advantage of the new criteria is that they allow other distributions than uniformity and they are fast to compute.