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Space-filling designs with a Dirichlet distribution for mixture experiments

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  • Astrid Jourdan

    (ETIS UMR 8051, CY Paris University)

Abstract

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.

Suggested Citation

  • Astrid Jourdan, 2024. "Space-filling designs with a Dirichlet distribution for mixture experiments," Statistical Papers, Springer, vol. 65(5), pages 2667-2686, July.
    • Handle: RePEc:spr:stpapr:v:65:y:2024:i:5:d:10.1007_s00362-023-01493-2
    DOI: 10.1007/s00362-023-01493-2
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    References listed on IDEAS

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    1. Yan Liu & Min-Qian Liu, 2016. "Construction of uniform designs for mixture experiments with complex constraints," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(8), pages 2172-2180, April.
    2. Chuang, S.C. & Hung, Y.C., 2010. "Uniform design over general input domains with applications to target region estimation in computer experiments," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 219-232, January.
    3. Harry Joe, 1989. "Estimation of entropy and other functionals of a multivariate density," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 41(4), pages 683-697, December.
    4. A. Jourdan & J. Franco, 2010. "Optimal Latin hypercube designs for the Kullback–Leibler criterion," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 94(4), pages 341-351, December.
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