A repulsion-based method for the definition and the enrichment of optimized space filling designs in constrained input spaces

Perrin, Guillaume; Cannamela, Claire

Journal de la société française de statistique, Special Issue : Computer Experiments, Uncertainty and Sensitivity Analysis, Tome 158 (2017) no. 1, pp. 37-67.

Profitant de l’essor considérable des puissances de calcul disponibles et de progrès importants en modélisation des phénomènes physiques, le rôle de la simulation n’est plus seulement descriptif, mais prédictif. Pour garantir cette capacité prédictive, il est alors nécessaire de développer des méthodes permettant d’associer à tout résultat numérique une précision, qui intègre les différentes sources d’incertitudes. Un des enjeux de ces méthodes de quantification des incertitudes concerne l’optimisation de l’exploration du domaine de variation des entrées de modélisation. Cette tâche peut s’avèrer difficile, en particulier lorsque le coût numérique associé à une simulation est élevé, ou lorsque le domaine d’entrée présente un certain nombre de contraintes, si bien qu’il ne peut plus être transformé en un hypercube via une bijection. Dans ce contexte, ce travail présente une méthode basée sur des répulsions permettant la définition de plans d’expériences optimisés dans des domaines contraints, dont les projections sur chaque paramètre d’entrée présentent de bonnes propriétés statistiques. Enfin, on montre que cette méthode permet également l’enrichissement de plans d’expériences déjà définis, tout en préservant un bon remplissage global du domaine de définition des entrées.

Due to increasing available computational resources and to a series of breakthroughs in the solving of nonlinear equations and in the modeling of complex mechanical systems, simulation nowadays becomes more and more predictive. Methods that could quantify the uncertainties associated with the results of the simulation are therefore needed to complete these predictions and widen the possibilities of simulation. One key step of these methods is the exploration of the whole space of the input variables, especially when the computational cost associated with one run of the simulation is high, and when there exists constraints on the inputs, such that the input space cannot be transformed into a hypercube through a bijection. In this context, the present work proposes an adaptive method to generate initial designs of experiments in any bounded convex input space, which are distributed as uniformly as possible on their definition space, while preserving good projection properties for each scalar input. Finally, it will be shown how this method can be used to add new elements to an initial design of experiments while preserving very interesting space filling properties.

[1] Auffray, Yves; Barbillon, Pierre; Marin, Jean Michel Maximin design on non hypercube domains and kernel interpolation, Statistics and Computing, Volume 22 (2012) no. 3, pp. 703-712 | DOI | MR | Zbl

[2] Bornert, M; Bretheau, T; Gilormini, P Homogenization in Mechanics of Materials, Wiley-Iste, 2008

[3] Bhadeshia, H K Neural Networks in Materials Science, ISIJ International, Volume 39 (1999) no. 10 | Zbl

[4] Bishop, C M Neural Networks for Pattern Recognition, Oxford: Oxford University Press, 1995 | MR | Zbl

[5] Damblin, G; Couplet, M; Ioss, B Numerical studies of space filling designs: optimization of Latin hypercube samples and subprojection properties., Journal of Simulation, Volume 7 (2013), pp. 276-289

[6] Devroye, L Non-uniform random variate generation, Springer, 1986 | MR | Zbl

[7] Dupuy, Delphine; Helbert, Céline; Franco, Jessica $\setminus$pkgDiceDesign and $\setminus$pkgDiceEval: Two R Packages for Design and Analysis of Computer Experiments, Journal of Statistical Software, Volume 65 (2015) no. 11, pp. 1-38 http://www.jstatsoft.org/v65/i11

[8] Draguljić, Danel; Santner, Thomas J.; Dean, Angela M. Noncollapsing Space-Filling Designs for Bounded Nonrectangular Regions, Technometrics, Volume 54 (2012) no. 2, pp. 169-178 | DOI | MR

[9] Franco, J; Corre, B; Dupuy, D Strauss processes: A new space-filling design for computer experiments, In Proceedings of Joint Meeting of the Statistical Society of Canada and the SFdS (2008) no. 11

[10] Fang, K T; Lin, D K Uniform experimental designs and their applications in industry, Handbook of Statistics, Volume 22 (2003), pp. 131-178 | MR

[11] Fang, K T; Li, R; Sudjianto, A Design and modeling for computer experiments, Chapman & Hall, Computer Science and Data Analysis Series, London, 2006 | MR | Zbl

[12] Flury, B A Principal Points, Biometrika, Volume 77 (1990) no. 1, pp. 34-41 | MR | Zbl

[13] Golchi, Shirin; Loeppky, Jason L. Space Filing Designs for Constrained Domains, Volume 7 (2015), pp. 1-25 | arXiv

[14] Hortado, J Structural reliability - Statistical learning perspectives, 17, Lectures notes in applied and computational mechanics, Springer, 2002 | MR | Zbl

[15] Jin, Ruichen; Chen, Wei; Sudjianto, Agus An efficient algorithm for constructing optimal design of computer experiments, Journal of Statistical Planning and Inference, Volume 134 (2005) no. 1, pp. 268-287 | DOI | MR | Zbl

[16] Joseph, V. Roshan; Gul, Evren; Ba, Shan Maximum projection designs for computer experiments, Biometrika, Volume 102 (2015) no. 2, pp. 371-380 | DOI | MR | Zbl

[17] Johnson, M E; Moore, L M; Ylvisaker, D Minimax and maximin distance designs., Journal of Statistical Planning and Inference, Volume 26 (1990), pp. 131-148 | MR

[18] Lekivetz, Ryan; Jones, Bradley Fast Flexible Space-Filling Designs for Nonrectangular Regions, Quality and Reliability Engineering International, Volume 31 (2015) no. 5, pp. 829-837 | DOI

[19] Likas, Aristidis; Vlassis, Nikos; J. Verbeek, Jakob The global k-means clustering algorithm, Pattern Recognition, Volume 36 (2003) no. 2, pp. 451-461 | DOI

[20] McKay, M; Beckman, R; Conover, W. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, Volume 21 (1979), pp. 239-245 | MR | Zbl

[21] Moon, H; Dean, A M; Santner, T J Algorithms for generating maximin Latin hypercube and orthogonal designs, Journal of Statistical Theory and Practice, Volume 5 (2011), pp. 81-88 | MR | Zbl

[22] Muehlenstaedt, Thomas; Gore, W L; Fruth, Jana; Roustant, Olivier; Nationale, E. Computer experiments with functional inputs and scalar outputs by a norm-based approach (2014), pp. 1-22 | arXiv | MR

[23] Mak, Simon; Joseph, V. Roshan Minimax designs using clustering (2016), pp. 1-24 | arXiv

[24] Morris, M; Mitchell, T Exploratory designs for computationnal experiments, Journal of Statistical Planning and Inference, Volume 43 (1995), pp. 381-402 | Zbl

[25] Osuolale, K.A.; Yahya, W.B.; Adeleke, B.L. Optimal orthogonal-array-based latin hypercubes, Journal of Applied Statistics, Volume 30 (2003) no. 5, pp. 585-598 | MR

[26] Park, J S Optimal Latin-hypercube designs for computer experiments, Journal of Statistical Planning and Inference, Volume 39 (1994), pp. 95-111 | MR | Zbl

[27] Pillards, T; Cools, R Transforming low-discrepancy sequences from a cube to a simplex, Katholieke Universiteit Leuven, 2003 | MR | Zbl

[28] Pratola, Matthew T.; Harari, Ofir; Bingham, Derek; Flowers, Gwenn E. Design and Analysis of Experiments on Non-Convex Regions, Technometrics, Volume 1706 (2015) no. November, pp. 1-37 | DOI | MR

[29] Schaback, R Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math., Volume 3 (1995) no. 3, pp. 251-264 | MR | Zbl

[30] Stinstra, E; den Hertog, D; Stehouwer, P; Vestjens, A Constrained maximin designs for computer experiments, Technometrics, Volume 45 (2003) no. 4, pp. 340-346 | MR

[31] Stinstra, E; den Hertog, D; Stehouwer, P; Vestjens, A Uniform designs over general input domains with applications to target region estimation in computer experiments, Computational Statisticsand Data Analysis, Volume 51 (2010) no. 1, pp. 219-232

[32] Scholkopf, B; Smola, A J Learning With Kernels: Support Vector Machines, Regularization, Optimization and Beyond, MIT Press, Boston, 2002

[33] Sacks, J; Welch, W; Mitchell, T; Wynn, H Design and analysis of computer experiments, Statistical Science, Volume 4 (1989), pp. 409-435 | MR | Zbl

[34] Santner, T J; Williams, B J; Notz, W I The design and analysis of computer experiments, Springer, New York, 2003 | MR | Zbl

[35] Tang, B Orthogonal array-based Latin hypercubes, Journal of the American Statistical Association, Volume 88 (1993), pp. 1392-1397 | MR | Zbl

[36] Vapnik, V Statistical Learning Theory, Wiley-Interscience, New York, 1998 | MR | Zbl