Abstract
When setting up a computer experiment, it has become a standard practice to select the inputs spread out uniformly across the available space. These so-called space-filling designs are now ubiquitous in corresponding publications and conferences. The statistical folklore is that such designs have superior properties when it comes to prediction and estimation of emulator functions. In this paper we want to review the circumstances under which this superiority holds, provide some new arguments and clarify the motives to go beyond space-filling. An overview over the state of the art of space-filling is introducing and complementing these results.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Abt, M.: Estimating the prediction mean squared error in Gaussian stochastic processes with exponential correlation structure. Scand. J. Stat. 26(4), 563–578 (1999)
Angelis, L., Senta, E.B., Moyssiadis, C.: Optimal exact experimental designs with correlated errors through a simulated annealing algorithm. Comput. Stat. Data Anal. 37(3), 275–296 (2001)
Ash, R.: Information Theory. Wiley, New York (1965) (republished by Dover, New York, 1990)
Audze, P., Eglais, V.: New approach for planning out experiments. Probl. Dyn. Strengths 35, 104–107 (1977)
Ball, K.: Eigenvalues of Euclidean distance matrices. J. Approx. Theory 68, 74–82 (1992)
Bates, R.A., Buck, R.J., Riccomagno, E., Wynn, H.P.: Experimental design and observation for large systems. J. R. Stat. Soc. B 58(1), 77–94 (1996)
Beirlant, J., Dudewicz, E., Györfi, L., van der Meulen, E.: Nonparametric entropy estimation; an overview. Int. J. Math. Stat. Sci. 6(1), 17–39 (1997)
Bellhouse, D.R., Herzberg, A.M.: Equally spaced design points in polynomial regression: a comparison of systematic sampling methods with the optimal design of experiments. Can. J. Stat. 12(2), 77–90 (1984)
Bettinger, R., Duchêne, P., Pronzato, L., Thierry, E.: Design of experiments for response diversity. In: Proc. 6th International Conference on Inverse Problems in Engineering (ICIPE). J. Phys. Conf. Ser., Dourdan (Paris) (2008)
Bettinger, R., Duchêne, P., Pronzato, L.: A sequential design method for the inversion of an unknown system. In: Proc. 15th IFAC Symposium on System Identification, Saint-Malo, France, pp. 1298–1303 (2009)
Bischoff, W., Miller, F.: Optimal designs which are efficient for lack of fit tests. Ann. Stat. 34(4), 2015–2025 (2006)
Boissonnat, J.D., Yvinec, M.: Algorithmic Geometry. Cambridge University Press, Cambridge (1998)
Bursztyn, D., Steinberg, D.: Comparison of designs for computer experiments. J. Stat. Plan. Inference 136(3), 1103–1119 (2006)
Carnell, R.: lhs: latin hypercube samples. R package version 0.5 (2009)
Chen, V.C.P., Tsui, K.L., Barton, R.R., Meckesheimer, M.: A review on design, modeling and applications of computer experiments. AIIE Trans. 38(4), 273–291 (2006)
Cignoni, P., Montani, C., Scopigno, R.: DeWall: a fast divide and conquer Delaunay triangulation algorithm in E d. Comput. Aided Des. 30(5), 333–341 (1998)
Cortés, J., Bullo, F.: Nonsmooth coordination and geometric optimization via distributed dynamical systems. SIAM Rev. 51(1), 163–189 (2009)
Cressie, N.: Statistics for Spatial Data. Wiley Series in Probability and Statistics. Wiley-Interscience, New York (1993). Rev. Sub. Ed.
den Hertog, D., Kleijnen, J.P.C., Siem, A.Y.D.: The correct kriging variance estimated by bootstrapping. J. Oper. Res. Soc. 57(4), 400–409 (2006)
Dette, H., Pepelyshev, A.: Generalized Latin hypercube design for computer experiments. Technometrics 52(4), 421–429 (2010)
Dette, H., Kunert, J., Pepelyshev, A.: Exact optimal designs for weighted least squares analysis with correlated errors. Stat. Sin. 18(1), 135–154 (2008)
Fang, K.T.: The uniform design: application of number theoretic methods in experimental design. Acta Math. Appl. Sin. 3, 363–372 (1980)
Fang, K.T., Li, R.: Uniform design for computer experiments and its optimal properties. Int. J. Mater. Prod. Technol. 25(1), 198–210 (2006)
Fang, K.T., Wang, Y.: Number-Theoretic Methods in Statistics, 1st edn. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. Chapman & Hall/CRC, London/Boca Raton (1993)
Fang, K.T., Lin, D.K.J., Winker, P., Zhang, Y.: Uniform design: Theory and application. Technometrics 42(3), 237–248 (2000)
Fang, K.T., Li, R., Sudjianto, A.: Design and Modeling for Computer Experiments. Chapman & Hall/CRC, London/Boca Raton (2005)
Fedorov, V.: Theory of Optimal Experiments. Academic Press, New York (1972)
Fedorov, V.V., Hackl, P.: Model-Oriented Design of Experiments. Lecture Notes in Statistics, vol. 125. Springer, Berlin (1997)
Franco, J.: Planification d’expériences numériques en phase exploratoire pour la simulation de phénomènes complexes. Ph.D. thesis, École Nationale Supérieure des Mines de Saint Etienne (2008)
Franco, J., Bay, X., Corre, B., Dupuy, D.: Planification d’expériences numériques à partir du processus ponctuel de Strauss. Preprint, Département 3MI, École Nationale Supérieure des Mines de Saint-Etienne. http://hal.archives-ouvertes.fr/hal-00260701/fr/ (2008)
Franco, J., Vasseur, O., Corre, B., Sergent, M.: Minimum Spanning Tree: a new approach to assess the quality of the design of computer experiments. Chemom. Intell. Lab. Syst. 97, 164–169 (2009)
Gensane, T.: Dense packings of equal spheres in a cube. Electron. J. Comb. 11 (2004)
Glover, F., Kelly, J., Laguna, M.: Genetic algorithms and tabu search: hybrids for optimization. Comput. Oper. Res. 22(1), 111–134 (1995)
Gramacy, R., Lee, H.: Cases for the nugget in modeling computer experiments. Tech. rep., http://arxiv.org/abs/1007.4580 (2010)
Gramacy, R.B., Lee, H.K.: Adaptive design and analysis of supercomputer experiments. Technometrics 51(2), 130–144 (2009)
Griffith, D.: Spatial Autocorrelation and Spatial Filtering: Gaining Understanding through Theory and Scientific Visualization. Springer, Berlin (2003)
Hall, P., Morton, S.: On the estimation of entropy. Ann. Inst. Stat. Math. 45(1), 69–88 (1993)
Harville, D.A., Jeske, D.R.: Mean squared error of estimation or prediction under a general linear model. J. Am. Stat. Assoc. 87(419), 724–731 (1992)
Havrda, M., Charvát, F.: Quantification method of classification processes: concept of structural α-entropy. Kybernetika 3, 30–35 (1967)
Herzberg, A.M., Huda, S.: A comparison of equally spaced designs with different correlation structures in one and more dimensions. Can. J. Stat. 9(2), 203–208 (1981)
Hoeting, J.A., Davis, R.A., Merton, A.A., Thompson, S.E.: Model selection for geostatistical models. Ecol. Appl. 16(1), 87–98 (2006)
Husslage, B., Rennen, G., van Dam, E., den Hertog, D.: Space-filling Latin hypercube designs for computer experiments. Discussion paper 2006-18, Tilburg University, Center for Economic Research (2006)
Iooss, B., Boussouf, L., Feuillard, V., Marrel, A.: Numerical studies of the metamodel fitting and validation processes. International Journal on Advances in Systems and Measurements 3(1–2), 11–21 (2010)
Irvine, K., Gitelman, A., Hoeting, J.: Spatial designs and properties of spatial correlation: effects on covariance estimation. J. Agric. Biol. Environ. Stat. 12(4), 450–469 (2007)
Jin, R., Chen, W., Sudjianto, A.: An efficient algorithm for constructing optimal design of computer experiments. J. Stat. Plan. Inference 134(1), 268–287 (2005)
Johnson, M., Moore, L., Ylvisaker, D.: Minimax and maximin distance designs. J. Stat. Plan. Inference 26, 131–148 (1990)
Johnson, R.T., Montgomery, D.C., Jones, B., Fowler, J.W.: Comparing designs for computer simulation experiments. In: WSC ’08: Proceedings of the 40th Conference on Winter Simulation, pp. 463–470 (2008)
Joseph, V.: Limit kriging. Technometrics 48(4), 458–466 (2006)
Jourdan, A., Franco, J.: Optimal Latin hypercube designs for the Kullback-Leibler criterion. AStA Adv. Stat. Anal. 94, 341–351 (2010)
Kiefer, J., Wolfowitz, J.: The equivalence of two extremum problems. Can. J. Math. 12, 363–366 (1960)
Kiseľák, J., Stehlík, M.: Equidistant and d-optimal designs for parameters of Ornstein–Uhlenbeck process. Stat. Probab. Lett. 78(12), 1388–1396 (2008)
Kleijnen, J.P.C.: Design and Analysis of Simulation Experiments. Springer, New York (2009)
Koehler, J., Owen, A.: Computer experiments. In: Ghosh, S., Rao, C.R. (eds.) Handbook of Statistics: Design and Analysis of Experiments, vol. 13, pp. 261–308. North-Holland, Amsterdam (1996)
Kozachenko, L., Leonenko, N.: On statistical estimation of entropy of random vector. Probl. Inf. Transm. 23(2), 95–101 (1987) (translated from Problemy Peredachi Informatsii, in Russian, vol. 23, No. 2, pp. 9–16, 1987)
Leary, S., Bhaskar, A., Keane, A.: Optimal orthogonal-array-based Latin hypercubes. J. Appl. Stat. 30(5), 585–598 (2003)
Leonenko, N., Pronzato, L., Savani, V.: A class of Rényi information estimators for multidimensional densities. Ann. Stat. 36(5), 2153–2182 (2008) (correction in Ann. Stat. 38(6), 3837–3838, 2010)
Li, X.S., Fang, S.C.: On the entropic regularization method for solving min-max problems with applications. Math. Methods Oper. Res. 46, 119–130 (1997)
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 21(2), 239–245 (1979)
Melissen, H.: Packing and covering with circles. Ph.D. thesis, University of Utrecht (1997)
Mitchell, T.: An algorithm for the construction of “D-optimal” experimental designs. Technometrics 16, 203–210 (1974)
Morris, M., Mitchell, T.: Exploratory designs for computational experiments. J. Stat. Plan. Inference 43, 381–402 (1995)
Müller, W.G.: Collecting Spatial Data: Optimum Design of Experiments for Random Fields, 3rd edn. Springer, Heidelberg (2007)
Müller, W.G., Stehlík, M.: Compound optimal spatial designs. Environmetrics 21(3–4), 354–364 (2010)
Müller, W.G., Pronzato, L., Waldl, H.: Relations between designs for prediction and estimation in random fields: an illustrative case. In: E. Porcu, J.M. Montero, M. Schlather (eds.) Advances and Challenges in Space-Time Modelling of natural Events. Springer Lecture Notes in Statistics (2011, to appear)
Nagy, B., Loeppky, J.L., Welch, W.J.: Fast Bayesian inference for Gaussian process models. Tech. rep., The University of British Columbia, Department of Statistics (2007)
Narcowich, F.: Norms of inverses and condition numbers for matrices associated with scattered data. J. Approx. Theory 64, 69–94 (1991)
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods (CBMS-NSF Regional Conference Series in Applied Mathematics). SIAM, Philadelphia (1992)
Okabe, A., Books, B., Sugihama, K.: Spatial Tessellations. Concepts and Applications of Voronoi Diagrams. Wiley, New York (1992)
Oler, N.: A finite packing problem. Can. Math. Bull. 4, 153–155 (1961)
Pebesma, E.J., Heuvelink, G.B.M.: Latin hypercube sampling of Gaussian random fields. Technometrics 41(4), 303–312 (1999)
Penrose, M., Yukich, J.: Laws of large numbers and nearest neighbor distances. In: Wells, M., Sengupta, A. (eds.) Advances in Directional and Linear Statistics. A Festschrift for Sreenivasa Rao Jammalamadaka, Chap. 13, pp. 189–199 (2011). arXiv:0911.0331v1
Petelet, M., Iooss, B., Asserin, O., Loredo, A.: Latin hypercube sampling with inequality constraints. AStA Adv. Stat. Anal. 94, 325–339 (2010)
Picheny, V., Ginsbourger, D., Roustant, O., Haftka, R.T., Kim, N.H.: Adaptive designs of experiments for accurate approximation of a target region. J. Mech. Des. 132(7), 071,008 (2010)
Pistone, G., Vicario, G.: Comparing and generating Latin hypercube designs in Kriging models. AStA Adv. Stat. Anal. 94, 353–366 (2010)
Pronzato, L.: Optimal experimental design and some related control problems. Automatica 44(2), 303–325 (2008)
Putter, H., Young, A.: On the effect of covariance function estimation on the accuracy of kriging predictors. Bernoulli 7(3), 421–438 (2001)
Qian, P.Z.G., Ai, M., Wu, C.F.J.: Construction of nested space-filling designs. Ann. Stat. 37(6A), 3616–3643 (2009)
Redmond, C., Yukich, J.: Asymptotics for Euclidian functionals with power-weighted edges. Stoch. Process. Appl. 61, 289–304 (1996)
Rennen, G., Husslage, B., van Dam, E., den Hertog, D.: Nested maximin Latin hypercube designs. Struct Multidisc Optiml 41, 371–395 (2010)
Rényi, A.: On measures of entropy and information. In: Proc. 4th Berkeley Symp. on Math. Statist. and Prob., pp. 547–561 (1961)
Riccomagno, E., Schwabe, R., Wynn, H.P.: Lattice-based D-optimum design for Fourier regression. Ann. Stat. 25(6), 2313–2327 (1997)
Royle, J., Nychka, D.: An algorithm for the construction of spatial coverage designs with implementation in SPLUS. Comput. Geosci. 24(5), 479–488 (1998)
Sacks, J., Welch, W., Mitchell, T., Wynn, H.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–435 (1989)
Santner, T., Williams, B., Notz, W.: The Design and Analysis of Computer Experiments. Springer, Heidelberg (2003)
Schaback, R.: Lower bounds for norms of inverses of interpolation matrices for radial basis functions. J. Approx. Theory 79, 287–306 (1994)
Schilling, M.F.: Spatial designs when the observations are correlated. Commun. Stat., Simul. Comput. 21(1), 243–267 (1992)
Scott, D.: Multivariate Density Estimation. Wiley, New York (1992)
Shewry, M., Wynn, H.: Maximum entropy sampling. Appl. Stat. 14, 165–170 (1987)
Sjöstedt-De-Luna, S., Young, A.: The bootstrap and kriging prediction intervals. Scand. J. Stat. 30(1), 175–192 (2003)
Stein, M.: Interpolation of Spatial Data: Some Theory for Kriging. Springer, Heidelberg (1999)
Stinstra, E., den Hertog, D., Stehouwer, P., Vestjens, A.: Constrained maximin designs for computer experiments. Technometrics 45(4), 340–346 (2003)
Sun, X.: Norm estimates for inverses of Euclidean distance matrices. J. Approx. Theory 70, 339–347 (1992)
Tang, B.: Orthogonal array-based latin hypercubes. J. Am. Stat. Assoc. 88(424), 1392–1397 (1993)
Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52(1/2), 479–487 (1988)
van Dam, E.: Two-dimensional minimax Latin hypercube designs. Discrete Appl. Math. 156(18), 3483–3493 (2007)
van Dam, E., Hussage, B., den Hertog, D., Melissen, H.: Maximin Latin hypercube designs in two dimensions. Oper. Res. 55(1), 158–169 (2007)
van Dam, E., Rennen, G., Husslage, B.: Bounds for maximin Latin hypercube designs. Oper. Res. 57(3), 595–608 (2009)
van Groenigen, J.: The influence of variogram parameters on optimal sampling schemes for mapping by kriging. Geoderma 97(3–4), 223–236 (2000)
Walvoort, D.J.J., Brus, D.J., de Gruijter, J.J.: An R package for spatial coverage sampling and random sampling from compact geographical strata by k-means. Comput. Geosci. 36, 1261–1267 (2010)
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)
Wolkowicz, H., Styan, G.: Bounds for eigenvalues using traces. Linear Algebra Appl. 29, 471–506 (1980)
Wynn, H.: Maximum entropy sampling and general equivalence theory. In: Di Bucchianico, A., Läuter, H., Wynn, H. (eds.) mODa’7—Advances in Model–Oriented Design and Analysis, Proceedings of the 7th Int. Workshop, Heeze, Netherlands, pp. 211–218. Physica Verlag, Heidelberg (2004)
Yfantis, E., Flatman, G., Behar, J.: Efficiency of kriging estimation for square, triangular, and hexagonal grids. Math. Geol. 19(3), 183–205 (1987)
Yukich, J.: Probability Theory of Classical Euclidean Optimization Problems. Springer, Berlin (1998)
Zagoraiou, M., Antognini, A.B.: Optimal designs for parameter estimation of the Ornstein-Uhlenbeck process. Appl. Stoch. Models Bus. Ind. 25(5), 583–600 (2009)
Zhang, H., Zimmerman, D.: Towards reconciling two asymptotic frameworks in spatial statistics. Biometrika 92(4), 921–936 (2005)
Zhu, Z., Stein, M.: Spatial sampling design for parameter estimation of the covariance function. J. Stat. Plan. Inference 134(2), 583–603 (2005)
Zhu, Z., Zhang, H.: Spatial sampling design under the infill asymptotic framework. Environmetrics 17(4), 323–337 (2006)
Zimmerman, D.L.: Optimal network design for spatial prediction, covariance parameter estimation, and empirical prediction. Environmetrics 17(6), 635–652 (2006)
Zimmerman, D., Cressie, N.: Mean squared prediction error in the spatial linear model with estimated covariance parameters. Ann. Inst. Stat. Math. 44(1), 27–43 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by a PHC Amadeus/OEAD Amadée grant FR11/2010.
Rights and permissions
About this article
Cite this article
Pronzato, L., Müller, W.G. Design of computer experiments: space filling and beyond. Stat Comput 22, 681–701 (2012). https://doi.org/10.1007/s11222-011-9242-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-011-9242-3