Construction of nested space-filling designs. (English) Zbl 1369.62195
Summary: New types of designs called nested space-filling designs have been proposed for conducting multiple computer experiments with different levels of accuracy. In this article, we develop several approaches to constructing such designs. The development of these methods also leads to the introduction of several new discrete mathematics concepts, including nested orthogonal arrays and nested difference matrices.
MSC:
| 62K99 | Design of statistical experiments |
| 62K15 | Factorial statistical designs |
| 05B15 | Orthogonal arrays, Latin squares, Room squares |
Keywords:
computer experiments; design of experiments; difference matrices; orthogonal arrays; OA-based Latin hypercubes; randomized orthogonal arrays; Wang-Wu methodReferences:
| [1] | Addelman, S. and Kempthorne, O. (1961). Some main-effect plans and orthogonal arrays of strength two. Ann. Math. Statist. 32 1167-1176. · Zbl 0107.36001 · doi:10.1214/aoms/1177704855 |
| [2] | Bayarri, M. J., Berger, J. O., Cafeo, J., Garcia-Donato, G., Liu, F., Palomo, J., Parthasarathy, R. J., Paulo, R., Sacks, J. and Walsh, D. (2007). Computer model validation with functional output. Ann. Statist. 35 1874-1906. · Zbl 1144.62368 · doi:10.1214/009053607000000163 |
| [3] | Bose, R. C. and Bush, K. A. (1952). Orthogonal arrays of strength two and three. Ann. Math. Statist. 23 508-524. · Zbl 0048.00803 · doi:10.1214/aoms/1177729331 |
| [4] | Butler, N. A. (2001). Optimal and orthogonal Latin hypercube designs for computer experiments. Biometrika 88 847-857. JSTOR: · Zbl 0985.62058 · doi:10.1093/biomet/88.3.847 |
| [5] | Dalal, S. R. and Mallows, C. L. (1998). Factor-covering designs for testing software. Technometrics 40 234-243. |
| [6] | Fang, K. T., Li, R. Z. and Sudjianto, A. (2006). Design and Modeling for Computer Experiments . Chapman & Hall, New York. · Zbl 1093.62117 |
| [7] | Fang, K. T., Lin, D. K. J., Winker, P. and Zhang, Y. (2000). Uniform design: Theory and application. Technometrics 42 237-248. JSTOR: · Zbl 0996.62073 · doi:10.2307/1271079 |
| [8] | Goldstein, M. and Rougier, J. (2004). Probabilistic formulations for transferring inferences from mathematical models to physical systems. SIAM J. Sci. Comput. 26 467-487. · Zbl 1138.62375 · doi:10.1137/S106482750342670X |
| [9] | Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications . Springer, New York. · Zbl 0935.05001 |
| [10] | Hedayat, A. S., Stufken, J. and Su, G. (1996). On difference schemes and orthogonal arrays of strength t . J. Statist. Plann. Inference 56 307-324. · Zbl 0873.05022 · doi:10.1016/S0378-3758(96)00026-2 |
| [11] | Higdon, D., Kennedy, M. C., Cavendish, J. C., Cafeo, J. A. and Ryne, R. D. (2004). Combining field data and computer simulations for calibration and prediction. SIAM J. Sci. Comput. 26 448-466. · Zbl 1072.62018 · doi:10.1137/S1064827503426693 |
| [12] | Johnson, D. M., Dulmage, A. L. and Mendelsohn, N. S. (1961). Orthomorphisms of groups and orthogonal Latin squares. I. Canad. J. Math. 13 356-372. · Zbl 0097.25102 · doi:10.4153/CJM-1961-031-7 |
| [13] | Kennedy, M. C. and O’Hagan, A. (2000). Predicting the output from a complex computer code when fast approximations are available. Biometrika 87 1-13. JSTOR: · Zbl 0974.62024 · doi:10.1093/biomet/87.1.1 |
| [14] | Kennedy, M. C. and O’Hagan, A. (2001). Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 425-464. JSTOR: · Zbl 1007.62021 · doi:10.1111/1467-9868.00294 |
| [15] | Lam, R. L. H., Welch, W. J. and Young, S. S. (2002). Uniform coverage designs for molecule selection. Technometrics 44 99-109. JSTOR: · doi:10.1198/004017002317375055 |
| [16] | Leary, S., Bhaskar, A. and Keane, A. (2003). Optimal orthogonal-array-based Latin hypercubes. J. Appl. Statist. 30 585-598. · Zbl 1121.62421 · doi:10.1080/0266476032000053691 |
| [17] | Loh, W. L. (1996a). A combinatorial central limit theorem for randomized orthogonal array sampling designs. Ann. Statist. 24 1209-1224. · Zbl 0869.62018 · doi:10.1214/aos/1032526964 |
| [18] | Loh, W. L. (1996b). On Latin hypercube sampling. Ann. Statist. 24 2058-2080. · Zbl 0867.62005 · doi:10.1214/aos/1069362310 |
| [19] | Loh, W. L. (2008). A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments. Ann. Statist. 36 1983-2023. · Zbl 1143.62044 · doi:10.1214/07-AOS530 |
| [20] | McKay, M. D., Beckman, R. J. and Conover, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21 239-245. JSTOR: · Zbl 0415.62011 · doi:10.2307/1268522 |
| [21] | Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods . SIAM, Philadelphia, PA. · Zbl 0761.65002 |
| [22] | Oberkampf, W.L. and Trucano, T. (2007). Verification and Validation Benchmarks . Sandia National Laboratories (SAND 2007-0853), Albuquerque, New Mexico. |
| [23] | Owen, A. B. (1992). Orthogonal arrays for computer experiments, integration and visualization. Statist. Sinica 2 439-452. · Zbl 0822.62064 |
| [24] | Owen, A. B. (1994a). Lattice sampling revisited: Monte Carlo variance of means over randomized orthogonal arrays. Ann. Statist. 22 930-945. · Zbl 0807.62059 · doi:10.1214/aos/1176325504 |
| [25] | Owen, A. B. (1994b). Controlling correlations in Latin hypercube samples. J. Amer. Statist. Assoc. 88 1517-1522. · Zbl 0813.65060 · doi:10.2307/2291014 |
| [26] | Qian, Z., Seepersad, C. C., Roshan, V. R., Allen, J. K. and Wu, C. F. J. (2006). Building surrogate models based on detailed and approximate simulations. ASME Transactions: J. Mechanical Design 128 668-677. |
| [27] | Qian, P. Z. G., Tang, B. and Wu, C. F. J. (2009). Nested space-filling designs for computer experiments with two levels of accuracy. Statist. Sinica 19 287-300. · Zbl 1153.62059 |
| [28] | Qian, P. Z. G. and Wu, C. F. J. (2008). Bayesian hierarchical modeling for integrating low-accuracy and high-accuracy experiments. Technometrics 50 192-204. · doi:10.1198/004017008000000082 |
| [29] | Reese, C. S., Wilson, A. G., Hamada, M., Martz, H. F. and Ryan, K. J. (2004). Integrated analysis of computer and physical experiments. Technometrics 46 153-164. · doi:10.1198/004017004000000211 |
| [30] | Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiments. Statist. Sci. 4 409-435. · Zbl 0955.62619 · doi:10.1214/ss/1177012413 |
| [31] | Santner, T. J., Williams, B. J. and Notz, W. I. (2003). The Design and Analysis of Computer Experiments . Springer, New York. · Zbl 1041.62068 |
| [32] | Seberry, J. (1979). Some remarks on generalised Hadamard matrices and theorems of Rajkundlia on SBIBD’s. In Combinatorial Mathematics . VI (A. F. Horadam and W. D. Wallis, eds.) 154-164. Springer, Berlin. · Zbl 0434.05019 |
| [33] | Shrikhande, S. S. (1964). Generalized Hadamard matrices and orthogonal arrays of strength two. Canad. J. Math. 16 736-740. · Zbl 0123.00301 · doi:10.4153/CJM-1964-070-1 |
| [34] | Stein, M. (1987). Large sample properties of simulations using Latin hypercube sampling. Technometrics 29 143-151. JSTOR: · Zbl 0627.62010 · doi:10.2307/1269769 |
| [35] | Steinberg, D. M. and Lin, D. K. J. (2006). A construction method for orthogonal Latin hypercube designs. Biometrika 93 279-288. · Zbl 1153.62349 · doi:10.1093/biomet/93.2.279 |
| [36] | Tang, B. (1993). Orthogonal array-based Latin hypercubes. J. Amer. Statist. Assoc. 88 1392-1397. JSTOR: · Zbl 0792.62066 · doi:10.2307/2291282 |
| [37] | Tang, B. (1994). A theorem for selecting OA-based Latin hypercubes using a distance criterion. Comm. Statist. Theory Methods 23 2047-2058. · Zbl 0825.62186 · doi:10.1080/03610929408831370 |
| [38] | Tang, B. (1998). Selecting Latin hypercubes using correlation criteria. Statist. Sinica 8 409-435. · Zbl 0905.62065 |
| [39] | Wang, J. C. (1996). Mixed difference matrices and the construction of orthogonal arrays. Statist. Probab. Lett. 28 121-126. · Zbl 0901.05020 · doi:10.1016/0167-7152(95)00105-0 |
| [40] | Wang, J. C. and Wu, C. F. J. (1991). An approach to the construction of asymmetrical orthogonal arrays. J. Amer. Statist. Assoc. 86 450-455. JSTOR: · doi:10.2307/2290593 |
| [41] | Welch, W. J., Buck, R. J., Sacks, J., Wynn, H. P., Mitchell, T. J. and Morris, M. D. (1992). Screening, predicting and computer experiments. Technometrics 34 15-25. |
| [42] | Wu, C. F. J. and Hamada, M. (2000). Experiments: Planning, Analysis, and Parameter Design Optimization . Wiley, New York. · Zbl 0964.62065 |
| [43] | Ye, K. Q. (1998). Orthogonal column Latin hypercubes and their application in computer experiments. J. Amer. Statist. Assoc. 88 1392-1397. JSTOR: · Zbl 1064.62553 · doi:10.2307/2670057 |
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