Source: Ann. Statist.
Volume 38, Number 3
We develop a new method for constructing “good” designs for computer experiments. The method derives its power from its basic structure that builds large designs using small designs. We specialize the method for the construction of orthogonal Latin hypercubes and obtain many results along the way. In terms of run sizes, the existence problem of orthogonal Latin hypercubes is completely solved. We also present an explicit result showing how large orthogonal Latin hypercubes can be constructed using small orthogonal Latin hypercubes. Another appealing feature of our method is that it can easily be adapted to construct other designs; we examine how to make use of the method to construct nearly orthogonal and cascading Latin hypercubes.
Bingham, D., Sitter, R. R. and Tang, B. (2009). Orthogonal and nearly orthogonal designs for computer experiments. Biometrika 96 51–65.
Cioppa, T. M. and Lucas, T. W. (2007). Efficient nearly orthogonal and space-filling Latin hypercubes. Technometrics 49 45–55.
Colbourn, C. J., Kløve, T. and Ling, A. C. H. (2004). Permutation arrays for powerline communication and mutually orthogonal Latin squares. IEEE Trans. Inform. Theory 50 1289–1291.
Dey, A. and Mukerjee, R. (1999). Fractional Factorial Plans. Wiley, New York.
Geramita, A. V. and Seberry, J. (1979). Orthogonal Designs: Quadratic Forms and Hadamard Matrices. Dekker, New York.
Mathematical Reviews (MathSciNet): MR534614
Handcock, M. S. (1991). On cascading Latin hypercube designs and additive models for experiments. Comm. Statist. Theory Methods 20 417–439.
Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications. Springer, New York.
Jones, D. R., Schonlau, M. and Welch, W. J. (1998). Efficient global optimization of expensive black-box functions. J. Global Optim. 13 455–492.
Joseph, V. R. and Hung, Y. (2008). Orthogonal-maximin Latin hypercube designs. Statist. Sinica 18 171–186.
Kennedy, M. C. and O’Hagan, A. (2001). Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 425–464.
Lin, C. D. (2008). New developments in designs for computer experiments and physical experiments. Ph.D. thesis, Simon Fraser Univ.
Linkletter, C., Bingham, D., Hengartner, N., Higdon, D. and Ye, K. Q. (2006). Variable selection for Gaussian process models in computer experiments. Technometrics 48 478–490.
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.
Mathematical Reviews (MathSciNet): MR533252
Morris, M. D. and Mitchell, T. J. (1995). Exploratory designs for computational experiments. J. Statist. Plann. Inference 43 381–402.
Mukerjee, R. and Wu, C. F. J. (2006). A Modern Theory of Factorial Designs. Springer, New York.
Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiments. Statist. Sci. 4 409–435.
Shewry, M. C. and Wynn, H. P. (1987). Maximum entropy sampling. J. Appl. Statist. 4 409–435.
Steinberg, D. M. and Lin, D. K. J. (2006). A construction method for orthogonal Latin hypercube designs. Biometrika 93 279–288.
Vartak, M. N. (1955). On an application of Kronecker product of matrices to statistical designs. Ann. Math. Statist. 26 420–438.
Mathematical Reviews (MathSciNet): MR72095
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.
Xu, H. (2002). An algorithm for constructing orthogonal and nearly-orthogonal arrays with mixed levels and small runs. Technometrics 44 356–368.
Ye, K. Q. (1998). Orthogonal column Latin hypercubes and their application in computer experiments. J. Amer. Statist. Assoc. 93 1430–1439.