Construction of Space-Filling Designs for Three Input Variables Computer Experiments

Latin hypercube designs (LHDs) have been applied in many computer experiments among the space-filling designs found in the literature. A LHD can be randomly generated but a randomly chosen LHD may have bad properties and thus act poorly in estimation and prediction. There is a connection between Latin squares and orthogonal arrays (OAs). A Latin square of order s involves an arrangement of s symbols in s rows and s columns, such that every symbol occurs once in each row and once in each column and this exists for every non-negative integer s. In this paper, a computer program was written to construct orthogonal array-based Latin hypercube designs (OA-LHDs). Orthogonal arrays (OAs) were constructed from Latin square of order s and the OAs constructed were afterward used to construct the desired Latin hypercube designs for three input variables for use in computer experiments. The LHDs constructed have better space-filling properties and they can be used in computer experiments that involve only three input factors. MATLAB 2012a computer package (www.mathworks.com/) was used for the development of the program that constructs the designs.

Authors:

• Kazeem A. Osuolale
• Waheed B. Yahya
• Babatunde L. Adeleke

Keywords:

• Computer Experiments
• Latin Squares
• Latin Hypercube Designs
• Orthogonal Array
• Space-filling Designs.

References:

[1] M. Bayarri, J. O. Berger, D. Higdon, M. Kennedy, A. Kottas, R. Paulo,
J. Sacks, J. Cafeo, J. Cavendish and J. Tu (2002). A Framework for the
Validation of Computer Models. Proceedings of the Workshop on
Foundations for V&V in the 21st Century, D. Pace and S. Stevenson,
eds., Society for Modelling and Simulation International.
[2] R. C. Bose and K. A. Bush (1952). Orthogonal Arrays of Strength Two
and Three. Annals of Mathematical Statistics 23, 508–524.
[3] A. S. Hedayat, N. J. A. Sloane and J. Stufken (1999). Orthogonal
Arrays: Theory and Applications. Springer-Verlag, New York.
[4] M. Johnson, L. Moore and D. Ylvisaker (1990). Minimax and Maximin
Distance Design. J. Statist. Plann. Inference 26, 131-148.
[5] W. F. Kuhfeld (2009). Orthogonal Arrays. Website courtesy of SAS
Institute.
[6] S. Leary, A. Bhaskar and A. Keane (2003). Optimal Orthogonal Aarray-
Based Latin Hypercubes. J. Applied Statist. 30, 585-598.
[7] M. D. McKay, R. J. Beckman and W. J. Conover (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.
[8] R. Mukerjee and C. F. Wu (2006). A Modern Theory of Factorial
Designs. Springer Verlag.
[9] K. A. Osuolale, W. B. Yahya and B. L. Adeleke (2014). An Algorithm
for constructing Space-Filling Designs for Hadamard Matrices of Orders
4λ and 8λ. Proceedings of the 49th Annual Conference of the Science
Association of Nigeria, 27 th April - 1st May.
[10] A. B. Owen (1992a). A Central Limit Theorem for Latin Hypercube
Sampling, Journal of the Royal Statistical Society. B54, 541-555.
[11] C. R. Rao (1946). Hypercube of Strength ‘d’ Leading to Confounded
Designs in Factorial Experiments. Bull. Calcutta Math. Soc., 38, 67-78.
[12] C. R. Rao (1947). Factorial Experiments Derivable from Combinatorial
Arrangements of Arrays. Journal of the Royal Statistical Society
(Suppl.). 9, 128-139.
[13] S. Strogatz (2003). The Real Scientific Hero of 1953, New York Times.
March 4, Editorial/Op-Ed.
[14] B. Tang (1991). Orthogonal Array-Based Latin Hypercubes. IIQP
Research Report, RR-91-10.
[15] B. Tang (1993). Orthogonal Array-Based Latin Hypercubes. Journal of
the American Statistical Association 88, 1392–1397.
[16] B. Tang (1994). A Theorem for selecting OA-Based Latin Hypercubes
using a Distance Criterion. Comm. Statist. : A Theory and Methods 23,
2047-2058.