# Flexible sliced designs for computer experiments

## Abstract

Sliced Latin hypercube designs are popularly adopted for computer experiments with qualitative factors. Previous constructions require the sizes of different slices to be identical. Here we construct sliced designs with flexible sizes of slices. Besides achieving desirable one-dimensional uniformity, flexible sliced designs (FSDs) constructed in this paper accommodate arbitrary sizes for different slices and cover ordinary sliced Latin hypercube designs as special cases. The sampling properties of FSDs are derived and a central limit theorem is established. It shows that any linear combination of the sample means from different models on slices follows an asymptotic normal distribution. Some simulations compare FSDs with other sliced designs in collective evaluations of multiple computer models.

## Keywords

Central limit theorem Latin hypercube design Sampling property Sliced design## Notes

### Acknowledgements

Mingyao Ai is the corresponding author. The authors thank the editor, the associate editor, and two referees for their comments, which have led to the improvement of the paper. Ai’s work is supported by NSFC Grants 11331011 and 11671019, BCMIIS and LMEQF. Tsui’s work is supported by NSFC Grant 11471275 and the Hong Kong Research Grant Council No. T32-101/15-R.

## References

- Ai, M., Jiang, B., Li, K. (2014). Construction of sliced space-filling designs based on balanced sliced orthogonal arrays.
*Statistica Sinica*,*24*, 1685–1702.Google Scholar - Ai, M., Kong, X., Li, K. (2016). A general theory for orthogonal array based Latin hypercube sampling.
*Statistica Sinica*,*26*, 761–777.Google Scholar - Ba, S., William, R. M., William, A. B. (2015). Optimal sliced Latin hypercube designs.
*Technometrics*,*57*, 479–487.Google Scholar - Drew, S., Homem-de Mello, T. (2005). Some large deviation results for latin hypercube sampling. In M. E. Kuhl, N. M. Steiger, F. B. Armstrong, J. A. Joines (Eds.),
*Proceedings of the 2005 winter simulation conference*(pp. 673–681). IEEE: Orlando, FL.Google Scholar - He, X., Qian, P. Z. G. (2015). A central limit theorem for nested or sliced Latin hypercube designs.
*Statistica Sinica*. doi: 10.5705/ss.202015.0240. - Hwang, Y. D., Qian, P. Z. G., He, X. (2016). Sliced orthogonal array based Latin hypercube designs.
*Technometrics*,*58*, 50–61.Google Scholar - Lehmann, E. L. (1966). Some concepts of dependence.
*Annals of Statistics*,*37*, 1137–1153.MathSciNetCrossRefzbMATHGoogle Scholar - McKay, M. D., Beckman, R. J., 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.Google Scholar - Morris, M. D., Mitchell, T. J. (1995). Exploratory designs for computer experiments.
*Journal of Statistical Planning and Inference*,*43*, 381–402.Google Scholar - Owen, A. B. (1992). A central limit theorem for Latin hypercube sampling.
*Journal of the Royal Statistical Society*:*Series B*,*54*, 541–551.Google Scholar - Owen, A. B. (1994). Controlling correlations in Latin hypercube samples.
*Journal of the Royal Statistical Society*:*Series B*,*89*, 1517–1522.Google Scholar - Qian, P. Z. G. (2012). Sliced Latin hypercube designs.
*Journal of the Royal Statistical Society*:*Series B*,*107*, 393–399.Google Scholar - Stein, M. (1987). Large sample properties of simulations using Latin hypercube sampling.
*Technometrics*,*29*, 142–151.MathSciNetCrossRefzbMATHGoogle Scholar - Williams, B., Morris, M., Santner, T. (2009). Using multiple computer models/multiple data sources simultaneously to inger calibration parameters, paper presented at the 2009 INFORMS Annual Conference, October 11–14. CA: San Diego.Google Scholar
- Yin, Y., Lin, D. K. J., Liu, M. (2014). Sliced Latin hypercube designs via orthogonal arrays.
*Journal of Statistical Planning and Inference*,*149*, 162–171.Google Scholar