Annals of Statistics

On the construction of nested space-filling designs

Fasheng Sun, Min-Qian Liu, and Peter Z. G. Qian

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Nested space-filling designs are nested designs with attractive low-dimensional stratification. Such designs are gaining popularity in statistics, applied mathematics and engineering. Their applications include multi-fidelity computer models, stochastic optimization problems, multi-level fitting of nonparametric functions, and linking parameters. We propose methods for constructing several new classes of nested space-filling designs. These methods are based on a new group projection and other algebraic techniques. The constructed designs can accommodate a nested structure with an arbitrary number of layers and are more flexible in run size than the existing families of nested space-filling designs. As a byproduct, the proposed methods can also be used to obtain sliced space-filling designs that are appealing for conducting computer experiments with both qualitative and quantitative factors.

Article information

Ann. Statist., Volume 42, Number 4 (2014), 1394-1425.

First available in Project Euclid: 25 June 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K15: Factorial designs
Secondary: 62K20: Response surface designs

Computer experiment difference matrix Galois field orthogonal array OA-based Latin hypercube Rao–Hamming construction sliced space-filling design


Sun, Fasheng; Liu, Min-Qian; Qian, Peter Z. G. On the construction of nested space-filling designs. Ann. Statist. 42 (2014), no. 4, 1394--1425. doi:10.1214/14-AOS1229.

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  • Bose, R. C. and Bush, K. A. (1952). Orthogonal arrays of strength two and three. Ann. Math. Statistics 23 508–524.
  • Choi, S., Alonso, J. J., Kroo, I. M. and Wintzer, M. (2008). Multifidelity design optimization of low-boom supersonic jets. Journal of Aircraft 45 106–118.
  • Dewettinck, K., Visscher, A. D., Deroo, L. and Huyghebaert, A. (1999). Modeling the steady-state thermodynamic operation point of top-spray fluidized bed processing. Journal of Food Engineering 39 131–143.
  • Fang, K.-T., Li, R. and Sudjianto, A. (2006). Design and Modeling for Computer Experiments. Chapman & Hall/CRC, Boca Raton, FL.
  • Fasshauer, G. E. (2007). Meshfree Approximation Methods with MATLAB. Interdisciplinary Mathematical Sciences 6. World Scientific, Hackensack, NJ.
  • Floater, M. S. and Iske, A. (1996). Multistep scattered data interpolation using compactly supported radial basis functions. J. Comput. Appl. Math. 73 65–78.
  • Haaland, B. and Qian, P. Z. G. (2010). An approach to constructing nested space-filling designs for multi-fidelity computer experiments. Statist. Sinica 20 1063–1075.
  • Haaland, B. and Qian, P. Z. G. (2011). Accurate emulators for large-scale computer experiments. Ann. Statist. 39 2974–3002.
  • Han, G., Santner, T. J., Notz, W. I. and Bartel, D. L. (2009). Prediction for computer experiments having quantitative and qualitative input variables. Technometrics 51 278–288.
  • Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications. Springer, New York.
  • Herstein, I. N. (1996). Abstract Algebra, 3rd ed. Prentice Hall, Upper Saddle River, NJ.
  • Husslage, B., Dam, E. V., Hertog, D. D., Stehouwer, P. and Stinstra, E. (2003). Collaborative metamodeling: Coordinating simulation-based product design. Concurrent Eng. 11 267–278.
  • 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.
  • Molina-Cristóbal, A., Palmer, P. R., Skinner, B. A. and Parks, G. T. (2010). Multi-fidelity simulation modelling in optimization of a submarine propulsion system. In Proceedings of the 2010 IEEE Vehicle Power and Propulsion Conference (VPPC). Lille, France.
  • Mukerjee, R., Qian, P. Z. G. and Jeff Wu, C. F. (2008). On the existence of nested orthogonal arrays. Discrete Math. 308 4635–4642.
  • Qian, P. Z. G. (2009). Nested Latin hypercube designs. Biometrika 96 957–970.
  • Qian, P. Z. G. (2012). Sliced Latin hypercube designs. J. Amer. Statist. Assoc. 107 393–399.
  • Qian, P. Z. G. and Ai, M. (2010). Nested lattice sampling: A new sampling scheme derived by randomizing nested orthogonal arrays. J. Amer. Statist. Assoc. 105 1147–1155.
  • Qian, P. Z. G., Ai, M. and Wu, C. F. J. (2009). Construction of nested space-filling designs. Ann. Statist. 37 3616–3643.
  • 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.
  • Qian, P. Z. G. and Wu, C. F. J. (2009). Sliced space-filling designs. Biometrika 96 945–956.
  • Qian, P. Z. G., Wu, H. and Wu, C. F. J. (2008). Gaussian process models for computer experiments with qualitative and quantitative factors. Technometrics 50 383–396.
  • Santner, T. J., Williams, B. J. and Notz, W. I. (2003). The Design and Analysis of Computer Experiments. Springer, New York.
  • Schmidt, R. R., Cruz, E. E. and Iyengar, M. K. (2005). Challenges of data center thermal management. IBM Journal of Research and Development 49 709–723.
  • Tang, B. (1993). Orthogonal array-based Latin hypercubes. J. Amer. Statist. Assoc. 88 1392–1397.
  • Williams, B., Morris, M. and Santner, T. (2009). Using multiple computer models/multiple data sources simultaneously to infer calibration parameters. Paper presented at the 2009 INFORMS Annual Conference, October 11–14, San Diego, CA.
  • Xu, X., Haaland, B. and Qian, P. Z. G. (2011). Sudoku-based space-filling designs. Biometrika 98 711–720.
  • Zhou, Q., Qian, P. Z. G. and Zhou, S. (2011). A simple approach to emulation for computer models with qualitative and quantitative factors. Technometrics 53 266–273.