Skip to main content
SpringerLink
Log in

Bayesian Cubic Spline in Computer Experiments

  • Reference work entry
  • First Online:
Handbook of Uncertainty Quantification

  • 9110 Accesses

Abstract

Cubic splines are commonly used in numerical analysis. It has also become popular in the analysis of computer experiments, thanks to its adoption by the software JMP 8.0.2 2010. In this chapter, a Bayesian version of the cubic spline method is proposed, in which the random function that represents prior uncertainty about y is taken to be a specific stationary Gaussian process and y is the output of the computer experiment. A Markov chain Monte Carlo (MCMC) procedure is developed for updating the prior given the observed y values. Simulation examples and a real data application are given to show that the proposed Bayesian method performs better than the frequentist cubic spline method and the standard method based on the Gaussian correlation function.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 1,099.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 1,399.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abrahamsen, P.: A Review of Gaussian Random Fields and Correlation Functions. Norsk Regnesentral/Norwegian Computing Center (1997)

    Google Scholar 

  2. Chen, Z., Gu, C., Wahba, G.: Comment on “linear smoothers and additive models”. Ann. Stat. 17(3), 515–521 (1989)

    Article  Google Scholar 

  3. Cressie, N.: Statistics for Spatial Data. Wiley, Chichester/New York (1992)

    MATH  Google Scholar 

  4. Currin, C., Mitchell, T., Morris, M., Ylvisaker, D.: A Bayesian approach to the design and analysis of computer experiments. ORNL-6498 (1988)

    Google Scholar 

  5. Currin, C., Mitchell, T., Morris, M., Ylvisaker, D.: Bayesian prediction of deterministic functions, with applications to the design and analysis of computer experiments. J. Am. Stat. Assoc. 86(416), 953–963 (1991)

    Article  MathSciNet  Google Scholar 

  6. Diggle, P.J., Ribeiro, P.J.: Model-Based Geostatistics. Springer, New York (2007)

    MATH  Google Scholar 

  7. DiMatteo, I., Genovese, C.R., Kass, R.E.: Bayesian curve-fitting with free-knot splines. Biometrika 88(4), 1055–1071 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fang, K.-T., Li, R., Sudjianto, A.: Design and Modeling for Computer Experiments. Chapman and Hall/CRC, Boca Raton (2010)

    MATH  Google Scholar 

  9. Geisser, S.: Predictive Inference: an Introduction, vol. 55. CRC Press, New York (1993)

    Book  MATH  Google Scholar 

  10. Geyer, C.J.: Introduction to MCMC. Chapman & Hall, Boca Raton (2011)

    Google Scholar 

  11. Gill, J.: Bayesian Methods: A Social and Behavioral Sciences Approach. CRC Press, Boca Raton (2002)

    MATH  Google Scholar 

  12. Gramacy, R.B., Apley, D.W.: Local Gaussian process approximation for large computer experiments. J. Comput. Graph. Stat. 24(2), 561–578 (2014)

    Article  MathSciNet  Google Scholar 

  13. Gramacy, R.B., Lee, H.K.H.: Cases for the nugget in modeling computer experiments. Stat. Comput. 22(3), 713–722 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  15. Joseph, V.R.: Limit kriging. Technometrics 48(4), 458–466 (2006)

    Article  MathSciNet  Google Scholar 

  16. Kaufman, C.G., Bingham, D., Habib, S., Heitmann, K., Frieman, J.A.: Efficient emulators of computer experiments using compactly supported correlation functions, with an application to cosmology. Ann. Appl. Stat. 5(4), 2470–2492 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kohavi, R.: A study of cross-validation and bootstrap for accuracy estimation and model selection. In: International Joint Conference on Artificial Intelligence, vol. 14, pp. 1137–1145. Lawrence Erlbaum Associates Ltd (1995)

    Google Scholar 

  18. Laslett, G.M.: Kriging and splines: an empirical comparison of their predictive performance in some applications. J. Am. Stat. Assoc. 89(426), 391–400 (1994)

    Article  MathSciNet  Google Scholar 

  19. Mardia, K.V., Marshall, R.J.: Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71(1), 135–146 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Matheron, G.: Principles of geostatistics. Econ. Geol. 58(8), 1246–1266 (1963)

    Article  Google Scholar 

  21. McKay, M.D., Beckman, R.J., Conover, W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239–245 (1979)

    MathSciNet  MATH  Google Scholar 

  22. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953)

    Article  Google Scholar 

  23. Morris, M.D., Mitchell, T.J., Ylvisaker, D.: Bayesian design and analysis of computer experiments: use of derivatives in surface prediction. Technometrics 35(3), 243–255 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. O’Hagan, A., Kingman, J.F.C.: Curve fitting and optimal design for prediction. J. R. Stat. Soc. Ser. B (Methodolog.) 2, 1–42 (1978)

    MathSciNet  MATH  Google Scholar 

  25. Patterson, H.D., Thompson, R.: Recovery of inter-block information when block sizes are unequal. Biometrika 58(3), 545–554 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  26. Peng, C.-Y., Wu, C.F.J.: On the choice of nugget in kriging modeling for deterministic computer experiments. J. Comput. Graph. Stat. 23(1), 151–168 (2014)

    Article  MathSciNet  Google Scholar 

  27. Robert, C.P., Casella, G.: Monte Carlo Statistical Methods, vol. 319. Citeseerx, New York (2004)

    Book  MATH  Google Scholar 

  28. Sacks, J., Schiller, S.: Spatial designs. Stat. Decis. Theory Relat. Topics IV 2(32), 385–399 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sacks, J., Schiller, S., Welch, W.J.: Designs for computer experiments. Technometrics 31(1), 41–47 (1989)

    Article  MathSciNet  Google Scholar 

  30. Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–423 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  31. Santner, T.J., Williams, B.J., Notz, W.I.: The Design and Analysis of Computer Experiments. Springer, New York (2003)

    Book  MATH  Google Scholar 

  32. Wahba, G.: Spline Models for Observational Data, vol. 59. Society for Industrial and Applied Mathematics, Philadelphia (1990)

    Book  MATH  Google Scholar 

  33. Wang, X.: Bayesian free-knot monotone cubic spline regression. J. Comput. Graph. Stat. 17(2), 518–527 (2008)

    Article  MathSciNet  Google Scholar 

  34. Wecker, W.E., Ansley, C.F.: The signal extraction approach to nonlinear regression and spline smoothing. J. Am. Stat. Assoc. 78(381), 81–89 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ylvisaker, D.: Designs on random fields. Surv. Stat. Des. Linear Models 37(6), 593–607 (1975)

    MathSciNet  MATH  Google Scholar 

  36. Ylvisaker, D.: Prediction and design. Ann. Stat. 52(17), 1–19 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Blizzard Entertainment, 765 Ferst Dr Mainbuilding 233, 92618, Irvine, CA, USA

    Yijie Dylan Wang

  2. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, ISyE Main Building, 30332, Atlanta, GA, USA

    C. F. Jeff Wu

Authors
  1. Yijie Dylan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. F. Jeff Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yijie Dylan Wang .

Editor information

Editors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, California, USA

    Roger Ghanem

  2. Social and Decision Analytics Laboratory, Virginia Bioinformatics Institute, Virginia Tech University, Arlington, Virginia, USA

    David Higdon

  3. Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California, USA

    Houman Owhadi

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this entry

Cite this entry

Wang, Y.D., Wu, C.F.J. (2017). Bayesian Cubic Spline in Computer Experiments. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-12385-1_69

Download citation

Publish with us

Policies and ethics

Search

Navigation