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Gaussian processes and limiting linear models

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  • Gramacy, Robert B.
  • Lee, Herbert K.H.

Abstract

Gaussian processes retain the linear model either as a special case, or in the limit. We show how this relationship can be exploited when the data are at least partially linear. However from the perspective of the Bayesian posterior, the Gaussian processes which encode the linear model either have probability of nearly zero or are otherwise unattainable without the explicit construction of a prior with the limiting linear model in mind. We develop such a prior, and show that its practical benefits extend well beyond the computational and conceptual simplicity of the linear model. For example, linearity can be extracted on a per-dimension basis, or can be combined with treed partition models to yield a highly efficient nonstationary model. Our approach is demonstrated on synthetic and real datasets of varying linearity and dimensionality.

Suggested Citation

  • Gramacy, Robert B. & Lee, Herbert K.H., 2008. "Gaussian processes and limiting linear models," Computational Statistics & Data Analysis, Elsevier, vol. 53(1), pages 123-136, September.
    • Handle: RePEc:eee:csdana:v:53:y:2008:i:1:p:123-136
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    References listed on IDEAS

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    1. Berger J.O. & De Oliveira V. & Sanso B., 2001. "Objective Bayesian Analysis of Spatially Correlated Data," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1361-1374, December.
    2. Harrison, David Jr. & Rubinfeld, Daniel L., 1978. "Hedonic housing prices and the demand for clean air," Journal of Environmental Economics and Management, Elsevier, vol. 5(1), pages 81-102, March.
    3. Marc C. Kennedy & Anthony O'Hagan, 2001. "Bayesian calibration of computer models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(3), pages 425-464.
    4. Gramacy, Robert B., 2007. "tgp: An R Package for Bayesian Nonstationary, Semiparametric Nonlinear Regression and Design by Treed Gaussian Process Models," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 19(i09).
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    Cited by:

    1. Paulo, Rui & García-Donato, Gonzalo & Palomo, Jesús, 2012. "Calibration of computer models with multivariate output," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 3959-3974.
    2. Chevalier, Clément & Picheny, Victor & Ginsbourger, David, 2014. "KrigInv: An efficient and user-friendly implementation of batch-sequential inversion strategies based on kriging," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 1021-1034.
    3. Michael Ludkovski & James Risk, 2017. "Sequential Design and Spatial Modeling for Portfolio Tail Risk Measurement," Papers 1710.05204, arXiv.org, revised May 2018.

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