Uncertainty in Gaussian Process Interpolation

In this article, we review a probabilistic method for multivariate interpolation based on Gaussian processes. This method is currently a standard approach for approximating complex computer models in statistics, and one of its advantages is the fact that it accompanies the predicted values with uncertainty statements. We focus on investigating the reliability of the method’s uncertainty statements in a simulation study. For this purpose we evaluate the effect of different objective priors and different computational approaches. We illustrate the interpolation method and the practical importance of uncertainty quantification in interpolation in a sequential design application in sheet metal forming. Here design points are added sequentially based on uncertainty statements.

[1]  Runze Li,et al.  Design and Modeling for Computer Experiments (Computer Science & Data Analysis) , 2005 .

[2]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[3]  William J. Welch,et al.  Screening the Input Variables to a Computer Model Via Analysis of Variance and Visualization , 2006 .

[4]  James O. Berger,et al.  A Framework for Validation of Computer Models , 2007, Technometrics.

[5]  P. Diaconis.  Bayesian Numerical Analysis , 1988 .

[6]  A. O'Hagan,et al.  Bayesian calibration of computer models , 2001 .

[7]  William I. Notz,et al.  DESIGNING COMPUTER EXPERIMENTS TO DETERMINE ROBUST CONTROL VARIABLES , 2004 .

[8]  George Michailidis,et al.  Sequential Experiment Design for Contour Estimation From Complex Computer Codes , 2008, Technometrics.

[9]  Robert B. Gramacy,et al.  Particle Learning of Gaussian Process Models for Sequential Design and Optimization , 2009, 0909.5262.

[10]  Rui Paulo.  Default priors for Gaussian processes , 2005 .

[11]  J. Ghosh,et al.  Posterior consistency of logistic Gaussian process priors in density estimation , 2007 .

[12]  D. Krige.  A statistical approach to some basic mine valuation problems on the Witwatersrand, by D.G. Krige, published in the Journal, December 1951 : introduction by the author , 1951 .

[13]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[14]  Robert B. Gramacy,et al.  Ja n 20 08 Bayesian Treed Gaussian Process Models with an Application to Computer Modeling , 2009 .

[15]  G. Wahba.  Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression , 1978 .

[16]  J. Rosenthal,et al.  Optimal scaling for various Metropolis-Hastings algorithms , 2001 .

[17]  Runze Li,et al.  Design and Modeling for Computer Experiments , 2005 .

[18]  Jeremy E. Oakley,et al.  Decision-Theoretic Sensitivity Analysis for Complex Computer Models , 2009, Technometrics.

[19]  Matthias W. Seeger,et al.  Gaussian Processes For Machine Learning , 2004, Int. J. Neural Syst..

[20]  D. Lindley.  Kendall's Advanced Theory of Statistics, volume 2B, Bayesian Inference, 2nd edn , 2005 .

[21]  Mark J. Schervish,et al.  Nonstationary Covariance Functions for Gaussian Process Regression , 2003, NIPS.

[22]  Chong Gu.  Smoothing Spline Anova Models , 2002 .

[23]  J. Berger,et al.  Objective Bayesian Analysis of Spatially Correlated Data , 2001 .

[24]  T. J. Mitchell,et al.  Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments , 1991 .

[25]  J. Q. Smith,et al.  1. Bayesian Statistics 4 , 1993 .

[26]  Stephen A. Freitas,et al.  Modern Industrial Statistics: Design and Control of Quality and Reliability , 1999, Technometrics.

[27]  Max D. Morris.  The Design and Analysis of Computer Experiments. Thomas J. Santner , Brian J. Williams , and William I. Notz , 2004 .

[28]  Wei Chen,et al.  A non‐stationary covariance‐based Kriging method for metamodelling in engineering design , 2007 .

[29]  M. Kendall,et al.  Kendall's advanced theory of statistics , 1995 .

[30]  G. Matheron.  Principles of geostatistics , 1963 .

[31]  M. Stein,et al.  A Bayesian analysis of kriging , 1993 .

[32]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[33]  Thomas J. Santner,et al.  The Design and Analysis of Computer Experiments , 2003, Springer Series in Statistics.

[34]  S. Gupta,et al.  Statistical decision theory and related topics IV , 1988 .

[35]  L. Wasserman,et al.  The Selection of Prior Distributions by Formal Rules , 1996 .

[36]  Thomas J. Santner,et al.  Sequential design of computer experiments to minimize integrated response functions , 2000 .