Classic Kriging versus Kriging with bootstrapping or conditional simulation: classic Kriging’s robust confidence intervals and optimization

Kriging is a popular method for estimating the global optimum of a simulated system. Kriging approximates the input/output function of the simulation model. Kriging also estimates the variances of the predictions of outputs for input combinations not yet simulated. These predictions and their variances are used by ‘efficient global optimization’ (EGO), to balance local and global search. This article focuses on two related questions: (1) How to select the next combination to be simulated when searching for the global optimum? (2) How to derive confidence intervals for outputs of input combinations not yet simulated? Classic Kriging simply plugs the estimated Kriging parameters into the formula for the predictor variance, so theoretically this variance is biased. This article concludes that practitioners may ignore this bias, because classic Kriging gives acceptable confidence intervals and estimates of the optimal input combination. This conclusion is based on bootstrapping and conditional simulation.

[1]  Herbert K. H. Lee,et al.  Bayesian Guided Pattern Search for Robust Local Optimization , 2009, Technometrics.

[2]  Victor Picheny,et al.  Quantile-Based Optimization of Noisy Computer Experiments With Tunable Precision , 2013, Technometrics.

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

[4]  Jack P. C. Kleijnen,et al.  Conditional simulation for efficient global optimization , 2013, 2013 Winter Simulations Conference (WSC).

[5]  H. Wackernagle,et al.  Multivariate geostatistics: an introduction with applications , 1998 .

[6]  Nicolas Gayton,et al.  AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation , 2011 .

[7]  Wei Xie,et al.  The influence of correlation functions on stochastic kriging metamodels , 2010, Proceedings of the 2010 Winter Simulation Conference.

[8]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[9]  Søren Nymand Lophaven,et al.  DACE - A Matlab Kriging Toolbox, Version 2.0 , 2002 .

[10]  Andy J. Keane,et al.  Engineering Design via Surrogate Modelling - A Practical Guide , 2008 .

[11]  Jack P. C. Kleijnen,et al.  Kriging Metamodeling in Simulation: A Review , 2007, Eur. J. Oper. Res..

[12]  C. J. Lawrence.  Robust estimates of location : survey and advances , 1975 .

[13]  Szu Hui Ng,et al.  A Bayesian metamodeling approach for stochastic simulations , 2010, Proceedings of the 2010 Winter Simulation Conference.

[14]  D. F. Andrews,et al.  Robust Estimates of Location: Survey and Advances. , 1975 .

[15]  Raphael T. Haftka,et al.  Performance Estimate and Simultaneous Application of Multiple Surrogates , 2006 .

[16]  Warren B. Powell,et al.  Calibrating simulation models using the knowledge gradient with continuous parameters , 2010, Proceedings of the 2010 Winter Simulation Conference.

[17]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[18]  Jack P. C. Kleijnen,et al.  The correct Kriging variance estimated by bootstrapping , 2006, J. Oper. Res. Soc..

[19]  Jerome Sacks,et al.  Choosing the Sample Size of a Computer Experiment: A Practical Guide , 2009, Technometrics.

[20]  Layne T. Watson,et al.  Efficient global optimization algorithm assisted by multiple surrogate techniques , 2012, Journal of Global Optimization.

[21]  Barry L. Nelson,et al.  Stochastic kriging for simulation metamodeling , 2008, 2008 Winter Simulation Conference.

[22]  Robert E. Shannon,et al.  Design and analysis of simulation experiments , 1978, WSC '78.

[23]  Susan A. Murphy,et al.  Monographs on statistics and applied probability , 1990 .