Thomas J. Santner Brian J. Williams William I. Notz The Design and Analysis of Computer Experiments February 18, 2014 Springer Use the template dedic.tex together with the Springer document class SVMono for monograph-type books or SVMult for contributed volumes to style a quotation or a dedication at the very beginning of your book in the Springer layout Preface Use the template preface.tex together with the Springer document class SVMono (monograph-type books) or SVMult (edited books) to style your preface in the Springer layout. A preface is a book’s preliminary statement, usually written by the author or ed- itor of a work, which states its origin, scope, purpose, plan, and intended audience, and which sometimes includes afterthoughts and acknowledgments of assistance. When written by a person other than the author, it is called a foreword. The preface or foreword is distinct from the introduction, which deals with the subject of the work. Customarily acknowledgments are included as last part of the preface. Place(s), Firstname Surname month year Firstname Surname vii Acknowledgements Use the template acknow.tex together with the Springer document class SVMono (monograph-type books) or SVMult (edited books) if you prefer to set your ac- knowledgement section as a separate chapter instead of including it as last part of your preface. ix Contents 1 Physical Experiments and Computer Experiments ................. 1 1.1 Introduction.................................... ........... 1 1.2 ExamplesofComputerModels ........................ ....... 3 1.3 Inputsand OutputsofComputerExperiments ........... ........ 12 1.4 ObjectivesofExperimentation..................... ........... 14 1.4.1 Introduction.................................. ....... 14 1.4.2 Research Goals for Homogeneous-InputCodes . .... ... ... 15 1.4.3 ResearchGoalsforMixed-Inputs.................. ..... 16 1.4.4 ExperimentswithMultipleOutputs ................ ..... 19 1.5 OrganizationoftheBook ........................... ......... 20 2 Stochastic Models for Computer Output .......................... 23 2.1 Introduction.................................... ........... 23 2.2 ModelsReal-ValuedOutput......................... ......... 26 2.2.1 ThestationaryGPmodel.......................... .... 26 2.2.2 Non-stationary Model 1: Regression + stationary GP model 26 2.2.3 Non-stationary Model 2: Regression + var(x) stationary GPmodel?? ........................................× 27 2.2.4 TreedGPmodel .................................. ... 27 2.2.5 CompositeGP(Convolution)Models ................ ... 27 2.3 Models for Output having Mixed Qualitative and Quantitative Inputs 27 2.4 Models for Multivariate and Functional Computer Output ......... 38 2.4.1 Reducing Functional Data to Multivariate Data . ..... 38 2.4.2 ConstructiveModels ............................ ..... 38 2.4.3 SeparableModels(ContiandO’Hagan) .............. ... 38 2.4.4 Basis Representations of Multivariate Output .. ....... 38 2.4.5 Using the Correlation Function to Specify a GRF with GivenSmoothnessProperties .......................... 45 2.4.6 Hierarchical Gaussian Random Field Models ... .... .. .... 53 2.5 ChapterNotes .................................... ......... 54 xi xii Contents 3 Predicting Output from Computer Experiments ................... 55 3.1 Introduction.................................... ........... 55 3.2 PredictionBasics................................ ........... 56 3.2.1 ClassesofPredictors ........................... ...... 56 3.2.2 BestMSPEPredictors ............................ .... 57 3.2.3 Best LinearUnbiasedMSPE Predictors.............. .... 64 3.3 EmpiricalBest Linear UnbiasedPrediction ........... .......... 67 3.3.1 Introduction.................................. ....... 67 3.3.2 Prediction When the Correlation Function Is Unknown........................................... 69 3.4 ASimulationComparisonofEBLUPs ................... ...... 74 3.5 Prediction for Multivariate Output Simulators .. ............ 87 3.6 ChapterNotes .................................... ......... 88 3.6.1 ProofThat(3.2.21)IsaBLUP(page66).............. ... 88 3.6.2 ProofThat(3.3.4)IsaBLUP(page68)............... ... 89 3.6.3 ImplementationIssues .......................... ...... 90 3.6.4 AlternatePredictors ........................... ....... 92 4 Bayesian Prediction of Computer Simulation Output ............... 93 4.1 PredictiveDistributions ......................... ............ 93 4.1.1 Introduction.................................. ....... 93 2 4.1.2 Predictive Distributions When σZ , R, and r0 AreKnown ... 94 4.1.3 Predictive Distributions When R and r0 AreKnown .......100 4.1.4 Prediction Distributions When Correlation Parameters Are Unknown...........................................102 5 Space-Filling Designs for Computer Experiments ..................107 5.1 Introduction.................................... ...........107 5.1.1 Some Basic Principles of ExperimentalDesign . .... .. ....107 5.1.2 Design Strategies for Computer Experiments ... .... .....110 5.2 Designs Based on Methods for Selecting Random Samples. ......112 5.2.1 Designs Generated by Elementary Methods for SelectingSamples ................................... 113 5.2.2 Designs Generated by Latin Hypercube Sampling . ...114 5.2.3 PropertiesofSampling-BasedDesigns ............. .....119 5.2.4 ExtensionsofLatinHypercubeDesigns ............. ....122 5.3 Latin Hypercube Designs Satisfying Additional Criteria ..........125 5.3.1 Orthogonal Array-Based Latin Hypercube Designs . ....125 5.3.2 OrthogonalLatinHypercubeDesigns ............... ....127 5.3.3 SymmetricLatinHypercubeDesigns................ ....130 5.4 DesignsBasedonMeasuresofDistance ................ .......132 5.5 Distance-based Designs for Non-rectangular Regions . ..........138 5.6 Designs Obtained from Quasi-Random Sequences . .......141 5.7 UniformDesigns.................................. .........146 5.8 ChapterNotes .................................... .........152 Contents xiii 5.8.1 ProofThat TL isUnbiasedandofTheorem5.1 ...........152 5.8.2 TheUseofLHDsinaRegressionSetting .............. .. 157 5.8.3 OtherSpace-FillingDesigns ..................... ......158 7 Sensitivity Analysis and Variable Screening .......................161 7.1 Introduction.................................... ...........161 7.2 Classical Approaches to Sensitivity Analysis. ............162 7.2.1 Sensitivity Analysis Based on Scatterplots and Correlations. 162 7.2.2 Sensitivity Analysis Based on Regression Modeling . .....163 7.3 Sensitivity Analysis Based on Elementary Effects................166 7.4 Global Sensitivity Analysis Based on a Functional ANOVA Decomposition...................................... .......169 7.4.1 MainEffect and Joint EffectFunctions ..................171 7.4.2 FunctionalANOVADecomposition.................. ...175 7.4.3 GlobalSensitivityIndices ...................... .......178 7.5 Estimating Effect Plots and Global Sensitivity Indices .. .... ..... 185 7.5.1 Estimated EffectPlots ................................186 7.5.2 EstimationofSensitivityIndices ................ .......189 7.5.3 Process-based Estimators of Sensitivity Indices . ........192 7.5.4 Process-based estimators of sensitivity indices . .........194 7.5.5 Formulae for the Gaussian correlation function . ......197 7.5.6 Formulae using the Bohman correlation function .. .....198 7.6 VariableSelection ............................... ...........200 7.7 ChapterNotes .................................... .........200 7.7.1 Elementary Effects...................................200 7.7.2 OrthogonalityofSobol´Terms .................... .....201 7.7.3 Sensitivity Index Estimators for Regression Means . ......203 A List of Notation ................................................205 A.1 Abbreviations................................... ...........205 A.2 Symbols ......................................... .........206 B Mathematical Facts ............................................209 B.1 The MultivariateNormalDistribution............... ...........209 B.2 The Non-Central Student t Distribution ........................211 B.3 SomeResultsfromMatrixAlgebra.................... ........212 References......................................... ............214 To Gail, Aparna, and Claudia for their encouragement and patience 1 Preface Use the template preface.tex together with the Springer document class SVMono (monograph-type books) or SVMult (edited books) to style your preface in the Springer layout. A preface is a book’s preliminary statement, usually written by the author or ed- itor of a work, which states its origin, scope, purpose, plan, and intended audience, and which sometimes includes afterthoughts and acknowledgments of assistance. When written by a person other than the author, it is called a foreword. The preface or foreword is distinct from the introduction, which deals with the subject of the work. Customarily acknowledgments are included as last part of the preface. Place(s), Firstname Surname month year Firstname Surname – 1 Chapter 1 Physical Experiments and Computer Experiments 1.1 Introduction Historically, there has been extensive use of both physical experiments and, later, stochastic simulation experiments in order to determine the impact of input variables on outputs of scientific, engineering,or teachnological importance. In the past 15 to 20 years, there has been an increasing use of computer codes to infer the effect of input variables on such outputs. To explain their genesis, sup- pose that a mathematical theory