Design and Modeling for Computer Experiments (Chapman & Hall/CRC Computer Science & Data Analysis)

Computer simulations based on mathematical models have become ubiquitous across the engineering disciplines and throughout the physical sciences. Successful use of a simulation model, however, requires careful interrogation of the model through systematic computer experiments. While specific theoretical/mathematical examinations of computer experiment design are available, those interested in applying proposed methodologies need a practical presentation and straightforward guidance on analyzing and interpreting experiment results. Written by authors with strong academic reputations and real-world practical experience, Design and Modeling for Computer Experiments is exactly the kind of treatment you need. The authors blend a sound, modern statistical approach with extensive engineering applications and clearly delineate the steps required to successfully model a problem and provide an analysis that will help find the solution. Part I introduces the design and modeling of computer experiments and the basic concepts used throughout the book. Part II focuses on the design of computer experiments. The authors present the most popular space-filling designs - like Latin hypercube sampling and its modifications and uniform design - including their definitions, properties, construction and related generating algorithms. Part III discusses the modeling of data from computer experiments. Here the authors present various modeling techniques and discuss model interpretation, including sensitivity analysis. An appendix reviews the statistics and mathematics concepts needed, and numerous examples clarify the techniques and their implementation. The complexity of real physical systems means that there is usually no simple analytic formula that sufficiently describes the phenomena. Useful both as a textbook and professional reference, this book presents the techniques you need to design and model computer experiments for practical problem solving. FM c5467_fm.pdf 1 Design and Modeling for Computer Experiments 3 Preface 5 Contents 8 C5467_CH01 12 Contents -1 Part I An Overview 12 Chapter 1 Introduction 13 1.1 Experiments and Their Statistical Designs 13 1.2 Some Concepts in Experimental Design 14 1.3 Computer Experiments 20 1.3.1 Motivations 20 1.3.2 Metamodels 22 1.3.3 Computer experiments in engineering 26 1.4 Examples of Computer Experiments 30 1.5 Space-Filling Designs 34 1.6 Modeling Techniques 36 1.7 Sensitivity Analysis 41 1.8 Strategies for Computer Experiments and an Illustration Case Study 43 1.9 Remarks on Computer Experiments 48 1.10 Guidance for Reading This Book 50 C5467_CH02 54 Contents -1 Part II Designs for Computer Experiments 54 Chapter 2 Latin Hypercube Sampling and Its Modifications 55 2.1 Latin Hypercube Sampling 55 2.2 Randomized Orthogonal Array 59 2.3 Symmetric and Orthogonal Column Latin Hypercubes 62 2.4 Optimal Latin Hypercube Designs 68 2.4.1 IMSE criterion 68 2.4.2 Entropy criterion 70 2.4.3 Minimax and maximin distance criteria and their extension 72 2.4.4 Uniformity criterion 73 C5467_CH03 75 Contents -1 Chapter 3 Uniform Experimental Design 75 3.1 Introduction 75 3.2 Measures of Uniformity 76 3.2.1 The star Lp-discrepancy 76 3.2.2 Modified L2-discrepancy 78 3.2.3 The centered discrepancy 79 3.2.4 The wrap-around discrepancy 80 3.2.5 A unified definition of discrepancy 81 3.2.6 Descrepancy for categorical factors 83 3.2.7 Applications of uniformity in experimental designs 84 3.3 Construction of Uniform Designs 86 3.3.1 One-factor uniform designs 86 3.3.2 Symmetrical uniform designs 87 3.3.3 Good lattice point method 88 3.3.4 Latin square method 93 3.3.5 Expanding orthogonal array method 94 3.3.6 The cutting method 94 3.3.7 Construction of uniform designs by optimization 98 3.4 Characteristics of the Uniform Design: Admissibility, Minimaxity, and Robustness 98 3.5 Construction of Uniform Designs via Resolvable Balanced Incomplete Block Designs 101 3.5.1 Resolvable balanced incomplete block designs 101 3.5.2 RBIBD construction method 102 3.5.3 New uniform designs 102 3.6 Construction of Asymmetrical Uniform Designs 105 3.6.1 Pseudo-level technique 105 3.6.2 Collapsing method 105 3.6.3 Combinatorial method 108 3.6.4 Miscellanea 111 C5467_CH04 113 Contents -1 Chapter 4 Optimization in Construction of Designs for Computer Experiments 113 4.1 Optimization Problem in Construction of Designs 113 4.1.1 Algorithmic construction 114 4.1.2 Neighborhood 114 4.1.3 Replacement rule 115 4.1.4 Iteration formulae 117 4.2 Optimization Algorithms 121 4.2.1 Algorithms 121 4.2.2 Local search algorithm 122 4.2.3 Simulated annealing algorithm 123 4.2.4 Threshold accepting algorithm 123 4.2.5 Stochastic evolutionary algorithm 124 4.3 Lower bounds of the discrepancy and related algorithm 125 4.3.1 Lower bounds of the categorical discrepancy 127 4.3.2 Lower bounds of the wrap-around L2-discrepancy 127 4.3.3 Lower bounds of the centered L2-discrepancy 129 4.3.4 Balance-pursuit heuristic algorithm 130 C5467_CH05 133 Contents -1 Part III Modeling for Computer Experiments 133 Chapter 5 Metamodeling 134 5.1 Basic Concepts 134 5.1.1 Mean square error and prediction error 134 5.1.2 Regularization 137 5.2 Polynomial Models 140 5.3 Spline Method 146 5.3.1 Construction of spline basis 147 5.3.2 An illustration 149 5.3.3 Other bases of global approximation 151 5.4 Gaussian Kriging Models 152 5.4.1 Prediction via Kriging 153 5.4.2 Estimation of parameters 154 5.4.3 A case study 160 5.5 Bayesian Approach 166 5.5.1 Gaussian processes 166 5.5.2 Bayesian prediction of deterministic functions 167 5.5.3 Use of derivatives in surface prediction 169 5.5.4 An example: borehole model 172 5.6 Neural Network 174 5.6.1 Multi-layer perceptron networks 175 5.6.2 A case study 179 5.6.3 Radial basis functions 184 5.7 Local Polynomial Regression 187 5.7.1 Motivation of local polynomial regression 187 5.7.2 Metamodeling via local polynomial regression 190 5.8 Some Recommendations 191 5.8.1 Connections 191 5.8.2 Recommendations 192 C5467_CH06 194 Contents -1 Chapter 6 Model Interpretation 194 6.1 Introduction 194 6.2 Sensitivity Analysis Based on Regression Analysis 195 6.2.1 Criteria 195 6.2.2 An example 198 6.3 Sensitivity Analysis Based on Variation Decomposition 200 6.3.1 Functional ANOVA representation 200 6.3.2 Computational issues 202 6.3.3 Example of Sobol’ global sensitivity 205 6.3.4 Correlation ratios and extension of Sobol’ indices 206 6.3.5 Fourier amplitude sensitivity test 209 6.3.6 Example of FAST application 212 C5467_CH07 214 Contents -1 Chapter 7 Functional Response 214 7.1 Computer Experiments with Functional Response 214 7.2 Spatial Temporal Models 222 7.2.1 Functional response with sparse sampling rate 222 7.2.2 Functional response with intensive sampling rate 225 7.3 Penalized Regression Splines 226 7.4 Functional Linear Models 229 7.4.1 A graphical tool 230 7.4.2 E.cient estimation procedure 231 7.4.3 An illustration 233 7.5 Semiparametric Regression Models 237 7.5.1 Partially linear model 237 7.5.2 Partially functional linear models 241 7.5.3 An illustration 243 C5467_Acron 248 Contents -1 Acronyms 248 C5467_Ref 250 Contents -1 References 250 C5467_APP 269 Contents -1 Appendix 269 A.1 Some Basic Concepts in Matrix Algebra 269 A.2 Some Concepts in Probability and Statistics 272 A.2.1 Random variables and random vectors 272 A.2.2 Some statistical distributions and Gaussian process 275 A.3 Linear Regression Analysis 277 A.3.1 Linear models 278 A.3.2 Method of least squares 279 A.3.3 Analysis of variance 280 A.3.4 An illustration 281 A.4 Variable Selection for Linear Regression Models 284 A.4.1 Nonconvex penalized least squares 285 A.4.2 Iteratively ridge regression algorithm 286 A.4.3 An illustration 287 "Computer simulations based on mathematical models have become ubiquitous across the engineering disciplines and throughout the physical sciences. Accuracy in a simulation, however, requires careful interrogation of the model through systematic computer experiments. This book provides the practical presentation of the techniques and straightforward guidance on analyzing experiment results needed by those interested in applying the methodologies discussed in other, more theoretical treatments."--BOOK JACKET