Design And Modeling for Computer Experiments

by Fang; Kai-Tai

• ISBN13: 9781584885467
• ISBN10: 1584885467
• Edition: 1st
• Format: Hardcover
• Copyright: 2005-10-14
• Publisher: Chapman & Hall/

Summary

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 andmathematics 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.

Table of Contents

Part I An Overview	1	(44)
1 Introduction	3	(42)
1.1 Experiments and Their Statistical Designs	3	(1)
1.2 Some Concepts in Experimental Design	4	(6)
1.3 Computer Experiments	10	(10)
1.3.1 Motivations	10	(2)
1.3.2 Metamodels	12	(4)
1.3.3 Computer Experiments in Engineering	16	(4)
1.4 Examples of Computer Experiments	20	(4)
1.5 Space-Filling Designs	24	(2)
1.6 Modeling Techniques	26	(5)
1.7 Sensitivity Analysis	31	(2)
1.8 Strategies for Computer Experiments and an Illustration Case Study	33	(5)
1.9 Remarks on Computer Experiments	38	(2)
1.10 Guidance for Reading This Book	40	(5)
Part II Designs for Computer Experiments	45	(80)
2 Latin Hypercube Sampling and Its Modifications	47	(20)
2.1 Latin Hypercube Sampling	47	(4)
2.2 Randomized Orthogonal Array	51	(3)
2.3 Symmetric and Orthogonal Column Latin Hypercubes	54	(6)
2.4 Optimal Latin Hypercube Designs	60	(7)
2.4.1 IMSE Criterion	60	(2)
2.4.2 Entropy Criterion	62	(2)
2.4.3 Minimax and Maximin Distance Criteria and Their extension	64	(1)
2.4.4 Uniformity Criterion	65	(2)
3 Uniform Experimental Design	67	(38)
3.1 Introduction	67	(1)
3.2 Measures of Uniformity	68	(10)
3.2.1 The Star Lp-Discrepancy	68	(2)
3.2.2 Modified L2-Discrepancy	70	(1)
3.2.3 The Centered Discrepancy	71	(1)
3.2.4 The Wrap-Around Discrepancy	72	(1)
3.2.5 A Unified Definition of Discrepancy	73	(2)
3.2.6 Descrepancy for Categorical Factors	75	(1)
3.2.7 Applications of Uniformity in Experimental Designs	76	(2)
3.3 Construction of Uniform Designs	78	(12)
3.3.1 One-Factor Uniform Designs	78	(1)
3.3.2 Symmetrical Uniform Designs	79	(1)
3.3.3 Good Lattice Point Method	80	(5)
3.3.4 Latin Square Method	85	(1)
3.3.5 Expanding Orthogonal Array Method	86	(1)
3.3.6 The Cutting Method	86	(4)
3.3.7 Construction of Uniform Designs by Optimization	90	(1)
3.4 Characteristics of the Uniform Design: Admissibility, Minimaxity, and Robustness	90	(3)
3.5 Construction of Uniform Designs via Resolvable Balanced Incomplete Block Designs	93	(4)
3.5.1 Resolvable Balanced Incomplete Block Designs	93	(1)
3.5.2 RBIBD Construction Method	94	(1)
3.5.3 New Uniform Designs	94	(3)
3.6 Construction of Asymmetrical Uniform Designs	97	(8)
3.6.1 Pseudo-Level Technique	97	(1)
3.6.2 Collapsing Method	97	(3)
3.6.3 Combinatorial Method	100	(3)
3.6.4 Miscellanea	103	(2)
4 Optimization in Construction of Designs for Computer Experiments	105	(20)
4.1 Optimization Problem in Construction of Designs	105	(8)
4.1.1 Algorithmic Construction	106	(1)
4.1.2 Neighborhood	106	(1)
4.1.3 Replacement Rule	107	(2)
4.1.4 Iteration Formulae	109	(4)
4.2 Optimization Algorithms	113	(4)
4.2.1 Algorithms	113	(1)
4.2.2 Local Search Algorithm	114	(1)
4.2.3 Simulated Annealing Algorithm	115	(1)
4.2.4 Threshold Accepting Algorithm	115	(1)
4.2.5 Stochastic Evolutionary Algorithm	116	(1)
4.3 Lower Bounds of the Discrepancy and Related Algorithm	117	(10)
4.3.1 Lower Bounds of the Categorical Discrepancy	119	(1)
4.3.2 Lower Bounds of the Wrap-Around L2-Discrepancy	119	(2)
4.3.3 Lower Bounds of the Centered L2-Discrepancy	121	(1)
4.3.4 Balance-Pursuit Heuristic Algorithm	122	(3)
Part III Modeling for Computer Experiments	125	(116)
5 Metamodeling	127	(60)
5.1 Basic Concepts	127	(6)
5.1.1 Mean Square Error and Prediction Error	127	(3)
5.1.2 Regularization	130	(3)
5.2 Polynomial Models	133	(6)
5.3 Spline Method	139	(6)
5.3.1 Construction of Spline Basis	140	(2)
5.3.2 An Illustration	142	(2)
5.3.3 Other Bases of Global Approximation	144	(1)
5.4 Gaussian Kriging Models	145	(14)
5.4.1 Prediction via Kriging	146	(1)
5.4.2 Estimation of Parameters	147	(6)
5.4.3 A Case Study	153	(6)
5.5 Bayesian Approach	159	(8)
5.5.1 Gaussian Processes	159	(1)
5.5.2 Bayesian Prediction of Deterministic Functions	160	(2)
5.5.3 Use of Derivatives in Surface Prediction	162	(3)
5.5.4 An Example: Borehole Model	165	(2)
5.6 Neural Network	167	(13)
5.6.1 Multi-Layer Perceptron Networks	168	(4)
5.6.2 A Case Study	172	(5)
5.6.3 Radial Basis Functions	177	(3)
5.7 Local Polynomial Regression	180	(4)
5.7.1 Motivation of Local Polynomial Regression	180	(3)
5.7.2 Metamodeling via Local Polynomial Regression	183	(1)
5.8 Some Recommendations	184	(3)
5.8.1 Connections	184	(1)
5.8.2 Recommendations	185	(2)
6 Model Interpretation	187	(20)
6.1 Introduction	187	(1)
6.2 Sensitivity Analysis Based on Regression Analysis	188	(5)
6.2.1 Criteria	188	(3)
6.2.2 An Example	191	(2)
6.3 Sensitivity Analysis Based on Variation Decomposition	193	(14)
6.3.1 Functional ANOVA Representation	193	(2)
6.3.2 Computational Issues	195	(3)
6.3.3 Example of Sobol' Global Sensitivity	198	(1)
6.3.4 Correlation Ratios and Extension of Sobol' Indices	199	(3)
6.3.5 Fourier Amplitude Sensitivity Test	202	(3)
6.3.6 Example of FAST Application	205	(2)
7 Functional Response	207	(34)
7.1 Computer Experiments with Functional Response	207	(8)
7.2 Spatial Temporal Models	215	(4)
7.2.1 Functional Response with Sparse Sampling Rate	215	(3)
7.2.2 Functional Response with Intensive Sampling Rate	218	(1)
7.3 Penalized Regression Splines	219	(3)
7.4 Functional Linear Models	222	(8)
7.4.1 A Graphical Tool	223	(1)
7.4.2 Efficient Estimation Procedure	224	(2)
7.4.3 An Illustration	226	(4)
7.5 Semiparametric Regression Models	230	(14)
7.5.1 Partially Linear Model	230	(4)
7.5.2 Partially Functional Linear Models	234	(2)
7.5.3 An Illustration	236	(5)
Appendix	241	(20)
A.1 Some Basic Concepts in Matrix Algebra	241	(3)
A.2 Some Concepts in Probability and Statistics	244	(5)
A.2.1 Random Variables and Random Vectors	244	(3)
A.2.2 Some Statistical Distributions and Gaussian Process	247	(2)
A.3 Linear Regression Analysis	249	(7)
A.3.1 Linear Models	250	(1)
A.3.2 Method of Least Squares	251	(1)
A.3.3 Analysis of Variance	252	(1)
A.3.4 An Illustration	253	(3)
A.4 Variable Selection for Linear Regression Models	256	(5)
A.4.1 Nonconvex Penalized Least Squares	257	(1)
A.4.2 Iteratively Ridge Regression Algorithm	258	(1)
A.4.3 An Illustration	259	(2)
Acronyms	261	(2)
References	263	(20)
Index	283	(4)
Author Index	287