Theory and Application of Uniform Experimental Designs (eBook)

Kai-Tai Fang is a professor at the BNU-HKBU United International college and is a research professor at the Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China. Min-Qian Liu is a professor at the Institute of Statistics, Nankai University, Tianjin, China Hong Qin is a professor at the Faculty of Mathematics and Statistics, Central China Normal University, Wuhan, China Yong-Dao Zhou is a professor at the Institute of Statistics, Nankai University, Tianjin, China

Foreword 6
Preface 9
References 11
Contents 12
1 Introduction 16
1.1 Experiments 16
1.1.1 Examples 17
1.1.2 Experimental Characteristics 20
1.1.3 Type of Experiments 22
1.2 Basic Terminologies Used 24
1.3 Statistical Models 27
1.3.1 Factorial Designs and ANOVA Models 28
1.3.2 Fractional Factorial Designs 31
1.3.3 Linear Regression Models 34
1.3.4 Nonparametric Regression Models 38
1.3.5 Robustness of Regression Models 40
1.4 Word-Length Pattern: Resolution and Minimum Aberration 41
1.4.1 Ordering 41
1.4.2 Defining Relation 42
1.4.3 Word-Length Pattern and Resolution 44
1.4.4 Minimum Aberration Criterion and Its Extension 45
1.5 Implementation of Uniform Designs for Multifactor Experiments 47
1.6 Applications of the Uniform Design 52
References 55
2 Uniformity Criteria 58
2.1 Overall Mean Model 58
2.2 Star Discrepancy 61
2.2.1 Definition 61
2.2.2 Properties 63
2.3 Generalized L2-Discrepancy 67
2.3.1 Definition 68
2.3.2 Centered L2-Discrepancy 69
2.3.3 Wrap-around L2-Discrepancy 71
2.3.4 Some Discussion on CD and WD 72
2.3.5 Mixture Discrepancy 76
2.4 Reproducing Kernel for Discrepancies 79
2.5 Discrepancies for Finite Numbers of Levels 85
2.5.1 Discrete Discrepancy 86
2.5.2 Lee Discrepancy 88
2.6 Lower Bounds of Discrepancies 89
2.6.1 Lower Bounds of the Centered L2-Discrepancy 91
2.6.2 Lower Bounds of the Wrap-around L2-Discrepancy 94
2.6.3 Lower Bounds of Mixture Discrepancy 101
2.6.4 Lower Bounds of Discrete Discrepancy 106
2.6.5 Lower Bounds of Lee Discrepancy 109
References 114
3 Construction of Uniform Designs—Deterministic Methods 116
3.1 Uniform Design Tables 117
3.1.1 Background of Uniform Design Tables 117
3.1.2 One-Factor Uniform Designs 122
3.2 Uniform Designs with Multiple Factors 124
3.2.1 Complexity of the Construction 124
3.2.2 Remarks 125
3.3 Good Lattice Point Method and Its Modifications 130
3.3.1 Good Lattice Point Method 130
3.3.2 The Leave-One-Out glpm 132
3.3.3 Good Lattice Point with Power Generator 136
3.4 The Cutting Method 137
3.5 Linear Level Permutation Method 139
3.6 Combinatorial Construction Methods 144
3.6.1 Connection Between Uniform Designs and Uniformly Resolvable Designs 144
3.6.2 Construction Approaches via Combinatorics 148
3.6.3 Construction Approach via Saturated Orthogonal Arrays 160
3.6.4 Further Results 162
References 167
4 Construction of Uniform Designs—Algorithmic Optimization Methods 170
4.1 Numerical Search for Uniform Designs 170
4.2 Threshold-Accepting Method 173
4.3 Construction Method Based on Quadratic Form 181
4.3.1 Quadratic Forms of Discrepancies 182
4.3.2 Complementary Design Theory 183
4.3.3 Optimal Frequency Vector 187
4.3.4 Integer Programming Problem Method 192
References 195
5 Modeling Techniques 198
5.1 Basis Functions 199
5.1.1 Polynomial Regression Models 199
5.1.2 Spline Basis 203
5.1.3 Wavelets Basis 204
5.1.4 Radial Basis Functions 205
5.1.5 Selection of Variables 206
5.2 Modeling Techniques: Kriging Models 206
5.2.1 Models 207
5.2.2 Estimation 209
5.2.3 Maximum Likelihood Estimation 210
5.2.4 Parametric Empirical Kriging 211
5.2.5 Examples and Discussion 212
5.3 A Case Study on Environmental Data—Model Selection 215
References 222
6 Connections Between Uniformity and Other Design Criteria 224
6.1 Uniformity and Isomorphism 224
6.2 Uniformity and Orthogonality 229
6.3 Uniformity and Confounding 233
6.4 Uniformity and Aberration 236
6.5 Projection Uniformity and Related Criteria 243
6.5.1 Projection Discrepancy Pattern and Related Criteria 243
6.5.2 Uniformity Pattern and Related Criteria 246
6.6 Majorization Framework 247
6.6.1 Based on Pairwise Coincidence Vector 247
6.6.2 Minimum Aberration Majorization 249
References 254
7 Applications of Uniformity in Other Design Types 257
7.1 Uniformity in Block Designs 257
7.1.1 Uniformity in BIBDs 257
7.1.2 Uniformity in PRIBDs 258
7.1.3 Uniformity in POTBs 259
7.2 Uniformity in Supersaturated Designs 261
7.2.1 Uniformity in Two-Level SSDs 262
7.2.2 Uniformity in Mixed-Level SSDs 263
7.3 Uniformity in Sliced Latin Hypercube Designs 264
7.3.1 A Combined Uniformity Measure 265
7.3.2 Optimization Algorithms 266
7.3.3 Determination of the Weight ? 267
7.4 Uniformity Under Errors in the Level Values 269
References 274
8 Uniform Design for Experiments with Mixtures 276
8.1 Introduction to Design with Mixture 276
8.1.1 Some Types of Designs with Mixtures 278
8.1.2 Criteria for Designs with Mixtures 281
8.2 Uniform Designs of Experiments with Mixtures 283
8.2.1 Discrepancy for Designs with Mixtures 283
8.2.2 Construction Methods for Uniform Mixture Design 286
8.2.3 Uniform Design with Restricted Mixtures 289
8.2.4 Uniform Design on Irregular region 293
8.3 Modeling Technique for Designs with Mixtures 298
References 308
Subject Index 309