Contents
1 Introduction 1
1.1 Experiments 1
1.1.1 Examples 2
1.1.2 Experimental Characteristics 5
1.1.3 Typeof Experiments 7
1.2 Basic TerminologiesUsed 9
1.3 StatisticalModels 12
1.3.1 Factorial DesignsandANOVA Models 13
1.3.2 FractionalFactorialDesigns 16
1.3.3 LinearRegressionModels 19
1.3.4 NonparametricRegression Models 23
1.3.5 RobustnessofRegressionModels 25
1.4 Word-Length Pattern: Resolution and Minimum Aberration 26
1.4.1 Ordering 26
1.4.2 De.ning Relation 27
1.4.3 Word-Length PatternandResolution 29
1.4.4 Minimum Aberration Criterion and Its Extension 30
1.5 Implementation of Uniform Designs for Multifactor Experiments 32
1.6 Applicationsofthe UniformDesign 37
Exercises 37
References 40
2 Uniformity Criteria 43
2.1 OverallMeanModel 43
2.2 StarDiscrepancy 46
2.2.1 De.nition 46
2.2.2 Properties 48
2.3 Generalized L2-Discrepancy 52
2.3.1 De.nition 53
2.3.2 Centered L2-Discrepancy 54
2.3.3 Wrap-around L2-Discrepancy 56
2.3.4 Some Discussion onCDandWD 57
2.3.5 Mixture Discrepancy 61
2.4 Reproducing Kernelfor Discrepancies 64
2.5 Discrepanciesfor FiniteNumbersof Levels 70
2.5.1 DiscreteDiscrepancy 71
2.5.2 Lee Discrepancy 73
2.6 LowerBoundsof Discrepancies 74
2.6.1 Lower Bounds of the Centered L2-Discrepancy 76
2.6.2 Lower Bounds of the Wrap-around L2-Discrepancy 79
2.6.3 LowerBoundsof Mixture Discrepancy 86
2.6.4 Lower Bounds of Discrete Discrepancy 91
2.6.5 LowerBoundsofLee Discrepancy 94
Exercises 97
References 99
3 Construction of Uniform Designs—Deterministic Methods 101
3.1 UniformDesignTables 102
3.1.1 Backgroundof Uniform Design Tables 102
3.1.2 One-Factor UniformDesigns 107
3.2 UniformDesignswithMultiple Factors 109
3.2.1 Complexityofthe Construction 109
3.2.2 Remarks 110
3.3 Good Lattice Point Method and Its Modi.cations 115
3.3.1 GoodLatticePoint Method 115
3.3.2 The Leave-One-Out glpm 117
3.3.3 Good Lattice Point with Power Generator 121
3.4 The CuttingMethod 122
3.5 LinearLevelPermutationMethod 124
3.6 CombinatorialConstructionMethods 129
3.6.1 Connection Between Uniform Designs and Uniformly ResolvableDesigns 129
3.6.2 Construction Approaches via Combinatorics 133
3.6.3 Construction Approach via Saturated Orthogonal Arrays 145
3.6.4 FurtherResults 147
Exercises 149
References 152
4 Construction of Uniform Designs—Algorithmic Optimization Methods 155
4.1 NumericalSearchforUniformDesigns 155
4.2 Threshold-AcceptingMethod 158
4.3 Construction Method Based on Quadratic Form 166
4.3.1 Quadratic Formsof Discrepancies 167
4.3.2 ComplementaryDesignTheory 168
4.3.3 OptimalFrequencyVector 172
4.3.4 Integer Programming Problem Method 177
Exercises 179
References 180
5 Modeling Techniques 183
5.1 BasisFunctions 184
5.1.1 PolynomialRegressionModels 184
5.1.2 SplineBasis 188
5.1.3 WaveletsBasis 189
5.1.4 RadialBasis Functions 190
5.1.5 SelectionofVariables 191
5.2 ModelingTechniques:KrigingModels 191
5.2.1 Models 192
5.2.2 Estimation 194
5.2.3 Maximum Likelihood Estimation 195
5.2.4 Parametric EmpiricalKriging 196
5.2.5 Examplesand Discussion 197
5.3 A Case Study on Environmental Data—Model Selection 200
Exercises 205
References 207
6 Connections Between Uniformity and Other Design Criteria 209
6.1 UniformityandIsomorphism 209
6.2 UniformityandOrthogonality 214
6.3 UniformityandConfounding 218
6.4 UniformityandAberration 221
6.5 Projection Uniformity and Related Criteria 228
6.5.1 Projection Discrepancy Pattern and Related Criteria 228
6.5.2 Uniformity Pattern and Related Criteria 231
6.6 MajorizationFramework 232
6.6.1 Based on Pairwise Coincidence Vector 232
6.6.2 MinimumAberration Majorization 234
Exercises 238
References 239
7 Applications of Uniformity in Other Design Types 243
7.1 UniformityinBlockDesigns 243
7.1.1 UniformityinBIBDs 243
7.1.2 Uniformityin PRIBDs 244
7.1.3 UniformityinPOTBs 245
7.2 UniformityinSupersaturatedDesigns 247
7.2.1 Uniformityin Two-LevelSSDs 248
7.2.2 UniformityinMixed-LevelSSDs 249
7.3 Uniformity in Sliced Latin Hypercube Designs 250
7.3.1 ACombined UniformityMeasure 251
7.3.2 OptimizationAlgorithms 252
7.3.3 Determination of the Weight x 253
7.4 Uniformity Under Errorsinthe Level Values 255
Exercises 258
References 260
8 Uniform Design for Experiments with Mixtures 263
8.1 IntroductiontoDesignwith Mixture 263
8.1.1 Some Typesof Designs with Mixtures 265
8.1.2 CriteriaforDesignswithMixtures 268
8.2 Uniform Designs of Experiments with Mixtures 270
8.2.1 Discrepancy for Designs with Mixtures 270
8.2.2 Construction Methods for Uniform Mixture Design 273
8.2.3 Uniform Design with Restricted Mixtures 276
8.2.4 UniformDesignon Irregular region 280
8.3 Modeling Technique for Designs with Mixtures 285
Exercises 292
References 295
Subject Index 297