Theory and application of uniform experimental designs

Table of contents

1.Introduction
1.1.Experiments
1.1.1.Examples
1.1.2.Experimental Characteristics
1.1.3.Type of Experiments
1.2.Basic Terminologies Used
1.3.Statistical Models
1.3.1.Factorial Designs and ANOVA Models
1.3.2.Fractional Factorial Designs
1.3.3.Linear Regression Models
1.3.4.Nonparametric Regression Models
1.3.5.Robustness of Regression Models
1.4.Word-Length Pattern: Resolution and Minimum Aberration
1.4.1.Ordering
1.4.2.Defining Relation
1.4.3.Word-Length Pattern and Resolution
1.4.4.Minimum Aberration Criterion and Its Extension
1.5.Implementation of Uniform Designs for Multifactor Experiments
1.6.Applications of the Uniform Design
Exercises
References
2.Uniformity Criteria
2.1.Overall Mean Model
2.2.Star Discrepancy
2.2.1.Definition
2.2.2.Properties
2.3.Generalized L2-Discrepancy
2.3.1.Definition
2.3.2.Centered La -Discrepancy
2.3.3.Wrap-around L2-Discrepancy
Contents note continued: 2.3.4.Some Discussion on CD and WD
2.3.5.Mixture Discrepancy
2.4.Reproducing Kernel for Discrepancies
2.5.Discrepancies for Finite Numbers of Levels
2.5.1.Discrete Discrepancy
2.5.2.Lee Discrepancy
2.6.Lower Bounds of Discrepancies
2.6.1.Lower Bounds of the Centered L2-Discrepancy
2.6.2.Lower Bounds of the Wrap-around L2-Discrepancy
2.6.3.Lower Bounds of Mixture Discrepancy
2.6.4.Lower Bounds of Discrete Discrepancy
2.6.5.Lower Bounds of Lee Discrepancy
3.Construction of Uniform Designs
-Deterministic Methods
3.1.Uniform Design Tables
3.1.1.Background of Uniform Design Tables
3.1.2.One-Factor Uniform Designs
3.2.Uniform Designs with Multiple Factors
3.2.1.Complexity of the Construction
3.2.2.Remarks
3.3.Good Lattice Point Method and Its Modifications
3.3.1.Good Lattice Point Method
3.3.2.The Leave-One-Out glpm
3.3.3.Good Lattice Point with Power Generator
Contents note continued: 3.4.The Cutting Method
3.5.Linear Level Permutation Method
3.6.Combinatorial Construction Methods
3.6.1.Connection Between Uniform Designs and Uniformly Resolvable Designs
3.6.2.Construction Approaches via Combinatorics
3.6.3.Construction Approach via Saturated Orthogonal Arrays
3.6.4.Further Results
4.Construction of Uniform Designs
-Algorithmic Optimization Methods
4.1.Numerical Search for Uniform Designs
4.2.Threshold-Accepting Method
4.3.Construction Method Based on Quadratic Form
4.3.1.Quadratic Forms of Discrepancies
4.3.2.Complementary Design Theory
4.3.3.Optimal Frequency Vector
4.3.4.Integer Programming Problem Method
5.Modeling Techniques
5.1.Basis Functions
5.1.1.Polynomial Regression Models
5.1.2.Spline Basis
5.1.3.Wavelets Basis
5.1.4.Radial Basis Functions
5.1.5.Selection of Variables
Contents note continued: 5.2.Modeling Techniques: Kriging Models
5.2.1.Models
5.2.2.Estimation
5.2.3.Maximum Likelihood Estimation
5.2.4.Parametric Empirical Kriging
5.2.5.Examples and Discussion
5.3.A Case Study on Environmental Data
-Model Selection
6.Connections Between Uniformity and Other Design Criteria
6.1.Uniformity and Isomorphism
6.2.Uniformity and Orthogonality
6.3.Uniformity and Confounding
6.4.Uniformity and Aberration
6.5.Projection Uniformity and Related Criteria
6.5.1.Projection Discrepancy Pattern and Related Criteria
6.5.2.Uniformity Pattern and Related Criteria
6.6.Majorization Framework
6.6.1.Based on Pairwise Coincidence Vector
6.6.2.Minimum Aberration Majorization
7.Applications of Uniformity in Other Design Types
7.1.Uniformity in Block Designs
7.1.1.Uniformity in BIBDs
7.1.2.Uniformity in PRIBDs
7.1.3.Uniformity in POTBs
Contents note continued: 7.2.Uniformity in Supersaturated Designs
7.2.1.Uniformity in Two-Level SSDs
7.2.2.Uniformity in Mixed-Level SSDs
7.3.Uniformity in Sliced Latin Hypercube Designs
7.3.1.A Combined Uniformity Measure
7.3.2.Optimization Algorithms
7.3.3.Determination of the Weight ω
7.4.Uniformity Under Errors in the Level Values
8.Uniform Design for Experiments with Mixtures
8.1.Introduction to Design with Mixture
8.1.1.Some Types of Designs with Mixtures
8.1.2.Criteria for Designs with Mixtures
8.2.Uniform Designs of Experiments with Mixtures
8.2.1.Discrepancy for Designs with Mixtures
8.2.2.Construction Methods for Uniform Mixture Design
8.2.3.Uniform Design with Restricted Mixtures
8.2.4.Uniform Design on Irregular region
8.3.Modeling Technique for Designs with Mixtures
References.

The book provides necessary knowledge for readers interested in developing the theory of uniform experimental design. It discusses measures of uniformity, various construction methods of uniform designs, modeling techniques, design and modeling for experiments with mixtures, and the usefulness of the uniformity in block, factorial and supersaturated designs.

Experimental design is an important branch of statistics with a long history, and is extremely useful in multi-factor experiments. Involving rich methodologies and various designs, it has played a key role in industry, technology, sciences and various other fields. A design that chooses experimental points uniformly scattered on the domain is known as uniform experimental design, and uniform experimental design can be regarded as a fractional factorial design with model uncertainty, a space-filling design for computer experiments, a robust design against the model specification, and a supersaturated design and can be applied to experiments with mixtures.

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