Minimum moment aberration for nonregular designs and supersaturated designs
01 Jan 2001-
TL;DR: In this paper, a new combinatorial criterion, called minimum moment aberration, is proposed for assessing the goodness of nonregular designs and supersaturated designs, which is a good surrogate with tremendous computational advantages for many statistically justified criteria, such as minimum G2-aberrration, generalized minimum aberration and E(s2).
Abstract: Nonregular designs are used widely in experiments due to their run size economy and flexibility. These designs include the Plackett-Burman designs and many other symmetrical and asymmetrical orthogonal arrays. Supersaturated designs have become increasingly popular in recent years because of the potential in saving run size and its technical novelty. In this paper, a novel combinatorial criterion, called minimum moment aberration, is proposed for assessing the goodness of nonregular designs and supersaturated designs. The new criterion, which is to sequentially minimize the power moments of the number of coincidence among runs, is a good surrogate with tremendous computational advantages for many statistically justified criteria, such as minimum G2-aberrration, generalized minimum aberration and E(s2). In addition, the minimum moment aberration is conceptually simple and convenient for theoretical development. The general theory developed here not only unifies several separate results, but also provides many novel results on nonregular designs and supersaturated designs.
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TL;DR: Experimental design is reviewed here for broad classes of data collection and analysis problems, including: fractioning techniques based on orthogonal arrays, Latin hypercube designs and their variants for computer experimentation, efficient design for data mining and machine learning applications, and sequential design for active learning.
Abstract: Maximizing data information requires careful selection, termed design, of the points at which data are observed. Experimental design is reviewed here for broad classes of data collection and analysis problems, including: fractioning techniques based on orthogonal arrays, Latin hypercube designs and their variants for computer experimentation, efficient design for data mining and machine learning applications, and sequential design for active learning. © 2012 Wiley Periodicals, Inc. © 2012 Wiley Periodicals, Inc.
1,025 citations
01 Jan 2001
TL;DR: In this paper, a two-stage anal- ysis that employs factor screening, projection and response surface exploration is proposed to achieve the two objectives on the same experiment, based on one design.
Abstract: Standard practice in response surface methodology performs factor screen- ing and response surface exploration sequentially, using different designs. A novel approach is proposed to achieve the two objectives on the same experiment, based on one design. Running a uni-stage experiment has the advantages of saving ex- perimentation time and run size. The approach is based on a two-stage anal- ysis that employs factor screening, projection and response surface exploration. Projection-efficiency criteria are defined to evaluate the performance of the pro- jected designs. The projection-efficiency properties of the 3 n−k designs and three nonregular designs are studied, and comparisons with central composite designs are made. Nonregular designs appear to enjoy better projection properties. The strategy is illustrated with the analysis of a PVC insulation experiment.
91 citations
TL;DR: Important developments in optimality criteria and comparison are reviewed, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.
Abstract: Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main e�ffects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.
73 citations
Cites background or methods from "Minimum moment aberration for nonre..."
...Xu (2003) gave several sufficient conditions for a design to have minimum moment aberration and generalized minimum aberration among all possible designs....
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...For mixed-level designs, Xu (2003) suggested to weight each column according to its level, called natural weights, and replace δij(D) in (8) with the number of weighted coincidences....
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...As a theoretical tool, Xu (2003) developed a unified theory for nonregular and supersaturated designs....
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...These identities can be derived easily from the generalized Pless power moment identities developed by Xu (2003)....
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...This leads to the minimum moment aberration criterion (Xu (2003)) that is to sequentially minimize the power moments K1(D),K2(D), . . . ,Km(D)....
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TL;DR: In this article, a collection of useful fractional factorial designs with 27, 81, 243, and 729 runs is given, and a catalogue of designs would help experimenters choose the best design.
Abstract: A common problem that experimenters face is the choice of fractional factorial designs. Minimum aberration designs are commonly used in practice. There are situations in which other designs meet practical needs better. A catalogue of designs would help experimenters choose the best design. Based on coding theory, new methods are proposed to classify and rank fractional factorial designs efficiently. We have completely enumerated all 27 and 81-run designs, 243-run designs of resolution IV or higher, and 729-run designs of resolution V or higher. A collection of useful fractional factorial designs with 27, 81, 243 and 729 runs is given. This extends the work of Ch93, who gave a collection of fractional factorial designs with 16, 27, 32 and 64 runs.
72 citations
Cites background or methods from "Minimum moment aberration for nonre..."
...Xu (2001, 2003) proposed the minimum moment aberration criterion which sequentially minimizes K1,K2, . . . ,Kn....
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...The definition of Kt here differs from that in Xu (2001, 2003)....
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...In the following we introduce another convenient approach due to Xu (2001, 2003) that uses the Pless power moment identities (4)....
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...See Suen, Chen and Wu (1997), Xu and Wu (2001) and Xu (2003) for characterizing MA designs in terms of their complements....
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...(6) By applying the Pless power moment identities (4), Xu (2001, 2003) showed that the power moments Kt are linear combinations of A1, ..., At as follows....
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01 Jan 2002
TL;DR: In this paper, a set of optimality criteria is proposed to assess the performance of designs for factor screening, classification, and interaction detection, and a three-step approach was proposed to search for opti-mal designs.
Abstract: Orthogonal arrays are widely used in industrial experiments for factor screening. Suppose only a few of the factors are important. An orthogonal array can be used not only for screening factors but also for detecting interactions among a subset of active factors. In this article, a set of optimality criteria is proposed to assess the performance of designs for factor screening, pro jection, and interaction detection, and a three-step approach is proposed to search for opti- mal designs. Combinatorial and algorithmic construction methods are proposed for generating new designs. Level permutation methods are used for improving the eligibility and estimation efficiency of the pro jected designs. The techniques are then applied to search for best three-level designs with 18 and 27 runs. Many new, efficient and practically useful nonregular designs are found and their properties discussed.
67 citations
References
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Book•
01 Jan 1977
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
Abstract: Linear Codes. Nonlinear Codes, Hadamard Matrices, Designs and the Golay Code. An Introduction to BCH Codes and Finite Fields. Finite Fields. Dual Codes and Their Weight Distribution. Codes, Designs and Perfect Codes. Cyclic Codes. Cyclic Codes: Idempotents and Mattson-Solomon Polynomials. BCH Codes. Reed-Solomon and Justesen Codes. MDS Codes. Alternant, Goppa and Other Generalized BCH Codes. Reed-Muller Codes. First-Order Reed-Muller Codes. Second-Order Reed-Muller, Kerdock and Preparata Codes. Quadratic-Residue Codes. Bounds on the Size of a Code. Methods for Combining Codes. Self-dual Codes and Invariant Theory. The Golay Codes. Association Schemes. Appendix A. Tables of the Best Codes Known. Appendix B. Finite Geometries. Bibliography. Index.
10,083 citations
Book•
01 Jan 2000
TL;DR: This book discusses Factorial and Fractional Factorial Experiments at Three Levels, Robust Parameter Design for Signal-Response Systems, and other Design and Analysis Techniques for Experiments for Improving Reliability.
Abstract: Basic Principles and Experiments with a Single Factor. Experiments With More Than One Factor. Full Factorial Experiments at Two Levels. Fractional Factorial Experiments at Two Levels. Full Factorial and Fractional Factorial Experiments at Three Levels. Other Design and Analysis Techniques for Experiments at More Than Two Levels. Nonregular Designs: Construction and Properties. Experiments with Complex Aliasing. Response Surface Methodology. Introduction to Robust Parameter Design. Robust Parameter Design for Signal-Response Systems. Experiments for Improving Reliability. Experiments With Nonnormal Data. Appendices. Indexes.
1,302 citations
Book•
22 Jun 1999
TL;DR: The Rao Inequalities for Mixed Orthogonal Arrays., 9.2 The Rao InEqualities for mixed Orthogonic Arrays.- 9.4 Construction X4.- 10.1 Constructions Inspired by Coding Theory.
Abstract: 1 Introduction.- 1.1 Problems.- 2 Rao's Inequalities and Improvements.- 2.1 Introduction.- 2.2 Rao's Inequalities.- 2.3 Improvements on Rao's Bounds for Strength 2 and 3.- 2.4 Improvements on Rao's Bounds for Arrays of Index Unity.- 2.5 Orthogonal Arrays with Two Levels.- 2.6 Concluding Remarks.- 2.7 Notes on Chapter 2.- 2.8 Problems.- 3 Orthogonal Arrays and Galois Fields.- 3.1 Introduction.- 3.2 Bush's Construction.- 3.3 Addelman and Kempthorne's Construction.- 3.4 The Rao-Hamming Construction.- 3.5 Conditions for a Matrix.- 3.6 Concluding Remarks.- 3.7 Problems.- 4 Orthogonal Arrays and Error-Correcting Codes.- 4.1 An Introduction to Error-Correcting Codes.- 4.2 Linear Codes.- 4.3 Linear Codes and Linear Orthogonal Arrays.- 4.4 Weight Enumerators and Delsarte's Theorem.- 4.5 The Linear Programming Bound.- 4.6 Concluding Remarks.- 4.7 Notes on Chapter 4.- 4.8 Problems.- 5 Construction of Orthogonal Arrays from Codes.- 5.1 Extending a Code by Adding More Coordinates.- 5.2 Cyclic Codes.- 5.3 The Rao-Hamming Construction Revisited.- 5.4 BCH Codes.- 5.5 Reed-Solomon Codes.- 5.6 MDS Codes and Orthogonal Arrays of Index Unity.- 5.7 Quadratic Residue and Golay Codes.- 5.8 Reed-Muller Codes.- 5.9 Codes from Finite Geometries.- 5.10 Nordstrom-Robinson and Related Codes.- 5.11 Examples of Binary Codes and Orthogonal Arrays.- 5.12 Examples of Ternary Codes and Orthogonal Arrays.- 5.13 Examples of Quaternary Codes and Orthogonal Arrays.- 5.14 Notes on Chapter 5.- 5.15 Problems.- 6 Orthogonal Arrays and Difference Schemes.- 6.1 Difference Schemes.- 6.2 Orthogonal Arrays Via Difference Schemes.- 6.3 Bose and Bush's Recursive Construction.- 6.4 Difference Schemes of Index 2.- 6.5 Generalizations and Variations.- 6.6 Concluding Remarks.- 6.7 Notes on Chapter 6.- 6.8 Problems.- 7 Orthogonal Arrays and Hadamard Matrices.- 7.1 Introduction.- 7.2 Basic Properties of Hadamard Matrices.- 7.3 The Connection Between Hadamard Matrices and Orthogonal Arrays.- 7.4 Constructions for Hadamard Matrices.- 7.5 Hadamard Matrices of Orders up to 200.- 7.6 Notes on Chapter 7.- 7.7 Problems.- 8 Orthogonal Arrays and Latin Squares.- 8.1 Latin Squares and Orthogonal Latin Squares.- 8.2 Frequency Squares and Orthogonal Frequency Squares.- 8.3 Orthogonal Arrays from Pairwise Orthogonal Latin Squares.- 8.4 Concluding Remarks.- 8.5 Problems.- 9 Mixed Orthogonal Arrays.- 9.1 Introduction.- 9.2 The Rao Inequalities for Mixed Orthogonal Arrays.- 9.3 Constructing Mixed Orthogonal Arrays.- 9.4 Further Constructions.- 9.5 Notes on Chapter 9.- 9.6 Problems.- 10 Further Constructions and Related Structures.- 10.1 Constructions Inspired by Coding Theory.- 10.2 The Juxtaposition Construction.- 10.3 The (u, u + ?) Construction.- 10.4 Construction X4.- 10.5 Orthogonal Arrays from Union of Translates of a Linear Code.- 10.6 Bounds on Large Orthogonal Arrays.- 10.7 Compound Orthogonal Arrays.- 10.8 Orthogonal Multi-Arrays.- 10.9 Transversal Designs, Resilient Functions and Nets.- 10.10 Schematic Orthogonal Arrays.- 10.11 Problems.- 11 Statistical Application of Orthogonal Arrays.- 11.1 Factorial Experiments.- 11.2 Notation and Terminology.- 11.3 Factorial Effects.- 11.4 Analysis of Experiments Based on Orthogonal Arrays.- 11.5 Two-Level Fractional Factorials with a Defining Relation.- 11.6 Blocking for a 2k-n Fractional Factorial.- 11.7 Orthogonal Main-Effects Plans and Orthogonal Arrays.- 11.8 Robust Design.- 11.9 Other Types of Designs.- 11.10 Notes on Chapter 11.- 11.11 Problems.- 12 Tables of Orthogonal Arrays.- 12.1 Tables of Orthogonal Arrays of Minimal Index.- 12.2 Description of Tables 12.1?12.3.- 12.3 Index Tables.- 12.4 If No Suitable Orthogonal Array Is Available.- 12.5 Connections with Other Structures.- 12.6 Other Tables.- Appendix A: Galois Fields.- A.1 Definition of a Field.- A.2 The Construction of Galois Fields.- A.3 The Existence of Galois Fields.- A.4 Quadratic Residues in Galois Fields.- A.5 Problems.- Author Index.
1,029 citations
"Minimum moment aberration for nonre..." refers background in this paper
...and Mukerjee (1999), Hedayat, Sloane and Stufken (1999) and Wu and Hamada (2000)....
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01 Jan 1999
TL;DR: This third edition has been revised and expanded, including new chapters on algebraic geometry, new classes of codes, and the essentials of the most recent developments on binary codes.
920 citations
"Minimum moment aberration for nonre..." refers background in this paper
...MacWilliams and Sloane (1977) and van Lint (1999) for details....
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501 citations