A graph-aided method for planning two-level experiments when certain interactions are important
TL;DR: In this paper, a graph-aided method is proposed to solve the problem of fractional factorial factorial experiment planning, where prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects.
Abstract: In planning a fractional factorial experiment prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects. In this article, we propose a graph-aided method to solve this problem for two-level experiments. First, we choose the defining relations for a 2 n–k design according to a goodness criterion such as the minimum aberration criterion. Then we construct all of the nonisomorphic graphs that represent the solutions to the problem of simultaneous estimation of main effects and two-factor interactions for the given defining relations. In each graph a vertex represents a factor and an edge represents the interaction between the two factors. For the experiment planner, the job is simple: Draw a graph representing the specified interactions and compare it with the list of graphs obtained previously. Our approach is a substantial improvement over Taguchi's linear graphs.
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Dissertation•
01 Jan 2006TL;DR: A graph-based partitioning method combines the graph and sparse matrix decomposition methods used by the electrical engineering community with the results of a screening test to create a quantitative method for partitioning large scale, black-box systems.
Abstract: The primary contributions of this thesis are associated with the development of a new method for exploring the relationships between inputs and outputs for large scale computer simulations. Primarily, the proposed design space exploration procedure uses a hierarchical partitioning method to help mitigate the curse of dimensionality often associated with the analysis of large scale systems. Closely coupled with the use of a partitioning approach, is the problem of how to partition the system. This thesis also introduces and discusses a quantitative method developed to aid the user in finding a set of good partitions for creating partitioned metamodels of large scale systems.
The new hierarchically partitioned metamodeling scheme, the lumped parameter model (LPM), was developed to address two primary limitations to the current partitioning methods for large scale metamodeling. First the LPM was formulated to negate the need to rely on variable redundancies between partitions to account for potentially important interactions. While variable redundancies improve the accuracy of the partitioned metamodel, they do so at the cost of significantly reducing the efficiency of the partitioned metamodeling process. By using a hierarchical structure, the LPM addresses the impact of neglected, direct interactions by indirectly accounting for these interactions via the interactions that occur between the lumped parameters in intermediate to top-level mappings. Secondly, the LPM was developed to allow for hierarchical modeling of black-box analyses that do not have available intermediaries with which to partition the system around.
The second contribution of this thesis is a graph-based partitioning method for large scale, black-box systems. The graph-based partitioning method combines the graph and sparse matrix decomposition methods used by the electrical engineering community with the results of a screening test to create a quantitative method for partitioning large scale, black-box systems. An ANOVA analysis of the results of a screening test can be used to determine the sparse nature of the large scale system. With this information known, the sparse matrix and graph theoretic partitioning schemes can then be used to create potential sets of partitions to use with the lumped parameter model.
19 citations
Cites background from "A graph-aided method for planning t..."
...Wu and Chen (1992)[189] used graph theory to plan two-level fractional factorial experiments when interactions are important....
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TL;DR: The proposed partially replicated designs are highly efficient in estimating the possibly active effects and provide a replication-based estimate of the error variance, and they provide a practical compromise between the power in identifying truly activeeffects and the number of runs in experiments.
Abstract: In a two-level factorial experiment, we consider construction of parallel-flats designs with two identical parallel flats that allow estimation of a set of specified possibly active effects and the pure error variance. A set of sufficient conditions is presented for the designs to be D-optimal for the specified effects, assuming that the other effects are negligible, over the class of competing parallel-flats designs. In addition, an algorithm is developed to generate the D-optimal designs with a choice of flexible degrees of freedom for the pure error variance. Because the proposed partially replicated designs are highly efficient in estimating the possibly active effects and provide a replication-based estimate of the error variance, they provide a practical compromise between the power in identifying truly active effects and the number of runs in experiments. This property is verified through a simulation study.
19 citations
Cites background from "A graph-aided method for planning t..."
...allowing one to take advantage of this information to design an appropriate plan, not necessarily a minimum aberration design, to ensure estimation of the specified possibly active effects with high efficiency (see, e.g., Franklin and Bailey 1977; Wu and Chen 1992; Liao, Iyer, and Vecchia 1996)....
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...…one to take advantage of this information to design an appropriate plan, not necessarily a minimum aberration design, to ensure estimation of the specified possibly active effects with high efficiency (see, e.g., Franklin and Bailey 1977; Wu and Chen 1992; Liao, Iyer, and Vecchia 1996)....
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TL;DR: In this paper, the authors considered fractional factorial split-plot (FFSP) designs with resolution III or IV under the clear effects criterion and derived the upper and lower bounds on the maximum number of clear WP2fi, WS2fi and SP2fi for FFSP designs.
Abstract: Fractional factorial split-plot (FFSP) designs have an important value of investigation for their special structures There are two types of factors in an FFSP design: the whole-plot (WP) factors and sub-plot (SP) factors, which can form three types of two-factor interactions: WP2fi, WS2fi and SP2fi This paper considers FFSP designs with resolution III or IV under the clear effects criterion It derives the upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for FFSP designs, and gives some methods for constructing the desired FFSP designs It further examines the performance of the construction methods
18 citations
TL;DR: In this article, a finite projective geometry is used to obtain fractional factorial plans for m-level symmetrical factorial experiments, where m is a prime or a prime power.
Abstract: Finite projective geometry is used to obtain fractional factorial plans for m-level symmetrical factorial experiments, where m is a prime or a prime power. Under a model that includes the mean, all main effects and a specified set of two-factor interactions, the plans are shown to be universally optimal within the class of all plans involving the same number of runs.
18 citations
TL;DR: In this article, a unified approach to the construction of regular factorial designs with randomization restrictions using randomization defining contrast subspaces for the representation of randomization restriction is presented.
Abstract: Regular factorial designs with randomization restrictions are widely used in practice. This paper provides a unified approach to the construction of such designs using randomization defining contrast subspaces for the representation of randomization restrictions. We use finite projective geometry to determine the existence of designs with the required structure and develop a systematic approach for their construction. An attractive feature is that commonly used factorial designs with randomization restrictions are special cases of this general representation. Issues related to the use of these designs for particular factorial experiments are also addressed.
17 citations
References
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Book•
01 Jan 2015TL;DR: This book offers a complete blueprint for structuring projects to achieve rapid completion with high engineering productivity during the research and development phase to ensure that high quality products can be made quickly and at the lowest possible cost.
Abstract: From the Publisher:
Phadke was trained in robust design techniques by Genichi Taguchi, the mastermind behind Japanese quality manufacturing technologies and the father of Japanese quality control. Taguchi's approach is currently under consideration to be adopted as a student protocol with the US govrnment. The foreword is written by Taguchi. This book offers a complete blueprint for structuring projects to achieve rapid completion with high engineering productivity during the research and development phase to ensure that high quality products can be made quickly and at the lowest possible cost. Some topics covered are: orthogonol arrays, how to construct orthogonal arrays, computer-aided robutst design techniques, dynamic systems design methods, and more.
3,928 citations
"A graph-aided method for planning t..." refers methods in this paper
...For a review on efficient algorithms for testing graph isomorphism, see Read and Corneil (1977) and Hoffman (1982)....
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...For simplicity we do not include in this article the column numbers for the lines, which can be easily read from the interaction tables given by Phadke (1989) and Taguchi (1987)....
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TL;DR: The present state of the art of isomorphism testing is surveyed, its relationship to NP-completeness is discussed, and some of the difficulties inherent in this particularly elusive and challenging problem are indicated.
Abstract: The graph isomorphism problem—to devise a good algorithm for determining if two graphs are isomorphic—is of considerable practical importance, and is also of theoretical interest due to its relationship to the concept of NP-completeness. No efficient (i.e., polynomial-bound) algorithm for graph isomorphism is known, and it has been conjectured that no such algorithm can exist. Many papers on the subject have appeared, but progress has been slight; in fact, the intractable nature of the problem and the way that many graph theorists have been led to devote much time to it, recall those aspects of the four-color conjecture which prompted Harary to rechristen it the “four-color disease.” This paper surveys the present state of the art of isomorphism testing, discusses its relationship to NP-completeness, and indicates some of the difficulties inherent in this particularly elusive and challenging problem. A comprehensive bibliography of papers relating to the graph isomorphism problem is given.
519 citations
TL;DR: In this article, the concept of aberration is proposed as a way of selecting the best designs from those with maximum resolution, and algorithms are presented for constructing these minimum aberration designs.
Abstract: For studying k variables in N runs, all 2 k–p designs of maximum resolution are not equally good. In this paper the concept of aberration is proposed as a way of selecting the best designs from those with maximum resolution. Algorithms are presented for constructing these minimum aberration designs.
420 citations
01 Jan 1980
TL;DR: The concept of resolution was introduced by Box and Hunter as discussed by the authors, who defined the resolution of a two-level fractional factorial design as the length of the shortest word in the defining relation.
Abstract: Fractional factorial designs-especially the twolevel designs-are useful in a variety of experimental situations, for example, (i) screening studies in which only a subset of the variables is expected to be important, (ii) research investigations in which certain interactions are expected to be negligible and (iii) experimental programs in which groups of runs are to be performed sequentially, ambiguities being resolved as the investigation evolves (see Box, Hunter and Hunter, 1978). The literature on fractional factorial designs is extensive. For references before 1969, see the comprehensive bibliography of Herzberg and Cox (1969). For more recent references, see Daniel (1976) and Joiner (1975-79). A useful concept associated with 2k-P fractional factorial designs is that of resolution (Box and Hunter, 1961). A design is of resolution R if no cfactor effect is confounded with any other effect containing less than R c factors. For example, a design of resolution III does not confound main effects with one another but does confound main effects with two-factor interactions, and a design of resolution IV does not confound main effects with two-factor interactions but does confound two-factor interactions with one another. The resolution of a two-level fractional factorial design is the length of the shortest word in the defining relation. Usually an experimenter will prefer to use a design which has the highest
354 citations