Minimum Aberration 2 k–p Designs
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.
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TL;DR: In this paper, the construction of minimum aberration 2n − k: 2p designs with respect to some existing combined wordlength patterns is considered. But the construction is restricted to the case where n = 2q runs with q = n − k and n > n/2.
Abstract: In this article, we consider the construction of minimum aberration 2n − k: 2p designs with respect to some existing combined wordlength patterns, where a 2n − k: 2p design is a blocked two-level design with n treatment factors, 2p blocks, and N = 2q runs with q = n − k. Two methods are proposed for two situations: n ⩽ 2q − p − 1 and n > N/2. These methods enable us to obtain some new minimum aberration 2n − k: 2p designs from existing minimum aberration unblocked and blocked designs. Examples are included to illustrate the theory.
7 citations
Additional excerpts
...For the unblocked case, theminimum aberration criterion sequentially minimizes the treatment wordlength pattern (Fries and Hunter, 1980)....
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Dissertation•
01 Jan 2007
Abstract: Two-level factorial and fract,ional factsot-id designs have playcd a prominent role in the theory and pract,ice of experimental design. Though commonly used in indust.ria1 experiments to identify the significant effects, it is often undesimble to perform t,he trials of a. factorial design (or, fractional factorial design) in a complet,ely random order. Instmead, restrictions are imposed on tJhe randomization of experirne~it~al runs. In recent years, considerable attentlion has been devot,ed to fact(oria1 and fractional fa~t~orial plans with different randomization restrict,ions (e.g., nested designs, split,-plot designs, split-split-plot designs, strip-plot designs, split-lot designs, and combinatiorls thereof). Bingham et al. (2006) proposed an approach to represent. t,he randomization structlure of factorial designs with randomization restri~t~ions. This thesis introduces a related, but more general, rcpresent,ation referred to as randomization defining contrast subspaces (RDCSS). The RDCSS is a projective geometric f~rmulat~ion of mndomization defining contrast subg~oups (RDCSG) defined in Bingham et al. (2006) and allows for t,heoretical st,udy. For factorial designs with different randomization struckures, the mere existence of a design is not straightforward. Here, the t'heoretical results are developed for the existence of fact,orial designs wit,h randomization restrictions within this unified framework. Our theory brings t,ogether results from finite projective geomet,ry to establish the existence and construction of such designs. Specifically, for the existence of a set of disjoint, RDCSSs, several results are proposed using ( t 1)-spreads and partial (t 1)-spreads of PG(pI , ? ) . Furthermore, t'he t'heory developed here offers a sy~t~emat~ic approach for the const,ructtion of t,wo-level full factorial designs and regular fractional factsorial designs with randomization restrictions. Finally, when t,he ~ondit~ions for the existmeme of a set of disjoint RDCSSs are violated, the data analysis is highly influenced fro111 the overlapping pat,tern among the RDCSSs. Under t,hese circumstances, a geometric structure called star is proposed for a set of (t 1)-dimensional subspaces of PG(p 1, q) , wherc 1 < t < p. This c~periment~al p an permits the assessment of a relatively larger nnmber of fact,orial effects. The necessary and sufficient conditions for the exist,ence of stars and a collection of stars are d so developed here. In particular, stars ~onsti t~ute useful designs for practitioners because of their flexith structure and easy construction.
7 citations
TL;DR: A first demonstration of the idea that matrix-based algorithms for nonlinear combinatorial optimization problems can be efficiently implemented is provided, by developing an implementation on a massively parallel architecture, and exploiting scalable and efficient parallel implementations of algorithms for ultra high-precision linear algebra.
Abstract: We provide a first demonstration of the idea that matrix-based algorithms for nonlinear combinatorial optimization problems can be efficiently implemented. Such algorithms were mainly conceived by theoretical computer scientists for proving efficiency. We are able to demonstrate the practicality of our approach by developing an implementation on a massively parallel architecture, and exploiting scalable and efficient parallel implementations of algorithms for ultra high-precision linear algebra. Additionally, we have delineated and implemented the necessary algorithmic and coding changes required in order to address problems several orders of magnitude larger, dealing with the limits of scalability from memory footprint, computational efficiency, reliability, and interconnect perspectives.
7 citations
TL;DR: In this article, the authors proposed and studied the clear effects problem for the asymmetrical case, and derived the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic's) in 4.
Abstract: Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and it has become an active research issue in recent years. Tang et al. derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi’s) in 2
n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2
n−(n−k) design for given k, which are only restricted for the symmetrical case. This paper proposes and studies the clear effects problem for the asymmetrical case. It improves the construction method of Tang et al. for 2
n−(n−k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic’s) in 4
m
2
n
designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic’s in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct 4
m
2
n
designs under the clear effects criterion.
7 citations
01 Jan 2000
TL;DR: The final author version and the galley proof are versions of the publication after peer review and the final published version features the final layout of the paper including the volume, issue and page numbers.
Abstract: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.
7 citations
Cites background from "Minimum Aberration 2 k–p Designs"
...The particular evaluation criterion proposed in Study 2.1 is the aberration criterion of Fries and Hunter (1980) as modified by Wu and Zhang (1993) for mixed four-and-two level designs....
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...All designs of the collection in Box et al. (1978) have minimum aberration (Fries and Hunter, 1980): the number of words with minimum length is minimal....
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References
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471 citations
Book•
23 Jun 1976
TL;DR: In conclusion, the size of Industrial Experiments, Fractional Replication--Elementary, and Incomplete Factorials are found to be about the same as that of conventional comparison experiments.
Abstract: Introduction. Simple Comparison Experiments. Two Factors, Each at Two Levels. Two Factors, Each at Three Levels. Unreplicated Three--Factor, Two--Level Experiments. Unreplicated Four--Factor, Two--Level Experiments. Three Five--Factor, Two--Level Unreplicated Experiments. Larger Two--Way Layouts. The Size of Industrial Experiments. Blocking Factorial Experiments, Fractional Replication--Elementary. Fractional Replication--Intermediate. Incomplete Factorials. Sequences of Fractional Replicates. Trend--Robust Plans. Nested Designs. Conclusions and Apologies.
311 citations
09 Jun 1976
271 citations
TL;DR: Incomplete Factorials, Fractional Replication, Intermediate Factorial, and Nested Designs as discussed by the authors are some of the examples of incomplete Factorial Experiments and incomplete fractional replicates.
Abstract: Introduction. Simple Comparison Experiments. Two Factors, Each at Two Levels. Two Factors, Each at Three Levels. Unreplicated Three--Factor, Two--Level Experiments. Unreplicated Four--Factor, Two--Level Experiments. Three Five--Factor, Two--Level Unreplicated Experiments. Larger Two--Way Layouts. The Size of Industrial Experiments. Blocking Factorial Experiments, Fractional Replication--Elementary. Fractional Replication--Intermediate. Incomplete Factorials. Sequences of Fractional Replicates. Trend--Robust Plans. Nested Designs. Conclusions and Apologies.
252 citations