Indicator function and complex coding for mixed fractional factorial designs

Pistone, Giovanni; Rogantin, Maria Piera

doi:10.1016/j.jspi.2007.02.007

Cited by 48 publications

(67 citation statements)

References 21 publications

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“…An indicator polynomial function is associated with a fraction with no replications. This approach was developed in Fontana et al (1996), Fontana et al (2000), Ye (2003) for 2-level design and generalized in Ye (2004), Pistone and Rogantin (2006).…”

Section: Introductionmentioning

confidence: 99%

Hilbert Bases for Orthogonal Arrays

Carlini

Pistone

2007

Journal of Statistical Theory and Practice

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Abstract. In this paper, we relate the problem of generating all 2-level orthogonal arrays of given dimension and force, i.e. elements in OA(n, m), where n is the number of factors and m the force, to the solution of an Integer Programming problem involving rational convex cones. We do not restrict the number of points in the array, i.e. we admit any number of replications. This problem can be theoretically solved by means of Hilbert bases which form a finite generating set for all the elements in in the infinite set OA(n, m). We discuss some examples which are explicitly solved with a software performing Hilbert bases computation.

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Section: Introductionmentioning

confidence: 99%

Hilbert Bases for Orthogonal Arrays

Carlini

Pistone

2007

Journal of Statistical Theory and Practice

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“…Some papers, such as [61,112,141,213], call an orthogonal array regular if and only if it is made from an Abelian group in this way. There are two problems with this.…”

Section: Orthogonal Arraysmentioning

confidence: 99%

Relations among partitions

Bailey¹

2017

Surveys in Combinatorics 2017

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Combinatorialists often consider a balanced incomplete-block design to consist of a set of points, a set of blocks, and an incidence relation between them which satisfies certain conditions. To a statistician, such a design is a set of experimental units with two partitions, one into blocks and the other into treatments; it is the relation between these two partitions which gives the design its properties. The most common binary relations between partitions that occur in statistics are refinement, orthogonality and balance. When there are more than two partitions, the binary relations may not suffice to give all the properties of the system. I shall survey work in this area, including designs such as double Youden rectangles.

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“…C4 possible relationships coming from desired properties of the fraction (for instance, the fact that a term is centered and/or some terms are mutually orthogonal, see Proposition 3 of Pistone and Rogantin (2008)). …”

Section: Convolution Formula With Countsmentioning

confidence: 99%

Aberration in qualitative multilevel designs

Fontana

Rapallo

Rogantin

2016

Journal of Statistical Planning and Inference

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Generalized Word Length Pattern (GWLP) is an important and widelyused tool for comparing fractional factorial designs. We consider qualitative factors, and we code their levels using the roots of the unity. We write the GWLP of a fraction F using the polynomial indicator function, whose coefficients encode many properties of the fraction. We show that the coefficient of a simple or interaction term can be written using the counts of its levels. This apparently simple remark leads to major consequence, including a convolution formula for the counts. We also show that the mean aberration of a term over the permutation of its levels provides a connection with the variance of the level counts. Moreover, using mean aberrations for symmetric s m designs with s prime, we derive a new formula for computing the GWLP of F. It is computationally easy, does not use complex numbers and also provides a clear way to interpret the GWLP. As case studies, we consider non-isomorphic orthogonal arrays that have the same GWLP. The different distributions of the mean aberrations suggest that they could be used as a further tool to discriminate between fractions.

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Indicator function and complex coding for mixed fractional factorial designs

Cited by 48 publications

References 21 publications

Hilbert Bases for Orthogonal Arrays

Hilbert Bases for Orthogonal Arrays

Relations among partitions

Aberration in qualitative multilevel designs

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