44,213 research outputs found
Counting Configurations in Designs
AbstractGiven a t-(v, k, λ) design, form all of the subsets of the set of blocks. Partition this collection of configurations according to isomorphism and consider the cardinalities of the resulting isomorphism classes. Generalizing previous results for regular graphs and Steiner triple systems, we give linear equations relating these cardinalities. For any fixed choice of t and k, the coefficients in these equations can be expressed as functions of v and λ and so depend only on the design's parameters, and not its structure. This provides a characterization of the elements of a generating set for m-line configurations of an arbitrary design
Aberration in qualitative multilevel designs
Generalized Word Length Pattern (GWLP) is an important and widely-used 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 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
designs with prime, we derive a new formula for computing the GWLP of
. 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.Comment: 16 pages, 1 figur
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