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TitleBetti numbers of polynomial hierarchical models for experimental designs.
Author(s) Hugo Maruri-Aguilar, Eduardo Sáenz-de-Cabezón, Henry P. Wynn
TypeArticle in Journal
AbstractPolynomial models, in statistics, interpolation and other fields, relate an output η to a set of input variables (factors), x = (x 1,..., x d ), via a polynomial η(x 1,...,x d ). The monomials terms in η(x) are sometimes referred to as “main effect” terms such as x 1, x 2, ..., or “interactions” such as x 1 x 2, x 1 x 3, ... Two theories are related in this paper. First, when the models are hierarchical, in a well-defined sense, there is an associated monomial ideal generated by monomials not in the model. Second, the so-called “algebraic method in experimental design” generates hierarchical models which are identifiable when observations are interpolated with η(x) based at a finite set of points: the design. We study conditions under which ideals associated with hierarchical polynomial models have maximal Betti numbers in the sense of Bigatti (Commun Algebra 21(7):2317–2334, 1993). This can be achieved for certain models which also have minimal average degree in the design theory, namely “corner cut models”.
KeywordsExperimental design, Hilbert function, Gröbner fan, Betti numbers
ISSN1012-2443; 1573-7470/e
URL http://link.springer.com/article/10.1007%2Fs10472-012-9295-9
JournalAnn. Math. Artif. Intell.
PublisherSpringer International Publishing, Cham
Translation No
Refereed No