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Betti numbers of polynomial hierarchical models for experimental designs

Maruri-Aguilar, Hugo, Saenz-de-Cabezon, Eduardo and Wynn, Henry P. (2012) Betti numbers of polynomial hierarchical models for experimental designs. Annals of Mathematics and Artificial Intelligence, 64 (4). pp. 411-426. ISSN 1012-2443

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Abstract

Polynomial 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 1x 2, x 1x 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".

Item Type: Article
Official URL: http://www.springer.com/computer/ai/journal/10472
Additional Information: © 2012 Springer
Library of Congress subject classification: H Social Sciences > HA Statistics
Q Science > QA Mathematics
Sets: Research centres and groups > Decision Support and Risk Group (DSRG)
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Date Deposited: 23 May 2012 11:06
URL: http://eprints.lse.ac.uk/43870/

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