Betti numbers of polynomial hierarchical models for experimental designs

  1. Maruri-Aguilar, H. 2
  2. Sáenz-de-Cabezón, E. 3
  3. Wynn, H.P. 1
  1. 1 London School of Economics and Political Science
    info

    London School of Economics and Political Science

    Londres, Reino Unido

    ROR https://ror.org/0090zs177

  2. 2 Queen Mary University of London
    info

    Queen Mary University of London

    Londres, Reino Unido

    ROR https://ror.org/026zzn846

  3. 3 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Revista:
Annals of Mathematics and Artificial Intelligence

ISSN: 1012-2443

Año de publicación: 2012

Volumen: 64

Número: 4

Páginas: 411-426

Tipo: Artículo

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DOI: 10.1007/S10472-012-9295-9 SCOPUS: 2-s2.0-84862331216 WoS: WOS:000305229800006 GOOGLE SCHOLAR

Otras publicaciones en: Annals of Mathematics and Artificial Intelligence

Resumen

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". © 2012 Springer Science+Business Media B.V.