id: 06119068
dt: j
an: 06119068
au: Maruri-Aguilar, Hugo; Sáenz-de-Cabezón, Eduardo; Wynn, Henry P.
ti: Betti numbers of polynomial hierarchical models for experimental designs.
so: Ann. Math. Artif. Intell. 64, No. 4, 411-426 (2012).
py: 2012
pu: Springer, Dordrecht
la: EN
cc: 
ut: experimental design; Hilbert function; Gröbner fan; Betti numbers
ci: 
li: doi:10.1007/s10472-012-9295-9
ab: Summary: Polynomial models, in statistics, interpolation and other fields,
    relate an output $η$ to a set of input variables (factors), $x = (x
    _{1},\dots,x _{d })$, via a polynomial $η(x _{1},\dots ,x _{d })$. The
    monomials terms in $η(x)$ are sometimes referred to as “main
    effect” terms such as $x _{1}, x _{2}, \dots $, or “interactions”
    such as $x _{1} x _{2}, x _{1} x _{3}, \dots $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”.
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