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 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”.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Babson, E., Onn, S., Thomas, R.: The Hilbert zonotope and a polynomial time algorithm for universal Gröbner bases. Adv. Appl. Math. 30(3), 529–544 (2003)
Basu, S.: On bounding the Betti numbers and computing the Euler characteristics of semi-algebraic sets. Discrete Comput. Geom. 22, 1–18 (1999)
Bayer, D., Morrison, I.: Standard bases and geometric invariant theory. I. Initial ideals and state polytopes. J. Symb. Comput. 6, 209–217 (1988)
Bernstein, Y., Maruri-Aguilar, H., Onn, S., Riccomagno, E., Wynn, H.: Minimal average degree aberration and the state polytope for experimental designs. Ann. Inst. Stat. Math. 62, 673–698 (2010)
Berrington de Gonzalez, A., Cox, D.: Additive and multiplicative models for the joint effect of two risk factors. Biostatistics 6, 1–9 (2005)
Berrington de Gonzalez, A., Cox, D.: Interpretation of interaction: a review. Ann. Appl. Stat. 1(2), 371–385 (2007)
Bigatti, A.: Upper bounds for the Betti numbers of a given Hilbert function. Commun. Algebra 21(7), 2317–2334 (1993)
Bigatti, A.M., Geramita, A.V.: Level algebras, lex segments, and minimal Hilbert functions. Commun. Algebra 31, 1427–1451 (2003)
Buchberger, B.: Ein algorithmus zum asussfinden der basielemente des restklassenringes nach eniem multidimensionalen polynomideal. Ph.D. thesis, U. Innsbruck, Austria (1965)
Buchberger, B., Winkler, F.: Gröbner Bases and Applications. Cambridge University Press, Cambridge (1998)
CoCoATeam: CoCoA: A System for Doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it (2009)
Corteel, S., Rémond, G., Schaeffer, G., Thomas, H.: The number of plane corner cuts. Adv. Appl. Math. 23(1), 49–53 (1999)
Edelsbrunner, H., Harer, J.L.: Computational Topology. American Mathematical Society, Providence (2010)
Friedman, J.: Linear transformations in Boolean complexity theory. In: Computation and Logic in the Real World. Lecture Notes in Computer Science, vol. 4497, pp. 307–315. Springer, Berlin (2007)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/ (2009)
Greuel, G.M., Pfister, G., Schönemann, H.: Singular 3-1-0—a computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2009)
Hulett, H.A.: Maximum Betti numbers of homogeneous ideals with a given Hilbert function. Commun. Algebra 21(7), 2335–2350 (1993)
Macaulay, F.: Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc. 26, 531–555 (1927)
Maruri-Aguilar, H.: Algebraic statistics in experimental design. Ph.D. thesis, U. Warwick, UK (2007)
Miller, E., Sturmfels, B.: Combinatorial commutative algebra. In: Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)
Milnor, J.: On the Betti numbers of real varities. Proc. Am. Math. Soc. 15, 275–280 (1964)
Mora, T., Robbiano, L.: The Gröbner fan of an ideal. J. Symb. Comput. 6(2/3), 183–208 (1988)
Müller, I.: Corner cuts and their polytopes. Beiträge Algebra Geom. 44(2), 323–333 (2003)
Onn, S., Sturmfels, B.: Cutting corners. Adv. Appl. Math. 23(1), 29–48 (1999)
Peeva, I.: Syzygies and Hilbert functions. Chapman and Hall/CRC, Boca Raton (2007)
Petrović, S., Stokes, E.: Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees. Preprint available at arXiv:0910.1610v3 (2011)
Pistone, G., Wynn, H.: Generalised confounding with Gröbner bases. Biometrika 83, 656–666 (1996)
Pistone, G., Riccomagno, E., Wynn, H.: Algebraic statistics. In: Monographs on Statistics and Applied Probability, vol. 89. Chapman & Hall/CRC, Boca Raton (2001)
Wagner, U.: On the number of corner cuts. Adv. Appl. Math. 29(2), 152–161 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Maruri-Aguilar, H., Sáenz-de-Cabezón, E. & Wynn, H.P. Betti numbers of polynomial hierarchical models for experimental designs. Ann Math Artif Intell 64, 411–426 (2012). https://doi.org/10.1007/s10472-012-9295-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10472-012-9295-9