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BM algorithms for noisy data and implicit regression modeling. (English) Zbl 1411.13040

Hibi, Takayuki (ed.), The 50th anniversary of Gröbner bases. Proceedings of the 8th Mathematical Society of Japan-Seasonal Institute (MSJ-SI 2015), Osaka, Japan, July 1–10, 2015. Tokyo: Mathematical Society of Japan (MSJ). Adv. Stud. Pure Math. 77, 87-107 (2018).
Summary: In this paper we consider the problem of finding a set of monomials \(\mathcal{O}\) and a polynomial \(f\) whose support is contained in \(\mathcal{O}\), such that (1) \(f\) is almost vanishing at a set of points \(\mathbb{X}\) whose coordinates are not known exactly and (2) \(\mathcal{O}\) exhibits structural stability, that is the model/design matrix associated to \(\mathbb{O}\) is full rank for each set of points differing only slightly from \(\mathbb{X}\).
We review some numerical versions of the Buchberger-Möller (BM) algorithm for computing the set \(\mathcal{O}\) and the polynomial \(f\) and we present a variant, called LDP-LP, which integrates one of these methods with a classical statistical least-squares algorithm for implicit regression from [H. I. Britt and R. H. Luecke, Technometrics 15, 233–247 (1973; Zbl 0257.62041)]. To illustrate the usefulness of these numerical BM algorithms, we review some of their application in the analyses of data sets for which standard techniques did not yield satisfactory results.
For the entire collection see [Zbl 1404.13003].

MSC:

13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
62J02 General nonlinear regression
65C60 Computational problems in statistics (MSC2010)
65F20 Numerical solutions to overdetermined systems, pseudoinverses

Citations:

Zbl 0257.62041
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