Almost vanishing polynomials for sets of limited precision points

Abstract

From the numerical point of view, given a set X, subset of R^n of s points whose coordinates are known with only limited precision, each set XP of s points whose elements differ from those of X of a quantity less than the data uncertainty can be considered equivalent to X. We present an algorithm that, given X and a tolerance on the data error, computes a set G of polynomials such that each element of G ``almost vanishing'' at X and at all its equivalent sets XP. Even if G is not, in the general case, a basis of the vanishing ideal I(X), we show that, differently from the basis of I(X) that can be greatly influenced by the data uncertainty, G can determine a geometrical configuration simultaneously characterizing the set X and all its equivalent sets XP

Numerical algebraic fan of a design for statistical model building

“…The novelty is the use of approximating ideals in order to deal with the instability in the observed or predicted designs. We employ an algorithm from Fassino [2010], whose use in statistics is completely new.…”

“…We seek a set of polynomials which "almost vanish" at the design points, namely evaluated at the design points are close enough to zero. To do that, we use the numerical Buchberger-Möller (NBM) algorithm in Fassino [2010] and its implementation in CoCoA4 (CoCoATeam). The NBM algorithm is from the field of approximate computational algebraic geometry and is based on a least square approximation.…”

“…Because of the fact that data in the second stage are very noisy, we extend existing theory by switching from symbolic, exact computations to numerical computations in the calculation of the design ideal and of its fan. Specifically, instead of polynomials whose solution are the design points, we identify a design with a set of polynomials which "almost vanish" at the design points using results and algorithms from Fassino [2010].…”

An Euclidean norm based criterion to assess robots’ 2D path-following performance

“…An interesting class of recently developed algorithms relies on tools from Numerical Commutative Algebra [17,2,10,6,7]. For all these algorithms the input is a set of points possibly in n-dimensions and the output is a polynomial f in n-variables whose zero locus (which is a curve, or a surface, or more generally an algebraic variety) gives an approximation for the input points and can be interpreted as an implicit polynomial regression model [12,Ch 2].…”

Gröbner bases for everyone with CoCoA-5 and CoCoALib

“…provided a certain polynomial does not vanish). So long as the approximate points are not too few nor too imprecise the NBM (Numerical Buchberger-Möller ) algorithm can compute at least a partial Border Basis, and this should identify any "approximate polynomial conditions" which the points the almost satisfy (see Abbott, Fassino, Torrente [11] and Fassino [16]). We can ask CoCoA to allow a certain approximation on the coordinates of the points:…”