BM algorithms for noisy data and implicit regression modelling

In this paper we consider the problem of finding a set of monomials O and a polynomial f whose support is contained in O, such that (1) f is almost vanishing at a set of points X whose coordinates are not known exactly and (2) O exhibits structural stability, that is the model/design matrix associated to O is full rank for each set of points differing only slightly from X. We review some numerical versions of the Buchberger- Moller (BM) algorithm for computing the set 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 [1]. 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.