Adjusted least squares fitting of algebraic hypersurfaces

TL;DR

This paper introduces an adjusted least squares method for fitting algebraic hypersurfaces to noisy data, correcting bias in traditional estimators using a quasi-Hankel matrix and polynomial eigenvalue problems.

Contribution

It presents a novel bias-corrected estimator based on quasi-Hankel matrices and improves algorithms for fitting algebraic hypersurfaces with unknown noise variance.

Findings

Bias correction improves fitting accuracy.

New invariance properties of the estimator are established.

Enhanced algorithms enable fitting with arbitrary monomials.

Abstract

We consider the problem of fitting a set of points in Euclidean space by an algebraic hypersurface. We assume that points on a true hypersurface, described by a polynomial equation, are corrupted by zero mean independent Gaussian noise, and we estimate the coefficients of the true polynomial equation. The adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator. The adjusted least squares estimator is based on constructing a quasi-Hankel matrix, which is a bias-corrected matrix of moments. For the case of unknown noise variance, the estimator is defined as a solution of a polynomial eigenvalue problem. In this paper, we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation.