Linear minimum-variance approximants for noisy data

TL;DR

This paper introduces linear minimum-variance approximants for noisy data using weighted least squares, optimizing for correlated noise and outperforming existing subdivision methods in such scenarios.

Contribution

It develops a new class of approximants based on annihilation-operators that are optimal for correlated non-uniform noise, extending previous uncorrelated noise solutions.

Findings

Proposed approximants outperform existing methods for correlated noise.

Derived formulas are optimal for general correlated non-uniform noise.

Numerical experiments confirm improved performance over current subdivision rules.

Abstract

Inspired by recent developments in subdivision schemes founded on the Weighted Least Squares technique, we construct linear approximants for noisy data in which the weighting strategy minimizes the output variance, thereby establishing a direct correspondence with the Generalized Least Squares and the Minimum-Variance Formulas methodologies. By introducing annihilation-operators for polynomial spaces, we derive usable formulas that are optimal for general correlated non-uniform noise. We show that earlier subdivision rules are optimal for uncorrelated non-uniform noise and, finally, we present numerical evidence to confirm that, in the correlated case, the proposed approximants are better than those currently used in the subdivision literature.