Moving Taylor Bayesian Regression for nonparametric multidimensional function estimation with possibly correlated errors

TL;DR

This paper introduces MOTABAR, a nonparametric Bayesian method for estimating functions and derivatives from noisy, correlated data, unifying several existing methods and demonstrating effectiveness on complex dynamical systems.

Contribution

The paper proposes MOTABAR, a novel Bayesian regression approach that handles correlated errors and generalizes multiple existing interpolation and regression techniques.

Findings

MOTABAR effectively reconstructs complex dynamical systems from noisy data.

The method unifies several regression and interpolation techniques.

Demonstrated successful application to Lorenz attractor reconstruction.

Abstract

We present a nonparametric method for estimating the value and several derivatives of an unknown, sufficiently smooth real-valued function of real-valued arguments from a finite sample of points, where both the function arguments and the corresponding values are known only up to measurement errors having some assumed distribution and correlation structure. The method, Moving Taylor Bayesian Regression (MOTABAR), uses Bayesian updating to find the posterior mean of the coefficients of a Taylor polynomial of the function at a moving position of interest. When measurement errors are neglected, MOTABAR becomes a multivariate interpolation method. It contains several well-known regression and interpolation methods as special or limit cases. We demonstrate the performance of MOTABAR using the reconstruction of the Lorenz attractor from noisy observations as an example.