Automatic numerical differentiation by maximum likelihood estimation of state-space model

TL;DR

This paper introduces a stable, maximum likelihood-based state-space smoothing algorithm for automatic numerical differentiation from noisy data, capable of handling various measurement conditions and providing accurate derivative estimates.

Contribution

It presents a novel linear Gaussian state-space smoothing method with maximum likelihood parameter estimation for automatic differentiation from noisy measurements.

Findings

Comparable or superior accuracy to existing methods on synthetic biomechanics data

Handles non-uniform sampling and multiple measurements simultaneously

Provides derivative estimates at arbitrary points between data samples

Abstract

A linear Gaussian state-space smoothing algorithm is presented for estimation of derivatives from a sequence of noisy measurements. The algorithm uses numerically stable square-root formulas, can handle simultaneous independent measurements and non-equally spaced abscissas, and can compute state estimates at points between the data abscissas. The state space model's parameters, including driving noise intensity, measurement variance, and initial state, are determined from the given data sequence using maximum likelihood estimation computed using a expectation maximisation iteration. In tests with synthetic biomechanics data, the algorithm has equivalent or better accuracy compared to other automatic numerical differentiation algorithms.