HOUND: High-Order Universal Numerical Differentiator for a Parameter-free Polynomial Online Approximation

TL;DR

This paper presents a parameter-free, high-order numerical differentiator that guarantees error convergence for polynomial signals with noise, operates online without tuning, and effectively handles interpolation and extrapolation tasks.

Contribution

The paper introduces a novel high-order nonlinear differential system for numerical differentiation that is parameter-free and suitable for online polynomial approximation.

Findings

Error converges to zero for polynomial signals with noise

Method operates online without parameter tuning

Handles interpolation and extrapolation without data fitting

Abstract

This paper introduces a scalar numerical differentiator, represented as a system of nonlinear differential equations of any high order. We derive the explicit solution for this system and demonstrate that, with a suitable choice of differentiator order, the error converges to zero for polynomial signals with additive white noise. In more general cases, the error remains bounded, provided that the highest estimated derivative is also bounded. A notable advantage of this numerical differentiation method is that it does not require tuning parameters based on the specific characteristics of the signal being differentiated. We propose a discretization method for the equations that implements a cumulative smoothing algorithm for time series. This algorithm operates online, without the need for data accumulation, and it solves both interpolation and extrapolation problems without fitting any coefficients to the data.