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

TL;DR

A discretization method for the equations that implements a cumulative smoothing algorithm for time series that operates online, without the need for data accumulation, and it solves both interpolation and extrapolation problems without fitting any coefficients to the data.

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.

Figures (7)

• Figure 1: The differentiable signal $f(t)$ and its interpolation $\hat{f}(t)$
• Figure 2: The differentiable signal $f(t)$ and its interpolation $\hat{f}(t)$, extrapolation of the signal $f_0(t)$ and its estimates $\hat{f_0}(t)$ beyond the beginning of the range
• Figure 3: The differentiable signal $f(t)$ and its interpolation $\hat{f}(t)$, extrapolation of the signal $f_0(t)$ and its estimates $\hat{f_0}(t)$ beyond the end of the range
• Figure 4: Errors of estimation of the signal and its derivatives in logarithmic scale (absolute values)
• Figure 5: Errors of estimation of signal and its derivatives in logarithmic scale (absolute values), initial range