Convergence Rate of the Causal Jacobi Derivative Estimator

TL;DR

This paper analyzes the convergence rate of causal Jacobi derivative estimators for noisy data, providing bounds on the estimation error and demonstrating their effectiveness through examples.

Contribution

It introduces a convergence rate analysis for causal Jacobi derivative estimators, linking error bounds to noise level and polynomial truncation order.

Findings

Error bound of order δ^{(q+1)/(n+1+q)} for the estimators

Parameter selection guidelines based on noise level and polynomial degree

Validation of the method's efficiency through numerical examples

Abstract

Numerical causal derivative estimators from noisy data are essential for real time applications especially for control applications or fluid simulation so as to address the new paradigms in solid modeling and video compression. By using an analytical point of view due to Lanczos \cite{C. Lanczos} to this causal case, we revisit $n^{th}$\ order derivative estimators originally introduced within an algebraic framework by Mboup, Fliess and Join in \cite{num,num0}. Thanks to a given noise level $\delta$ and a well-suitable integration length window, we show that the derivative estimator error can be $\mathcal{O}(\delta ^{\frac{q+1}{n+1+q}})$ where $q$\ is the order of truncation of the Jacobi polynomial series expansion used. This so obtained bound helps us to choose the values of our parameter estimators. We show the efficiency of our method on some examples.