Modelling Periodic Measurement Data Having a Piecewise Polynomial Trend Using the Method of Variable Projection

TL;DR

This paper introduces a novel method using variable projection and B-Splines to model periodic signals with complex, time-varying backgrounds, improving accuracy over traditional polynomial approaches in industrial and synthetic data.

Contribution

It extends the IEEE-standard 1057 by allowing time-varying backgrounds and multiple harmonics, providing a more flexible and accurate modeling approach for complex signals.

Findings

B-Splines outperform higher order polynomials for complex backgrounds

The method enables covariance and confidence interval calculations

Effective on both real industrial and synthetic data

Abstract

This paper presents a new method for modelling periodic signals having an aperiodic trend, using the method of variable projection. It is a major extension to the IEEE-standard 1057 by permitting the background to be time varying; additionally, any number of harmonics of the periodic portion can be modelled. This paper focuses on using B-Splines to implement a piecewise polynomial model for the aperiodic portion of the signal. A thorough algebraic derivation of the method is presented, as well as a comparison to using global polynomial approximation. It is proven that B-Splines work better for modelling a more complicated aperiodic portion when compared to higher order polynomials. Furthermore, the piecewise polynomial model is capable of modelling the local signal variations produced by the interaction of a control system with a process in industrial applications. The method of variable projection reduces the problem to a one-dimensional nonlinear optimization, combined with a linear least-squares computation. An added benefit of using the method of variable projection is the possibility to calculate the covariances of the linear coefficients of the model, enabling the calculation of confidence and prediction intervals. The method is tested on both real measurement data acquired in industrial processes, as well as synthetic data. The method shows promising results for the precise characterization of periodic signals embedded in highly complex aperiodic backgrounds. Finally, snippets of the m-code are provided, together with a toolbox for B-Splines, which permit the implementation of the complete computation.