Abstract

This paper presents a new approach to the detection of discontinuities in the n-th derivative of observational data. This is achieved by performing two polynomial approximations at each interstitial point. The polynomials are coupled by constraining their coefficients to ensure continuity of the model up to the (n − 1)-th derivative; while yielding an estimate for the discontinuity of the n-th derivative. The coefficients of the polynomials correspond directly to the derivatives of the approximations at the interstitial points through the prudent selection of a common coordinate system. The approximation residual and extrapolation errors are investigated as measures for detecting discontinuity. This is necessary since discrete observations of continuous systems are discontinuous at every point. It is proven, using matrix algebra, that positive extrema in the combined approximation-extrapolation error correspond exactly to extrema in the difference of the Taylor coefficients. This provides a relative measure for the severity of the discontinuity in the observational data. The matrix algebraic derivations are provided for all aspects of the methods presented here; this includes a solution for the covariance propagation through the computation. The performance of the method is verified with a Monte Carlo simulation using synthetic piecewise polynomial data with known discontinuities. It is also demonstrated that the discontinuities are suitable as knots for B-spline modelling of data. For completeness, the results of applying the method to sensor data acquired during the monitoring of heavy machinery are presented.

Highlights

  • In the recent past physics informed data science has become a focus of research activities, e.g., [9]

  • The major difference is that neither of these methods use constraints to ensure that the local polynomial approximations enforce continuity of the lower derivatives, which is done in this paper

  • In the method presented here, which falls under the category of bottomup approaches, the selection criterion is based on calculus and statistics, which allows for incorporation of the fundamental physical laws governing the system, in the model, and ensures mathematical relevance and rigour

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Summary

Introduction

In the recent past physics informed data science has become a focus of research activities, e.g., [9]. It appears under different names e.g., physics informed [12]; hybrid learning [13]; physics-based [17], etc.; but with the same basic idea of embedding physical principles into the data science algorithms. Discontinuities in the observations of continuous c The Author(s) 2020 M. Detecting discontinuities is of fundamental importance in physics based data science. Due to the discrete and finite nature of the observational data, only jump discontinuities in the n-th derivative are considered; asymptotic discontinuities are not considered. In more classical data modelling, Cn jump discontinuities form the basis for the locations of knots in B-Spline models of observational data [15]

State of the Art
The New Approach
Detecting Cn Discontinuities
Constrained and Coupled Polynomial Approximation
Covariance Propagation
Combined Error
Extrapolation Error
Synthetic Data
Sensor Data
Conclusion and Future Work
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