Detection of Derivative Discontinuities in Observational Data

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.

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

“…Note that when using splines, determining the locations of the breakpoints (knots) is very important and often very challenging. Here, however, the focus is on the variable projection method and for further information on the placement of knots, the reader is referred to [29], [30]. Fortunately, in the application fields driving this work, the locations of the knots are known from the control system data.…”