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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.

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.

This paper presents a novel method for polynomial approximation (Hermite approximation) using the fusion of value and derivative information. Therefore, the least-squares error in both domains is simultaneously minimized. A covariance weighting is used to introduce a metric between the value and derivative domain, to handle different noise behaviour. Based on a recurrence relation with full re-orthogonalization, a weighted polynomial basis function set is generated. This basis is numerically more stable compared to other algorithms, making it suitable for the approximation of data with high degree polynomials. With the new method, the fitting problem can be solved using inner products instead of matrix-inverses, yielding a computational more efficient method, e.g., for real-time approximation. A Monte Carlo simulation is performed on synthetic data, demonstrating the validity of the method. Additionally, various tests on the basis function set are presented, showing the improvement on the numerical stability., Comment: 7 pages, 7 figures

This paper presents advanced symbolic time series analysis (ASTSA) for large data sets emanating from cyber physical systems (CPS). The definition of CPS most pertinent to this paper is: A CPS is a system with a coupling of the cyber aspects of computing and communications with the physical aspects of dynamics and engineering that must abide by the laws of physics. This includes sensor networks, real-time and hybrid systems. To ensure that the computation results conform to the laws of physics a linear differential operator (LDO) is embedded in the processing channel for each sensor. In this manner the dynamics of the system can be incorporated prior to performing symbolic analysis. A non-linear quantization is used for the intervals corresponding to the symbols. The intervals are based on observed modes of the system, which can be determined either during an exploratory phase or online during operation of the system. A complete processing channel is called a single channel lexical analyser; one is made available for each sensor on the machine being observed. The implementation of LDO in the system is particularly important since it enables the establishment of a causal link between the observations of the dynamic system and their cause. Without causality there can be no semantics and without semantics no knowledge acquisition based on the physical background of the system being observed. Correlation alone is not a guarantee for causality. This work was originally motivated from the observation of large bulk mate- rial handling systems. Typically, there are $n = 150\dots250$ sensors per machine, and data is collected in a multi rate manner; whereby general sensors are sampled with $f_s = 1Hz$ and vibration data being sampled in the kilo-hertz range., Comment: Conference: International Work-Conference on TImeSEries Analysis (ITISE 2017)

This paper proposes a system for the ingestion and analysis of real-time sensor and actor data of bulk materials handling plants and machinery. It references issues that concern mining sensor data in cyber physical systems (CPS). The advance of cyber physical systems has created a significant change in the architecture of sensor and actor data. It affects the complexity of the observed systems in general, the number of signals being processed, the spatial distribution of the signal sources on a machine or plant and the global availability of the data. There are different definitions for what constitutes cyber physical systems: the most succinct and pertinent to the work shown in this paper is the definition given by the IEEE: A CPS is a system with a coupling of the cyber aspects of computing and communications with the physical aspects of dynamics and engineering that must abide by the laws of physics. This includes sensor networks, real-time and hybrid systems. Results computed from sensor and actor data must obey the equations used for modelling the physics of the observed system - this fundamentally poses an inverse problem. Such problems are not covered sufficiently by literature addressing mining of sensor data. Even available standard books on mining sensor data do not discuss the special nature of sensor data. Typically, present approaches of mining data rely on correlation as being a sole, reliable measure for significance. It is not taken into account that the inverse solutions to the model-describing equations are required to establish a semantic link between a sensor observation and its precedent cause. Without this link - without causality - there can be no physics based knowledge discovery., Comment: Conference on Innovation on Raw Material Extraction - Real Time Mining - RTM2017

This article presents a new approach to the real-time solution of inverse problems on embedded systems. The class of problems addressed corresponds to ordinary differential equations (ODEs) with generalized linear constraints, whereby the data from an array of sensors forms the forcing function. The solution of the equation is formulated as a least squares (LS) problem with linear constraints. The LS approach makes the method suitable for the explicit solution of inverse problems where the forcing function is perturbed by noise. The algebraic computation is partitioned into a initial preparatory step, which precomputes the matrices required for the run-time computation; and the cyclic run-time computation, which is repeated with each acquisition of sensor data. The cyclic computation consists of a single matrix-vector multiplication, in this manner computation complexity is known a-priori, fulfilling the definition of a real-time computation. Numerical testing of the new method is presented on perturbed as well as unperturbed problems; the results are compared with known analytic solutions and solutions acquired from state-of-the-art implicit solvers. The solution is implemented with model based design and uses only fundamental linear algebra; consequently, this approach supports automatic code generation for deployment on embedded systems. The targeting concept was tested via software- and processor-in-the-loop verification on two systems with different processor architectures. Finally, the method was tested on a laboratory prototype with real measurement data for the monitoring of flexible structures. The problem solved is: the real-time overconstrained reconstruction of a curve from measured gradients. Such systems are commonly encountered in the monitoring of structures and/or ground subsidence., Comment: 24 pages, journal article

This paper presents several new algorithms for the regularized reconstruction of a surface from its measured gradient field. By taking a matrix-algebraic approach, we establish general framework for the regularized reconstruction problem based on the Sylvester Matrix Equation. Specifically, Spectral Regularization via Generalized Fourier Series (e.g., Discrete Cosine Functions, Gram Polynomials, Haar Functions, etc.), Tikhonov Regularization, Constrained Regularization by imposing boundary conditions, and regularization via Weighted Least Squares can all be solved expediently in the context of the Sylvester Equation framework. State-of-the-art solutions to this problem are based on sparse matrix methods, which are no better than $\mathcal{O}!(n^6)$ algorithms for an $m\times n$ surface. In contrast, the newly proposed methods are based on the global least squares cost function and are all $\mathcal{O}!(n^3)$ algorithms. In fact, the new algorithms have the same computational complexity as an SVD of the same size. The new algorithms are several orders of magnitude faster than the state-of-the-art; we therefore present, for the first time, Monte-Carlo simulations demonstrating the statistical behaviour of the algorithms when subject to various forms of noise. We establish methods that yield the lower bound of their respective cost functions, and therefore represent the "Gold-Standard" benchmark solutions for the various forms of noise. The new methods are the first algorithms for regularized reconstruction on the order of megapixels, which is essential to methods such as Photometric Stereo.

A matrix framework is presented for the solution of ODEs, including initial-, boundary and inner-value problems. The framework enables the solution of the ODEs for arbitrary nodes. There are four key issues involved in the formulation of the framework: the use of a Lanczos process with complete reorthogonalization for the synthesis of discrete orthonormal polynomials (DOP) orthogonal over arbitrary nodes within the unit circle on the complex plane; a consistent definition of a local differentiating matrix which implements a uniform degree of approximation over the complete support --- this is particularly important for initial and boundary value problems; a method of computing a set of constraints as a constraining matrix and a method to generate orthonormal admissible functions from the constraints and a DOP matrix; the formulation of the solution to the ODEs as a least squares problem. The computation of the solution is a direct matrix method. The worst case maximum number of computations required to obtain the solution is known a-priori. This makes the method, by definition, suitable for real-time applications. The functionality of the framework is demonstrated using a selection of initial value problems, Sturm-Liouville problems and a classical Engineering boundary value problem. The framework is, however, generally formulated and is applicable to countless differential equation problems.