Abstract
This paper addresses the issue of hierarchical approximation and decomposition of long time series emerging from the observation of physical systems. The first level of the decomposition uses spatial weighted polynomial approximation to obtain local estimates for the state vectors of a system, i.e., values and derivatives. Covariance weighted Hermite approximation is used to approximate the next hierarchy of state vectors by using value and derivative information from the previous hierarchy to improve the approximation. This is repeated until a certain rate of compression and/or smoothing is reached. For further usage, methods for interpolation between the state vectors are presented to reconstruct the signal at arbitrary points. All derivations needed for the presented approach are provided in this paper along with derivations needed for covariance propagation. Additionally, numerical tests reveal the benefits of the single steps. The proposed hierarchical method is successfully tested on synthetic data, proving the validity of the concept.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
To simplify readability, \(x_i\) is used for points in \(L_{0}\), \(x_j\) for points in \(L_{1}\) and \(x_k\) for points in \(L_{2}\) and above if not defined in another way. The subscripts (0), (1) and (2) denote the different levels.
- 2.
The \(\hat{\text { }}\) notation is used here, since the given state vector is the ‘noisy’ input for the next level of approximation.
References
Bajaj, C.L.: Multi-dimensional hermite interpolation and approximation for modelling and visualization. In: Proceedings of the IFIP TC5/WG5.2/WG5.10 CSI International Conference on Computer Graphics: Graphics, Design and Visualization, pp. 335–348. North-Holland Publishing Co., Amsterdam (1993). http://dl.acm.org/citation.cfm?id=645465.653690
Burden, R.L., Faires, J.D.: Numerical Analysis, 872 edn. Cengage Learning. Thomson Brooks/Cole (2005). http://books.google.at/books?id=wmcL0y2avuUC
Cleveland, W.S.: Robust locally weighted regression and smoothing scatterplots. J. Am. Stat. Assoc. 74(368), 829 (1979). https://doi.org/10.2307/2286407. http://www.jstor.org/stable/2286407
Crochiere, R.E., Rabiner, L.R.: Multirate Digital Signal Processing. Prentice-Hall (1983). https://de.scribd.com/doc/82733332/Multirate-Digital-Signal-Processing-Crochiere-Rabiner
Eilers, P.H.C.: A perfect smoother. Anal. Chem. 75(14), 3631–3636 (2003). https://doi.org/10.1021/ac034173t
Epperson, J.F.: On the runge example. Am. Math. Monthly 94(4), 329 (1987). https://doi.org/10.2307/2323093. https://www.jstor.org/stable/2323093?origin=crossref
Grabocka, J., Wistuba, M., Schmidt-Thieme, L.: Scalable classification of repetitive time series through frequencies of local polynomials. IEEE Trans. Knowl. Data Eng. 27(6), 1683–1695 (2015). https://doi.org/10.1109/TKDE.2014.2377746. http://ieeexplore.ieee.org/document/6975152/
Gupta, S., Ray, A., Keller, E.: Symbolic time series analysis of ultrasonic data for early detection of fatigue damage. Mech. Syst. Signal Process. 21(2), 866–884 (2007). https://doi.org/10.1016/j.ymssp.2005.08.022. http://linkinghub.elsevier.com/retrieve/pii/S0888327005001329
Hermite, C.: Sur la formule d’interpolation de Lagrange. (Extrait d’une lettre de M. Ch. Hermite à M. Borchardt). Journal für die reine und angewandte Mathematik 84, 70–79 (1877). http://eudml.org/doc/148345
Jerri, A.J.: The Gibbs Phenomenon in Fourier Analysis, Splines, and Wavelet Approximations. Kluwer Academic Publishers, Boston (1998)
Joldes, G.R., Chowdhury, H.A., Wittek, A., Doyle, B., Miller, K.: Modified moving least squares with polynomial bases for scattered data approximation. Appl. Math. Comput. 266, 893–902 (2015). https://doi.org/10.1016/j.amc.2015.05.150
Komargodski, Z., Levin, D.: Hermite type moving-least-squares approximations. Comput. Math. Appl. 51(8), 1223–1232 (2006). https://doi.org/10.1016/j.camwa.2006.04.005. http://linkinghub.elsevier.com/retrieve/pii/S0898122106000757
Marron, J.S., Hill, C.: Local polynomial smoothing under qualitative constraints. Computing Science and Statistics, pp. 647–652 (1997)
Mennig, J., Auerbach, T., Hälg, W.: Two point hermite approximations for the solution of linear initial value and boundary value problems. Comput. Methods Appl. Mech. Eng. 39(2), 199–224 (1983). https://doi.org/10.1016/0045-7825(83)90021-X
Moore, A.W., Schneider, J., Deng, K.: Efficient locally weighted polynomial regression predictions. In: International Conference on Machine Learning, pp. 236–244 (1997)
O’Leary, P., Harker, M.: Discrete polynomial moments and Savitzky-Golay smoothing. World Acad. Sci. Eng. Technol. Int. J. Comput. Inf. Eng. 72, 439–443 (2010). https://waset.org/publications/12268/discrete-polynomial-moments-and-savitzky-golay-smoothing
O’Leary, P., Harker, M.: Surface modelling using discrete basis functions for real-time automatic inspection. In: 3-D Surface Geometry and Reconstruction, pp. 216–264. IGI Global (2010). https://doi.org/10.4018/978-1-4666-0113-0.ch010. http://services.igi-global.com/resolvedoi/resolve.aspx?doi=10.4018/978-1-4666-0113-0.ch010
O’Leary, P., Harker, M.: A framework for the evaluation of inclinometer data in the measurement of structures. IEEE Trans. Instrum. Meas. 61(5), 1237–1251 (2012). https://doi.org/10.1109/TIM.2011.2180969. http://ieeexplore.ieee.org/document/6162983/
O’Leary, P., Harker, M.: Inverse boundary value problems with uncertain boundary values and their relevance to inclinometer measurements. In: 2014 IEEE International Instrumentation and Measurement Technology Conference (I2MTC) Proceedings, pp. 165–169. IEEE (2014). https://doi.org/10.1109/I2MTC.2014.6860725. http://ieeexplore.ieee.org/document/6860725/
O’Leary, P., Harker, M., Neumayr, R.: Savitzky-Golay smoothing for multivariate cyclic measurement data. In: 2010 IEEE International Instrumentation and Measurement Technology Conference, I2MTC 2010 - Proceedings, pp. 1585–1590 (2010). https://doi.org/10.1109/IMTC.2010.5488242
Poincaré, H.: Sur le probleme des trois corps et les équations de la dynamique. Acta Math. 13(1), 5–7 (1890). https://doi.org/10.1007/BF02392506
Proietti, T., Luati, A.: Low-pass filter design using locally weighted polynomial regression and discrete prolate spheroidal sequences. J. Stat. Plan. Inference 141(2), 831–845 (2011). https://doi.org/10.1016/j.jspi.2010.08.006. http://linkinghub.elsevier.com/retrieve/pii/S0378375810003769
Racine, J.S.: Local polynomial derivative estimation: analytic or taylor? In: Essays in Honor of Aman Ullah (Advances in Econometrics), Chap. 18, vol. 36, pp. 617–633. Emerald Group Publishing Limited (2016). https://doi.org/10.1108/S0731-905320160000036027. http://www.emeraldinsight.com/doi/10.1108/S0731-905320160000036027
Rajagopalan, V., Ray, A., Samsi, R., Mayer, J.: Pattern identification in dynamical systems via symbolic time series analysis. Pattern Recogn. 40(11), 2897–2907 (2007). https://doi.org/10.1016/j.patcog.2007.03.007
Runge, C.: Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Zeitschrift für Mathematik und Physik 46, 224–243 (1901)
Acknowledgments
Partial funding for this work was provided by the Austrian research funding association (FFG) under the scope of the COMET program within the K2 center “IC-MPP” (contract number 859480). This programme is promoted by BMVIT, BMDW and the federal states of Styria, Upper Austria and Tyrol.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Ritt, R., O’Leary, P., Rothschedl, C.J., Almasri, A., Harker, M. (2019). Hierarchical Decomposition and Approximation of Sensor Data. In: Abdel Wahab, M. (eds) Proceedings of the 1st International Conference on Numerical Modelling in Engineering . NME 2018. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-13-2273-0_27
Download citation
DOI: https://doi.org/10.1007/978-981-13-2273-0_27
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-2272-3
Online ISBN: 978-981-13-2273-0
eBook Packages: EngineeringEngineering (R0)