“Comparison of Nonlinearity Measures based on Time Series Analysis for Nonlinearity Detection”

Authors: Seyed M. Hosseini, Tor A. Johansen and Alireza Fatehi,
Affiliation: Toosi University of Technology (Iran) and NTNU, Department of Engineering Cybernetics
Reference: 2011, Vol 32, No 4, pp. 123-140.

Keywords: System Identification, Nonlinearity Measure, Higher Order Spectra, Volterra Series

Abstract: The main purpose of this paper is a study of the efficiency of different nonlinearity detection methods based on time-series data from a dynamic process as a part of system identification. A very useful concept in measuring the nonlinearity is the definition of a suitable index to measure any deviation from linearity. To analyze the properties of such an index, the observed time series is assumed to be the output of Volterra series driven by a Gaussian input. After reviewing these methods, some modifications and new indices are proposed, and a benchmark simulation study is made. Correlation analysis, harmonic analysis and higher order spectrum analysis are selected methods to be investigated in our simulations. Each method has been validated with its own advantages and disadvantages.

PDF PDF (1031 Kb)        DOI: 10.4173/mic.2011.4.1

DOI forward links to this article:
[1] Jizhen Liu, Qingwei Meng and Fang Fang (2013), doi:10.1016/j.jprocont.2013.06.012
[2] SeyedMehrdad Hosseini, Alireza Fatehi, Tor Arne Johansen and Ali Khaki Sedigh (2012), doi:10.1016/j.jprocont.2012.07.006
[3] M. W. Zhang, Z. K. Peng, X. J. Dong, W. M. Zhang and G. Meng (2016), doi:10.1007/s11071-016-2609-4
[4] S. Afshar Khamseh, A. Khaki Sedigh, B. Moshiri and A. Fatehi (2016), doi:10.1016/j.conengprac.2016.01.008
[5] V. Ondra, I.A. Sever and C.W. Schwingshackl (2016), doi:10.1016/j.ymssp.2016.06.008
[6] Davood Shaghaghi, Alireza Fatehi and Ali Khaki-Sedigh (2017), doi:10.1016/j.isatra.2017.01.021
[7] SeyedM Hosseini, Alireza Fatehi, Ali K Sedigh and Tor A Johansen (2013), doi:10.1177/0959651813479863
[8] Jingjing Du and Tor Arne Johansen (2017), doi:10.1016/j.jprocont.2017.07.001
[9] Saman Saki and Alireza Fatehi (2019), doi:10.1016/j.isatra.2019.08.001
[10] Kiril Alexiev (2020), doi:10.1007/978-3-030-39237-6_5
[11] Henrik Skyvulstad, Tommaso Argentini, Alberto Zasso and Ole Oiseth (2021), doi:10.1016/j.jweia.2020.104491
[12] Erica L. Jenson and Daniel J. Scheeres (2022), doi:10.2514/1.G006760
[1] Ashley, R., Patterson, D., Hinich, M. (1986). A diagnostic test for nonlinear serial dependence in time series fitting errors, Journal of Time Series Analysis, 7:165--178 doi:10.1111/j.1467-9892.1986.tb00500.x
[2] Barnett, A. Wolff, R. (2005). A time-domain test for some types of nonlinearity, IEEE Transactions on Signal Processing, 53:26--33 doi:10.1109/TSP.2004.838942
[3] Bedrosian, E. Rice, S. (1971). The output properties of volterra systems, nonlinear systems with memory driven by harmonic and gaussian inputs. In Proc. of the IEEE, volume59. pp. 1688--1707 doi:10.1109/PROC.1971.8525
[4] Bendat, J. Piersol, A. (1980). Engineering applications of correlation and spectral analysis, John Wiley.
[5] Billings, S. Chen, S. (1980). Identification of nonlinear systems - a survey, IEE Proceeding, 127:272--285 doi:10.1049/ip-d:19800047
[6] Billings, S. Fadzil, M. (1985). The practical identification of system with nonlinearities, In IFAC Identification and System Parameter Estimation. pp. 117--130.
[7] Billings, S. Voon, W. (1983). Structure detection and model validity tests in the identification of nonlinear systems, IEE Proceeding, 130:193--199 doi:10.1049/ip-d:19830034
[8] Billings, S. Voon, W. (1986). Correlation based model validity tests for nonlinear models, International Journal of Control, 44:235--244 doi:10.1080/00207178608933593
[9] Brillinger, D. (1965). An introduction to polyspectra, Annals of Mathematical Statistics, 36:1351--1374 doi:10.1214/aoms/1177699896
[10] Caillec, J. Garello, R. (2004). Comparison of statistical indices using third order statistics for nonlinearity detection, Journal of Signal Processing, 84:499--525 doi:10.1016/j.sigpro.2003.11.013
[11] Choudhury, M., Shah, S., Thornhill, N. (2008). Diagnosis of process nonlinearities and valve stiction, Springer.
[12] Collis, W., White, P., Hammond, J. (1998). Higher order spectra: The bispectrum and trispectrum, Mechanical Systems and Signal Processing, 12:375--394 doi:10.1006/mssp.1997.0145
[13] Evans, C., Rees, D., Jones, D. (1995). Identifying linear models of systems suffering nonlinear distortions, with a gas turbine application, In IEE Proc. Control Theory and Application, volume 142. pp. 229--240 doi:10.1049/ip-cta:19951841
[14] Evans, C., Rees, D., Jones, D., Weiss, M. (1995). Probing signals for measuring nonlinear volterra kernels, In IEEE Proc. of Instrumentation and Measurement Technology Conf. pp. 10--15 doi:10.1109/IMTC.1995.515094
[15] Evans, C., Rees, D., Jones, L. (1994). Nonlinear disturbance errors in system identification using multisine test signals, IEEE Transactions on Instrumentation and Measurements, 43:238--244 doi:10.1109/19.293427
[16] Fackrell, J. (1997). Bispectral analysis of speech signals, Ph.D. thesis, University of Edinburgh.
[17] Haber, R. (1979). Parametric identification of nonlinear dynamic systems based on correlation functions, In IFAC 5th Identification and System Parameter Estimation.
[18] Haber, R. (1985). Nonlinearity test for dynamic process, In IFAC Identification and system Parameter Estimation.
[19] Hinich, M. (1982). Testing for gaussianity and linearity of a stationary time series, Journal of Time Series Analysis, 3:169--176 doi:10.1111/j.1467-9892.1982.tb00339.x
[20] Hjellvik, V. Tjøstheim, D. (1995). Nonparametric tests of linearity for time series, Biometrika, 82:351--368 doi:10.1093/biomet/82.2.351
[21] Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika, 12:134--139 doi:10.1093/biomet/12.1-2.134
[22] Kreyzsig, E. (1978). Introductory functional analysis with application, John Wiley.
[23] Ljung, L. (1999). System identification-theory for the user, Prentice-Hall, 2nd ed.
[24] Marzocca, P., Nicholas, J., Milanese, A., Seaver, M., Trickey, S. (2008). Second-order spectra for quadratic nonlinear systems by volterra function series: analytical description and numerical simulation, Mechanical Systems and Signal Processing, 22:1882--1895 doi:10.1016/j.ymssp.2008.02.002
[25] McCormack, A., Godfrey, K., Flower, J. (1994). The detection of and compensation for nonlinear effects using periodic input signals, In Intl. Conf. on Control. pp. 297--302 doi:10.1049/cp:19940148
[26] Nicholas, J., Oslon, C., Michalowicz, J., Bucholtz, F. (2009). The bispectrum and bicoherence for quadratically nonlinear systems subject to non-gaussian inputs, IEEE Transaction of Signal Processing, 57:3879--3890 doi:10.1109/TSP.2009.2024267
[27] Nijmeijer, H. Schaft, A. (1990). Nonlinear dynamical control systems, Springer.
[28] Nuttal, A. (1958). Theory and application of the separable class of random processes, MIT Technical Report.
[29] Pintelon, R. Schoukens, J. (2001). System identification - a frequency approach, Wiley-IEEE Press.
[30] Priestley, M. (1988). Non-linear and non-stationary time series analysis, Academic Press.
[31] Rao, T. Gabr, M. (1980). A test for linearity of time series analysis, International Journal of Time Series, 1:145--158.
[32] Rusticelli, E., Ashley, R., Dagum, E., Patterson, D. (2008). A new bisepectral tests for nonlinear serial dependence, Econometric Reviews, 28:279--293 doi:10.1080/07474930802388090
[33] Saikkonen, P. Luukkonen, R. (1988). Lagrange multiplier tests for testing non-linearities in time series models, Scandinavian Journal of Statistics, 15:55--68.
[34] Schetzen, M. (1980). The volterra and wiener theories of nonlinear systems, John Wiley.
[35] Solomou, M. Rees, D. (2003). Frequency domain analysis distortions on linear frf measurements, In Instrumentation and Measurement Technology Conf., volume2. pp. 1653--1658 doi:10.1109/IMTC.2003.1208030
[36] Szücs, B., Monos, E., Czaki, F. (1975). New aspects of blood pressure control, In IFAC 6th Triannial World Congress. pp. 1--10.
[37] Varlaki, P., Terdik, G., Lototsky, V. (1985). Test for linearity and bilinearity of dynamic systems, In IFAC 7th Symposium on System Identification and Parameter Estimation. pp. 427--432.
[38] Worden, K. Tomlinson, G. (2001). Nonlinearity in structural dynamics: detection, identification and modelling, Inst. of Physics Publ. Bristol and Philadelphia.IOP.

  title={{Comparison of Nonlinearity Measures based on Time Series Analysis for Nonlinearity Detection}},
  author={Hosseini, Seyed M. and Johansen, Tor A. and Fatehi, Alireza},
  journal={Modeling, Identification and Control},
  publisher={Norwegian Society of Automatic Control}