Modeling a nonlinear process using the exponential autoregressive time series model


The parameter estimation methods for the nonlinear exponential autoregressive (ExpAR) model are investigated in this work. Combining the hierarchical identification principle with the negative gradient search, we derive a hierarchical stochastic gradient algorithm. Inspired by the multi-innovation identification theory, we develop a hierarchical-based multi-innovation identification algorithm for the ExpAR model. Introducing two forgetting factors, a variant of the hierarchical-based multi-innovation identification algorithm is proposed. Moreover, to compare and demonstrate the serviceability of these algorithms, a nonlinear ExpAR process is taken as an example in the simulation.

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  1. 1.

    Gan, M., Li, H.X., Peng, H.: A variable projection approach for efficient estimation of RBF-ARX model. IEEE Trans. Cybern. 45(3), 462–471 (2015)

    Article  Google Scholar 

  2. 2.

    Ozaki, T.: Non-linear time series models for non-linear random vibrations. J. Appl. Probab. 17(1), 84–93 (1980)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ozaki, T.: The statistical analysis of perturbed limit cycle processes using nonlinear time series models. J. Time Ser. Anal. 3(1), 29–41 (1982)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Teräsvirta, T.: Specification, estimation, and evaluation of smooth transition autoregressive models. J. Am. Stat. Assoc. 89(425), 208–218 (1994)

    MATH  Google Scholar 

  5. 5.

    Merzougui, M., Dridi, H., Chadli, A.: Test for periodicity in restrictive EXPAR models. Commun. Stat. Theory Methods 45(9), 2770–2783 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chen, G.Y., Gan, M., Chen, G.L.: Generalized exponential autoregressive models for nonlinear time series: stationarity, estimation and applications. Inf. Sci. 438, 46–57 (2018)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Zhou, Z.P., Liu, X.F.: State and fault estimation of sandwich systems with hysteresis. Int. J. Robust Nonlinear Control 28(13), 3974–3986 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Yu, C.P., Verhaegen, M., Hansson, A.: Subspace identification of local systems in one-dimensional homogeneous networks. IEEE Trans. Autom. Control 63(4), 1126–1131 (2018)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Pan, J., Ma, H., Jiang, X., et al.: Adaptive gradient-based iterative algorithm for multivariate controlled autoregressive moving average systems using the data filtering technique. Complexity (2018).

  10. 10.

    Schoukens, M., Tiels, K.: Identification of block-oriented nonlinear systems starting from linear approximations: a survey. Automatica 85, 272–292 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Arqub, O.A., Abo-Hammour, Z.: Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inf. Sci. 279, 396–415 (2014)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Yu, C.P., Verhaegen, M.: Blind multivariable ARMA subspace identification. Automatica 66, 3–14 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Yu, C.P., Verhaegen, M.: Data-driven fault estimation of non-minimum phase LTI systems. Automatica 92, 181–187 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Chen, J., Jiang, B.: Modified stochastic gradient parameter estimation algorithms for a nonlinear two-variable difference system. Int. J. Control Autom. Syst. 14(6), 1493–1500 (2016)

    Article  Google Scholar 

  15. 15.

    Chen, F.W., Garnier, H., Gilson, M.: Robust identification of continuous-time models with arbitrary time-delay from irregularly sampled data. J. Process Control 25, 19–27 (2015)

    Article  Google Scholar 

  16. 16.

    Ding, F., Xu, L., Liu, X.M.: Signal modeling—part F: hierarchical iterative parameter estimation for multi-frequency signal models. J. Qingdao Univ. Sci. Technol. (Nat. Sci Ed.) 38(6), 1–13 (2017)

    Google Scholar 

  17. 17.

    Ding, F.: Several multi-innovation identification methods. Digit. Signal Process. 20(4), 1027–1039 (2010)

    Article  Google Scholar 

  18. 18.

    Li, L.W., Ren, X.M., Guo, F.M.: Modified multi-innovation stochastic gradient algorithm for Wiener–Hammerstein systems with backlash. J. Franklin Inst. 355(9), 4050–4075 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Cheng, S.S., Wei, Y.H., Sheng, D., Chen, Y.Q., Wang, Y.: Identification for Hammerstein nonlinear ARMAX systems based on multi-innovation fractional order stochastic gradient. Signal Process. 142, 1–10 (2018)

    Article  Google Scholar 

  20. 20.

    Ding, F., Xu, L., Liu, X.M.: Signal modeling—part E: hierarchical parameter estimation for multi-frequency signal models. J. Qingdao Univ. Sci. Technol. (Nat. Sci Ed.) 38(5), 1–15 (2017)

    Google Scholar 

  21. 21.

    Ding, F.: System Identification—Multi-Innovation Identification Theory and Methods. Science Press, Beijing (2016)

    Google Scholar 

  22. 22.

    Zhang, B., Billings, S.A.: Identification of continuous-time nonlinear systems: the nonlinear difference equation with moving average noise (NDEMA) framework. Mech. Syst. Signal Process. 60–61, 810–835 (2015)

    Article  Google Scholar 

  23. 23.

    El-Ajou, A., Arqub, O.A., Al-Smadi, M.: A general form of the generalized Taylor’s formula with some applications. Appl. Math. Comput. 256, 851–859 (2015)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Li, X., Zhu, D.Q.: An improved SOM neural network method to adaptive leader-follower formation control of AUVs. IEEE Trans. Ind. Electron. 65(10), 8260–8270 (2018)

    Google Scholar 

  25. 25.

    Chen, M.Z., Zhu, D.Q.: A workload balanced algorithm for task assignment and path planning of inhomogeneous autonomous underwater vehicle system. IEEE Trans. Cognit. Dev. Syst. (2018).

  26. 26.

    Geng, F.Z., Qian, S.P.: An optimal reproducing kernel method for linear nonlocal boundary value problems. Appl. Math. Lett. 77, 49–56 (2018)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Li, X.Y., Wu, B.Y.: A new reproducing kernel collocation method for nonlocal fractional boundary value problems with non-smooth solutions. Appl. Math. Lett. 86, 194–199 (2018)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Pan, J., Li, W., Zhang, H.P.: Control algorithms of magnetic suspension systems based on the improved double exponential reaching law of sliding mode control. Int. J. Control Autom. Syst. 16(6), 2878–2887 (2018)

    Article  Google Scholar 

  29. 29.

    Xu, L.: The parameter estimation algorithms based on the dynamical response measurement data. Adv. Mech. Eng. 9(11), 1–12 (2017).

    Article  Google Scholar 

  30. 30.

    Xu, L., Ding, F.: Parameter estimation for control systems based on impulse responses. Int. J. Control Autom. Syst. 15(6), 2471–2479 (2017)

    Article  Google Scholar 

  31. 31.

    Xu, L., Ding, F.: Iterative parameter estimation for signal models based on measured data. Circuits Syst. Signal Process. 37(7), 3046–3069 (2018)

    MathSciNet  Article  Google Scholar 

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This work was supported by the 111 Project (B12018), the National Natural Science Foundation of China (No. 61273194) and the National First-Class Discipline Program of Light Industry Technology and Engineering (LITE2018-26).

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Correspondence to Feng Ding.

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Xu, H., Ding, F. & Yang, E. Modeling a nonlinear process using the exponential autoregressive time series model. Nonlinear Dyn 95, 2079–2092 (2019).

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  • Nonlinear ExpAR model
  • Parameter estimation
  • Hierarchical identification
  • Multi-innovation identification
  • Negative gradient search