Detecting the Nonlinearity in Time Series from Continuous Dynamic Systems Based on Delay Vector Variance Method

Authors: Shumin Hou, Yourong Li, Sanxing Zhao

Abstract:

Much time series data is generally from continuous dynamic system. Firstly, this paper studies the detection of the nonlinearity of time series from continuous dynamics systems by applying the Phase-randomized surrogate algorithm. Then, the Delay Vector Variance (DVV) method is introduced into nonlinearity test. The results show that under the different sampling conditions, the opposite detection of nonlinearity is obtained via using traditional test statistics methods, which include the third-order autocovariance and the asymmetry due to time reversal. Whereas the DVV method can perform well on determining nonlinear of Lorenz signal. It indicates that the proposed method can describe the continuous dynamics signal effectively.

Keywords: Nonlinearity, Time series, continuous dynamics system, DVV method

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1071043

References:

[1] T. Schreiber and A. Schmitz, Surrogate time series, Phys. D 142(2000) 346-382.
[2] J. Timmer, The power of surrogate data testing with respect to non-stationarity, Phys. Rev. E 58 (1998) 5153?5156.
[3] A.M. Albano, P.E. Rapp, Phase-randomised surrogates can produce spurious identification of non-random structure, Phys. Lett. A 192(1994) 27-33.
[4] D. Kugiumtzis, Test your surrogate data before you test for nonlinearity, Phys. Rev. E 60 (1999) 2808?2816.
[5] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian and J.D. Farmer, Testing for nonlinearity in time series: The method of surrogate data, Physica D 58 (1992) 77-94.
[6] J. Theiler, B. Galdrikian, A. Longtin, S. Eubank and J.D. Farmer, Using surrogate data to detect nonlinearity in time series, in: Nonlinear Modeling and Forecasting, SFI Studies in the Sciences of Complexity, Proc. Vol. XII, eds. Casdagli and S. Eubank (Addision-Wesley, Reading, MA, 1992) pp. 163-188.
[7] C.J. Stam, J.RM. Pijn , W.S. Pritchard ,Reliable detection of nonlinearity in experimental time series with strong periodic components ,Physica D 112 (1998) 361-380
[8] J. M. Le Caillec, R. Garello, Comparison of statistical indices using third order statistics for nonlinearity detection, Signal Processing 84 (2004) 499 ?525
[9] Temujin Gautama,Danilo P. Mandic ,Marc M. Van Hulle, The delay vector variance method for detecting determinism and nonlinearity in time series, Physica D 190 (2004) 167?176
[10] Kanty H, Schreiber T. Nonlinear Time Series Analysis. Cambridge: Cambridge University Press, 1997
[11] T. Schreiber and A. Schmitz, Improved surrogate data for nonlinearity tests, Phys.Rev. Lett. 77(1996) 635-638.
[12] Wu shanyuan, Wang zhaojun Nonparametric statistic method, Publishing House of Higher education, Beijing (1996) 144150.
[13] D.J. Christini, F.M. Bennett, K.R. Lutchen, H.M. Ahmed, J.M. Hausdorff, and N. Oriol, ?Applications of linear and nonlinear time series modeling to heart rate dynamics analysis,? IEEE Trans. Biomed. Eng., vol. 42, pp. 411?415, 1995.
[14] S. Guzzetti, M.G. Signorini, C. Cogliati, S. Mezzetti, A. Porta, S. Cerutti, and A. Malliani, ?Non-linear dynamics and chaotic indices in heart rate World Academy of Science, Engineering and Technology 2 2007474 variability of normal subjects and heart-transplanted patients,? Cardiovasc. Res., vol. 31, pp. 441?446, 1996.
[15] K.K. Ho, G.B. Moody, C.K. Peng, J.E. Mietus, M.G. Larson, D. Levy, and A.L. Goldberger, ?Predicting survival in heart
[16] failure case and control subjects by use of fully automated methods for deriving nonlinear and conventional indicaes of heart rate dynamics,? Circulation, vol. 96, pp. 842?848, 1997.
[17] C.-S. Poon and C.K. Merrill, ?Decrease of cardiac choas in congestive heart failure,? Nature, vol. 389, pp. 492?495, 1997.
[18] D.P. Mandic and J.A. Chambers, Recurrent Neural Networks for Prediction: Learning Algorithms Architecture and Stability, Wiley: Chichester, 2001.
[19] M.C. Casdagli and A.S. Weigend, ?Exploring the continuum between deterministic and stochastic modeling,? in Time Series Prediction: Forecasting the Future and Understanding the Past, A.S. Weigend and N.A. Gershenfield, Eds. Addison-Wesley, 1994.
[20] H.O.A. Wold, A Study in the Analysis of Stationary Time Series, Almqvist and Wiksell: Uppsala, 1938.
[21] A.S. Weigend and N.A. Gershenfeld, Eds., Time Series Prediction: Forecasting the Future and Understanding the Past, Addison-Wesley, Reading, MA, 1994.