×

Testing for nonlinearity in time series: the method of surrogate data. (English) Zbl 1194.37144

Summary: We describe a statistical approach for identifying nonlinearity in time series. The method first specifies some linear process as a null hypothesis, then generates surrogate data sets which are consistent with this null hypothesis, and finally computes a discriminating statistic for the original and for each of the surrogate data sets. If the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data, then the null hypothesis is rejected and nonlinearity is detected. We discuss various null hypotheses and discriminating statistics. The method is demonstrated for numerical data generated by known chaotic systems, and applied to a number of experimental time series which arise in the measurement of superfluids, brain waves, and sunspots; we evaluate the statistical significance of the evidence for nonlinear structure in each case, and illustrate aspects of the data which this approach identifies.

MSC:

37M10 Time series analysis of dynamical systems
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Nicolis, C.; Nicolis, G., Nature, 311, 529 (1984)
[2] Grassberger, P., Nature, 323, 609 (1986)
[3] Nicolis, C.; Nicolis, G., Nature, 326, 523 (1987)
[4] Grassberger, P., Nature, 326, 524 (1987)
[5] Osborne, A. R.; Kirwin, A. D.; Provenzale, A.; Bergamasco, L., Physica D, 23, 75 (1986)
[6] Mayer-Kress, G., (Hao, B.-L., Directions In Chaos, Vol. I (1988), World Scientific: World Scientific Singapore), 122-147
[7] Osborne, A. R.; Provenzale, A., Physica D, 35, 357 (1989)
[8] Provenzale, A.; Osborne, A. R.; Soj, R., Physica D, 47, 361 (1991)
[9] Theiler, J., Phys. Lett. A, 155, 480 (1991)
[10] Ramsey, J. B.; Yuan, J.-J., Phys. Lett. A, 134, 287 (1989)
[11] Theiler, J., Phys. Rev. A, 41, 3038 (1990)
[12] (Braaksma, B. L.J.; Broer, H. W.; Takens, F., Lecture Notes in Mathematics, Vol. 1125 (1985), Springer: Springer Berlin), 99-106
[13] Caswell, W. E.; Yorke, J. A., Dimensions and Entropies in Chaotic Systems - Quantification of Complex Behavior, (Springer Series in Synergetics, Vol. 32 (1986), Springer: Springer Berlin), 123-136
[14] Holzfuss, J.; Mayer-Kress, G., Dimensions and Entropies in Chaotic Systems - Quantification of Complex Behavior, (Springer Series in Synergetics, Vol. 32 (1986), Springer: Springer Berlin), 114-122
[15] Theiler, J., Quantifying Chaos: Practical Estimation of the Correlation Dimension, (Ph.D. thesis (1988), Caltech)
[16] Möller, M.; Lange, W.; Mitschke, F.; Abraham, N. B.; Hübner, U., Phys. Lett. A, 138, 176 (1989)
[17] Theiler, J., J. Opt. Soc. Am. A, 7, 1055 (1990)
[18] Smith, R. L., Nonlinear Modeling and Forecasting, (Casdagli, M.; Eubank, S., SFI Studies in the Sciences of Complexity, Vol. XII (1992), Addison-Wesley: Addison-Wesley Reading, MA), 115-136
[19] Theiler, J.; Galdrikian, B.; Longtin, A.; Eubank, S.; Farmer, J. D., Nonlinear Modeling and Forecasting, (Casdagli, M.; Eubank, S., SFI Studies in the Sciences of Complexity, Vol. XII (1992), Addison-Wesley: Addison-Wesley Reading, MA), 163-188
[20] Subba Rao, T.; Gabr, M. M., J. Time Series Anal., 1, 145 (1980) · Zbl 0499.62078
[21] Hinich, M. J., J. Time Series Anal., 3, 169 (1982)
[22] McLeod, A. I.; Li, W. K., J. Time Series Anal., 4, 269 (1983) · Zbl 0536.62067
[23] Keenan, D. M., Biometrika, 72, 39 (1985)
[24] Tsay, R. S., Biometrika, 73, 461 (1986)
[25] Tsay, R. S., Stat. Sin., 1, 431 (1991)
[26] Tong, H., Non-linear Time Series: A Dynamical System Approach (1990), Clarendon Press: Clarendon Press Oxford
[27] Efron, B., SIAM Rev., 21, 460 (1979)
[28] Tsay, R. S., Appl. Stat., 41, 1 (1992)
[29] Scheinkman, J. A.; LeBaron, B., J. Business, 62, 311 (1989)
[30] Breeden, J. L.; Packard, N. H., Nonlinear analysis of data sampled nonuniformly in time, Physica D, 58, 273 (1992), these Proceedings
[31] (Wax, N., Noise and Stochastic Processes (1954), Dover: Dover New York), reprinted in:
[32] Kaplan, D. T.; Cohen, R. J., Circulation Res., 67, 886 (1990)
[33] Brock, W. A.; Dechert, W. D.; Scheinkman, J., A test for independence based on the correlation dimension (1986), Social Systems Research Institute, University of Wisconsin at Madison, technical report 8702 · Zbl 0893.62034
[34] T.-H. Lee, H. White and C.W.J. Granger, Testing for neglected nonlinearity in time series models: A comparison of neural network methods and alternative tests J. Econometrics, to appear.; T.-H. Lee, H. White and C.W.J. Granger, Testing for neglected nonlinearity in time series models: A comparison of neural network methods and alternative tests J. Econometrics, to appear. · Zbl 0766.62055
[35] Kostelich, E. J.; Swinney, H. L., (Procaccia, I.; Shapiro, M., Chaos and Related Natural Phenomena (1987), Plenum: Plenum New York), 141
[36] Theiler, J., Phys. Rev. A, 36, 4456 (1987)
[37] Grassberger, P., Phys. Lett. A, 148, 63 (1990)
[38] Grassberger, P.; Procaccia, I., Phys. Rev. Lett., 50, 346 (1983)
[39] Grassberger, P.; Procaccia, I., Physica D, 9, 189 (1983)
[40] Takens, F., Invariants related to dimension and entropy, Atas do 13° Colóqkio Brasiliero de Matemática (1983) · Zbl 0532.58017
[41] Ellner, S., Phys. Lett. A, 133, 128 (1988)
[42] Farmer, J. D.; Sidorowich, J. J., (Lee, Y. C., Evolution, Learning and Cognition (1988), World Scientific: World Scientific Singapore), 277-330
[43] Casdagli, M., (Lam, L.; Naroditsky, V., Modeling Complex Phenomena (1992), Springer: Springer New York), 131
[44] Casdagli, M., Chaos and deterministic versus stochastic nonlinear modeling, J. R. Stat. Soc. B, 54, 303 (1992)
[45] Sano, M.; Sawada, Y., Phys. Rev. Lett., 55, 1082 (1985)
[46] Eckmann, J.-P.; Ruelle, D., Rev. Mod. Phys., 57, 617 (1985)
[47] Eckmann, J.-P.; Kamphorst, S. O.; Ruelle, D.; Ciliberto, S., Phys. Rev. A, 34, 4971 (1986)
[48] Ellner, S.; Gallant, A. R.; McCaffrey, D.; Nychka, D., Phys. Lett. A, 153, 357 (1991)
[49] D. McCaffrey, S. Ellner, A.R. Gallant and D. Nychka, Estimating the Lyapunov exponent of a chaotic system with nonparametric regression, J. Am. Stat. Assoc., to appear.; D. McCaffrey, S. Ellner, A.R. Gallant and D. Nychka, Estimating the Lyapunov exponent of a chaotic system with nonparametric regression, J. Am. Stat. Assoc., to appear. · Zbl 0782.62045
[50] Nychka, D.; Ellner, S.; McCaffrey, D.; Gallant, A. R., J. R. Stat. Soc. B, 54, 399 (1992)
[51] Schreiber, T.; Grassberger, P., Phys. Lett. A, 160, 411 (1991)
[52] W.A. Brock, J. Lakonishok and B. LeBaron, Simple technical trading rules and the stochastic properties of stock returns, J. Finance, to appear.; W.A. Brock, J. Lakonishok and B. LeBaron, Simple technical trading rules and the stochastic properties of stock returns, J. Finance, to appear.
[53] Blackman, R. B.; Tukey, J. W., The Measurement of Power Spectra (1959), Dover: Dover New York · Zbl 0084.21703
[54] Hénon, M., Commun. Math. Phys., 50, 69 (1976)
[55] Mackey, M. C.; Glass, L., Science, 197, 287 (1977)
[56] Theiler, J., Phys. Rev. A, 34, 2427 (1986)
[57] Haucke, H.; Ecke, R., Physica D, 25, 307 (1987)
[58] Rapp, P. E.; Zimmerman, I. D.; Albano, A. M.; de Guzman, G. C.; Greenbaum, N. N.; Bashore, T. R., (Othmer, H. G., Nonlinear Oscillations in Biology and Chemistry (1986), Springer: Springer Berlin), 175-205
[59] Rapp, P. E.; Bashore, T. R.; Martinerie, J. M.; Albano, A. M.; Zimmerman, I. D.; Mees, A. I., Brain Topography, 2, 99 (1989)
[60] Yule, G. U., Philos. Trans. R. Soc. London A, 226, 267 (1927)
[61] Tong, H.; Lim, K. S., J. R. Stat. Soc. B, 42, 245 (1980)
[62] Weiss, N. O., Phil. Trans. R. Soc. London A, 330, 617 (1990)
[63] Kurths, J.; Ruzmaikin, A. A., Solar Phys., 126, 407 (1990)
[64] Weigend, A.; Huberman, B.; Rummelhart, D., Intern. J. Neural Systems, 1, 193 (1990)
[65] Mundt, M.; Maguire, W. B.; Chase, R. B.P., J. Geophys. Res., 96, 1705 (1991)
[66] Weigend, A.; Huberman, B. A.; Rummelhart, D. E., Predicting sunspots and exchange rates with connectionist networks, (Casdagli, M.; Eubank, S., Nonlinear Modeling and Forecasting. Nonlinear Modeling and Forecasting, SFI Studies in the Sciences of Complexity, Vol. XII (1992), Addison-Wesley: Addison-Wesley Reading, MA), 397-434
[67] L. Smith, Identification and prediction of deterministic dynamical systems, this volume.; L. Smith, Identification and prediction of deterministic dynamical systems, this volume.
[68] Ellner, S., Detecting low-dimensional chaos in population dynamics data: a critical review, (Logan, J. A.; Hain, F. P., Chaos and Insect Ecology (1991), University of Virginia Press: University of Virginia Press Blacksburg, VA), 65-92
[69] Brock, W. A.; Sayers, C. L., J. Monetary Econ., 22, 71 (1988)
[70] Brock, W. A.; Dechert, W. D., Statistical inference theory for measures of complexity in chaos theory and nonlinear science, (Abraham, N.; etal., Measures of Complexity and Chaos (1989), Plenum: Plenum New York), 79-98
[71] Brock, W. A.; Potter, S. M., Nonlinear Modeling and Forecasting, (Casdagli, M.; Eubank, S., SFI Studies in the Sciences of Complexity, Vol. XII (1992), Addison-Wesley: Addison-Wesley Reading, MA), 137-162
[72] Hsieh, D. A., J. Business, 62, 339 (1989)
[73] Hsieh, D. A., J. Finance, 46, 1839 (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.