Dynamics from multivariate time series. (English) Zbl 0933.62083

Summary: Multivariate time series data are common in experimental and industrial systems. If the generating system has nonlinear dynamics, we may be able to construct a model that reproduces the dynamics and can be used for control and other purposes. In principle, multivariate time series are not necessary for recovering dynamics: according to the embedding theorem, only one time series should be needed. However, for real data, there may be large gains in using all of the measurements. We examine the issues of how to use multiple data streams most effectively for modeling and prediction. For example, perhaps the data are redundant in that only a subset of the data streams is useful. And how should we embed the data, if indeed embedding is required at all? We show how these questions can be answered, and describe some numerical experiments which show that using multivariate time series can significantly improve predictability. We also demonstrate a somewhat surprising synchronization between different reconstructions.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
37M10 Time series analysis of dynamical systems
62M20 Inference from stochastic processes and prediction
Full Text: DOI


[1] Abarbanel, H. D.I., Analysis of Observed Chaotic Data (1996), Springer: Springer New York · Zbl 0875.70114
[2] Abarbanel, H. D.I.; Carroll, T. A.; Pecora, L. M.; Sidorowich, J. J.; Tsimring, L. S., Predicting physical variables in time-delay embedding, Phys. Rev. E, 49, 1840 (1994)
[3] Albano, A. M.; Mees, A. I.; deGuzman, G. C.; Rapp, P. E., Data requirements for reliable estimation of correlation dimensions, (Degn, H.; Holden, A. V.; Olsen, L. F., Chaos in Biological Systems (1987), Plenum: Plenum New York), 207-220
[4] Broomhead, D. S.; King, G. P., Extracting qualitative dynamics from experimental data, Physica D, 20, 217-236 (1985) · Zbl 0603.58040
[5] Casdagli, M., Nonlinear prediction of chaotic time series, Physica D, 35, 335-356 (1989) · Zbl 0671.62099
[6] Farmer, J. D.; Sidorowich, J. J., Predicting chaotic time series, Phys. Rev. Letters, 59, 845-848 (1987)
[7] Fraser, A. M.; Swinney, H. L., Independent coordinates for strange attractors from mutual information, Phys. Rev. A, 33, 1134-1140 (1986) · Zbl 1184.37027
[8] Grassberger, P.; Hegger, R.; Kantz, H.; Schaffrath, C.; Schreiber, T., On noise reduction methods for chaotic data, Chaos, 3, 127 (1993) · Zbl 1055.37585
[9] Grebogi, C.; Kostelich, E.; Ott, E.; Yorke, J. A., Multi-dimensioned intertwined basin boundaries: Basin structure of the kicked double rotor, Physica D, 25, 347-360 (1986)
[10] Judd, K.; Mees, A. I., On selecting models for nonlinear time series, Physica D, 82, 426-444 (1995) · Zbl 0888.58034
[11] Judd, K.; Mees, A. I., Embedding as a modeling problem, Physica D (1998), in press · Zbl 0965.37061
[12] Kadtke, J., Classification of highly noisy signals using global dynamical models, Phys. Lett. A, 203, 196 (1995) · Zbl 1020.94505
[13] Kostelich, E. J.; Schreiber, T., Noise reduction in chaotic time-series data: A survey of common methods, Phys. Rev. E, 48, 1752 (1993)
[14] Lorenz, E. N., Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130-141 (1963) · Zbl 1417.37129
[15] Mees, A. I., Dynamical systems and tesselations: Detecting determinism in data, Int. J. Bifur. Chaos, 1, 777-794 (1991) · Zbl 0876.58045
[16] Mees, A. I.; Rapp, P. E.; Jennings, L. S., Singular value decomposition and embedding dimension, Phys. Rev. A, 36, 340-346 (1987)
[17] Ott, E.; Grebogi, C.; Yorke, J., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990) · Zbl 0964.37501
[18] Ott, E.; Sauer, T.; Yorke, J. A., Coping With Chaos (1994), Wiley: Wiley New York
[19] Romeiras, F. J.; Grebogi, C.; Ott, E.; Dayawansa, W. P., Controlling chaotic dynamical systems, Physica D, 58, 165-192 (1992) · Zbl 1194.37140
[20] Rossler, O. E., An equation for continuous chaos, Phys. Lett., 57A, 397-398 (1976) · Zbl 1371.37062
[21] Sauer, T.; Yorke, J. A.; Casdagli, M., Embedology, J. Stat. Phys., 65, 579-616 (1992) · Zbl 0943.37506
[22] Takens, F., Detecting strange attractors in turbulence, (Rand, D. A.; Young, L. S., Dynamical Systems and Turbulence, vol. 898 (1981), Springer: Springer Berlin), 365-381 · Zbl 0513.58032
[23] Weigend, A. S.; Gershenfeld, N. A., Time Series Prediction: Forecasting the Future and Understanding the Past (1994), Addison-Wesley: Addison-Wesley New York
[24] Zaslavsky, G. M., The simplest case of a strange attractor, Phys. Lett. A, 69, 145 (1978)
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