×

zbMATH — the first resource for mathematics

State space reconstruction in the presence of noise. (English) Zbl 0736.62075
Takens’ theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series. This theorem gives little guidance, however, about practical considerations for reconstructing a good state space. We extend Takens’ treatment, applying statistical methods to incorporate the effects of observational noise and estimation error. We define the distortion matrix, which is proportional to the conditional covariance of a state, given a series of noisy measurements, and the noise amplification, which is proportional to root-mean-square time series prediction errors with an ideal model. We derive explicit formulae for these quantities, and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
93E99 Stochastic systems and control
62M99 Inference from stochastic processes
93E10 Estimation and detection in stochastic control theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abarbanel, H.D.I.; Brown, R.; Kadtke, J.B., Prediction and system identification in chaotic nonlinear systems: time series with broadband spectra, Phys. lett. A, 18, 401-408, (1989)
[2] Z. Aleksić, Estimating the embedding dimension, Physica D, to appear.
[3] Badii, R.; Broggi, G.; Derighetti, B.; Ravani, M.; Ciliberto, S.; Politi, A.; Rubio, M.A., Dimension increase in filtered chaotic signals, Phys. rev. lett., 60, 979, (1988)
[4] Breeden, J.L.; Hübler, A., Reconstructing equations of motion from experimental data with unobserved variables, Phys. rev. A, 42, 5817-5826, (1990)
[5] Broomhead, D.S.; Indik, R.; Newell, A.C.; Rand, D.A., Local adaptive Galerkin bases for large dimensional dynamical systems, Nonlinearity, (1991), to appear · Zbl 0729.58034
[6] Broomhead, D.S.; Jones, R.; King, G.P., Topological dimension and local coordinates from time series data, J. phys. A, 20, L563-L569, (1987) · Zbl 0644.58030
[7] Broomhead, D.S.; King, G.P., Extracting qualitative dynamics from experimental data, Physica D, 20, 217, (1987) · Zbl 0603.58040
[8] Casdagli, M., Nonlinear prediction of chaotic time series, Physica D, 35, 335-356, (1989) · Zbl 0671.62099
[9] Čenys, A.; Pyragas, K., Estimation of the number of degrees of freedom from chaotic time series, Phys. lett. A, 129, 227, (1988)
[10] Cremers, J.; Hübler, A., Construction of differential equations from experimental data, Z. naturforsch., 42a, 797-802, (1987)
[11] Crutchfield, J.P.; McNamara, B.S., Equations of motion from a data series, Complex systems, 1, 417-452, (1987) · Zbl 0675.58026
[12] Farmer, J.D., Chaotic attractors of an infinite-dimensional dynamical system, Physica D, 4, 366-393, (1982) · Zbl 1194.37052
[13] Farmer, J.D.; Ott, E.; Yorke, J.A., The dimension of chaotic attractors, Physica D, 7, 153-180, (1983)
[14] Farmer, J.D.; Sidorowich, J.J., Predicting chaotic time series, Phys. rev. lett., 59, 845-848, (1987)
[15] Farmer, J.D.; Sidorowich, J.J., Exploiting chaos to predict the future and reduce noise, ()
[16] Farmer, J.D.; Sidorowich, J.J., Optimal shadowing and noise reduction, Physica D, 47, 373, (1991) · Zbl 0729.65501
[17] Fraser, A.M., Information and entropy in strange attractors, IEEE transactions on information theory, IT-35, (1989) · Zbl 0712.58038
[18] Fraser, A.M., Reconstructing attractors from scalar time series: A comparison of singular system and redundancy criteria, Physica D, 34, 391-404, (1989) · Zbl 0709.94626
[19] Fraser, A.M.; Swinney, H.L., Independent coordinates for strange attractors from mutual information, Phys. rev. A, 33, 1134-1140, (1986) · Zbl 1184.37027
[20] Froehling, H.; Crutchfield, J.P.; Farmer, J.P.; Packard, N.H.; Shaw, R.S., On determining the dimension of chaotic flows, Physica D, 3, 605, (1981) · Zbl 1194.37053
[21] J. Geweke, Inference and forecasting for chaotic nonlinear time series, in preparation.
[22] Gibson, J.F.; Casdagli, M.; Eubank, S.; Farmer, J.D., Principal component analysis and derivatives of time series, ()
[23] Grassberger, P., Information content and predictability of lumped and distributed dynamical systems, ()
[24] Guckenheimer, J., Noise in chaotic systems, Nature, 298, 358-361, (1982)
[25] Hammel, S.M.; Kostelich, E.J.; Yorke, J.A., Noise reduction in dynamical systems, Phys. lett. A, Phys. rev. A, 38, 3, 421-428, (1988)
[26] Hunter, N.F., Pleistocene climate as a dynamic system, ()
[27] Kaplan, J.L.; Yorke, J.A., Chaotic behavior of multidimensional difference equations, (), 204
[28] Lapedes, A.S.; Farber, R., Nonlinear signal processing using neural networks: prediction and system modeling, ()
[29] Larimore, W., System identification, reduced order filtering, and modeling via canonical analysis, ()
[30] Liebert, W.; Pawelzik, K.; Schuster, H.G., Optimal embedding of chaotic attractors from topological considerations, (1989)
[31] Liebert, W.; Schuster, H.G., Proper choice of the time delay for the analysis of chaotic time series, Phys. lett. A, 142, 107-111, (1988)
[32] Mess, A.I., (), 104-124
[33] Packard, N.H.; Crutchfield, J.P.; Farmer, J.D.; Shaw, R.S., Geometry from a time series, Phys. rev. lett., 45, 712-716, (1980)
[34] Priestley, M.B., State dependent models: A general approach to nonlinear time series analysis, J. time series anal., 1, 47-71, (1980) · Zbl 0496.62076
[35] Priestley, M.B., Spectral analysis of time series, (1981), Academic Press New York · Zbl 0537.62075
[36] Sauer, T.; Yorke, J.; Casdagli, M.; Kostelich, E., Embedology, (1990), University of Maryland, Technical report · Zbl 0943.37506
[37] Savit, R.; Green, M., Time series and independent variables, Physica D, 50, 95-116, (1991) · Zbl 0728.62089
[38] Shaw, R.S., Strange attractors, chaotic behavior, and information flow, Z. naturforsch., 36a, 80-112, (1981) · Zbl 0599.58033
[39] Shaw, R.S., The dripping faucet as a model dynamical system, (1984), Aerial Press Santa Cruz
[40] Silverman, B.W., Kernel density estimation techniques for statistics and data analysis, (1986), Chapman Hall London · Zbl 0617.62042
[41] Takens, F., Detecting strange attractors in fluid turbulence, () · Zbl 0513.58032
[42] Tong, H.; Lim, K.S., Threshold autoregression, limit cycles and cyclical data, J. R. stat. soc. B, 42, 3, 245-292, (1980) · Zbl 0473.62081
[43] Townshend, B., Nonlinear prediction of speech signals, (), to appear
[44] Yule, G.U., Phil. trans. R. soc. London A, 226, 267, (1927)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.