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Zbl 0736.62075
Casdagli, Martin; Eubank, Stephen; Farmer, J.Doyne; Gibson, John
State space reconstruction in the presence of noise. (English)
[J] Physica D 51, No.1-3, 52-98 (1991). ISSN 0167-2789

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 2000:: *62M10 Time series, etc. (statistics)
93E99 Stochastic systems and stochastic control
62M99 Inference from stochastic processes
93E10 Estimation and detection in stochastic control

Keywords: state space reconstruction; Takens' theorem; time series; observational noise; estimation error; distortion matrix; root-mean-square time series prediction errors; noise amplification; asymptotic scaling laws; Lyapunov exponents; low noise limit

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