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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:
62M10Time series, auto-correlation, regression, etc. (statistics)
93E99Stochastic systems and stochastic control
62M99Inference from stochastic processes
93E10Estimation and detection in stochastic control