A theory of state space reconstruction in the presence of noise
- Los Alamos National Lab., NM (USA) Santa Fe Inst., NM (USA)
Takens' theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a single time series. This theorem does not treat the effect of noise, however, and so gives no guidance about practical considerations for reconstructing a good state space. We study the problem of reconstructing a state space with observational noise, examining the likelihood for a particular state given a series of noisy observations. We define a quantity called distortion, which is proportional to the covariance of the likelihood function in a reconstructed state space. This is related to the noise amplification, which corresponds to the root-mean-square errors for time series prediction with an ideal model. We prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification. We derive several asymptotic scaling laws for distortion and noise amplification. They depend on properties of the state space reconstruction, such as the sampling time and the reconstruction dimension, and properties of the dynamical system, such as the dimension and Lyapunov exponents. When the dimension and Lyapunov exponents are sufficiently large these scaling laws show that, no matter how the state space is reconstructed, there is an explosion in the noise amplification -- from a practical point of view all determinism is lost, even for short times, so that the time series is effectively a random process. In the low noise, large data limit we show that the technique of local principal value decomposition (PVD) is an optimal method of state space reconstruction, in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension. 20 refs., 12 figs.
- Research Organization:
- Los Alamos National Lab., NM (USA)
- Sponsoring Organization:
- DOE/AD
- DOE Contract Number:
- W-7405-ENG-36
- OSTI ID:
- 6223934
- Report Number(s):
- LA-UR-90-4342; CONF-9007195-1; ON: DE91005851
- Resource Relation:
- Conference: Information dynamics conference, Kaufbeuren (Germany, F.R.), Jul 1990
- Country of Publication:
- United States
- Language:
- English
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