Generalized theorems for nonlinear state space reconstruction

Takens' theorem (1981) shows how lagged variables of a single time series can be used as proxy variables to reconstruct an attractor for an underlying dynamic process. State space reconstruction (SSR) from single time series has been a powerful approach for the analysis of the complex, non-line...

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Bibliographic Details
Published in:PloS one 2011-03, Vol.6 (3), p.e18295-e18295
Main Authors: Deyle, Ethan R, Sugihara, George
Format: Article
Language:English
Subjects:
Biology
Dynamical systems
Economic models
Fractals
Geophysics
Linear systems
Mathematics
Models, Theoretical
Noise
Nonlinear Dynamics
Nonlinear systems
Oceanography
Physics
Reconstruction
Science
Theorems
Time series
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Summary:Takens' theorem (1981) shows how lagged variables of a single time series can be used as proxy variables to reconstruct an attractor for an underlying dynamic process. State space reconstruction (SSR) from single time series has been a powerful approach for the analysis of the complex, non-linear systems that appear ubiquitous in the natural and human world. The main shortcoming of these methods is the phenomenological nature of attractor reconstructions. Moreover, applied studies show that these single time series reconstructions can often be improved ad hoc by including multiple dynamically coupled time series in the reconstructions, to provide a more mechanistic model. Here we provide three analytical proofs that add to the growing literature to generalize Takens' work and that demonstrate how multiple time series can be used in attractor reconstructions. These expanded results (Takens' theorem is a special case) apply to a wide variety of natural systems having parallel time series observations for variables believed to be related to the same dynamic manifold. The potential information leverage provided by multiple embeddings created from different combinations of variables (and their lags) can pave the way for new applied techniques to exploit the time-limited, but parallel observations of natural systems, such as coupled ecological systems, geophysical systems, and financial systems. This paper aims to justify and help open this potential growth area for SSR applications in the natural sciences.
ISSN:1932-6203
1932-6203
DOI:10.1371/journal.pone.0018295