Generalized Theorems for Nonlinear State Space Reconstruction

Ethan R Deyle,George Sugihara,Matej Oresic

doi:10.1371/journal.pone.0018295

Ethan R Deyle, George Sugihara + Show 1 more

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https://doi.org/10.1371/journal.pone.0018295

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Abstract

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.

Highlights

A growing realization in many natural sciences is that simple idealized notions of linearly decomposable, fixed equilibrium systems often do not accord with reality
The functions y(w(x)),y(w2(x)), . . ., corresponding to the lags of the time series are technically observation functions as well. This bears mention, because in the more general case of mappings W : M?R2mz1, the observation functions making up the components of W are not all derived from a single time series, but can be various lags of multiple time-series
Theorem 1 and the more general result presented in Theorem 2 were given proofs intended to follow those presented by Takens

Summary

Introduction

A growing realization in many natural sciences is that simple idealized notions of linearly decomposable, fixed equilibrium systems often do not accord with reality. Quantitative measurements show many systems to be fundamentally nonequilibrium and unstable, in a manner more consistent with nonlinear (state dependent) dynamics occurring on a strange attractor manifold M, where relationships between state variables cannot be studied independently of the overall system state [17,18,19,20,21,22,23,24,25,26,27]. Mirage correlations are commonplace in nonlinear systems where the manifold may contain trajectories that can temporarily exhibit positive correlations between variables for surprisingly long time periods (and in some regions of the state space) and can subsequently and rapidly exhibit negative correlations or no relationship in other time periods (and other regions of M) This transient property of apparent non-stationarity in correlations is one of the confounding phenomena faced by traditional linear models that require continual refitting and exhibit little or no predictive power

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