A unified and automated approach to attractor reconstruction

Hauke Kraemer,Norbert Marwan,George Datseris,Istvan Kiss,Jorge L Ocampo-Espindola,Juergen Kurths

doi:10.5194/egusphere-egu21-1495

Abstract

&lt;p&gt;Since acquisition costs for sensors and data collection decrease rapidly especially in the geo-scientific fields, researchers often have to deal with a large amount of multivariable data, which they would need to automatically analyze in an appropriate way. In nonlinear time series analysis, phase space reconstruction often makes the very first step of any sophisticated analysis, but the established methods are either unable to reliably automate the process or they can not handle multivariate time series input. Here we present a fully automated method for the optimal state space reconstruction from univariate and multivariate time series. The proposed methodology generalizes the time delay embedding procedure by unifying two promising ideas in a symbiotic fashion. Using non-uniform delays allows the successful reconstruction of systems inheriting different time scales. In contrast to the established methods, the minimization of an appropriate cost function determines the embedding dimension without using a threshold parameter. Moreover, the method is capable of detecting stochastic time series and, thus, can handle noise contaminated input without adjusting parameters. The superiority of the proposed method is shown on some paradigmatic models and experimental data.&lt;/p&gt;

Highlights

State space reconstruction from observational time series often marks the first and basic step in nonlinear time series analysis
The reconstruction problem starts with the unknown system u(t) with a mapping f : RDB → RDB, which is observed via a measurement function h and lead to M observational time series {si(t)|i = 1, . . . , M} (Fig. 1)
We compare our method to the standard time delay embedding (TDE)

Summary

Introduction

State space reconstruction from observational time series often marks the first and basic step in nonlinear time series analysis. Several methods addressed the reconstruction problem, but none of them allow for a fully automatized and reliable way of embedding a uni- or multivariate set of observed time series with no, or at least very few, free parameters. The embedding theorems of Whitney [1], Mañé [2], and Takens [3] among with their extension by Sauer et al [4] allow several approaches to tackle the reconstruction problem. Since Takens’ theorem [3] is based on noise-free and infinitely long data, it does not give any guidance to choose the proper time delay(s) τ in practice. There can be different measurement functions h forming the multivariate dataset si(t) and the combination of Whitney’s and Takens’ embedding theorems allow for constructing u(t) from more than one time series (multivariate embedding) [4, 9]. For a detailed introduction into the reconstruction problem we refer to Casdagli et al [10], Gibson et al [8], Uzal et al [11] or Nichkawde [12]

Objectives

Results

Conclusion