A Unified Approach to Attractor Reconstruction

Abstract

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e. attractor reconstruction. The process has focused primarily on heuristic and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several long-standing, but common problems unresolved in which the standard approaches produce inferior results or give no guidance at all. We view the current reconstruction process as unnecessarily broken into separate problems. We propose an alternative approach that views the problem of choosing all embedding parameters as being one and the same problem addressable using a single statistical test formulated directly from the reconstruction theorems. This allows for varying time delays appropriate to the data and simultaneously helps decide on embedding dimension. A second new statistic, undersampling, acts as a check against overly long time delays and overly large embedding dimension. Our approach is more flexible than those currently used, but is more directly connected with the mathematical requirements of embedding. In addition, the statistics developed guide the user by allowing optimization and warning when embedding parameters are chosen beyond what the data can support. We demonstrate our approach on uni- and multivariate data, data possessing multiple time scales, and chaotic data. This unified approach resolves all the main issues in attractor reconstruction.

22 pages, revised version as submitted to CHAOS. Manuscript is currently under review. 4 Figures, 31 references

References in corpus (2)

• Synchronous Behavior of Two Coupled Biological Neurons
• Nearest neighbor embedding with different time delays

Cited by in corpus (25)

• Nonlinear time-series analysis revisited
• Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems
• Non-uniform state space reconstruction and coupling detection
• Limits to causal inference with state-space reconstruction for infectious disease
• Persistent topological features of dynamical systems
• Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology
• A unified and automated approach to attractor reconstruction
• Optimal reconstruction of dynamical systems: A noise amplification approach
• Data-driven prediction and prevention of extreme events in a spatially extended excitable system
• Detecting Hidden Units and Network Size from Perceptible Dynamics
• Phase space reconstruction from a biological time series. A PhotoPlethysmoGraphic signal a case study
• Deep reconstruction of strange attractors from time series
• A new method for choosing parameters in delay reconstruction-based forecast strategies
• Prediction in Projection
• Finding nonlinear system equations and complex network structures from data: a sparse optimization approach
• Data-assimilation by delay-coordinate nudging
• Optimal Markov Approximations and Generalized Embeddings
• Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
• Testing Dynamical System Variables for Reconstruction
• Using Curvature to Select the Time Lag for Delay Reconstruction
• A study on dynamical complexity of noise induced blood flow
• Data-driven reduced order models using invariant foliations, manifolds and autoencoders
• Using scaling-region distributions to select embedding parameters
• Recurrence flow measure of nonlinear dependence
• A high dimensional delay selection for the reconstruction of proper Phase Space with Cross auto-correlation