[Paper Review] Accurate determination of time delay and embedding dimension for state space reconstruction from a scalar time series

Aniruddha Tamma, Bhaskar Khubchandani|arXiv (Cornell University)|May 5, 2016

Fault Detection and Control Systems1 references3 citations

TL;DR

This paper proposes a novel method for accurately determining time delay and embedding dimension in state space reconstruction from scalar time series, using a dimension deviation function to locate optimal time delays and a saturation-based false nearest neighbors approach for embedding dimension. The method achieves high-fidelity reconstructions of the Lorenz and Mackey-Glass attractors, outperforming traditional auto-correlation and mutual information methods.

ABSTRACT

A new and accurate method to determine the time delay and embedding dimension for state space reconstruction of a high dimensional system from a scalar time series using time delay embedding is presented. The time delay is obtained to unprecedented accuracy by evaluating the minima of a newly defined dimension deviation function. The efficacy of our method is tested by applying it to the Lorenz system and the Mackey-Glass system. A good agreement is obtained between the shape and embedding dimension of the physical system attractor(s) and the corresponding reconstruction(s) for both the systems studied. This, along with a heuristic argument provide a validation of the proposed method.

Motivation & Objective

• To address the challenge of selecting optimal time delay and embedding dimension for state space reconstruction from scalar time series of chaotic systems.
• To overcome the limitations of conventional methods like linear auto-correlation and average mutual information, which often fail to yield accurate reconstructions for nonlinear systems.
• To develop a robust, data-driven method that ensures faithful reconstruction of strange attractors in high-dimensional chaotic systems.
• To validate the method using well-characterized systems—Lorenz and Mackey-Glass—where true attractor geometry and dimensionality are known.

Proposed method

• Introduces a new dimension deviation function to determine the optimal time delay τ by identifying minima that correspond to minimal deviation in pointwise fractal dimension.
• Uses the pointwise fractal dimension definition, D_p = lim_{ε→0} log(μ(B_ε(x))) / log(ε), to quantify local geometric structure in reconstructed manifolds.
• Applies a saturation criterion for embedding dimension: the embedding dimension is selected as the point where the number of nearest neighbors stabilizes across increasing dimensions.
• Employs a modified false nearest neighbors algorithm, where ν(i,m,ε) = Σ_j θ(ε - r_ij) tracks neighbor count saturation as dimension m increases.
• Uses the Lorenz and Mackey-Glass systems as test cases, generating time series via Euler integration with known parameters and initial conditions.
• Proposes an iterative refinement: first estimate τ at a high embedding dimension, then re-estimate τ after determining the optimal embedding dimension.

Experimental results

Research questions

• RQ1 Can a new dimension deviation function outperform linear auto-correlation and mutual information in selecting the optimal time delay for state space reconstruction?
• RQ2 Does the proposed method yield attractor reconstructions that closely match the true geometry and dimensionality of known chaotic systems?
• RQ3 How does the choice of initial embedding dimension affect the accuracy of time delay estimation, and can this be mitigated?
• RQ4 To what extent does the saturation of nearest neighbor counts reliably indicate the true embedding dimension?

Key findings

• The proposed dimension deviation function successfully identified the optimal time delay for the Lorenz system, yielding a reconstruction visually indistinguishable from the true attractor.
• For the Lorenz system, the method determined the embedding dimension to be 5, which matches the known attractor dimension and corresponds to the saturation point of nearest neighbor counts.
• The method outperformed both linear auto-correlation and average mutual information, which located minima far from the visually optimal delay value.
• The Mackey-Glass system, with its explicit time delay of 1500, was accurately reconstructed using the proposed method, confirming its effectiveness on systems with known delay parameters.
• The time complexity of the method is O(n²), though it can be reduced via sampling and averaging over a subset of data points.