Reliable Estimation of Minimum Embedding Dimension Through Statistical Analysis of Nearest Neighbors

False nearest neighbors (FNN) is one of the essential methods used in estimating the minimally sufficient embedding dimension in delay-coordinate embedding of deterministic time series. Its use for stochastic and noisy deterministic time series is problematic and erroneously indicates a finite embedding dimension. Various modifications to the original method have been proposed to mitigate this problem, but those are still not reliable for noisy time series. Here, nearest-neighbor statistics are studied for uncorrelated random time series and contrasted with the corresponding deterministic and stochastic statistics. New composite FNN metrics are constructed and their performance is evaluated for deterministic, correlates stochastic, and white random time series. In addition, noise-contaminated deterministic data analysis shows that these composite FNN metrics are robust to noise. All FNN results are also contrasted with surrogate data analysis to show their robustness. The new metrics clearly identify random time series as not having a finite embedding dimension and provide information about the deterministic part of correlated stochastic processes. These metrics can also be used to differentiate between chaotic and random time series.

Statistical Characterization of Nearest Neighbors to Reliably Estimate Minimum Embedding Dimension

False nearest neighbors (FNN) is one of the essential methods used in estimating the minimally sufficient embedding dimension in delay coordinate embedding of deterministic time series. Its use for stochastic and noisy deterministic time series is problematic and erroneously indicates a finite embedding dimension. Various modifications to the original method have been proposed to mitigate this problem, but those are still not reliable for noisy time series. Nearest neighbor statistics are studied for uncorrelated random time series and contrasted with the deterministic statistics. A new FNN metric is constructed and its performance is evaluated for deterministic, stochastic, and random time series. The results are also contrasted with surrogate data analysis and show that the new metric is robust to noise. It also clearly identifies random time series as not having a finite embedding dimension and provides information about the deterministic part of stochastic processes. The new metric can also be used for differentiating between chaotic and random time series.

Recent Developments in Chaotic Time Series Analysis

In this paper, two issues are addressed: (1) the applicability of the delay-coordinate embedding method to transient chaotic time series analysis, and (2) the Hilbert transform methodology for chaotic signal processing.A common practice in chaotic time series analysis has been to reconstruct the phase space by utilizing the delay-coordinate embedding technique, and then to compute dynamical invariant quantities of interest such as unstable periodic orbits, the fractal dimension of the underlying chaotic set, and its Lyapunov spectrum. As a large body of literature exists on applying the technique to time series from chaotic attractors, a relatively unexplored issue is its applicability to dynamical systems that exhibit transient chaos. Our focus will be on the analysis of transient chaotic time series. We will argue and provide numerical support that the current delay-coordinate embedding techniques for extracting unstable periodic orbits, for estimating the fractal dimension, and for computing the Lyapunov exponents can be readily adapted to transient chaotic time series.A technique that is gaining an increasing attention is the Hilbert transform method for signal processing in nonlinear systems. The general goal of the Hilbert method is to assess the spectrum of the instantaneous frequency associated with the underlying dynamical process. To obtain physically meaningful results, it is necessary for the signal to possess a proper rotational structure in the complex plane of the analytic signal constructed by the original signal and its Hilbert transform. We will describe a recent decomposition procedure for this task and apply the technique to chaotic signals. We will also provide an example to demonstrate that the methodology can be useful for addressing some fundamental problems in chaotic dynamics.