Coarse-grained embeddings of time series: random walks, Gaussian random processes, and deterministic chaos

A new method for studying time series is described based on a statistic indicating the degree to which trajectories passing through a small region of an embedding space are parallel. The method is particularly suited to time series with a significant correlation time. Analytic results are presented for Brownian motion and Gaussian random processes. These are generally different from the results for chaotic systems, allowing a test for deterministic dynamics in a time series. A variety of examples are presented of the application of the method to low- and high-dimensional systems.

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