Estimating fractal dimensions: A comparative review and open source implementations.

The fractal dimension is a central quantity in nonlinear dynamics and can be estimated via several different numerical techniques. In this review paper, we present a self-contained and comprehensive introduction to the fractal dimension. We collect and present various numerical estimators and focus on the three most promising ones: generalized entropy, correlation sum, and extreme value theory.

Ordinal pattern-based complexity analysis of high-dimensional chaotic time series.

The ordinal pattern-based complexity-entropy plane is a popular tool in nonlinear dynamics for distinguishing stochastic signals (noise) from deterministic chaos. Its performance, however, has mainly been demonstrated for time series from low-dimensional discrete or continuous dynamical systems. In order to evaluate the usefulness and power of the complexity-entropy (CE) plane approach for data representing high-dimensional chaotic dynamics, we applied this method to time series generated by the Lorenz-96 system, the generalized Hénon map, the Mackey-Glass equation, the Kuramoto-Sivashinsky equation, and to phase-randomized surrogates of these data.

Extracting Robust Biomarkers From Multichannel EEG Time Series Using Nonlinear Dimensionality Reduction Applied to Ordinal Pattern Statistics and Spectral Quantities.

In this study, ordinal pattern analysis and classical frequency-based EEG analysis methods are used to differentiate between EEGs of different age groups as well as individuals. As characteristic features, functional connectivity as well as single-channel measures in both the time and frequency domain are considered. We compare the separation power of each feature set after nonlinear dimensionality reduction using t-distributed stochastic neighbor embedding and demonstrate that ordinal pattern-based measures yield results comparable to frequency-based measures applied to preprocessed data, and outperform them if applied to raw data.