Markov modeling via ordinal partitions: An alternative paradigm for network-based time-series analysis

Analysis of induced dynamic biceps EMG signal complexity using Markov transition networks

Purpose

Surface electromyography (sEMG) is a non-invasive technique to characterize muscle electrical activity. The analysis of sEMG signals under muscle fatigue play a crucial part in the branch of neurorehabilitation, sports medicine, biomechanics, and monitoring neuromuscular pathologies. In this work, a method to transform sEMG signals to complex networks under muscle fatigue conditions using Markov transition field (MTF) is proposed. The importance of normalization to a constant Maximum voluntary contraction (MVC) is also considered.

Methods

For this, dynamic signals are recorded using two different experimental protocols one under constant load and another referenced to 50% MVC from Biceps brachii of 50 and 45 healthy subjects respectively. MTF is generated and network graph is constructed from preprocesses signals. Features such as average self-transition probability, average clustering coefficient and modularity are extracted.

Results

All the extracted features showed statistical significance for the recorded signals. It is found that during the transition from non-fatigue to fatigue, average clustering coefficient decreases while average self-transition probability and modularity increases.

Conclusion

The results indicate higher degree of signal complexity during non-fatigue condition. Thus, the MTF approach may be used to indicate the complexity of sEMG signals. Although both datasets showed same trend in results, sEMG signals under 50% MVC exhibited higher separability for the features. The inter individual variations of the MTF features is found to be more for the signals recorded using constant load. The proposed study can be adopted to study the complex nature of muscles under various neuromuscular conditions.

Ordinal Poincaré sections: Reconstructing the first return map from an ordinal segmentation of time series

We propose a robust algorithm for constructing first return maps of dynamical systems from time series without the need for embedding. A first return map is typically constructed using a convenient heuristic (maxima or zero-crossings of the time series, for example) or a computationally nuanced geometric approach (explicitly constructing a Poincaré section from a hyper-surface normal to the flow and then interpolating to determine intersections with trajectories). Our method is based on ordinal partitions of the time series, and the first return map is constructed from successive intersections with specific ordinal sequences. We can obtain distinct first return maps for each ordinal sequence in general. We define entropy-based measures to guide our selection of the ordinal sequence for a "good" first return map and show that this method can robustly be applied to time series from classical chaotic systems to extract the underlying first return map dynamics. The results are shown for several well-known dynamical systems (Lorenz, Rössler, and Mackey-Glass in chaotic regimes).

Generalized relational tensors for chaotic time series

The article deals with a generalized relational tensor, a novel discrete structure to store information about a time series, and algorithms (1) to fill the structure, (2) to generate a time series from the structure, and (3) to predict a time series. The algorithms combine the concept of generalized z-vectors with ant colony optimization techniques. To estimate the quality of the storing/re-generating procedure, a difference between the characteristics of the initial and regenerated time series is used. For chaotic time series, a difference between characteristics of the initial time series (the largest Lyapunov exponent, the auto-correlation function) and those of the time series re-generated from a structure is used to assess the effectiveness of the algorithms in question. The approach has shown fairly good results for periodic and benchmark chaotic time series and satisfactory results for real-world chaotic data.

Phase space partition with Koopman analysis

Symbolic dynamics is a powerful tool to describe topological features of a nonlinear system, where the required partition, however, remains a challenge for some time due to the complications involved in determining the partition boundaries. In this article, we show that it is possible to carry out interesting symbolic partitions for chaotic maps based on properly constructed eigenfunctions of the finite-dimensional approximation of the Koopman operator. The partition boundaries overlap with the extrema of these eigenfunctions, the accuracy of which is improved by including more basis functions in the numerical computation. The validity of this scheme is demonstrated in well-known 1D and 2D maps.

Complex network analysis of spatiotemporal dynamics of premixed flame in a Hele-Shaw cell: A transition from chaos to stochastic state

We study the dynamical state of a noisy nonlinear evolution equation describing flame front dynamics in a Hele-Shaw cell from the viewpoint of complex networks. The high-dimensional chaos of flame front fluctuations at a negative Rayleigh number retains the deterministic nature for sufficiently small additive noise levels. As the strength of the additive noise increases, the flame front fluctuations begin to coexist with stochastic effects, leading to a fully stochastic state. The additive noise significantly promotes the irregular appearance of the merge and divide of small-scale wrinkles of the flame front at a negative Rayleigh number, resulting in the transition of high-dimensional chaos to a fully stochastic state.

ordpy: A Python package for data analysis with permutation entropy and ordinal network methods

Since Bandt and Pompe's seminal work, permutation entropy has been used in several applications and is now an essential tool for time series analysis. Beyond becoming a popular and successful technique, permutation entropy inspired a framework for mapping time series into symbolic sequences that triggered the development of many other tools, including an approach for creating networks from time series known as ordinal networks. Despite increasing popularity, the computational development of these methods is fragmented, and there were still no efforts focusing on creating a unified software package. Here, we present ordpy (http://github.com/arthurpessa/ordpy), a simple and open-source Python module that implements permutation entropy and several of the principal methods related to Bandt and Pompe's framework to analyze time series and two-dimensional data. In particular, ordpy implements permutation entropy, Tsallis and Rényi permutation entropies, complexity-entropy plane, complexity-entropy curves, missing ordinal patterns, ordinal networks, and missing ordinal transitions for one-dimensional (time series) and two-dimensional (images) data as well as their multiscale generalizations. We review some theoretical aspects of these tools and illustrate the use of ordpy by replicating several literature results.

Asymptotic distribution of sample Shannon entropy in the case of an underlying finite, regular Markov chain

The inference of Shannon entropy out of sample histograms is known to be affected by systematic and random errors that depend on the finite size of the available data set. This dependence was mostly investigated in the multinomial case, in which states are visited in an independent fashion. In this paper the asymptotic behavior of the distribution of the sample Shannon entropy, also referred to as plug-in estimator, is investigated in the case of an underlying finite Markov process characterized by a regular stochastic matrix. As the size of the data set tends to infinity, the plug-in estimator is shown to become asymptotically normal, though in a way that substantially deviates from the known multinomial case. The asymptotic behavior of bias and variance of the plug-in estimator are expressed in terms of the spectrum of the stochastic matrix and of the related covariance matrix. Effects of initial conditions are also considered. By virtue of the formal similarity with Shannon entropy, the results are directly applicable to the evaluation of permutation entropy.

Symbolic partition in chaotic maps

In this work, we only use data on the unstable manifold to locate the partition boundaries by checking folding points at different levels, which practically coincide with homoclinic tangencies. The method is then applied to the classic two-dimensional Hénon map and a well-known three-dimensional map. Comparison with previous results is made in the Hénon case, and Lyapunov exponents are computed through the metric entropy based on the partition to show the validity of the current scheme.

Estimating topological entropy using ordinal partition networks

We propose a computationally simple and efficient network-based method for approximating topological entropy of low-dimensional chaotic systems. This approach relies on the notion of an ordinal partition. The proposed methodology is compared to the three existing techniques based on counting ordinal patterns-all of which derive from collecting statistics about the symbolic itinerary-namely (i) the gradient of the logarithm of the number of observed patterns as a function of the pattern length, (ii) direct application of the definition of topological permutation entropy, and (iii) the outgrowth ratio of patterns of increasing length. In contrast to these alternatives, our method involves the construction of a sequence of complex networks that constitute stochastic approximations of the underlying dynamics on an increasingly finer partition. An ordinal partition network can be computed using any scalar observable generated by multidimensional ergodic systems, provided the measurement function comprises a monotonic transformation if nonlinear. Numerical experiments on an ensemble of systems demonstrate that the logarithm of the spectral radius of the connectivity matrix produces significantly more accurate approximations than existing alternatives-despite practical constraints dictating the selection of low finite values for the pattern length.