Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics

Fetal QRS extraction from single-channel abdominal ECG using adaptive improved permutation entropy

Fetal electrocardiogram (fECG) monitoring is crucial for assessing fetal condition during pregnancy. However, current fECG extraction algorithms are not suitable for wearable devices due to their high computational cost and multi-channel signal requirement. The paper introduces a novel and efficient algorithm called Adaptive Improved Permutation Entropy (AIPE), which can extract fetal QRS from a single-channel abdominal ECG (aECG). The proposed algorithm is robust and computationally efficient, making it a reliable and effective solution for wearable devices. To evaluate the performance of the proposed algorithm, we utilized our clinical data obtained from a pilot study with 10 subjects, each recording lasting 20 min. Additionally, data from the PhysioNet 2013 Challenge bank with labeled QRS complex annotations were simulated. The proposed methodology demonstrates an average positive predictive value ( + P ) of 91.0227%, sensitivity (Se) of 90.4726%, and F1 score of 90.6525% from the PhysioNet 2013 Challenge bank, outperforming other methods. The results suggest that AIPE could enable continuous home-based monitoring of unborn babies, even when mothers are not engaging in any hard physical activities.

Slope Entropy Characterisation: An Asymmetric Approach to Threshold Parameters Role Analysis

Slope Entropy (SlpEn) is a novel method recently proposed in the field of time series entropy estimation. In addition to the well-known embedded dimension parameter, m, used in other methods, it applies two additional thresholds, denoted as δ and γ, to derive a symbolic representation of a data subsequence. The original paper introducing SlpEn provided some guidelines for recommended specific values of these two parameters, which have been successfully followed in subsequent studies. However, a deeper understanding of the role of these thresholds is necessary to explore the potential for further SlpEn optimisations. Some works have already addressed the role of δ, but in this paper, we extend this investigation to include the role of γ and explore the impact of using an asymmetric scheme to select threshold values. We conduct a comparative analysis between the standard SlpEn method as initially proposed and an optimised version obtained through a grid search to maximise signal classification performance based on SlpEn. The results confirm that the optimised version achieves higher time series classification accuracy, albeit at the cost of significantly increased computational complexity.

Quantifying the diversity of multiple time series with an ordinal symbolic approach

The main motivation of this paper is to introduce the ordinal diversity, a symbolic tool able to quantify the degree of diversity of multiple time series. Analytical, numerical, and experimental analyses illustrate the utility of this measure to quantify how diverse, from an ordinal perspective, a set of many time series is. We have shown that ordinal diversity is able to characterize dynamical richness and dynamical transitions in stochastic processes and deterministic systems, including chaotic regimes. This ordinal tool also serves to identify optimal operating conditions in the machine learning approach of reservoir computing. These results allow us to envision potential applications for the handling and characterization of large amounts of data, paving the way for addressing some of the most pressing issues facing the current big data paradigm.

Interpolation and sampling effects on recurrence quantification measures

The recurrence plot and the recurrence quantification analysis (RQA) are well-established methods for the analysis of data from complex systems. They provide important insights into the nature of the dynamics, periodicity, regime changes, and many more. These methods are used in different fields of research, such as finance, engineering, life, and earth science. To use them, the data have usually to be uniformly sampled, posing difficulties in investigations that provide non-uniformly sampled data, as typical in medical data (e.g., heart-beat based measurements), paleoclimate archives (such as sediment cores or stalagmites), or astrophysics (supernova or pulsar observations). One frequently used solution is interpolation to generate uniform time series. However, this preprocessing step can introduce bias to the RQA measures, particularly those that rely on the diagonal or vertical line structure in the recurrence plot. Using prototypical model systems, we systematically analyze differences in the RQA measure average diagonal line length for data with different sampling and interpolation. For real data, we show that the course of this measure strongly depends on the choice of the sampling rate for interpolation. Furthermore, we suggest a correction scheme, which is capable of correcting the bias introduced by the prepossessing step if the interpolation ratio is an integer.

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. We find that both the high-dimensional deterministic time series and the stochastic surrogate data may be located in the same region of the complexity-entropy plane, and their representations show very similar behavior with varying lag and pattern lengths. Therefore, the classification of these data by means of their position in the CE plane can be challenging or even misleading, while surrogate data tests based on (entropy, complexity) yield significant results in most cases.

Power spectral estimate for discrete data

The identification of cycles in periodic signals is a ubiquitous problem in time series analysis. Many real-world datasets only record a signal as a series of discrete events or symbols. In some cases, only a sequence of (non-equidistant) times can be assessed. Many of these signals are furthermore corrupted by noise and offer a limited number of samples, e.g., cardiac signals, astronomical light curves, stock market data, or extreme weather events. We propose a novel method that provides a power spectral estimate for discrete data. The edit distance is a distance measure that allows us to quantify similarities between non-equidistant event sequences of unequal lengths. However, its potential to quantify the frequency content of discrete signals has so far remained unexplored. We define a measure of serial dependence based on the edit distance, which can be transformed into a power spectral estimate (EDSPEC), analogous to the Wiener-Khinchin theorem for continuous signals. The proposed method is applied to a variety of discrete paradigmatic signals representing random, correlated, chaotic, and periodic occurrences of events. It is effective at detecting periodic cycles even in the presence of noise and for short event series. Finally, we apply the EDSPEC method to a novel catalog of European atmospheric rivers (ARs). ARs are narrow filaments of extensive water vapor transport in the lower troposphere and can cause hazardous extreme precipitation events. Using the EDSPEC method, we conduct the first spectral analysis of European ARs, uncovering seasonal and multi-annual cycles along different spatial domains. The proposed method opens new research avenues in studying of periodic discrete signals in complex real-world systems.

Statistics and contrasts of order patterns in univariate time series

Order patterns apply well to many fields, because of minimal stationarity assumptions. Here, we fix the methodology of patterns of length 3 by introducing an orthogonal system of four pattern contrasts, that is, weighted differences of pattern frequencies. These contrasts are statistically independent and turn up as eigenvectors of a covariance matrix both in the independence model and the random walk model. The most important contrast is the turning rate. It can be used to evaluate sleep depth directly from EEG (electroencephalographic brain data). The paper discusses fluctuations of permutation entropy, statistical tests, and the need of new models for noises like EEG.

Information-Based Similarity of Ordinal Pattern Sequences as a Novel Descriptor in Obstructive Sleep Apnea Screening Based on Wearable Photoplethysmography Bracelets

Obstructive sleep apnea (OSA) is a common respiratory disorder associated with autonomic nervous system (ANS) dysfunction, resulting in abnormal heart rate variability (HRV). Capable of acquiring heart rate (HR) information with more convenience, wearable photoplethysmography (PPG) bracelets are proven to be a potential surrogate for electrocardiogram (ECG)-based devices. Meanwhile, bracelet-type PPG has been heavily marketed and widely accepted. This study aims to investigate the algorithm that can identify OSA with wearable devices. The information-based similarity of ordinal pattern sequences (OP_IBS), which is a modified version of the information-based similarity (IBS), has been proposed as a novel index to detect OSA based on wearable PPG signals. A total of 92 PPG recordings (29 normal subjects, 39 mild-moderate OSA subjects and 24 severe OSA subjects) were included in this study. OP_IBS along with classical indices were calculated. For severe OSA detection, the accuracy of OP_IBS was 85.9%, much higher than that of the low-frequency power to high-frequency power ratio (70.7%). The combination of OP_IBS, IBS, CV and LF/HF can achieve 91.3% accuracy, 91.0% sensitivity and 91.5% specificity. The performance of OP_IBS is significantly improved compared with our previous study based on the same database with the IBS method. In the Physionet database, OP_IBS also performed exceptionally well with an accuracy of 91.7%. This research shows that the OP_IBS method can access the HR dynamics of OSA subjects and help diagnose OSA in clinical environments.

Estimating Permutation Entropy Variability via Surrogate Time Series

In the last decade permutation entropy (PE) has become a popular tool to analyze the degree of randomness within a time series. In typical applications, changes in the dynamics of a source are inferred by observing changes of PE computed on different time series generated by that source. However, most works neglect the crucial question related to the statistical significance of these changes. The main reason probably lies in the difficulty of assessing, out of a single time series, not only the PE value, but also its uncertainty. In this paper we propose a method to overcome this issue by using generation of surrogate time series. The analysis conducted on both synthetic and experimental time series shows the reliability of the approach, which can be promptly implemented by means of widely available numerical tools. The method is computationally affordable for a broad range of users.

Beyond randomness: Evaluating measures of information entropy in binary series

The enormous amount of currently available data demands efforts to extract meaningful information. For this purpose, different measurements are applied, including Shannon's entropy, permutation entropy, and the Lempel-Ziv complexity. These methods have been used in many applications, such as pattern recognition, series classification, and several other areas (e.g., physical, financial, and biomedical). Data in these applications are often presented in binary series with temporal correlations. Herein, we compare the measures of information entropy in binary series conveying short- and long-range temporal correlations characterized by the Hurst exponent H. Combining numerical and analytical approaches, we scrutinize different methods that were not efficient in detecting temporal correlations. To surpass this limitation, we propose a measure called the binary permutation index (BPI). We will demonstrate that BPI efficiently discriminates patterns embedded in the series, offering advantages over previous methods. Subsequently, we collect stock market time series and rain precipitation data as well as perform in vivo electrophysiological recordings in the hippocampus of an experimental animal model of temporal lobe epilepsy, in which the BPI application in both public open source and experimental data is demonstrated. An index is proposed to evaluate information entropy, allowing the ability to discriminate randomness and extract meaningful information in binary time series.

Permutation Jensen-Shannon distance: A versatile and fast symbolic tool for complex time-series analysis

The main motivation of this paper is to introduce the permutation Jensen-Shannon distance, a symbolic tool able to quantify the degree of similarity between two arbitrary time series. This quantifier results from the fusion of two concepts, the Jensen-Shannon divergence and the encoding scheme based on the sequential ordering of the elements in the data series. The versatility and robustness of this ordinal symbolic distance for characterizing and discriminating different dynamics are illustrated through several numerical and experimental applications. Results obtained allow us to be optimistic about its usefulness in the field of complex time-series analysis. Moreover, thanks to its simplicity, low computational cost, wide applicability, and less susceptibility to outliers and artifacts, this ordinal measure can efficiently handle large amounts of data and help to tackle the current big data challenges.

Evaluating Temporal Correlations in Time Series Using Permutation Entropy, Ordinal Probabilities and Machine Learning

Time series analysis comprises a wide repertoire of methods for extracting information from data sets. Despite great advances in time series analysis, identifying and quantifying the strength of nonlinear temporal correlations remain a challenge. We have recently proposed a new method based on training a machine learning algorithm to predict the temporal correlation parameter, α, of flicker noise (FN) time series. The algorithm is trained using as input features the probabilities of ordinal patterns computed from FN time series, xαFN(t), generated with different values of α. Then, the ordinal probabilities computed from the time series of interest, x(t), are used as input features to the trained algorithm and that returns a value, αe, that contains meaningful information about the temporal correlations present in x(t). We have also shown that the difference, Ω, of the permutation entropy (PE) of the time series of interest, x(t), and the PE of a FN time series generated with α=αe, xαeFN(t), allows the identification of the underlying determinism in x(t). Here, we apply our methodology to different datasets and analyze how αe and Ω correlate with well-known quantifiers of chaos and complexity. We also discuss the limitations for identifying determinism in highly chaotic time series and in periodic time series contaminated by noise. The open source algorithm is available on Github.

Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks

Extracting relevant properties of empirical signals generated by nonlinear, stochastic, and high-dimensional systems is a challenge of complex systems research. Open questions are how to differentiate chaotic signals from stochastic ones, and how to quantify nonlinear and/or high-order temporal correlations. Here we propose a new technique to reliably address both problems. Our approach follows two steps: first, we train an artificial neural network (ANN) with flicker (colored) noise to predict the value of the parameter, [Formula: see text], that determines the strength of the correlation of the noise. To predict [Formula: see text] the ANN input features are a set of probabilities that are extracted from the time series by using symbolic ordinal analysis. Then, we input to the trained ANN the probabilities extracted from the time series of interest, and analyze the ANN output. We find that the [Formula: see text] value returned by the ANN is informative of the temporal correlations present in the time series. To distinguish between stochastic and chaotic signals, we exploit the fact that the difference between the permutation entropy (PE) of a given time series and the PE of flicker noise with the same [Formula: see text] parameter is small when the time series is stochastic, but it is large when the time series is chaotic. We validate our technique by analysing synthetic and empirical time series whose nature is well established. We also demonstrate the robustness of our approach with respect to the length of the time series and to the level of noise. We expect that our algorithm, which is freely available, will be very useful to the community.

Time-Delay Identification Using Multiscale Ordinal Quantifiers

Time-delayed interactions naturally appear in a multitude of real-world systems due to the finite propagation speed of physical quantities. Often, the time scales of the interactions are unknown to an external observer and need to be inferred from time series of observed data. We explore, in this work, the properties of several ordinal-based quantifiers for the identification of time-delays from time series. To that end, we generate artificial time series of stochastic and deterministic time-delay models. We find that the presence of a nonlinearity in the generating model has consequences for the distribution of ordinal patterns and, consequently, on the delay-identification qualities of the quantifiers. Here, we put forward a novel ordinal-based quantifier that is particularly sensitive to nonlinearities in the generating model and compare it with previously-defined quantifiers. We conclude from our analysis on artificially generated data that the proper identification of the presence of a time-delay and its precise value from time series benefits from the complementary use of ordinal-based quantifiers and the standard autocorrelation function. We further validate these tools with a practical example on real-world data originating from the North Atlantic Oscillation weather phenomenon.

Short-term deceleration capacity of heart rate: a sensitive marker of cardiac autonomic dysfunction in idiopathic Parkinson's disease

PURPOSE

Cardiac autonomic dysfunction in idiopathic Parkinson's disease (PD) manifests as reduced heart rate variability (HRV). In the present study, we explored the deceleration capacity of heart rate (DC) in patients with idiopathic PD, an advanced HRV marker that has proven clinical utility.

METHODS

Standard and advanced HRV measures derived from 7-min electrocardiograms in 20 idiopathic PD patients and 27 healthy controls were analyzed. HRV measures were compared using regression analysis, controlling for age, sex, and mean heart rate.

RESULTS

Significantly reduced HRV was found only in the subcohort of PD patients older than 60 years. Low- frequency power and global HRV measures were lower in patients than in controls, but standard beat-to-beat HRV markers (i.e., rMSSD and high-frequency power) were not significantly different between groups. DC was significantly reduced in the subcohort of PD patients older than 60 years compared to controls.

CONCLUSIONS

Deceleration-related oscillations of HRV were significantly reduced in the older PD patients compared to healthy controls, suggesting that short-term DC may be a sensitive marker of cardiac autonomic dysfunction in PD. DC may be complementary to traditional markers of short-term HRV for the evaluation of autonomic modulation in PD. Further study to examine the association between DC and cardiac adverse events in PD is needed to clarify the clinical relevance of DC in this population.

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.

Can the application of certain music information retrieval methods contribute to the machine learning classification of electrocardiographic signals?

The electrocardiogram is traditionally used to diagnose a large number of heart pathologies. Research to improve the readability and classification of cardiac signals includes studies geared toward sonification of the electrocardiographic signal and others involving features related to music processing, such as Mel-frequency cepstral coefficients. In terms of music processing features, this study seeks to use music information retrieval (MIR) features as electrocardiographic signal descriptors. The study compares the discriminatory capability of the introduced features in relation to standard groups such as heart rate variability, wavelet transform, descriptive statistics, Mel coefficients and fractal analysis, evaluated using classification algorithms; the signals analyzed were extracted from public databases. The group of features extracted from wavelet transform and the MIR group showed a high level of discrimination; the best representation of the ECG signals in the study was achieved in most cases by the MIR features. Moreover, a correlation coefficient higher than 0.8 was found between a number of MIR and other feature groups, indicating a likely relationship between the electrocardiographic signals and MIR features. These results suggest the feasibility of representing the analyzed signals by music information retrieval descriptors, giving the potential to consider these electrocardiographic signals as analogues to musical signals.

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.

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. Our analysis yields no significant differences in performance between single-channel features and functional connectivity features regarding the question of age group separation.

Ordinal patterns in the Duffing oscillator: Analyzing powers of characterization

Ordinal patterns are a time-series data analysis tool used as a preliminary step to construct the permutation entropy, which itself allows the same characterization of dynamics as chaotic or regular as more theoretical constructs such as the Lyapunov exponent. However, ordinal patterns store strictly more information than permutation entropy or Lyapunov exponents. We present results working with the Duffing oscillator showing that ordinal patterns reflect changes in dynamical symmetry that is invisible to other measures, even permutation entropy. We find that these changes in symmetry at given parameter values are correlated with a change in stability at neighboring parameters, which suggests a novel predictive capability for this analysis technique.

A generalized permutation entropy for noisy dynamics and random processes

Permutation entropy measures the complexity of a deterministic time series via a data symbolic quantization consisting of rank vectors called ordinal patterns or simply permutations. Reasons for the increasing popularity of this entropy in time series analysis include that (i) it converges to the Kolmogorov-Sinai entropy of the underlying dynamics in the limit of ever longer permutations and (ii) its computation dispenses with generating and ad hoc partitions. However, permutation entropy diverges when the number of allowed permutations grows super-exponentially with their length, as happens when time series are output by dynamical systems with observational or dynamical noise or purely random processes. In this paper, we propose a generalized permutation entropy, belonging to the class of group entropies, that is finite in that situation, which is actually the one found in practice. The theoretical results are illustrated numerically by random processes with short- and long-term dependencies, as well as by noisy deterministic signals.

Complexity of heart rate variability during moral judgement of actions and omissions

Recent research strongly supports the idea that cardiac activity is involved in the organisation of behaviour, including social behaviour and social cognition. The aim of this work was to explore the complexity of heart rate variability, as measured by permutation entropy, while individuals were making moral judgements about harmful actions and omissions. Participants (N = 58, 50% women, age 21-52 years old) were presented with a set of moral dilemmas describing situations when sacrificing one person resulted in saving five other people. In line with previous studies, our participants consistently judged harmful actions as less permissible than equivalently harmful omissions (phenomenon known as the "omission bias"). Importantly, the response times were significantly longer and permutation entropy of the heart rate was higher when participants were evaluating harmful omissions, as compared to harmful actions. These results may be viewed as a psychophysiological manifestation of differences in causal attribution between actions and omissions. We discuss the obtained results from the positions of the system-evolutionary theory and propose that heart rate variability reflects complexity of the dynamics of neurovisceral activity within the organism-environment interactions, including their social aspects. This complexity can be described in terms of entropy and our work demonstrates the potential of permutation entropy as a tool of analyzing heart rate variability in relation to current behaviour and observed cognitive processes.

Contrasting chaotic with stochastic dynamics via ordinal transition networks

We introduce a representation space to contrast chaotic with stochastic dynamics. Following the complex network representation of a time series through ordinal pattern transitions, we propose to assign each system a position in a two-dimensional plane defined by the permutation entropy of the network (global network quantifier) and the minimum value of the permutation entropy of the nodes (local network quantifier). The numerical analysis of representative chaotic maps and stochastic systems shows that the proposed approach is able to distinguish linear from non-linear dynamical systems by different planar locations. Additionally, we show that this characterization is robust when observational noise is considered. Experimental applications allow us to validate the numerical findings and to conclude that this approach is useful in practical contexts.

Neuronal Transmission of Subthreshold Periodic Stimuli Via Symbolic Spike Patterns

We study how sensory neurons detect and transmit a weak external stimulus. We use the FitzHugh-Nagumo model to simulate the neuronal activity. We consider a sub-threshold stimulus, i.e., the stimulus is below the threshold needed for triggering action potentials (spikes). However, in the presence of noise the neuron that perceives the stimulus fires a sequence of action potentials (a spike train) that carries the stimulus' information. To yield light on how the stimulus' information can be encoded and transmitted, we consider the simplest case of two coupled neurons, such that one neuron (referred to as neuron 1) perceives a subthreshold periodic signal but the second neuron (neuron 2) does not perceive the signal. We show that, for appropriate coupling and noise strengths, both neurons fire spike trains that have symbolic patterns (defined by the temporal structure of the inter-spike intervals), whose frequencies of occurrence depend on the signal's amplitude and period, and are similar for both neurons. In this way, the signal information encoded in the spike train of neuron 1 propagates to the spike train of neuron 2. Our results suggest that sensory neurons can exploit the presence of neural noise to fire spike trains where the information of a subthreshold stimulus is encoded in over expressed and/or in less expressed symbolic patterns.

Order patterns, their variation and change points in financial time series and Brownian motion

Order patterns and permutation entropy have become useful tools for studying biomedical, geophysical or climate time series. Here we study day-to-day market data, and Brownian motion which is a good model for their order patterns. A crucial point is that for small lags (1 up to 6 days), pattern frequencies in financial data remain essentially constant. The two most important order parameters of a time series are turning rate and up-down balance. For change points in EEG brain data, turning rate is excellent while for financial data, up-down balance seems the best. The fit of Brownian motion with respect to these parameters is tested, providing a new version of a forgotten test by Bienaymé.

Characterizing signal encoding and transmission in class I and class II neurons via ordinal time-series analysis

Neurons encode and transmit information in spike sequences. However, despite the effort devoted to understand the encoding and transmission of information, the mechanisms underlying the neuronal encoding are not yet fully understood. Here, we use a nonlinear method of time-series analysis (known as ordinal analysis) to compare the statistics of spike sequences generated by applying an input signal to the neuronal model of Morris-Lecar. In particular, we consider two different regimes for the neurons which lead to two classes of excitability: class I, where the frequency-current curve is continuous and class II, where the frequency-current curve is discontinuous. By applying ordinal analysis to sequences of inter-spike-intervals (ISIs) our goals are (1) to investigate if different neuron types can generate spike sequences which have similar symbolic properties; (2) to get deeper understanding on the effects that electrical (diffusive) and excitatory chemical (i.e., excitatory synapse) couplings have; and (3) to compare, when a small-amplitude periodic signal is applied to one of the neurons, how the signal features (amplitude and frequency) are encoded and transmitted in the generated ISI sequences for both class I and class II type neurons and electrical or chemical couplings. We find that depending on the frequency, specific combinations of neuron/class and coupling-type allow a more effective encoding, or a more effective transmission of the signal.

Decreased electrocortical temporal complexity distinguishes sleep from wakefulness

In most mammals, the sleep-wake cycle is constituted by three behavioral states: wakefulness (W), non-REM (NREM) sleep, and REM sleep. These states are associated with drastic changes in cognitive capacities, mostly determined by the function of the thalamo-cortical system. The intra-cranial electroencephalogram or electocorticogram (ECoG), is an important tool for measuring the changes in the thalamo-cortical activity during W and sleep. In the present study we analyzed broad-band ECoG recordings of the rat by means of a time-series complexity measure that is easy to implement and robust to noise: the Permutation Entropy (PeEn). We found that PeEn is maximal during W and decreases during sleep. These results bring to light the different thalamo-cortical dynamics emerging during sleep-wake states, which are associated with the well-known spectral changes that occur when passing from W to sleep. Moreover, the PeEn analysis allows us to determine behavioral states independently of the electrodes' cortical location, which points to an underlying global pattern in the signal that differs among the cycle states that is missed by classical methods. Consequently, our data suggest that PeEn analysis of a single EEG channel could allow for cheap, easy, and efficient sleep monitoring.

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

Mapping time series to complex networks to analyze observables has recently become popular, both at the theoretical and the practitioner's level. The intent is to use network metrics to characterize the dynamics of the underlying system. Applications cover a wide range of problems, from geoscientific measurements to biomedical data and financial time series. It has been observed that different dynamics can produce networks with distinct topological characteristics under a variety of time-series-to-network transforms that have been proposed in the literature. The direct connection, however, remains unclear. Here, we investigate a network transform based on computing statistics of ordinal permutations in short subsequences of the time series, the so-called ordinal partition network. We propose a Markovian framework that allows the interpretation of the network using ergodic-theoretic ideas and demonstrate, via numerical experiments on an ensemble of time series, that this viewpoint renders this technique especially well-suited to nonlinear chaotic signals. The aim is to test the mapping's faithfulness as a representation of the dynamics and the extent to which it retains information from the input data. First, we show that generating networks by counting patterns of increasing length is essentially a mechanism for approximating the analog of the Perron-Frobenius operator in a topologically equivalent higher-dimensional space to the original state space. Then, we illustrate a connection between the connectivity patterns of the networks generated by this mapping and indicators of dynamics such as the hierarchy of unstable periodic orbits embedded within a chaotic attractor. The input is a scalar observable and any projection of a multidimensional flow suffices for reconstruction of the essential dynamics. Additionally, we create a detailed guide for parameter tuning. We argue that there is no optimal value of the pattern length m, rather it admits a scaling region akin to traditional embedding practice. In contrast, the embedding lag and overlap between successive patterns can be chosen exactly in an optimal way. Our analysis illustrates the potential of this transform as a complementary toolkit to traditional time-series methods.

Slope Entropy: A New Time Series Complexity Estimator Based on Both Symbolic Patterns and Amplitude Information

The development of new measures and algorithms to quantify the entropy or related concepts of a data series is a continuous effort that has brought many innovations in this regard in recent years. The ultimate goal is usually to find new methods with a higher discriminating power, more efficient, more robust to noise and artifacts, less dependent on parameters or configurations, or any other possibly desirable feature. Among all these methods, Permutation Entropy (PE) is a complexity estimator for a time series that stands out due to its many strengths, with very few weaknesses. One of these weaknesses is the PE’s disregarding of time series amplitude information. Some PE algorithm modifications have been proposed in order to introduce such information into the calculations. We propose in this paper a new method, Slope Entropy (SlopEn), that also addresses this flaw but in a different way, keeping the symbolic representation of subsequences using a novel encoding method based on the slope generated by two consecutive data samples. By means of a thorough and extensive set of comparative experiments with PE and Sample Entropy (SampEn), we demonstrate that SlopEn is a very promising method with clearly a better time series classification performance than those previous methods.

Permutation Entropy: Enhancing Discriminating Power by Using Relative Frequencies Vector of Ordinal Patterns Instead of Their Shannon Entropy

Many measures to quantify the nonlinear dynamics of a time series are based on estimating the probability of certain features from their relative frequencies. Once a normalised histogram of events is computed, a single result is usually derived. This process can be broadly viewed as a nonlinear IRn mapping into IR, where n is the number of bins in the histogram. However, this mapping might entail a loss of information that could be critical for time series classification purposes. In this respect, the present study assessed such impact using permutation entropy (PE) and a diverse set of time series. We first devised a method of generating synthetic sequences of ordinal patterns using hidden Markov models. This way, it was possible to control the histogram distribution and quantify its influence on classification results. Next, real body temperature records are also used to illustrate the same phenomenon. The experiments results confirmed the improved classification accuracy achieved using raw histogram data instead of the PE final values. Thus, this study can provide a very valuable guidance for the improvement of the discriminating capability not only of PE, but of many similar histogram-based measures.

Monitoring Autonomic and Central Nervous System Activity by Permutation Entropy during Short Sojourn in Antarctica

The aim of this study was to monitor acute response patterns of autonomic and central nervous system activity during an encounter with Antarctica by synchronously recording heart rate variability (HRV) and electroencephalography (EEG). On three different time-points during the two-week sea journey, the EEG and HRV were recorded from nine male scientists who participated in “The First Turkish Antarctic Research Expedition”. The recordings were performed in a relaxed state with the eyes open, eyes closed, and during a space quantity perception test. For the EEG recordings, the wireless 14 channel EPOC-Emotiv device was used, and for the HRV recordings, a Polar heart rate monitor S810i was used. The HRV data were analyzed by time/frequency domain parameters and ordinal pattern statistics. For the EEG data, spectral band power in the conventional frequency bands, as well as permutation entropy values were calculated. Regarding HRV, neither conventional nor permutation entropy calculations produced significant differences for the different journey time-points, but only permutation entropy was able to differentiate between the testing conditions. During the cognitive test, permutation entropy values increased significantly, whereas the conventional HRV parameters did not show any significant differences. In the EEG analysis, the ordinal pattern statistics revealed significant transitions in the course of the sea voyage as permutation entropy values decreased, whereas spectral band power analysis could not detect any significant difference. Permutation entropy analysis was further able to differentiate between the three testing conditions as well between the brain regions. In the conventional spectral band power analysis, alpha band power could separate the three testing conditions and brain regions, and beta band power could only do so for the brain regions. This superiority of permutation entropy in discerning subtle differences in the autonomic and central nervous system’s responses to an overwhelming subjective experience renders it suitable as an analysis tool for biomonitoring in extreme environments.

Adaptive symbolic transfer entropy and its applications in modeling for complex industrial systems

Directed coupling between variables is the foundation of studying the dynamical behavior of complex systems. We propose an adaptive symbolic transfer entropy (ASTE) method based on the principle of equal probability division. First, the adaptive kernel density method is used to obtain an accurate probability density function for an observation series. Second, the complete phase space of the system can be obtained by using the multivariable phase space reconstruction method. This provides common parameters for symbolizing a time series, including delay time and embedding dimension. Third, an optimization strategy is used to select the appropriate symbolic parameters of a time series, such as the symbol set and partition intervals, which can be used to convert the time series to a symbol sequence. Then the transfer entropy between the symbolic sequences can be carried out. Finally, the proposed method is analyzed and validated using the chaotic Lorenz system and typical complex industrial systems. The results show that the ASTE method is superior to the existing transfer entropy and symbolic transfer entropy methods in terms of measurement accuracy and noise resistance, and it can be applied to the network modeling and performance safety analysis of complex industrial systems.

Increased markers of cardiac vagal activity in leucine-rich repeat kinase 2-associated Parkinson's disease

PURPOSE

Cardiac autonomic dysfunction manifests as reduced heart rate variability (HRV) in idiopathic Parkinson's disease (PD), but no significant reduction has been found in PD patients who carry the LRRK2 mutation. Novel HRV features have not been investigated in these individuals. We aimed to assess cardiac autonomic modulation through standard and novel approaches to HRV analysis in individuals who carry the LRRK2 G2019S mutation.

METHODS

Short-term electrocardiograms were recorded in 14 LRRK2-associated PD patients, 25 LRRK2-non-manifesting carriers, 32 related non-carriers, 20 idiopathic PD patients, and 27 healthy controls. HRV measures were compared using regression modeling, controlling for age, sex, mean heart rate, and disease duration. Discriminant analysis highlighted the feature combination that best distinguished LRRK2-associated PD from controls.

RESULTS

Beat-to-beat and global HRV measures were significantly increased in LRRK2-associated PD patients compared with controls (e.g., deceleration capacity of heart rate: p = 0.006) and idiopathic PD patients (e.g., 8th standardized moment of the interbeat interval distribution: p = 0.0003), respectively. LRRK2-associated PD patients also showed significantly increased irregularity of heart rate dynamics, as quantified by Rényi entropy, when compared with controls (p = 0.002) and idiopathic PD patients (p = 0.0004). Ordinal pattern statistics permitted the identification of LRRK2-associated PD individuals with 93% sensitivity and 93% specificity. Consistent results were found in a subgroup of LRRK2-non-manifesting carriers when compared with controls.

CONCLUSIONS

Increased beat-to-beat HRV in LRRK2 G2019S mutation carriers compared with controls and idiopathic PD patients may indicate augmented cardiac autonomic cholinergic activity, suggesting early impairment of central vagal feedback loops in LRRK2-associated PD.

Small Order Patterns in Big Time Series: A Practical Guide

The study of order patterns of three equally-spaced values xt,xt+d,xt+2d in a time series is a powerful tool. The lag d is changed in a wide range so that the differences of the frequencies of order patterns become autocorrelation functions. Similar to a spectrogram in speech analysis, four ordinal autocorrelation functions are used to visualize big data series, as for instance heart and brain activity over many hours. The method applies to real data without preprocessing, and outliers and missing data do not matter. On the theoretical side, we study the properties of order correlation functions and show that the four autocorrelation functions are orthogonal in a certain sense. An analysis of variance of a modified permutation entropy can be performed with four variance components associated with the functions.

Using Permutations for Hierarchical Clustering of Time Series

Two distances based on permutations are considered to measure the similarity of two time series according to their strength of dependency. The distance measures are used together with different linkages to get hierarchical clustering methods of time series by dependency. We apply these distances to both simulated theoretical and real data series. For simulated time series the distances show good clustering results, both in the case of linear and non-linear dependencies. The effect of the embedding dimension and the linkage method are also analyzed. Finally, several real data series are properly clustered using the proposed method.

Correlations Preceding High-Intensity Events in the Chaotic Dynamics of a Raman Fiber Laser

We study the time series of the output intensity of a Raman fiber laser with an ordinal patterns analysis in the laminar-turbulent transition. We look for signatures among consecutive events that indicate when the system changes from triggering low-intensity to high-intensity events. We set two thresholds, a low one and a high one, to distinguish between low intensity versus high-intensity events. We find that when the time series is performing low-intensity events (below the low threshold), it shows some preferred temporal patterns before triggering high-intensity events (above a high threshold). The preferred temporal patterns remain the same all through the pump current range studied, even though two clearly different dynamical regimes are covered (laminar regime for low pump currents and turbulent regime for high pump currents). We also find that the turbulent regime shows clearer signatures of determinism than the laminar regime.

Fusion of Motif- and Spectrum-Related Features for Improved EEG-Based Emotion Recognition

Emotion recognition is a burgeoning field allowing for more natural human-machine interactions and interfaces. Electroencephalography (EEG) has shown to be a useful modality with which user emotional states can be measured and monitored, particularly primitives such as valence and arousal. In this paper, we propose the use of ordinal pattern analysis, also called motifs, for improved EEG-based emotion recognition. Motifs capture recurring structures in time series and are inherently robust to noise, thus are well suited for the task at hand. Several connectivity, asymmetry, and graph-theoretic features are proposed and extracted from the motifs to be used for affective state recognition. Experiments with a widely used public database are conducted, and results show the proposed features outperforming benchmark spectrum-based features, as well as other more recent nonmotif-based graph-theoretic features and amplitude modulation-based connectivity/asymmetry measures. Feature and score-level fusion suggest complementarity between the proposed and benchmark spectrum-based measures. When combined, the fused models can provide up to 9% improvement relative to benchmark features alone and up to 16% to nonmotif-based graph-theoretic features.

Differentiating resting brain states using ordinal symbolic analysis

Symbolic methods of analysis are valuable tools for investigating complex time-dependent signals. In particular, the ordinal method defines sequences of symbols according to the ordering in which values appear in a time series. This method has been shown to yield useful information, even when applied to signals with large noise contamination. Here, we use ordinal analysis to investigate the transition between eyes closed (EC) and eyes open (EO) resting states. We analyze two electroencephalography datasets (with 71 and 109 healthy subjects) with different recording conditions (sampling rates and the number of electrodes in the scalp). Using as diagnostic tools the permutation entropy, the entropy computed from symbolic transition probabilities, and an asymmetry coefficient (that measures the asymmetry of the likelihood of the transitions between symbols), we show that the ordinal analysis applied to the raw data distinguishes the two brain states. In both datasets, we find that, during the EC-EO transition, the EO state is characterized by higher entropies and lower asymmetry coefficient, as compared to the EC state. Our results thus show that these diagnostic tools have the potential for detecting and characterizing changes in time-evolving brain states.

Strange nonchaotic and chaotic attractors in a self-excited thermoacoustic oscillator subjected to external periodic forcing

We experimentally investigate the synchronization dynamics of a self-excited thermoacoustic system forced beyond its phase-locked state. The system consists of a laminar premixed flame in a tube combustor subjected to periodic acoustic forcing. On increasing the forcing amplitude above that required for phase locking, we find that the system can transition out of phase locking and into chaos, which is consistent with the Afraimovich-Shilnikov theorem for the breakdown of a phase-locked torus. However, we also find some unexpected behavior, most notably the emergence of a strange nonchaotic attractor (SNA) before the onset of chaos. We verify the existence of the SNA and chaotic attractor by examining the correlation dimension, the autocorrelation function, the power-law scaling in the Fourier amplitude spectrum, the permutation entropy in a pseudoperiodic surrogate test, and the permutation spectrum. In summary, this study explores the SNA and chaotic dynamics of a thermoacoustic system forced beyond its phase-locked state, opening up new pathways for the development of alternative strategies to control self-excited thermoacoustic oscillations in combustion devices such as gas turbines and rocket engines.

Lagrangian Cloud Tracking and the Precipitation-Column Humidity Relationship

The tropical, oceanic mean relationship between column relative humidity and precipitation is highly non-linear. Mean precipitation remains weak until it rapidly picks up and grows at high column humidity. To investigate the origin of this relationship, a Lagrangian cloud tracking code, RAMStracks, is developed, which can follow the evolution of clouds. RAMStracks can record the morphological properties of convective clouds, the meteorological environment of clouds, and their effects. RAMStracks is applied to a large-domain radiative convective equilibrium simulation, which produces a complex population of convective clouds. RAMStracks records the lifecycle of 501 clouds through growth, splits, mergers, and decay. The mean evolution of all these clouds is examined. It is shown that the column humidity evolves non-monotonically, but that lower-level and upper-level contributions to total moisture do evolve monotonically. The precipitation efficiency of tropical storms tends to increase with cloud age. This is confirmed using a prototype testing method. The same method reveals that different tracked clouds with similar initial conditions evolve in very different ways. This makes drawing general conclusions from individual storms difficult. Finally, the causality of the precipitation-column humidity relationship is examined. A Granger Causality test, as well as regressions, suggest that moisture and precipitation are causally linked, but that the direction of causality is ambiguous. Much of this link appears to come from the lower-level moisture’s contribution to column humidity.

Assessing sustainability in North America's ecosystems using criticality and information theory

Sustainability is a key concept in economic and policy debates. Nevertheless, it is usually treated only in a qualitative way and has eluded quantitative analysis. Here, we propose a sustainability index based on the premise that sustainable systems do not lose or gain Fisher Information over time. We test this approach using time series data from the AmeriFlux network that measures ecosystem respiration, water and energy fluxes in order to elucidate two key sustainability features: ecosystem health and stability. A novel definition of ecosystem health is developed based on the concept of criticality, which implies that if a system's fluctuations are scale invariant then the system is in a balance between robustness and adaptability. We define ecosystem stability by taking an information theory approach that measures its entropy and Fisher information. Analysis of the Ameriflux consortium big data set of ecosystem respiration time series is contrasted with land condition data. In general we find a good agreement between the sustainability index and land condition data. However, we acknowledge that the results are a preliminary test of the approach and further verification will require a multi-signal analysis. For example, high values of the sustainability index for some croplands are counter-intuitive and we interpret these results as ecosystems maintained in artificial health due to continuous human-induced inflows of matter and energy in the form of soil nutrients and control of competition, pests and disease.

Forecasting Events in the Complex Dynamics of a Semiconductor Laser with Optical Feedback

Complex systems performing spiking dynamics are widespread in Nature. They cover from earthquakes, to neurons, variable stars, social networks, or stock markets. Understanding and characterizing their dynamics is relevant in order to detect transitions, or to predict unwanted extreme events. Here we study, under an ordinal patterns analysis, the output intensity of a semiconductor laser with feedback in a regime where it develops a complex spiking behavior. We unveil that, in the transitions towards and from the spiking regime, the complex dynamics presents two competing behaviors that can be distinguished with a thresholding method. Then we use time and intensity correlations to forecast different types of events, and transitions in the dynamics of the system.

Bandt-Pompe symbolization dynamics for time series with tied values: A data-driven approach

In 2002, Bandt and Pompe [Phys. Rev. Lett. 88, 174102 (2002)] introduced a successfully symbolic encoding scheme based on the ordinal relation between the amplitude of neighboring values of a given data sequence, from which the permutation entropy can be evaluated. Equalities in the analyzed sequence, for example, repeated equal values, deserve special attention and treatment as was shown recently by Zunino and co-workers [Phys. Lett. A 381, 1883 (2017)]. A significant number of equal values can give rise to false conclusions regarding the underlying temporal structures in practical contexts. In the present contribution, we review the different existing methodologies for treating time series with tied values by classifying them according to their different strategies. In addition, a novel data-driven imputation is presented that proves to outperform the existing methodologies and avoid the false conclusions pointed by Zunino and co-workers.

Sub-threshold signal encoding in coupled FitzHugh-Nagumo neurons

Despite intensive research, the mechanisms underlying the neural code remain poorly understood. Recent work has focused on the response of a single neuron to a weak, sub-threshold periodic signal. By simulating the stochastic FitzHugh-Nagumo (FHN) model and then using a symbolic method to analyze the firing activity, preferred and infrequent spike patterns (defined by the relative timing of the spikes) were detected, whose probabilities encode information about the signal. As not individual neurons but neuronal populations are responsible for sensory coding and information transfer, a relevant question is how a second neuron, which does not perceive the signal, affects the detection and the encoding of the signal, done by the first neuron. Through simulations of two stochastic FHN neurons we show that the encoding of a sub-threshold signal in symbolic spike patterns is a plausible mechanism. The neuron that perceives the signal fires a spike train that, despite having an almost random temporal structure, has preferred and infrequent patterns which carry information about the signal. Our findings could be relevant for sensory systems composed by two noisy neurons, when only one detects a weak external input.