Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics

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.

Characterizing Complex Dynamics in the Classical and Semi-Classical Duffing Oscillator Using Ordinal Patterns Analysis

The driven double-well Duffing oscillator is a well-studied system that manifests a wide variety of dynamics, from periodic behavior to chaos, and describes a diverse array of physical systems. It has been shown to be relevant in understanding chaos in the classical to quantum transition. Here we explore the complexity of its dynamics in the classical and semi-classical regimes, using the technique of ordinal pattern analysis. This is of particular relevance to potential experiments in the semi-classical regime. We unveil different dynamical regimes within the chaotic range, which cannot be detected with more traditional statistical tools. These regimes are characterized by different hierarchies and probabilities of the ordinal patterns. Correlation between the Lyapunov exponent and the permutation entropy is revealed that leads us to interpret dips in the Lyapunov exponent as transitions in the dynamics of the system.

Comparison of Machine Learning Methods for the Arterial Hypertension Diagnostics

The paper presents results of machine learning approach accuracy applied analysis of cardiac activity. The study evaluates the diagnostics possibilities of the arterial hypertension by means of the short-term heart rate variability signals. Two groups were studied: 30 relatively healthy volunteers and 40 patients suffering from the arterial hypertension of II-III degree. The following machine learning approaches were studied: linear and quadratic discriminant analysis, k-nearest neighbors, support vector machine with radial basis, decision trees, and naive Bayes classifier. Moreover, in the study, different methods of feature extraction are analyzed: statistical, spectral, wavelet, and multifractal. All in all, 53 features were investigated. Investigation results show that discriminant analysis achieves the highest classification accuracy. The suggested approach of noncorrelated feature set search achieved higher results than data set based on the principal components.

Detection and characterization of intermittent complexity variations in cardiac arrhythmia

OBJECTIVE

A frequent observation during cardiac fibrillation is a fluctuation in complexity where the irregular pattern of the fibrillation is interrupted by more regular phases of varying length.

APPROACH

We apply different measures to sliding windows of raw ECG signals for quantifying the temporal complexity. The methods include permutation entropy, power spectral entropy, a measure for the extent of the set of reconstructed states and several wavelet measures.

MAIN RESULTS

Using these methods, variations of fibrillation patterns over time are detected and visualized.

SIGNIFICANCE

These quantifications can be used to characterize different phases of the ECG during fibrillation and might improve diagnosis and treatment methods for heart diseases.

Multiscale ordinal network analysis of human cardiac dynamics

In this study, we propose a new information theoretic measure to quantify the complexity of biological systems based on time-series data. We demonstrate the potential of our method using two distinct applications to human cardiac dynamics. Firstly, we show that the method clearly discriminates between segments of electrocardiogram records characterized by normal sinus rhythm, ventricular tachycardia and ventricular fibrillation. Secondly, we investigate the multiscale complexity of cardiac dynamics with respect to age in healthy individuals using interbeat interval time series and compare our findings with a previous study which established a link between age and fractal-like long-range correlations. The method we use is an extension of the symbolic mapping procedure originally proposed for permutation entropy. We build a Markov chain of the dynamics based on order patterns in the time series which we call an ordinal network, and from this model compute an intuitive entropic measure of transitional complexity. A discussion of the model parameter space in terms of traditional time delay embedding provides a theoretical basis for our multiscale approach. As an ancillary discussion, we address the practical issue of node aliasing and how this effects ordinal network models of continuous systems from discrete time sampled data, such as interbeat interval time series.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Variance of permutation entropy and the influence of ordinal pattern selection

Permutation entropy (PE) is a widely used measure for complexity, often used to distinguish between complex systems (or complex systems in different states). Here, the PE variance for a stationary time series is derived, and the influence of ordinal pattern selection, specifically whether the ordinal patterns are permitted to overlap or not, is examined. It was found that permitting ordinal patterns to overlap reduces the PE variance, improving the ability of this statistic to distinguish between complex system states for both numeric (fractional Gaussian noise) and experimental (semiconductor laser with optical feedback) systems. However, with overlapping ordinal patterns, the precision to which the PE variance can be estimated becomes diminished, which can manifest as increased incidences of false positive and false negative errors when applying PE to statistical inference problems.

Complexity of cardiovascular rhythms during head-up tilt test by entropy of patterns

OBJECTIVE

The head-up tilt (HUT) test, which provokes transient dynamical alterations in the regulation of cardiovascular system, provides insights into complex organization of this system. Based on signals with heart period intervals (RR-intervals) and/or systolic blood pressure (SBP), differences in the cardiovascular regulation between vasovagal patients (VVS) and the healthy people group (CG) are investigated.

APPROACH

Short-term relations among signal data represented symbolically by three-beat patterns allow to qualify and quantify the complexity of the cardiovascular regulation by Shannon entropy. Four types of patterns: permutation, ordinal, deterministic and dynamical, are used, and different resolutions of signal values in the the symbolization are applied in order to verify how entropy of patterns depends on a way in which values of signals are preprocessed.

MAIN RESULTS

At rest, in the physiologically important signal resolution ranges, independently of the type of patterns used in estimates, the complexity of SBP signals in VVS is different from the complexity found in CG. Entropy of VVS is higher than CG what could be interpreted as substantial presence of noisy ingredients in SBP of VVS. After tilting this relation switches. Entropy of CG occurs significantly higher than VVS for SBP signals. In the case of RR-intervals and large resolutions, the complexity after the tilt becomes reduced when compared to the complexity of RR-intervals at rest for both groups. However, in the case of VVS patients this reduction is significantly stronger than in CG.

SIGNIFICANCE

Our observations about opposite switches in entropy between CG and VVS might support a hypothesis that baroreflex in VVS affects stronger the heart rate because of the inefficient regulation (possibly impaired local vascular tone alternations) of the blood pressure.

Counting forbidden patterns in irregularly sampled time series. II. Reliability in the presence of highly irregular sampling

We are motivated by real-world data that exhibit severe sampling irregularities such as geological or paleoclimate measurements. Counting forbidden patterns has been shown to be a powerful tool towards the detection of determinism in noisy time series. They constitute a set of ordinal symbolic patterns that cannot be realised in time series generated by deterministic systems. The reliability of the estimator of the relative count of forbidden patterns from irregularly sampled data has been explored in two recent studies. In this paper, we explore highly irregular sampling frequency schemes. Using numerically generated data, we examine the reliability of the estimator when the sampling period has been drawn from exponential, Pareto and Gamma distributions of varying skewness. Our investigations demonstrate that some statistical properties of the sampling distribution are useful heuristics for assessing the estimator's reliability. We find that sampling in the presence of large chronological gaps can still yield relatively accurate estimates as long as the time series contains sufficiently many densely sampled areas. Furthermore, we show that the reliability of the estimator of forbidden patterns is poor when there is a high number of sampling intervals, which are larger than a typical correlation time of the underlying system.

Computing algebraic transfer entropy and coupling directions via transcripts

Most random processes studied in nonlinear time series analysis take values on sets endowed with a group structure, e.g., the real and rational numbers, and the integers. This fact allows to associate with each pair of group elements a third element, called their transcript, which is defined as the product of the second element in the pair times the first one. The transfer entropy of two such processes is called algebraic transfer entropy. It measures the information transferred between two coupled processes whose values belong to a group. In this paper, we show that, subject to one constraint, the algebraic transfer entropy matches the (in general, conditional) mutual information of certain transcripts with one variable less. This property has interesting practical applications, especially to the analysis of short time series. We also derive weak conditions for the 3-dimensional algebraic transfer entropy to yield the same coupling direction as the corresponding mutual information of transcripts. A related issue concerns the use of mutual information of transcripts to determine coupling directions in cases where the conditions just mentioned are not fulfilled. We checked the latter possibility in the lowest dimensional case with numerical simulations and cardiovascular data, and obtained positive results.

Emergence of spike correlations in periodically forced excitable systems

In sensory neurons the presence of noise can facilitate the detection of weak information-carrying signals, which are encoded and transmitted via correlated sequences of spikes. Here we investigate the relative temporal order in spike sequences induced by a subthreshold periodic input in the presence of white Gaussian noise. To simulate the spikes, we use the FitzHugh-Nagumo model and to investigate the output sequence of interspike intervals (ISIs), we use the symbolic method of ordinal analysis. We find different types of relative temporal order in the form of preferred ordinal patterns that depend on both the strength of the noise and the period of the input signal. We also demonstrate a resonancelike behavior, as certain periods and noise levels enhance temporal ordering in the ISI sequence, maximizing the probability of the preferred patterns. Our findings could be relevant for understanding the mechanisms underlying temporal coding, by which single sensory neurons represent in spike sequences the information about weak periodic stimuli.

Permutation entropy of finite-length white-noise time series

Permutation entropy (PE) is commonly used to discriminate complex structure from white noise in a time series. While the PE of white noise is well understood in the long time-series limit, analysis in the general case is currently lacking. Here the expectation value and variance of white-noise PE are derived as functions of the number of ordinal pattern trials, N, and the embedding dimension, D. It is demonstrated that the probability distribution of the white-noise PE converges to a χ^{2} distribution with D!-1 degrees of freedom as N becomes large. It is further demonstrated that the PE variance for an arbitrary time series can be estimated as the variance of a related metric, the Kullback-Leibler entropy (KLE), allowing the qualitative N≫D! condition to be recast as a quantitative estimate of the N required to achieve a desired PE calculation precision. Application of this theory to statistical inference is demonstrated in the case of an experimentally obtained noise series, where the probability of obtaining the observed PE value was calculated assuming a white-noise time series. Standard statistical inference can be used to draw conclusions whether the white-noise null hypothesis can be accepted or rejected. This methodology can be applied to other null hypotheses, such as discriminating whether two time series are generated from different complex system states.

Symbolic features and classification via support vector machine for predicting death in patients with Chagas disease

This paper introduces a technique for predicting death in patients with Chagas disease using features extracted from symbolic series and time-frequency indices of heart rate variability (HRV). The study included 150 patients: 15 patients who died and 135 who did not. The HRV series were obtained from 24-h Holter monitoring. Sequences of symbols from 5-min epochs from series of RR intervals were generated using symbolic dynamics and ordinal pattern statistics. Fourteen features were extracted from symbolic series and four derived from clinical aspects of patients. For classification, the 18 features from each epoch were used as inputs in a support vector machine (SVM) with a radial basis function (RBF) kernel. The results showed that it is possible to distinguish between the two classes, patients with Chagas disease who did or did not die, with a 95% accuracy rate. Therefore, we suggest that the use of new features based on symbolic series, coupled with classic time-frequency and clinical indices, proves to be a good predictor of death in patients with Chagas disease.

Unveiling Temporal Correlations Characteristic of a Phase Transition in the Output Intensity of a Fiber Laser

We use advanced statistical tools of time-series analysis to characterize the dynamical complexity of the transition to optical wave turbulence in a fiber laser. Ordinal analysis and the horizontal visibility graph applied to the experimentally measured laser output intensity reveal the presence of temporal correlations during the transition from the laminar to the turbulent lasing regimes. Both methods unveil coherent structures with well-defined time scales and strong correlations both, in the timing of the laser pulses and in their peak intensities. Our approach is generic and may be used in other complex systems that undergo similar transitions involving the generation of extreme fluctuations.

Quantifying spatiotemporal complexity of cardiac dynamics using ordinal patterns

Analyzing the dynamics of complex excitation wave patterns in cardiac tissue plays a key role for understanding the origin of life-threatening arrhythmias and for devising novel approaches to control them. The quantification of spatiotemporal complexity, however, remains a challenging task. This holds in particular for the analysis of data from fluorescence imaging (optical mapping), which allows for the measurement of membrane potential and intracellular calcium at high spatial and temporal resolution. Hitherto methods, like dominant frequency maps and the analysis of phase singularities, address important aspects of cardiac dynamics, but they consider very specific properties of excitable media, only. This article focuses on the benchmark of spatial complexity measures over time in the context of cardiac cell cultures. Standard Shannon Entropy and Spatial Permutation Entropy, an adaption of [1], have been implemented and applied to optical mapping data from embryonic chicken cell culture experiments. We introduce spatial separation of samples when generating ordinal patterns and show its importance for Spatial Permutation Entropy. Results suggest that Spatial Permutation Entropies provide a robust and interpretable measure for detecting qualitative changes in the dynamics of this excitable medium.

Detecting deterministic nature of pressure measurements from a turbulent combustor

Identifying nonlinear structures in a time series, acquired from real-world systems, is essential to characterize the dynamics of the system under study. A single time series alone might be available in most experimental situations. In addition to this, conventional techniques such as power spectral analysis might not be sufficient to characterize a time series if it is acquired from a complex system such as a thermoacoustic system. In this study, we analyze the unsteady pressure signal acquired from a turbulent combustor with bluff-body and swirler as flame holding devices. The fractal features in the unsteady pressure signal are identified using the singularity spectrum. Further, we employ surrogate methods, with translational error and permutation entropy as discriminating statistics, to test for determinism visible in the observed time series. In addition to this, permutation spectrum test could prove to be a robust technique to characterize the dynamical nature of the pressure time series acquired from experiments. Further, measures such as correlation dimension and correlation entropy are adopted to qualitatively detect noise contamination in the pressure measurements acquired during the state of combustion noise. These ensemble of measures is necessary to identify the features of a time series acquired from a system as complex as a turbulent combustor. Using these measures, we show that the pressure fluctuations during combustion noise has the features of a high-dimensional chaotic data contaminated with white and colored noise.

Limits of permutation-based entropies in assessing complexity of short heart period variability

The study compares permutation-based and coarse-grained entropy approaches for the assessment of complexity of short heart period (HP) variability recordings. Shannon permutation entropy (SPE) and conditional permutation entropy (CPE) are computed as examples of permutation-based entropies, while the k-nearest neighbor conditional entropy (KNNCE) is calculated as an example of coarse-grained conditional entropy. SPE, CPE and KNNCE were applied to ad-hoc simulated autoregressive processes corrupted by increasing amounts of broad band noise and to real HP variability series recorded after complete vagal blockade obtained via administration of a high dose of atropine (AT) in nine healthy volunteers and during orthostatic challenge induced by 90° head-up tilt (T90) in 15 healthy individuals. Over the simulated series the performances of SPE and CPE degraded more rapidly with the amplitude of the superimposed broad band noise than those of KNNCE. Over real data KNNCE identified the expected decrease of the HP variability complexity both after AT and during T90. Conversely SPE and CPE detected the decrease of HP variability complexity solely during T90 as a likely result of the more favorable signal-to-noise ratio during T90 than after AT. Results derived from both simulations and real data indicated that permutation-based entropies had a larger susceptibility to broad band noise than KNNCE. We recommend caution in applying permutation-based entropies in presence of short HP variability series characterized by a low signal-to-noise ratio.

Ordinal symbolic analysis and its application to biomedical recordings

Ordinal symbolic analysis opens an interesting and powerful perspective on time-series analysis. Here, we review this relatively new approach and highlight its relation to symbolic dynamics and representations. Our exposition reaches from the general ideas up to recent developments, with special emphasis on its applications to biomedical recordings. The latter will be illustrated with epilepsy data.

Changes of sleep-stage transitions due to ageing and sleep disorder

Transition patterns between different sleep stages are analysed in terms of probability distributions of symbolic sequences for young and old subjects with and without sleep disorder. Changes of these patterns due to ageing are compared with variations of transition probabilities due to sleep disorder.

Discriminating chaotic and stochastic dynamics through the permutation spectrum test

In this paper, we propose a new heuristic symbolic tool for unveiling chaotic and stochastic dynamics: the permutation spectrum test. Several numerical examples allow us to confirm the usefulness of the introduced methodology. Indeed, we show that it is robust in situations in which other techniques fail (intermittent chaos, hyperchaotic dynamics, stochastic linear and nonlinear correlated dynamics, and deterministic non-chaotic noise-driven dynamics). We illustrate the applicability and reliability of this pragmatic method by examining real complex time series from diverse scientific fields. Taking into account that the proposed test has the advantages of being conceptually simple and computationally fast, we think that it can be of practical utility as an alternative test for determinism.

Unveiling the complex organization of recurrent patterns in spiking dynamical systems

Complex systems displaying recurrent spike patterns are ubiquitous in nature. Understanding the organization of these patterns is a challenging task. Here we study experimentally the spiking output of a semiconductor laser with feedback. By using symbolic analysis we unveil a nontrivial organization of patterns, revealing serial spike correlations. The probabilities of the patterns display a well-defined, hierarchical and clustered structure that can be understood in terms of a delayed model. Most importantly, we identify a minimal model, a modified circle map, which displays the same symbolic organization. The validity of this minimal model is confirmed by analyzing the output of the forced laser. Since the circle map describes many dynamical systems, including neurons and cardiac cells, our results suggest that similar correlations and hierarchies of patterns can be found in other systems. Our findings also pave the way for optical neurons that could provide a controllable set up to mimic neuronal activity.

Distinguishing signatures of determinism and stochasticity in spiking complex systems

We describe a method to infer signatures of determinism and stochasticity in the sequence of apparently random intensity dropouts emitted by a semiconductor laser with optical feedback. The method uses ordinal time-series analysis to classify experimental data of inter-dropout-intervals (IDIs) in two categories that display statistically significant different features. Despite the apparent randomness of the dropout events, one IDI category is consistent with waiting times in a resting state until noise triggers a dropout, and the other is consistent with dropouts occurring during the return to the resting state, which have a clear deterministic component. The method we describe can be a powerful tool for inferring signatures of determinism in the dynamics of complex systems in noisy environments, at an event-level description of their dynamics.

K-nearest-neighbor conditional entropy approach for the assessment of the short-term complexity of cardiovascular control

Complexity analysis of short-term cardiovascular control is traditionally performed using entropy-based approaches including corrective terms or strategies to cope with the loss of reliability of conditional distributions with pattern length. This study proposes a new approach aiming at the estimation of conditional entropy (CE) from short data segments (about 250 samples) based on the k-nearest-neighbor technique. The main advantages are: (i) the control of the loss of reliability of the conditional distributions with the pattern length without introducing a priori information; (ii) the assessment of complexity indexes without fixing the pattern length to an arbitrary low value. The approach, referred to as k-nearest-neighbor conditional entropy (KNNCE), was contrasted with corrected approximate entropy (CApEn), sample entropy (SampEn) and corrected CE (CCE), being the most frequently exploited approaches for entropy-based complexity analysis of short cardiovascular series. Complexity indexes were evaluated during the selective pharmacological blockade of the vagal and/or sympathetic branches of the autonomic nervous system. We found that KNNCE was more powerful than CCE in detecting the decrease of complexity of heart period variability imposed by double autonomic blockade. In addition, KNNCE provides indexes indistinguishable from those derived from CApEn and SampEn. Since this result was obtained without using strategies to correct the CE estimate and without fixing the embedding dimension to an arbitrary low value, KNNCE is potentially more valuable than CCE, CApEn and SampEn when the number of past samples most useful to reduce the uncertainty of future behaviors is high and/or variable among conditions and/or groups.

Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach

In this paper we introduce a multiscale symbolic information-theory approach for discriminating nonlinear deterministic and stochastic dynamics from time series associated with complex systems. More precisely, we show that the multiscale complexity-entropy causality plane is a useful representation space to identify the range of scales at which deterministic or noisy behaviors dominate the system's dynamics. Numerical simulations obtained from the well-known and widely used Mackey-Glass oscillator operating in a high-dimensional chaotic regime were used as test beds. The effect of an increased amount of observational white noise was carefully examined. The results obtained were contrasted with those derived from correlated stochastic processes and continuous stochastic limit cycles. Finally, several experimental and natural time series were analyzed in order to show the applicability of this scale-dependent symbolic approach in practical situations.