Bandt Pompe Research Articles

Statistical complexity and stochastic resonance of an underdamped bistable periodic potential system excited by Lévy noise

This paper focuses on the underdamped bistable potential system's stochastic resonance (SR) phenomenon under Lévy noise excitation. The statistical complexity of the system is analyzed under the changes of damping factor, amplitude and frequency of the periodic signal, stability index and skew parameter of Lévy noise, and the effects of the changes of above parameters on SR are investigated. The results show that the increase of the damping factor hinders the appearance of SR phenomenon; the larger amplitude and lower frequency of the periodic signal are beneficial to the appearance of SR; the larger stability index of the noise parameter, the more likely SR occurs, and the change of the skew parameter has little effect on the SR. Because the Bandt-Pompe (B-P) algorithm is used in the information-theoretic measure to model the statistical complexity of this system, the robustness of the method is analyzed by changing the embedding dimension and the length of residual time interval sequence in the B-P algorithm. Finally, the mean first passage time of the particle is calculated as a function of the noise intensity and the noise-enhanced stability effect is found.

Entropy-Based Informational Study of the COVID-19 Series of Data

Since the appearance in China of the first cases, the entire world has been deeply affected by the flagellum of the Coronavirus Disease (COVID-19) pandemic. There have been many mathematical approaches trying to characterize the data collected about this serious issue. One of the most important aspects for attacking a problem is knowing what information is really available. We investigate here the information contained in the COVID-19 data of infected and deceased people in all countries, using informational quantifiers such as entropy and statistical complexity. For the evaluation of these quantities, we use the Bandt–Pompe permutation methodology, as well as the wavelet transform, to obtain the corresponding probability distributions from the available series of data. The period analyzed covers from the appearance of the disease up to the massive use of anti-COVID vaccines.

A symbolic information approach to characterize response-related differences in cortical activity during a Go/No-Go task

How the brain processes information from external stimuli in order to perceive the world and act on it is one of the greatest questions in neuroscience. To address this question, different time series analyses techniques have been employed to characterize the statistical properties of brain signals during cognitive tasks. Typically, response-specific processes are addressed by comparing the time course of average event-related potentials in different trials type. Here, we analyze monkey local field potentials data during visual pattern discrimination called Go/No-Go task in the light of information theory quantifiers. We show that the Bandt–Pompe symbolization methodology to calculate entropy and complexity of data is a useful tool to distinguish response-related differences between Go and No-Go trials. We propose to use an asymmetry index to statistically validate trial-type differences. Moreover, by using the multi-scale approach and embedding time delays to downsample the data we can estimate the important time scales in which the relevant information has been processed.

Analysis and Classification of SAR Textures Using Information Theory

The use of Bandt–Pompe probability distributions and descriptors of information theory has been presenting satisfactory results with low computational cost in the time series analysis literature <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> – <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[3]</xref> . However, these tools have limitations when applied to data without time dependency. Given this context, we present a newly proposed technique for texture analysis and classification based on the Bandt–Pompe symbolization for SAR data. It consists of linearizing a 2-D patch of the image using the Hilbert–Peano curve, build an ordinal pattern transition graph that considers the data amplitude encoded into the weight of the edges, obtain a probability distribution function derived from this graph, and compute information theory descriptors (permutation entropy and statistical complexity) from this distribution and use them as features to feed a classifier. The ordinal pattern graph we propose considers that the edges’ weight is related to the absolute difference of observations, which encodes the information about the data amplitude. This modification considers the unfavorable signal-to-noise ratio of SAR images and leads to the characterization of several types of textures. Experiments with data from Munich urban areas, Guatemala forest regions, and Cape Canaveral ocean samples show the effectiveness of our technique in homogeneous areas, achieving satisfactory separability levels. The two descriptors chosen in this work are easy and quick to calculate and are used as input for a k-nearest neighbor classifier. Experiments show that this technique presents results similar to state-of-the-art techniques that employ a much larger number of features and, consequently, impose a higher computational cost.

Dynamical complexity of FitzHugh–Nagumo neuron model driven by Lévy noise and Gaussian white noise

In this paper, on the basis of information theory measures (statistical complexity and normalized Shannon entropy), the dynamical complexity of FitzHugh–Nagumo (FHN) neuron model under the co-excitation of Lévy noise and Gaussian white noise is studied. Because the potential function of the neuron system is asymmetric, we consider not only the total residence time interval of the system, but also the residence time interval of the left and right potential wells respectively. Here, we use Bandt–Pompe algorithm to calculate the three interval sequences, and obtain the statistical complexity and normalized Shannon entropy of the total system as well as the left and right potential wells. Finally, the effects of additive noise intensity, multiplicative noise intensity and system parameter on complexity of system are analyzed. We find that the total dynamical complexity of the system is obviously different from that of a single potential well. In addition, Gaussian white noise and Lévy noise have different effects on the complexity of the system.

Dynamical Complexity of FHN Neuron System Driven by Correlated Noises and Periodic Signal

In this paper, the dynamical complexity of FHN neuron system under the co-excitation of correlated noises and periodic signals is studied by means of information theory measures. Based on the definition of statistical complexity and normalized Shannon entropy, as well as the Bandt–Pompe algorithm, the total statistical complexity and the total normalized Shannon entropy of the FHN neuron system are calculated. Because the potential function of the system is asymmetric, we also calculate the statistical complexity and the normalized Shannon entropy in two different potential wells respectively. Moreover, the effects of additive noise intensity, multiplicative noise intensity, noise correlation time, cross-correlation strength and amplitude on dynamical complexity are analyzed.

Complexity and information measures in planar characterization of chaos and noise

In this work, we present a comprehensive assessment of the Fisher information measure and statistical complexity measures based on Euclidean distance, Wootters distance and Jensen–Shannon divergence, regarding their abilities to (planar-) distinguish between/among (1) chaos and periodicity; (2) different degrees of periodicities; (3) different chaotic regimes; and (4) chaos and noise, and characterize delay dynamics. The Bandt–Pompe approach is used to build up the probability space to generate the entropy-complexity/information plane. The effect of embedding parameters on the evaluation is also considered. Within this framework, complexity measures based on the Wootters distance and Jensen–Shannon divergence are superior to the Fisher information measure in capturing subtle details of chaotic dynamics. The Fisher information measure shows advantages in robustness to additive noises and in planar-behavior representation of chaos and noise. Moreover, all measures are able to properly characterize the intrinsic delay dynamics of chaotic and stochastic systems. Nevertheless, the complexity measure based on the Euclidean distance is not valid by definition, thus, not applicable at any cases.

Analysis of financial time series through forbidden patterns

The forbidden pattern is consistently considered to be an instrument of discriminating the potentially deterministic time series from random one. As a complexity parameter, the forbidden pattern is valuable for the real-world time series since noise eliminations are not needed—it focuses on the comparison of adjacent values. In some cases, based on the permutation entropy and the Bandt–Pompe (BP) methodology, it turns to analogous results with the Lyapunov exponent, with faster calculation and better robustness, however. In this work, we use the approach of the number of forbidden patterns to distinguish completely random behavior and potentially deterministic dynamic system in the real world. In particular, we pay attention to the performance of the method in the financial time series. In addition, we are concerned about the effect of time delays on determinism of stock indices and make a discussion of the sensitivity of different stock indices to time delays. Moreover, in this paper we focus on the evolution of deterministic behavior, pointing out that it is effective to use this method to quantify the degree of determinism in different time states. We can see that the approach of the number of forbidden patterns has excellent characteristics in terms of capturing potential properties of financial systems.

Model Selection: Using Information Measures from Ordinal Symbolic Analysis to Select Model Subgrid-Scale Parameterizations

Abstract The use of information measures for model selection in geophysical models with subgrid parameterizations is examined. Although the resolved dynamical equations of atmospheric or oceanic global numerical models are well established, the development and evaluation of parameterizations that represent subgrid-scale effects pose a big challenge. For climate studies, the parameters or parameterizations are usually selected according to a root-mean-square error criterion that measures the differences between the model-state evolution and observations along the trajectory. However, inaccurate initial conditions and systematic model errors contaminate root-mean-square error measures. In this work, information theory quantifiers, in particular Shannon entropy, statistical complexity, and Jensen–Shannon divergence, are evaluated as measures of the model dynamics. An ordinal analysis is conducted using the Bandt–Pompe symbolic data reduction in the signals. The proposed ordinal information measures are examined in the two-scale Lorenz-96 system. By comparing the two-scale Lorenz-96 system signals with a one-scale Lorenz-96 system with deterministic and stochastic parameterizations, the study shows that information measures are able to select the correct model and to distinguish the parameterizations, including the degree of stochasticity that results in the closest model dynamics to the two-scale Lorenz-96 system.

Permutation entropy and statistical complexity in characterising low-aspect-ratio reversed-field pinch plasma

Permutation entropy and statistical complexity are measures for complex time series. The Bandt–Pompe methodology evaluates probability distribution using permutation. The method is robust and effective to quantify information of time series data. Statistical complexity is the product of Jensen–Shannon divergence and permutation entropy. These physical parameters are introduced to analyse time series of emission and magnetic fluctuations in low-aspect-ratio reversed-field pinch (RFP) plasma. The observed time-series data aggregates in a region of the plane, the so-called C–H plane, determined by entropy versus complexity. The C–H plane is a representation space used for distinguishing periodic, chaos, stochastic and noisy processes of time series data. The characteristics of the emissions and magnetic fluctuation change under different RFP-plasma conditions. The statistical complexities of soft x-ray emissions and magnetic fluctuations depend on the relationships between reversal and pinch parameters.

Characterization of electric load with Information Theory quantifiers

Abstract This paper presents a study of the electric load behavior based on the Causality Complexity–Entropy Plane. We use a public data set, namely REDD , which contains detailed power usage information from several domestic appliances. In our characterization, we use the available power data of the circuit/devices of all houses. The Bandt–Pompe methodology combined with the Causality Complexity–Entropy Plane was used to identify and characterize regimes and behaviors over these data. The results showed that this characterization provides a useful insight into the underlying dynamics that govern the electric load.

Quantum remnants in the classical limit

We analyze here the common features of two dynamical regimes: a quantum and a classical one. We deal with a well known semi-classic system in its route towards the classical limit, together with its purely classic counterpart. We wish to ascertain i) whether some quantum remnants can be found in the classical limit and ii) the details of the quantum-classic transition. The so-called mutual information is the appropriate quantifier for this task. Additionally, we study the Bandt–Pompe's symbolic patterns that characterize dynamical time series (representative of the semi-classical system under scrutiny) in their evolution towards the classical limit.

Amplitude-aware permutation entropy: Illustration in spike detection and signal segmentation

Background and objectiveSignal segmentation and spike detection are two important biomedical signal processing applications. Often, non-stationary signals must be segmented into piece-wise stationary epochs or spikes need to be found among a background of noise before being further analyzed. Permutation entropy (PE) has been proposed to evaluate the irregularity of a time series. PE is conceptually simple, structurally robust to artifacts, and computationally fast. It has been extensively used in many applications, but it has two key shortcomings. First, when a signal is symbolized using the Bandt–Pompe procedure, only the order of the amplitude values is considered and information regarding the amplitudes is discarded. Second, in the PE, the effect of equal amplitude values in each embedded vector is not addressed. To address these issues, we propose a new entropy measure based on PE: the amplitude-aware permutation entropy (AAPE). MethodsAAPE is sensitive to the changes in the amplitude, in addition to the frequency, of the signals thanks to it being more flexible than the classical PE in the quantification of the signal motifs. To demonstrate how the AAPE method can enhance the quality of the signal segmentation and spike detection, a set of synthetic and realistic synthetic neuronal signals, electroencephalograms and neuronal data are processed. We compare the performance of AAPE in these problems against state-of-the-art approaches and evaluate the significance of the differences with a repeated ANOVA with post hoc Tukey's test. ResultsIn signal segmentation, the accuracy of AAPE-based method is higher than conventional segmentation methods. AAPE also leads to more robust results in the presence of noise. The spike detection results show that AAPE can detect spikes well, even when presented with single-sample spikes, unlike PE. For multi-sample spikes, the changes in AAPE are larger than in PE. ConclusionWe introduce a new entropy metric, AAPE, that enables us to consider amplitude information in the formulation of PE. The AAPE algorithm can be used in almost every irregularity-based application in various signal and image processing fields. We also made freely available the Matlab code of the AAPE.

Using forbidden ordinal patterns to detect determinism in irregularly sampled time series.

It is known that when symbolizing a time series into ordinal patterns using the Bandt-Pompe (BP) methodology, there will be ordinal patterns called forbidden patterns that do not occur in a deterministic series. The existence of forbidden patterns can be used to identify deterministic dynamics. In this paper, the ability to use forbidden patterns to detect determinism in irregularly sampled time series is tested on data generated from a continuous model system. The study is done in three parts. First, the effects of sampling time on the number of forbidden patterns are studied on regularly sampled time series. The next two parts focus on two types of irregular-sampling, missing data and timing jitter. It is shown that forbidden patterns can be used to detect determinism in irregularly sampled time series for low degrees of sampling irregularity (as defined in the paper). In addition, comments are made about the appropriateness of using the BP methodology to symbolize irregularly sampled time series.

Chaotic density fluctuations in L-mode plasmas of the DIII-D tokamak

Analysis of the time series obtained with the Doppler backscattering system (Hillsheim et al 2009 Rev. Sci. Instrum. 80 0835070) in the DIII-D tokamak (Luxon 2005 Fusion Sci. Technol. 48 828) shows that intermediate wave number plasma density fluctuations in low confinement (L-mode) tokamak plasmas are chaotic. The supporting evidence is based on the shape of the power spectrum; the location of the signal in the complexity-entropy plane (C-H plane) (Rosso et al 2007 Phys. Rev. Lett. 99 154102); and the population of the corresponding Bandt–Pompe (Bandt and Pompe 2002 Phys. Rev. Lett. 88 174102) probability distributions.

Stochastic resonance quantified by statistical complexity measures in a bistable system subject to colored noise

This work deals mainly with the problem of the stochastic resonance in a symmetric bistable system subject to colored noise by virtue of the statistical complexity measures. Firstly, based on Bandt–Pompe methodology, the statistical complexity measures are dedicated to quantify the stochastic resonance of the system. Then, the effects of the correlation time of the colored noise, the amplitude and the frequency of the signal on the stochastic resonance are discussed by using the statistical complexity measures. Meanwhile, the SNR is also given for the purpose of verifying the validity of this approach. It is turned out that the statistical complexity measures can be used to characterize some subtle signatures of noise-induced phenomena. Additionally, the robustness of the results is verified with reducing the embedding dimension and the total length of the time series.

On Extracting Probability Distribution Information from Time Series

Time-series (TS) are employed in a variety of academic disciplines. In this paper we focus on extracting probability density functions (PDFs) from TS to gain an insight into the underlying dynamic processes. On discussing this “extraction” problem, we consider two popular approaches that we identify as histograms and Bandt–Pompe. We use an information-theoretic method to objectively compare the information content of the concomitant PDFs.

The Tsallis-complexity of a semiclassical time-evolution

An investigation is undertaken of semiclassical time-evolutions and their classical limit with the intent of getting insights into the classical–quantum frontier. We deal with a system that represents the interaction between matter and a given field, and our main research tool is the so-called q-complexity quantifier, for which two different versions are considered. The probability distribution associated with the time-evolution process is determined by recourse to the Bandt–Pompe symbolic technique [C. Bandt, B. Pompe, Permutation entropy: a natural complexity measure for time series, Phys. Rev. Lett. 88 (2002) 174102:1–174102:4]. The most salient details of the quantum–classical transition turn out to be described not only well, but also in a better fashion than that of previous literature.

Ambiguities in Bandt–Pompe’s methodology for local entropic quantifiers

The Bandt–Pompe (BP) prescription for building up probability densities [C. Bandt, B. Pompe, Permutation entropy: a natural complexity measure for time series, Phys. Rev. Lett. 88 (2002) 174102] constituted a significant advance in the treatment of time-series. However, as we show here, ambiguities arise in applying the BP technique with reference to the permutation of ordinal patterns. This happens if one wishes to employ the BP-probability density to construct local entropic quantifiers that would characterize time-series generated by nonlinear dynamical systems. Explicit evidence of this fact is presented by comparing two different procedures, frequently found in the literature, that generate sequences of ordinal patterns. In opposition to the case of global quantifiers in the orthodox Shannon fashion, the proper order of the pertinent symbols turns out to be not uniquely predetermined for local entropic indicators. We advance the idea of employing the Fisher–Shannon information plane as a tool to resolve the ambiguity and give illustrative examples.

Bandt–Pompe–Tsallis quantifier and quantum-classical transition

We concern ourselves with statistical quantifiers of semiclassical time-evolutions and their classical limit. The system of interest represents the interaction between matter and a given field. Our tool here is the so-called Permutation Entropy, evaluated by recourse to the so-called Bandt-Pompe technique, within a Tsallis scenario. We encounter that the most salient details of the quantum-classical transition are well-described, indeed, in a better fashion than that of previous approaches.