Complexity-entropy causality plane as a complexity measure for two-dimensional patterns

Characterizing unstructured data with the nearest neighbor permutation entropy

Permutation entropy and its associated frameworks are remarkable examples of physics-inspired techniques adept at processing complex and extensive datasets. Despite substantial progress in developing and applying these tools, their use has been predominantly limited to structured datasets such as time series or images. Here, we introduce the k-nearest neighbor permutation entropy, an innovative extension of the permutation entropy tailored for unstructured data, irrespective of their spatial or temporal configuration and dimensionality. Our approach builds upon nearest neighbor graphs to establish neighborhood relations and uses random walks to extract ordinal patterns and their distribution, thereby defining the k-nearest neighbor permutation entropy. This tool not only adeptly identifies variations in patterns of unstructured data but also does so with a precision that significantly surpasses conventional measures such as spatial autocorrelation. Additionally, it provides a natural approach for incorporating amplitude information and time gaps when analyzing time series or images, thus significantly enhancing its noise resilience and predictive capabilities compared to the usual permutation entropy. Our research substantially expands the applicability of ordinal methods to more general data types, opening promising research avenues for extending the permutation entropy toolkit for unstructured data.

Permutation entropy analysis of EEG signals for distinguishing eyes-open and eyes-closed brain states: Comparison of different approaches

Developing reliable methodologies to decode brain state information from electroencephalogram (EEG) signals is an open challenge, crucial to implementing EEG-based brain-computer interfaces (BCIs). For example, signal processing methods that identify brain states could allow motor-impaired patients to communicate via non-invasive, EEG-based BCIs. In this work, we focus on the problem of distinguishing between the states of eyes closed (EC) and eyes open (EO), employing quantities based on permutation entropy (PE). An advantage of PE analysis is that it uses symbols (ordinal patterns) defined by the ordering of the data points (disregarding the actual values), hence providing robustness to noise and outliers due to motion artifacts. However, we show that for the analysis of multichannel EEG recordings, the performance of PE in discriminating the EO and EC states depends on the symbols' definition and how their probabilities are estimated. Here, we study the performance of PE-based features for EC/EO state classification in a dataset of N=107 subjects with one-minute 64-channel EEG recordings in each state. We analyze features obtained from patterns encoding temporal or spatial information, and we compare different approaches to estimate their probabilities (by averaging over time, over channels, or by "pooling"). We find that some PE-based features provide about 75% classification accuracy, comparable to the performance of features extracted with other statistical analysis techniques. Our work highlights the limitations of PE methods in distinguishing the eyes' state, but, at the same time, it points to the possibility that subject-specific training could overcome these limitations.

Feature-preserving synthesis of termite-mimetic spinodal nest morphology

Termite-built topology is complex due to group interactions and environmental feedback. Being interlinked with material characteristics and related to functionality, an accurate synthesis of termite mound topology has never been achieved. We scanned inner termite mound pieces via high-resolution micro-computed tomography. A wavelet scattering transform followed by optimization extracts features that are fed into a Gaussian Random Fields (GRFs) approach to synthesize termite-mimetic spinodal topology. Compared to natural structures the GRF topology is more regular. Irregularity is related to anisotropy, indicative of directionality caused by porous network connectivity of chambers and corridors. Since GRFs are related to diffusion, we assume that deterministic behavioral traits play a significant role in the development of these local differences. We pioneer a framework to reliably mimic termite mound spinodal features. Engineering termite-inspired structures will allow to inspect aspects of termite architectures and their behavior to manufacture novel material concepts with imprinted multi-functionality.

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.

Two new parameters for the ordinal analysis of images

Local patterns play an important role in statistical physics as well as in image processing. Two-dimensional ordinal patterns were studied by Ribeiro et al. who determined permutation entropy and complexity in order to classify paintings and images of liquid crystals. Here, we find that the 2 × 2 patterns of neighboring pixels come in three types. The statistics of these types, expressed by two parameters, contains the relevant information to describe and distinguish textures. The parameters are most stable and informative for isotropic structures.

Rao-Burbea centroids applied to the statistical characterization of time series and images through ordinal patterns

Divergences or similarity measures between probability distributions have become a very useful tool for studying different aspects of statistical objects, such as time series, networks, and images. Notably, not every divergence provides identical results when applied to the same problem. Therefore, it seems convenient to have the widest possible set of divergences to be applied to the problems under study. Besides this choice, an essential step in the analysis of every statistical object is the mapping of each one of their representing values into an alphabet of symbols conveniently chosen. In this work, we choose the family of divergences known as the Burbea-Rao centroids (BRCs). For the mapping of the original time series into a symbolic sequence, we work with the ordinal pattern scheme. We apply our proposals to analyze simulated and real time series and to real textured images. The main conclusion of our work is that the best BRC, at least in the studied cases, is the Jensen-Shannon divergence, besides the fact that it verifies some interesting formal properties.

Entropy-based early detection of critical transitions in spatial vegetation fields

In semiarid regions, vegetated ecosystems can display abrupt and unexpected changes, i.e., transitions to different states, due to drifting or time-varying parameters, with severe consequences for the ecosystem and the communities depending on it. Despite intensive research, the early identification of an approaching critical point from observations is still an open challenge. Many data analysis techniques have been proposed, but their performance depends on the system and on the characteristics of the observed data (the resolution, the level of noise, the existence of unobserved variables, etc.). Here, we propose an entropy-based approach to identify an upcoming transition in spatiotemporal data. We apply this approach to observational vegetation data and simulations from two models of vegetation dynamics to infer the arrival of an abrupt shift to an arid state. We show that the permutation entropy (PE) computed from the probabilities of two-dimensional ordinal patterns may provide an early warning indicator of an approaching tipping point, as it may display a maximum (or minimum) before decreasing (or increasing) as the transition approaches. Like other spatial early warning indicators, the spatial permutation entropy does not need a time series of the system dynamics, and it is suited for spatially extended systems evolving on long time scales, like vegetation plots. We quantify its performance and show that, depending on the system and data, the performance can be better, similar or worse than the spatial correlation. Hence, we propose the spatial PE as an additional indicator to try to anticipate regime shifts in vegetated ecosystems.

Multiscale two-dimensional permutation entropy to analyze encrypted images

Multiscale versions of weighted (and non-weighted) permutation entropy for two dimensions are considered in order to compare and analyze the results when different experiments are conducted. We propose the application of these measures to analyze encrypted images with different security levels and encryption methods.

Application of Two-Dimensional Entropy Measures to Detect the Radiographic Signs of Tooth Resorption and Hypercementosis in an Equine Model

Dental disorders are a serious health problem in equine medicine, their early recognition benefits the long-term general health of the horse. Most of the initial signs of Equine Odontoclastic Tooth Resorption and Hypercementosis (EOTRH) syndrome concern the alveolar aspect of the teeth, thus, the need for early recognition radiographic imaging. This study is aimed to evaluate the applicability of entropy measures to quantify the radiological signs of tooth resorption and hypercementosis as well as to enhance radiographic image quality in order to facilitate the identification of the signs of EOTRH syndrome. A detailed examination of the oral cavity was performed in eighty horses. Each evaluated incisor tooth was assigned to one of four grade-related EOTRH groups (0-3). Radiographs of the incisor teeth were taken and digitally processed. For each radiograph, two-dimensional sample (SampEn2D), fuzzy (FuzzEn2D), permutation (PermEn2D), dispersion (DispEn2D), and distribution (DistEn2D) entropies were measured after image filtering was performed using Normalize, Median, and LaplacianSharpening filters. Moreover, the similarities between entropy measures and selected Gray-Level Co-occurrence Matrix (GLCM) texture features were investigated. Among the 15 returned measures, DistEn2D was EOTRH grade-related. Moreover, DistEn2D extracted after Normalize filtering was the most informative. The EOTRH grade-related similarity between DistEn2D and Difference Entropy (GLCM) confirms the higher irregularity and complexity of incisor teeth radiographs in advanced EOTRH syndrome, demonstrating the greatest sensitivity (0.50) and specificity (0.95) of EOTRH 3 group detection. An application of DistEn2D to Normalize filtered incisor teeth radiographs enables the identification of the radiological signs of advanced EOTRH with higher accuracy than the previously used entropy-related GLCM texture features.

Statistical properties of the entropy from ordinal patterns

The ultimate purpose of the statistical analysis of ordinal patterns is to characterize the distribution of the features they induce. In particular, knowing the joint distribution of the pair entropy-statistical complexity for a large class of time series models would allow statistical tests that are unavailable to date. Working in this direction, we characterize the asymptotic distribution of the empirical Shannon's entropy for any model under which the true normalized entropy is neither zero nor one. We obtain the asymptotic distribution from the central limit theorem (assuming large time series), the multivariate delta method, and a third-order correction of its mean value. We discuss the applicability of other results (exact, first-, and second-order corrections) regarding their accuracy and numerical stability. Within a general framework for building test statistics about Shannon's entropy, we present a bilateral test that verifies if there is enough evidence to reject the hypothesis that two signals produce ordinal patterns with the same Shannon's entropy. We applied this bilateral test to the daily maximum temperature time series from three cities (Dublin, Edinburgh, and Miami) and obtained sensible results.

Application of the Two-Dimensional Entropy Measures in the Infrared Thermography-Based Detection of Rider: Horse Bodyweight Ratio in Horseback Riding

As obesity is a serious problem in the human population, overloading of the horse's thoracolumbar region often affects sport and school horses. The advances in using infrared thermography (IRT) to assess the horse's back overload will shortly integrate the IRT-based rider-horse fit into everyday equine practice. This study aimed to evaluate the applicability of entropy measures to select the most informative measures and color components, and the accuracy of rider:horse bodyweight ratio detection. Twelve horses were ridden by each of the six riders assigned to the light, moderate, and heavy groups. Thermal images were taken pre- and post-exercise. For each thermal image, two-dimensional sample (SampEn), fuzzy (FuzzEn), permutation (PermEn), dispersion (DispEn), and distribution (DistEn) entropies were measured in the withers and the thoracic spine areas. Among 40 returned measures, 30 entropy measures were exercise-dependent, whereas 8 entropy measures were bodyweight ratio-dependent. Moreover, three entropy measures demonstrated similarities to entropy-related gray level co-occurrence matrix (GLCM) texture features, confirming the higher irregularity and complexity of thermal image texture when horses worked under heavy riders. An application of DispEn to red color components enables identification of the light and heavy rider groups with higher accuracy than the previously used entropy-related GLCM texture features.

NNetEn2D: Two-Dimensional Neural Network Entropy in Remote Sensing Imagery and Geophysical Mapping

Measuring the predictability and complexity of 2D data (image) series using entropy is an essential tool for evaluation of systems’ irregularity and complexity in remote sensing and geophysical mapping. However, the existing methods have some drawbacks related to their strong dependence on method parameters and image rotation. To overcome these difficulties, this study proposes a new method for estimating two-dimensional neural network entropy (NNetEn2D) for evaluating the regularity or predictability of images using the LogNNet neural network model. The method is based on an algorithm for converting a 2D kernel into a 1D data series followed by NNetEn2D calculation. An artificial test image was created for the study. We demonstrate the advantage of using circular instead of square kernels through comparison of the invariance of the NNetEn2D distribution after image rotation. Highest robustness was observed for circular kernels with a radius of R = 5 and R = 6 pixels, with a NNetEn2D calculation error of no more than 10%, comparable to the distortion of the initial 2D data. The NNetEn2D entropy calculation method has two main geometric parameters (kernel radius and its displacement step), as well as two neural network hyperparameters (number of training epochs and one of six reservoir filling techniques). We evaluated our method on both remote sensing and geophysical mapping images. Remote sensing imagery (Sentinel-2) shows that brightness of the image does not affect results, which helps keep a rather consistent appearance of entropy maps over time without saturation effects being observed. Surfaces with little texture, such as water bodies, have low NNetEn2D values, while urban areas have consistently high values. Application to geophysical mapping of rocks to the northwest of southwest Australia is characterized by low to medium entropy and highlights aspects of the geology. These results indicate the success of NNetEn2D in providing meaningful entropy information for 2D in remote sensing and geophysical applications.

Entropy and complexity unveil the landscape of memes evolution

On the Internet, information circulates fast and widely, and the form of content adapts to comply with users’ cognitive abilities. Memes are an emerging aspect of the internet system of signification, and their visual schemes evolve by adapting to a heterogeneous context. A fundamental question is whether they present culturally and temporally transcendent characteristics in their organizing principles. In this work, we study the evolution of 2 million visual memes published on Reddit over ten years, from 2011 to 2020, in terms of their statistical complexity and entropy. A combination of a deep neural network and a clustering algorithm is used to group memes according to the underlying templates. The grouping of memes is the cornerstone to trace the growth curve of these objects. We observe an exponential growth of the number of new created templates with a doubling time of approximately 6 months, and find that long-lasting templates are associated with strong early adoption. Notably, the creation of new memes is accompanied with an increased visual complexity of memes content, in a continuous effort to represent social trends and attitudes, that parallels a trend observed also in painting art.

Parameter Analysis of Multiscale Two-Dimensional Fuzzy and Dispersion Entropy Measures Using Machine Learning Classification

Two-dimensional fuzzy entropy, dispersion entropy, and their multiscale extensions (MFuzzyEn2D and MDispEn2D, respectively) have shown promising results for image classifications. However, these results rely on the selection of key parameters that may largely influence the entropy values obtained. Yet, the optimal choice for these parameters has not been studied thoroughly. We propose a study on the impact of these parameters in image classification. For this purpose, the entropy-based algorithms are applied to a variety of images from different datasets, each containing multiple image classes. Several parameter combinations are used to obtain the entropy values. These entropy values are then applied to a range of machine learning classifiers and the algorithm parameters are analyzed based on the classification results. By using specific parameters, we show that both MFuzzyEn2D and MDispEn2D approach state-of-the-art in terms of image classification for multiple image types. They lead to an average maximum accuracy of more than 95% for all the datasets tested. Moreover, MFuzzyEn2D results in a better classification performance than that extracted by MDispEn2D as a majority. Furthermore, the choice of classifier does not have a significant impact on the classification of the extracted features by both entropy algorithms. The results open new perspectives for these entropy-based measures in textural analysis.

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.

Three-Dimensional Multiscale Fuzzy Entropy: Validation and Application to Idiopathic Pulmonary Fibrosis

Idiopathic Pulmonary Fibrosis (IPF) is a chronic, severe, and progressive lung disease with short life expectancy. Based on information theory and entropy measurement, a three-dimensional multiscale fuzzy entropy (MFE _3D) algorithm is proposed to identify IPF patients from their computed tomography (CT) volumetric data. First, the validation of the algorithm was performed by analyzing several volumetric synthetic noises (white, blue, brown, and pink), MIX(p) processes-based volumes, and texture-based volumes. The entropy values obtained by MFE _3D were consistent with the values obtained using the one, and two-dimensional versions, validating its use in biomedical data. Hence, MFE _3D was applied to CT scans to identify the existence of IPF within two different groups, one of healthy subjects (26) and another of IPF patients (26). Statistical differences were found (p < 0.05) between the entropy values of each group in 5 scale factors out of 10. These results demonstrate that MFE _3D could be an interesting metric to identify IPF in CT scans.

Structural Statistical Quantifiers and Thermal Features of Quantum Systems

This paper deals primarily with relatively novel thermal quantifiers called disequilibrium and statistical complexity, whose role is growing in different disciplines of physics and other sciences. These quantifiers are called L. Ruiz, Mancini, and Calvet (LMC) quantifiers, following the initials of the three authors who advanced them. We wish to establish information-theoretical bridges between LMC structural quantifiers and (1) Thermal Heisenberg uncertainties ΔxΔp (at temperature T); (2) A nuclear physics fermion model. Having achieved such purposes, we determine to what an extent our bridges can be extended to both the semi-classical and classical realms. In addition, we find a strict bound relating a special LMC structural quantifier to quantum uncertainties.

Mapping images into ordinal networks

An increasing abstraction has marked some recent investigations in network science. Examples include the development of algorithms that map time series data into networks whose vertices and edges can have different interpretations, beyond the classical idea of parts and interactions of a complex system. These approaches have proven useful for dealing with the growing complexity and volume of diverse data sets. However, the use of such algorithms is mostly limited to one-dimensional data, and there has been little effort towards extending these methods to higher-dimensional data such as images. Here we propose a generalization for the ordinal network algorithm for mapping images into networks. We investigate the emergence of connectivity constraints inherited from the symbolization process used for defining the network nodes and links, which in turn allows us to derive the exact structure of ordinal networks obtained from random images. We illustrate the use of this new algorithm in a series of applications involving randomization of periodic ornaments, images generated by two-dimensional fractional Brownian motion and the Ising model, and a data set of natural textures. These examples show that measures obtained from ordinal networks (such as average shortest path and global node entropy) extract important image properties related to roughness and symmetry, are robust against noise, and can achieve higher accuracy than traditional texture descriptors extracted from gray-level co-occurrence matrices in simple image classification tasks.

Gaze Information Channel in Van Gogh's Paintings

This paper uses quantitative eye tracking indicators to analyze the relationship between images of paintings and human viewing. First, we build the eye tracking fixation sequences through areas of interest (AOIs) into an information channel, the gaze channel. Although this channel can be interpreted as a generalization of a first-order Markov chain, we show that the gaze channel is fully independent of this interpretation, and stands even when first-order Markov chain modeling would no longer fit. The entropy of the equilibrium distribution and the conditional entropy of a Markov chain are extended with additional information-theoretic measures, such as joint entropy, mutual information, and conditional entropy of each area of interest. Then, the gaze information channel is applied to analyze a subset of Van Gogh paintings. Van Gogh artworks, classified by art critics into several periods, have been studied under computational aesthetics measures, which include the use of Kolmogorov complexity and permutation entropy. The gaze information channel paradigm allows the information-theoretic measures to analyze both individual gaze behavior and clustered behavior from observers and paintings. Finally, we show that there is a clear correlation between the gaze information channel quantities that come from direct human observation, and the computational aesthetics measures that do not rely on any human observation at all.

Complexity Analysis of Escher's Art

Art is the output of a complex system based on the human spirit and driven by several inputs that embed social, cultural, economic and technological aspects of a given epoch. A solid quantitative analysis of art poses considerable difficulties and reaching assertive conclusions is a formidable challenge. In this paper, we adopt complexity indices, dimensionality-reduction and visualization techniques for studying the evolution of Escher's art. Grayscale versions of 457 artworks are analyzed by means of complexity indices and represented using the multidimensional scaling technique. The results are correlated with the distinct periods of Escher's artistic production. The time evolution of the complexity and the emergent patterns demonstrate the effectiveness of the approach for a quantitative characterization of art.

Estimating physical properties from liquid crystal textures via machine learning and complexity-entropy methods

Imaging techniques are essential tools for inquiring a number of properties from different materials. Liquid crystals are often investigated via optical and image processing methods. In spite of that, considerably less attention has been paid to the problem of extracting physical properties of liquid crystals directly from textures images of these materials. Here we present an approach that combines two physics-inspired image quantifiers (permutation entropy and statistical complexity) with machine learning techniques for extracting physical properties of nematic and cholesteric liquid crystals directly from their textures images. We demonstrate the usefulness and accuracy of our approach in a series of applications involving simulated and experimental textures, in which physical properties of these materials (namely: average order parameter, sample temperature, and cholesteric pitch length) are predicted with significant precision. Finally, we believe our approach can be useful in more complex liquid crystal experiments as well as for probing physical properties of other materials that are investigated via imaging techniques.

History of art paintings through the lens of entropy and complexity

Art is the ultimate expression of human creativity that is deeply influenced by the philosophy and culture of the corresponding historical epoch. The quantitative analysis of art is therefore essential for better understanding human cultural evolution. Here, we present a large-scale quantitative analysis of almost 140,000 paintings, spanning nearly a millennium of art history. Based on the local spatial patterns in the images of these paintings, we estimate the permutation entropy and the statistical complexity of each painting. These measures map the degree of visual order of artworks into a scale of order-disorder and simplicity-complexity that locally reflects qualitative categories proposed by art historians. The dynamical behavior of these measures reveals a clear temporal evolution of art, marked by transitions that agree with the main historical periods of art. Our research shows that different artistic styles have a distinct average degree of entropy and complexity, thus allowing a hierarchical organization and clustering of styles according to these metrics. We have further verified that the identified groups correspond well with the textual content used to qualitatively describe the styles and the applied complexity-entropy measures can be used for an effective classification of artworks.

Astrocytic Coverage of Dendritic Spines, Dendritic Shafts, and Axonal Boutons in Hippocampal Neuropil

Distal astrocytic processes have a complex morphology, reminiscent of branchlets and leaflets. Astrocytic branchlets are rod-like processes containing mitochondria and endoplasmic reticulum, capable of generating inositol-3-phosphate (IP₃)-dependent Ca²⁺ signals. Leaflets are small and flat processes that protrude from branchlets and fill the space between synapses. Here we use three-dimensional (3D) reconstructions from serial section electron microscopy (EM) of rat CA1 hippocampal neuropil to determine the astrocytic coverage of dendritic spines, shafts and axonal boutons. The distance to the maximum of the astrocyte volume fraction (VF) correlated with the size of the spine when calculated from the center of mass of the postsynaptic density (PSD) or from the edge of the PSD, but not from the spine surface. This suggests that the astrocytic coverage of small and larger spines is similar in hippocampal neuropil. Diffusion simulations showed that such synaptic microenvironment favors glutamate spillover and extrasynaptic receptor activation at smaller spines. We used complexity and entropy measures to characterize astrocytic branchlets and leaflets. The 2D projections of astrocytic branchlets had smaller spatial complexity and entropy than leaflets, consistent with the higher structural complexity and less organized distribution of leaflets. The VF of astrocytic leaflets was highest around dendritic spines, lower around axonal boutons and lowest around dendritic shafts. In contrast, the VF of astrocytic branchlets was similarly low around these three neuronal compartments. Taken together, these results suggest that astrocytic leaflets preferentially contact synapses as opposed to the dendritic shaft, an arrangement that might favor neurotransmitter spillover and extrasynaptic receptor activation along dendritic shafts.

Shearlet-based measures of entropy and complexity for two-dimensional patterns

New spatial entropy and complexity measures for two-dimensional patterns are proposed. The approach is based on the notion of disequilibrium and is built on statistics of directional multiscale coefficients of the fast finite shearlet transform. Shannon entropy and Jensen-Shannon divergence measures are employed. Both local and global spatial complexity and entropy estimates can be obtained, thus allowing for spatial mapping of complexity in inhomogeneous patterns. The algorithm is validated in numerical experiments with a gradually decaying periodic pattern and Ising surfaces near critical state. It is concluded that the proposed algorithm can be instrumental in describing a wide range of two-dimensional imaging data, textures, or surfaces, where an understanding of the level of order or randomness is desired.

Characterizing time series via complexity-entropy curves

The search for patterns in time series is a very common task when dealing with complex systems. This is usually accomplished by employing a complexity measure such as entropies and fractal dimensions. However, such measures usually only capture a single aspect of the system dynamics. Here, we propose a family of complexity measures for time series based on a generalization of the complexity-entropy causality plane. By replacing the Shannon entropy by a monoparametric entropy (Tsallis q entropy) and after considering the proper generalization of the statistical complexity (q complexity), we build up a parametric curve (the q-complexity-entropy curve) that is used for characterizing and classifying time series. Based on simple exact results and numerical simulations of stochastic processes, we show that these curves can distinguish among different long-range, short-range, and oscillating correlated behaviors. Also, we verify that simulated chaotic and stochastic time series can be distinguished based on whether these curves are open or closed. We further test this technique in experimental scenarios related to chaotic laser intensity, stock price, sunspot, and geomagnetic dynamics, confirming its usefulness. Finally, we prove that these curves enhance the automatic classification of time series with long-range correlations and interbeat intervals of healthy subjects and patients with heart disease.

Diagnosing the Dynamics of Observed and Simulated Ecosystem Gross Primary Productivity with Time Causal Information Theory Quantifiers

Data analysis and model-data comparisons in the environmental sciences require diagnostic measures that quantify time series dynamics and structure, and are robust to noise in observational data. This paper investigates the temporal dynamics of environmental time series using measures quantifying their information content and complexity. The measures are used to classify natural processes on one hand, and to compare models with observations on the other. The present analysis focuses on the global carbon cycle as an area of research in which model-data integration and comparisons are key to improving our understanding of natural phenomena. We investigate the dynamics of observed and simulated time series of Gross Primary Productivity (GPP), a key variable in terrestrial ecosystems that quantifies ecosystem carbon uptake. However, the dynamics, patterns and magnitudes of GPP time series, both observed and simulated, vary substantially on different temporal and spatial scales. We demonstrate here that information content and complexity, or Information Theory Quantifiers (ITQ) for short, serve as robust and efficient data-analytical and model benchmarking tools for evaluating the temporal structure and dynamical properties of simulated or observed time series at various spatial scales. At continental scale, we compare GPP time series simulated with two models and an observations-based product. This analysis reveals qualitative differences between model evaluation based on ITQ compared to traditional model performance metrics, indicating that good model performance in terms of absolute or relative error does not imply that the dynamics of the observations is captured well. Furthermore, we show, using an ensemble of site-scale measurements obtained from the FLUXNET archive in the Mediterranean, that model-data or model-model mismatches as indicated by ITQ can be attributed to and interpreted as differences in the temporal structure of the respective ecological time series. At global scale, our understanding of C fluxes relies on the use of consistently applied land models. Here, we use ITQ to evaluate model structure: The measures are largely insensitive to climatic scenarios, land use and atmospheric gas concentrations used to drive them, but clearly separate the structure of 13 different land models taken from the CMIP5 archive and an observations-based product. In conclusion, diagnostic measures of this kind provide data-analytical tools that distinguish different types of natural processes based solely on their dynamics, and are thus highly suitable for environmental science applications such as model structural diagnostics.

Using Link Disconnection Entropy Disorder to Detect Fast Moving Nodes in MANETs

Mobile ad-hoc networks (MANETs) are dynamic by nature; this dynamism comes from node mobility, traffic congestion, and other transmission conditions. Metrics to evaluate the effects of those conditions shine a light on node's behavior in an ad-hoc network, helping to identify the node or nodes with better conditions of connection. In this paper, we propose a relative index to evaluate a single node reliability, based on the link disconnection entropy disorder using neighboring nodes as reference. Link disconnection entropy disorder is best used to identify fast moving nodes or nodes with unstable communications, this without the need of specialized sensors such as GPS. Several scenarios were studied to verify the index, measuring the effects of Speed and traffic density on the link disconnection entropy disorder. Packet delivery ratio is associated to the metric detecting a strong relationship, enabling the use of the link disconnection entropy disorder to evaluate the stability of a node to communicate with other nodes. To expand the utilization of the link entropy disorder, we identified nodes with higher speeds in network simulations just by using the link entropy disorder.

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.

The complexity of sequences generated by the arc-fractal system

We study properties of the symbolic sequences extracted from the fractals generated by the arc-fractal system introduced earlier by Huynh and Chew. The sequences consist of only a few symbols yet possess several nontrivial properties. First using an operator approach, we show that the sequences are not periodic, even though they are constructed from very simple rules. Second by employing the ϵ-machine approach developed by Crutchfield and Young, we measure the complexity and randomness of the sequences and show that they are indeed complex, i.e. neither periodic nor random, with the value of complexity measure being significant as compared to the known example of logistic map at the edge of chaos. The complexity and randomness of the sequences are then discussed in relation with the properties of associated fractal objects, such as their fractal dimension, symmetry and orientations of the arcs.