Ordinal methods: Concepts, applications, new developments and challenges

Frequencies of order patterns, first mentioned by Bienaymé in 1875, form a basic attribute of data series. They have long been disregarded, and they still lack a theory. Their success in applications is due to the mild stationarity assumptions required for their estimation. Beside permutation entropy, we discuss certain correlation parameters and distance to white noise. Using sums of squares as in classical statistics, contributions of different parameters and of patterns with different lengths can be compared. We introduce a martingale describing frequencies of order patterns of any length for a random process. Due to a statistical estimation error, we recommend to study order patterns of length three and four unless we have really huge data series or deal with mean frequencies like entropy. The delays between data points of the patterns should be varied, however, creating autocorrelation functions. For length three, we have a canonical choice of parameters with independent estimators, under the null hypothesis of white noise. In the last part of the talk we construct combinatorial models of order structure in a random process without referring to any numerical values. We show that any stationary measure on patterns of length n can be extended to stationary orders on the positive integers. Details are presented for a curious example which we call the coin-tossing order.

Coding time series of continuous variables into a sequence of discrete symbols using the ordinal patterns (OP thereafter) of C. Bandt and B. Pompe opened new research avenues in many areas of mathematics, physics, statistics and computer science. OP coding strongly influenced the intersection of information theory and dynamical systems due to the relation of the permutation entropy and the Kologorov-Sinai entropy of dynamical systems. Tools of information theory such as the conditional mutual information, a.k.a. transfer entropy, used in inference of causality, have also been formulated using the OP coding. We will discuss the suitability of OP coding in less usual applications of CMI/TE in such areas as the phase dynamics and cross-scale (cross-frequency) interactions and in computer science-based causality methods such as the compression complexity. The considerations will be illustrated by numerical and real-world data examples from climate and neuroscience. M. Paluš, A. Kathpalia, P. Manshour

Compression-Complexity Causality (CCC) is a recently proposed causality detection method for time series data. It employs complexity estimation techniques based on lossless data-compression algorithms. Along with being formulated as an ‘interventional’ scheme of causality estimation, it overcomes several limitations of traditional causality estimation methods based on Wiener-Granger’s idea, such as the requirement of long-term stationarity and long length data suitable for the estimation of joint probability distributions. It has been shown to be robust for bivariate systems with low temporal resolution, missing samples, long-term memory and finite length data [1]. However, a straightforward extension of CCC for multivariate systems is yet not available. Direct application of bivariate CCC for such systems can give spurious causal estimates, for example, in the presence of indirect and confounding causal variables. In the case when we are interested in determining causal interaction between two observed variables from coupled dynamical systems with multiple variables, the time-delay embedding of the two observables based on Taken’s theorem can help to determine the dynamics of the entire system. Further, the depiction of embedded variables using ordinal patterns-based quantization (symbolization) can help to reduce the dimensionality of the coupled systems to single variable systems. Since the use of ordinal patterns helps to preserve the dynamics of the systems, causal estimation can now be done for these symbolized variables. The technique of ordinal pattern-based coding has been employed in earlier studies for other causality estimation measures such as Transfer Entropy/ Conditional Mutual Information. When used with CCC, this technique combines the strengths of the data-compression and ordinal pattern methodologies, giving promising results on challenging data with irregular and low-resolution sampling, missing samples and short length time series from multivariate systems. We show the performance of this novel ‘permutation CCC’ approach on simulated dynamical systems data and real data from climate system. References: [1] Kathpalia, A., & Nagaraj, N. (2019). Data-based intervention approach for Complexity-Causality measure. PeerJ Computer Science, 5, e196. Authors: Aditi Kathpalia, Pouya Manshour, Milan Paluš Acknowledgements: This study is supported by the Czech Science Foundation, Project No.~GA19-16066S and by the Czech Academy of Sciences, Praemium Academiae awarded to M. Paluš.

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. In this contribution, the properties of several ordinal-based quantifiers for the identification of time-delays from time series are explored. To that end, artificial time series of stochastic and deterministic time-delay models are generated. The obtained results indicate 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. This contribution focuses on an ordinal-based quantifier that is particularly sensitive to nonlinearities in the generating model compared to other quantifiers. The analysis on artificially generated data illustrates 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.

Eye-tracking is becoming widespread due to the appearance of low-cost high-accuracy devices. The provided information may be extremely useful in several contexts, of which we concentrate in people monitoring during protocolized cognitive tasks. The aim is to assess the possible relation between eye behavior and neurocognitive conditions. Eye-tracking signals may be regarded as a special kind of {\em trajectory} since data is a sequence of positions sampled at regular time intervals. We propose a novel adaptation of permutation entropy analysis applied to this kind of data. To the best of our knowledge this kind of analysis has not been proposed elsewhere. We quantize the possible orientations of the strides arising between successive samples (having also "no movement" as a possible situation given the specific nature of eye behavior). In this way, eye movements arising during a cognitive experiment are characterized as sequences of orientation changes, with which the customary permutation entropy analysis can be performed. Preliminary results show a strong relationship among the kind of cognitive task and the permutation entropy of the eye-tracking trajectories. Moreover (and most interestingly) a PE analysis of the behavior of subjects along {\em all} the experiments show a tendency to cluster together into two groups. This may be an indication of common underlying cognitive "styles" among the subjects, which are expressed as eye-movement patterns with distinct features.

The objective of the present study was to optimize the design of a new data set structure for signal classification to distinguish two classes. Once the data set was formulated, based on informational measures such as Bandt and Pompe permutation entropy, statistical complexity and individualized embedding time delay for each signal, a random forest algorithm was used to elaborate a classification model. These selection of the components of the feature space was done upon the criteria of the higher quality classification parameters, after the performance evaluation of other magnitudes. To compute the embedding time delay for every signal several algorithms was implemented to select the one to obtain the best results. The method proposed in the present work was applied to differentiate ECGs from normal subjects and from those with a diagnosed cardiac arrhythmia. The Physionet ECG Data Bases were used, both for healthy individuals and patients suffering cardiac arrhythmia. The classification process begins with a Montecarlo simulation of 1000 trials to pick up 70 % of the samples of every group of interest, randomly selected to be used in the training process, and the remaining 30 % is used for classification. The resulting methodology contribute to differentiate ECG signals from healthy subjects from those from patients with diagnosed arrhythmia. To evaluate the performance of the present, propose the area under the ROC curve, F1 score and classification accuracy was calculated for every Montecarlo trial. The values obtained for these indicators have proved to be very stimulating to continue with the line of research developed. Thus, the results obtained constitutes a preliminary protocol for differentiating between healthy patients and those with cardiac arrhythmias, based on a very simple and economical electrophysiological recording such as a single-lead ECG.

Permutation entropy, ordinal networks, and other ordinal methods are essential for time series analysis. On the other hand, generalizations of these methods for image analysis remain much less explored. This talk presents some recent advances in using ordinal methods for characterizing images. First, we present a simple extension of the Bandt–Pompe symbolization algorithm together with generalizations of the complexity-entropy plane and ordinal networks for two-dimensional data. Next, we demonstrate the versatility and the discrimination power of these technics in applications ranging from the characterization of art paintings to predictions of liquid crystal properties. Finally, we briefly describe the ordpy package, a simple Python module that implements these image analysis tools and many other ordinal methods.

The Earth system is a very complex and dynamical one basing on various feedbacks. This makes predictions and risk analysis even of very strong (sometime extreme) events as floods, landslides, or heatwaves a challenging task. After introducing physical models for weather forecast already in 1922 by L.F. Richardson, a fundamental open problem has been the understanding of basic physical mechanisms and exploring anthropogenic influences on climate. In 2021 Hasselmann and Manabe got the Physics Nobel Price for their pioneering works on this. I will shortly review their main seminal contributions. Next, I will discuss a recently developed approach via complex networks mainly to analyze strong climate events. This leads to an inverse problem: Is there a backbone-like structure underlying the climate system? To treat this problem, we have proposed a method to reconstruct and analyze a complex network from spatio-temporal data. This approach enables us to uncover relations to global and regional circulation patterns in oceans and atmosphere, which leads to construct substantially better predictions of extreme climate events, in particular for the Indian Summer Monsoon, extreme rainfall in South America, the Indian Ocean Dipole, El Nino and tropical cyclones. References N. Boers, B. Bookhagen, H.M.J. Barbosa, N. Marwan, J. Kurths, J.A. Marengo, Nature Comm. 5, 5199 (2014) V. Stolbova, E. Surovyatkina, B. Bookhagen, and J. Kurths, Geophys. Res. Lett. (2016) D. Eroglu, F. McRobie, I. Ozken, T. Stemler, K. Wyrwoll, S. Breitenbach, N. Marwan, J. Kurths, Nature Comm. 7, 12929 (2016) N. Boers, B. Goswami, A. Rheinwalt, B. Bookhagen, B. Hoskins, J. Kurths, Nature 566, 373 (2019) J. Meng, J. Fan, J. Ludescher, A. Agarwal, X. Chen, A. Bunde, J. Kurths, H. Schellnhuber, PNAS 117, 177 (2020) Z. Lu, N. Yuan, Q. Yang, Z. Ma, Z., J. Kurths, Geophys. Res. L. 48, e2020GL091674 (2021). S. Gupta, N. Boers, F. Pappenberger, J. Kurths, Climate Dynamics 57, 3355 (2021) Z. Lu, N. Yuan, Q. Yang, Z. Ma, Z., J. Kurths, PNAS (2022, in press).

We seek for a better understanding of the statistical properties of points in the Entropy-Complexity plane by proposing a test for the white noise hypothesis. Our test is based on true white noise sequences obtained from physical devices. The proposed methodology provides consistent results: it assesses sequences of true random samples as random (adequate test size), rejects correlated and contaminated sequences (sound test power), and captures the randomness of generators previously analyzed in the literature.

Recent progress in the ordinal pattern-based analysis has opened up a new pathway as an important platform for understanding various dynamic behavior emerging in nonlinear systems, and has yielded significant success in the field of combustion physics and science, mainly achieving two aims: (i) an in-depth physical interpretation of nonlinear dynamics and (ii) the development of substitute detectors for capturing the onset of combustion instabilities. Gotoda and co-workers have adopted the ordinal pattern-based analysis for wide spectrum of combustion phenomena including fame instability induced by swirl/buoyancy coupling (H. Gotoda et al., Phys. Rev. E 95, 022201, 2017), a buoyant turbulent fire (K. Takagi et al., Phys. Rev. E 96, 052223, 2017; K. Takagi and H. Gotoda, Phys. Rev. E, 98, 032207, 2018), a propagating flame in a Hele-Shaw cell (Y. Nomi et al., Phys. Rev. E 103, 022218, 2021; Y. Nomi et al., Chaos 31, 123133, 2021), and thermoacoustic combustion oscillations (S. Murayama et al., Phys. Rev. E 97, 022223, 2018; T. Hachijo et al., Chaos 29, 103123, 2019; C. Aoki et al., J. Appl. Phys. 127, 224903, 2020). In this workshop, we present the usefulness of the ordinal pattern-based analysis for dealing with nonlinear dynamics of flame front and combustion instabilities.

Generalized entropies can play a crucial role in the characterization of the complexity of a large class of deterministic and random processes. In the first part of this talk, we will show that an intrinsic group-¬theoretical structure is at the heart of the notion of generalized entropy. This structure emerges when imposing the requirement of composability of an entropy with respect to the union of two statistically independent systems. A generalization of the celebrated Shannon-¬Khinchin set of axioms is proposed, obtained by replacing the additivity axiom with that of composability. This formulation, which makes use of the theory of formal groups of algebraic topology, leads to the new, infinite family of non-trace-form entropies called group entropies. The first example of this class is represented by the classical Renyi entropy. We will show that complex systems can be classified into universality classes, each characterized by a specific phase space growth rate, and described by a suitable group entropy, representing the complexity measure associated to the universality class. The volume of phase space may also grow super-exponentially with the number of degrees of freedom for certain types of complex systems such as those encountered in biology and neuroscience, where components interact and create new emergent states. We discuss how the axiomatically based group entropies are able to ensure extensivity in the micro-canonical ensemble for super-exponentially growing phase spaces and in fact for any monotonously increasing phase space volume. Group entropies are also relevant in information geometry, since they allow to define a new class of divergences generalizing the Kullback-Leibler one. The applications of the new theory of group entropies to the ordinal analysis of a wide class of random processes will be presented by J. M. Amigó in the second part of the talk. Joint work in collaboration with J. M. Amigó (CIO-Elche), R. Dale (CIO-Elche), H. Jensen (Imperial College, London).

Multimodal measurements in biomedical physics provide ample multivariate and spatio-temporal time series of cardiac and neural activities, including EEG and ECG time series, optical mapping (using fluorescent dyes), 4D Ultrasound, and real-time MRI. To analyze and classify these data in clinical diagnosis and preclinical experiments fast algorithms are required which are robust with respect to observational noise and other artifacts. In this presentation we will discuss how statistics of spatio-temporal ordinal patterns may address these (medical) needs using practical examples.

In recent years, the construction and analysis of ordinal pattern transition networks (OPTNs) from time series has provided new means for complexity characterization and coupling inference from observational time series. In my presentation, I will demonstrate an extension of the recently developed bivariate OPTN analysis framework to multivariate situations based on proper conditioning on the occurrence of patterns obeyed by any third time series. The presented methodology does not only allow identifying coupling directions and delays, but also distinguishing direct from indirect coupling configurations. The power of the proposed methodology is investigated based on numerical studies of different types of examples including coupled linear stochastic processes, Lorenz-63 systems and networks of neural mass models. Finally, I will discuss some recent ideas on combining concepts used in the analysis of ordinal patterns with alternative approaches, including the replacement of classical ordinal patterns by other types of subsequence specific discrete graphlets (e.g. visibility graphlets) and the exploitation of synchronous occurrences of specific (ordinal or graphlet) patterns in a time series using event coincidence analysis.

By: Inga Kottlarz, Sebastian Berg, Diana Toscano-Tejeida, Iris Steinmann, Mathias Bähr, Stefan Luther, Melanie Wilke, Ulrich Parlitz and Alexander Schlemmer We extract ordinal pattern based features from multichannel EEG time series to differentiate between different age groups and individuals. We consider functional connectivity in the form of ordinal pattern-based mutual information and single channel features in the form of the pattern distributions in each individual EEG channel. Both functional connectivity and single-channel features are subjected to nonlinear dimensionality reduction using t-distributed stochastic neighbor embedding. We analyze the separation of EEGs from different age groups and individuals 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. References: I. Kottlarz et al. “Extracting Robust Biomarkers From Multichannel EEG Time Series Using Nonlinear Dimensionality Reduction Applied to Ordinal Pattern Statistics and Spectral Quantities”. In: Front. Physiol. 11 (2021), p. 1790. doi: 10.3389/fphys.2020.614565.

In this presentation, the permutation Jensen-Shannon distance (PJSD) is introduced. It is a symbolic metric able to quantify the ordinal similarity between two arbitrary time series. Based on the fusion of two well-known concepts, namely, the Jensen-Shannon divergence and the ordinal encoding scheme, the PJSD inherits all the important advantages associated with them: simplicity, low computational cost, noise robustness and wide applicability. Consequently, large amounts of data with outliers and artifacts can be efficiently handled with this quantifier, making it especially suited to deal with the current big data challenges. It is worth emphasizing its versatility, since hypothesis tests related to the nature of an arbitrary time series can be easily carried out by estimating its PJSD to reference time series appropriately generated according to the null model. Several numerical and experimental analyses illustrate this versatility as well as the robustness for characterizing and classifying time series. For all these reasons, the proposed ordinal distance seems to be a promising addition to the repertoire of existing methods for complex signals analysis.

The multiple sensations experienced by our body are accompanied by exchanges of information in the form of electrical signals within the brain. These electrical signals can be waves, like the way radio waves are used to broadcast music from the transmitter to the receptors. In the brain, different areas can function as transmitters or receivers of signals depending on the situation. Neural oscillations, or brain waves, are rhythmic or repetitive patterns of neural activity in the cortex. The interaction between neurons can result in oscillations with a different frequency from the one associated with the firing of individual neurons as depends on the possible rules of synaptic strength between neurons. These neuronal oscillations are a fundamental mechanism that allows synchronization of neuronal activity within and between brain regions. The mechanism facilitates precise temporal coordination of the neural processes underlying cognition, memory, perception, and behavior. Synchronization in neural networks has attracted a lot of attention in recent years, focusing on the type of transitions of different oscillations’ patterns in the network. Whether the transition could appear as a continuous or a burst it could depend on the structure of the network, as well as synaptic plasticity rules. We consider the effect of synaptic interaction as well as structural connectivity on the synchronization transition in network models, of regularly spiking neurons, with different neuronal rhythms. Synaptic strength depresses low activation and enhances high activation of postsynaptic neurons. Importantly, triplet models of spike-timing-dependent plasticity (STDP) have shown to describe accurately plasticity experiments that the classical STDP rule, based on spiking pairs, failed to capture. We consider a triplet model of STDP that depends on the interactions of three synchronized spikes with a network of excitatory and inhibitory neurons, emulating the activity of the cortex. We study the dynamic evolution of the triplet model with STDP synchronization, contrasting it to a classical pairwise model. Then the electrical recordings in patients with refractory epilepsy is investigated to discern the underlying oscillatory mechanisms during the epileptic process. For this, neuronal activity is studied for basal (far from the seizure) and preictal (immediately before the seizure) periods through recordings of intracerebral electrodes implanted in patients to achieve a greater resolution of the local field potential. We explore how the combination of pairwise and triplets STDP models can reproduce the observed dynamics. The intrinsic dynamics of the two types of records is discerned by using a time windows analysis and studying the amplitude and phase couplings for each signal. The causality of these records is quantified through information theory tools and a symbolic method of analysis that accounts for the ordinal structure of the time series, showing an enhancing of information of brain oscillations in the range of high frequencies.

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.

Gait is a basic cognitive purposeful action, essential for the survival of the individual, and hence highly controlled by the Central Nervous System. When gait is altered, for instance due to neurodegenerative dementias or other lesions, a series of compensatory mechanisms are deployed to maintain this basic functionality. It is thus not surprising that gait study, and instrumental gait analysis (IGA) in particular, has been receiving increasing attention in the last few years, for being the complex result of the interactions between different brain motor areas, and thus a proxy in the understanding of the underlying neural dynamics. In spite of this, our understanding of gait, and especially of its potential use as a biomarker of initial cognitive decline, is hitherto limited. In this talk I will review some recent results on the use of ordinal patterns in the analysis of kinematic and spatiotemporal parameters obtained with IGA, in Mild Cognitive Impairment, mild Alzheimer’s Disease, and cerebral palsy.

Nowadays, Ordinal Patterns (OrdPatt) can be considered a cross-disciplinary field of research where scientists all along the globe work together to find new theoretical developments, challenges, and applications. Regarding the latter, applying OrdPatt for capturing the content of information of real and finite samples has been the hallmark in recent years. As an enthusiastic researcher, I have had the opportunity to work with OrdPatt in systems supposedly different from each other, and this talk goes in this line. This is just a simple talk where I want to exemplify how versatile OrdPatt are by tackling one of the main paradigms in complexity science: the bridge between topology and dynamics of a system, specifically, how the structure may influence its dynamics. I am going to talk about some common points between football and the brain from the viewpoint of complex systems, and how the OrdPatt were key at capturing both the inner dynamics of the networks’ nodes and the content of information of its evolving structure.

Advances in the technique of monitoring vital signs have made wide access to data characterizing the cardiovascular and respiratory systems. However, to gain better insight into physiology and improve diagnostic and treatment strategies requires facing a variety of problems related to recording, editing and analysis of the cardiorespiratory data. The talk will outline the possibilities and challenges of cardiorespiratory function assessment highlighting examples of successful applications of mathematical methods including ordinal patterns analysis.

In patients with cardiovascular diseases the cardiovascular regulation is often altered which might negatively impact the prognosis. We use various types of methods based on the concept of ordinal patterns to evaluate coupling and bidirectional influences of heart rate and blood pressure. This is a joint work with J. M. Amig\'o, B. Graff, K. Narkiewicz and K Tessmer.

Ordinal data analysis is an interesting direction in machine learning. It mainly deals with data for which only the relationships "<", "=", ">" between pairs of points are known. We do an attempt of formalizing structures behind ordinal data analysis by introducing the notion of ordinal spaces on the base of a strict axiomatic approach. For these spaces we study general properties as isomorphism conditions, connections with metric spaces, embeddability in Euclidean spaces, topological properties etc.

Sensory neurons use action potentials or spikes to encode and transmit information of external stimuli. The coding mechanisms used are not yet fully understood but it is known that neurons may use different mechanisms, which can be complementary or functional under different situations. An important open problem is to understand how neurons may exploit neural noise (stochastic electrical fluctuations which do not carry any information) for signal encoding and transmission. In this work, I focus on subthreshold inputs, that is, signals that are weak enough so that, by themselves (in the absence of noise), they do not trigger spikes. In this situation, neural noise triggers spikes, and I will discuss a plausible signal encoding mechanism based on ordinal spike patterns.

The family of cumulative paired $\phi$-entropies offers a wide variety of ordinal dispersion measures, covering many well-known dispersion measures as a special case. After a comprehensive analysis of this family of entropies, we consider the corresponding sample versions and derive their asymptotic distributions for stationary ordinal time series data. Based on an investigation of their asymptotic bias, we propose a family of signed serial dependence measures, which can be understood as weighted types of Cohen's $\kappa$ with the weights being related to the actual choice of~$\phi$. Again, the asymptotic distribution of the corresponding sample~$\kappa_\phi$ is derived and applied to test for serial dependence in ordinal time series. Using numerical computations and simulations, the practical relevance of the dispersion and dependence measures is investigated. We conclude with an environmental data example, where the novel $\phi$-entropy-related measures are applied to an ordinal time series on the daily level of air quality.

We consider change-point tests based on rank statistics to test for structural changes in long-range dependent observations. Under the hypothesis of stationary time series, the asymptotic distributions of the corresponding test statistics are derived. For this, we consider a uniform reduction principle for the empirical process in a two-parameter Skorohod space equipped with a weighted supremum norm. Moreover, special emphasis is laid on an application-oriented approach to the mathematical results by considering self-normalized statistics and an approximation of the distribution of test statistics by subsampling procedures.

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This talk provides examples of time series analysis with ordinal pattern statistics from the environmental sciences. Using examples from carbon fluxes, nutrient concentrations and hydrology, we investigate the ability of opd's and associated complexity measures to distinguish different ecosystems, as well as processes within them. Special attention will be given to the effect of varying temporal resolution of variables. Complexity and distance measures based on the opd also allow for a rather rigorous and challenging model-data comparison, pointing to potential model improvements with a sensitivity far beyond ordinary quality indicators such as Pearson correlation or model efficiency.

In the seminal paper of C. Bandt and B. Pomper [\emph{Permutation entropy: an natural complexity measure for time series} Physical review letters] in 2002, a valuable tool, permutation entropy, was introduced for the analysis of time series. The vast amount of references and related works around the notion of permutation entropy gives a slight idea about de the importance of using it to analyze time series. The expansion in the number of applications and versions of the permutation entropy support not only the results but also the information that can be extracted from the time series. Knowledge also makes it possible to anticipate or make probabilistic estimations about future events. Climatic networks, biological processes, finances, ..., are some of the subjects in which the application of these resources becomes possible, mainly because the amount of accumulated information is continuously increasing. However, there are many other questions that arise when conducting studies as the appropriate selection of parameters. In this talk we will focus on the application of order patterns to the analysis of economic time series.

We introduce various ordinal based methods to construct complex networks - the nodes of those networks correspond to particular ordinal encodings and the connections between them describe the dynamics. We use this to obtain a simple and efficient scheme to compute topological entropy and demonstrate the application of this method for low dimensional systems. In contrast to existing methods our approach proceeds by constructing a sequence of networks representing the underlying dynamics at different measurement resolution. We extend this method and show connections between networks constructed in this way and Markov modelling of nonlinear dynamics. This is joint work with Thomas Stemler and Konstantinos Sakellariou (PRE 103, 022214).

This paper expands previous studies on the informational efficiency of the new financial asset type, called generically cryptocurrency. In spite of the fact that for the general public Bitcoin is tantamount to cryptocurrency, this market is composed of more than one thousand assets that are daily traded. These assets are traded on a 27/7 basis, meaning that there are no predetermined market closures. In addition, considering a large number of transactions, they can be analyzed at a very high frequency. We analyze the time series of prices of the most important cryptocurrencies and compare them with traditional financial assets, such as stocks, bonds, and gold.

Since the beginning of the research on the dynamics of complex networks, the deep relationship between topology and dynamics has been thoroughly explored with regard to the synchronization between the nodes’ dynamics, and the knowledge gathered so far has driven the advance in crucial applications. However, there are cases in which the system performs its activity in a partial or weakly synchronization regime to preserve the balance between functional integration and segregation, whereas full synchronization is pathological. But even in this far from synchronized state, each coupled unit is encoding in its own dynamics the signature of its structural role. We use ordinal methods to explore how this prominent feature could be used to extract information about the network, without having the need to make any reference to pairwise correlations, even in those situations where the structure is unknown. The effect of the coupling strength against the dynamical complexity of the nodes is found to be a function of their topological roles, with nodes of higher degree displaying lower levels of complexity. Importantly, our results imply that it is possible to infer the degree distribution of a network only from individual dynamical measurements in a large variety of systems.

Causality analysis aims at estimating the interactions of the observed variables and subsequently the connectivity structure of the observed dynamical system or stochastic process. The partial mutual information from mixed embedding (PMIME) is an information based measure for causality analysis of high-dimensional continuous-valued time series. The PMIME applies dimension reduction by appropriate selection of the mixed embedding vector comprised the most relevant lag variables of all the observed variables to the response using conditional mutual information (CMI). The presence of lag components of the driving variable in this vector implies direct causal (driving-response) effect. In this work, the PMIME is appropriately adapted to discrete-valued multivariate time series (symbol sequences), called discrete PMIME (DPMIME). Appropriate estimation of discrete probability distributions and CMI for discrete variables is implemented in DPMIME. Further, asymptotic distribution of the estimated CMI is derived allowing for a parametric significance test for CMI in DPMIME, whereas for PMIME there is no parametric test for CMI and the test is done using resampling. The parametric significance test for CMI in the progressive algorithm of DPMIME is compared favorably to the corresponding resampling significance test. Further, performance of DPMIME is demonstrated in a simulation study, where the discrete-valued time series are (a) formed by discretizing time series from continuous-valued chaotic dynamical systems, (b) generated from a multivariate Markov chain with transition probabilities estimated on discretized time series in (a). The results of the simulations suggest the accuracy of DPMIME in the estimation of direct causality converges with the time series length to the accuracy of PMIME. Moreover, the DPMIME is found to perform well even in high dimensional systems of 25 subsystems provided the sparsity of the connectivity network. An application to connectivity networks of financial world market is finally presented.

Since its inception 20 years ago, the practical application of the Bandt-Pompe methodology to the study of time series has been very successful. However, the theoretical aspects have remained limited to noiseless deterministic series and dynamical systems, the main obstacle being the super-exponential growth of allowed permutations with length when randomness (also in form of observational noise) is present in the data. To overcome this difficulty, we take a new approach through complexity classes, which are precisely defined by the growth of allowed permutations with length, regardless of the deterministic or noisy nature of the data. We consider three major classes: exponential, sub-factorial and factorial. Our main results is a novel class of generalized permutation entropies which widely extend the standard notion of permutation entropy. They represent new complexity measures which possess many interesting group-theoretical and analytic properties. Besides, our information measures can be considered as the topological analogue of the family of so-called group entropies, useful to classify the different notions of generalized entropy used in statistical mechanics and information theory. The conceptual framework proposed provides us with a unified approach to the ordinal analysis of deterministic and random processes, from dynamical systems to white noise. This talk is the second part of the talk presented by P. Tempesta. Joint work in collaboration with R. Dale (Centro de Investigación Operativa, Elche) and P. Tempesta (Universidad Complutense and Instituto de Ciencias Matemáticas, Madrid).

This talk aims to present an example of application of ordinal pattern based entropies to electrocardiogram (ECG) beat classification. Indeed, the automatic and rapid analysis of long-term electrocardiogram (ECG) records still remains a challenging task. Given that most of the existing algorithms are time consuming, we present in this talk a two-level hierarchical model based on ordinal patterns for classifying ECG beats into three types, namely N (Normal) S (Supraventricular escape beat) and V (Ventricular escape beat). The classification rules include morphological and temporal properties of the ECG signal that are compared to R-R and QRS dependent thresholds derived from the beat conditional entropy of ordinal patterns (CEOP) or beat permutation entropy (PE) series. The experimental classification rates obtained from the MIT-BIH Arrythmia database ($93:66\%$) and the St. Petersburg Institute of Cardiological Technics (INCART) database ($95:43\%$) confirm the ability of the ordinal pattern based entropies for a multi-class ECG classification. Moreover, we evaluated the running speed of the algorithms as approximately 500 Mbps, which confirms the high efficiency of ordinal pattern methods for real-time applications.

The use of anaesthesia is a fundamental tool in the investigation of consciousness. Anesthesia procedures allow to investigate different states of consciousness from sedation to deep anesthesia within controlled scenarios. In this study we use information quantifiers to measure the complexity of electrocorticogram recordings in monkeys. We apply these metrics to compare different stages of general anesthesia for evaluating consciousness in several anesthesia protocols. We find that the complexity of brain activity can be used as a correlate of consciousness. For two of the anaesthetics used, propofol and medetomidine, we find that the anaesthetised state is accompanied by a reduction in the complexity of brain activity. On the other hand we observe that use of ketamine produces an increase in complexity measurements. We relate this observation with increase activity within certain brain regions associated with the ketamine used doses. Our measurements indicate that complexity of brain activity is a good indicator for a general evaluation of different levels of consciousness awareness, both in anesthetized and non anesthetizes states.

Complex systems are typically characterized as an intermediate situation between a complete regular structure and a random system. Brain signals can be studied as a striking example of such systems: cortical states can range from highly synchronous and ordered neuronal activity (with higher spiking variability) to desynchronized and disordered regimes (with lower spiking variability). It has been recently shown, by testing independent signatures of criticality, that a phase transition occurs in a cortical state of intermediate spiking variability. Here, we use a symbolic information approach to show that, despite the monotonical increase of the Shannon entropy between ordered and disordered regimes, we can determine an intermediate state of maximum complexity based on the Jensen disequilibrium measure. More specifically, we show that statistical complexity is maximized close to criticality for cortical spiking data of urethane-anesthetized rats, as well as for a network model of excitable elements that presents a critical point of a non-equilibrium phase transition

Ordinal patterns describe the order structure of data points over a small time horizon. Using a moving window approach we reduce the complexity of a time series by analyzing the sequence of ordinal patterns instead of the original data. The concepts have been established first in the theory of dynamical systems and have later been adapted by statisticians. Ordinal pattern dependence is a new way of measuring the degree of dependence between time series. Since it only relies on the ordinal structure of the data, it is robust against monotone transformations and measurement errors. This method has proved to be useful already in the context of hydrological, financial as well as medical data. Using this concept it is possible to analyze whether the dependence structure between two time series changes over time. We present limit theorems for ordinal pattern probabilities and tests for structural breaks in the short-range dependent setting. Furthermore, we give an outlook towards long-range dependence, extremal events and multivariate extensions.

In real-world applications, one often deals with multivariate time series, e.g., with medical measurements, which are stored as ECG data are usually not determined from a single electrode but from multiple electrodes. Since ordinal pattern representations are designed for univariate time series, the concept has to be extended to the multivariate case in order to apply it for multivariate time series as well. Numerous studies introduce multivariate extensions of permutation entropy under the generic name "Multivariate Permutation Entropy", which claim to be the general approach, although there are many differences between them with different strengths and weaknesses. In this talk, by multivariate permutation entropy (MPE) we denote the class of all multivariate extensions of permutation entropy. We divide them into four strategies. The first three are based on the concept of univariate ordinal patterns, while the fourth strategy is based on a new concept of multivariate ordinal patterns. We elaborate on the different characteristics of the strategies and discuss their possible applications.

Ordinal patterns, or permutations, have been used extensively for characterizing nonlinear deterministic systems after the seminal work of Bandt and Pompe, PRL (2002). But, only a limited number of papers have used the ordinal patterns for characterizing nonlinear stochastic systems. Thus, here, we summarize our recent works to use ordinal patterns for characterizing nonlinear stochastic systems. Ordinal patterns can be used for characterizing nonlinear stochastic systems because the number of ordinal patterns increases super-exponentially when the size of ordinal patterns is enlarged. Especially, we contrast the approach using ordinal patterns with that using recurrence plots for this purpose. References: Y. Hirata & M. Shiro, Detecting nonlinear stochastic systems using two independent hypothesis tests, Physical Review E 100, 022203 (2019) Y. Hirata, M. Shiro, & J. M. Amigó, Surrogate data preserving all the properties of ordinal patterns up to a certain length, Entropy 21, 713 (2019) Y. Hirata, Y. Sato, & D. Faranda, Permutations uniquely identify states and unknown external forces in non-autonomous dynamical systems, Chaos 30, 103103 (2020) Y. Hirata, Recurrence plots for characterizing random dynamical systems, Communications in Nonlinear Science and Numerical Simulation 94, 105552 (2021)

I discuss a complexity measure for coupling among multiple time series in terms of ordinal patterns. Here, ordinal patterns are thought of as total orders on the set of time points. One can consider their intersection that is not a total order but in general a partial order. This can be used to define the above-mentioned complexity measure. In this talk, first I review my previous work along this line of idea. Second, a new direction of study making use of the further algebraic property is presented.

A growing number of algorithms have been proposed to map a scalar time series into ordinal partition transition networks. However, most observable phenomena in the empirical sciences are of a multivariate nature. We construct ordinal pattern transition networks for multivariate time series. This approach yields weighted directed networks representing the pattern transition properties of time series in velocity space, which hence provides dynamic insights of the underling system. Furthermore, we propose a measure of entropy to characterize ordinal partition transition dynamics, which is sensitive to capturing the possible local geometric changes of phase space trajectories. In addition, we systematically show that cross and joint ordinal pattern transition networks are sensitive to phase synchronization. We will show some successful applications of ordinal pattern transition networks to identify: (i) coupling direction and delay, (ii) dynamical transitions in both model system and (iii) real-world time series of fluid experiments, electrocardiogram recordings, and teleconnections in climate system.

Data analysis methods based on ordinal patterns have been applied in different fields of research such as biomedicine, econophysics and engineering. Main reasons for the increasing success of these analysis methods are that ordinal patterns contain intrinsic information on the dynamical structure of a system and that ordinal pattern-based methods are robust and simple from a computational viewpoint. Whereas some nice asymptotic results have been found for pattern length going to infinity, in practical data analysis short patterns describing specific features of data and models behind them are of some special interest. We demonstrate this point by discussing some statistics based on counting monotone changes and on considering asymmetries in ordinal pattern distributions. The data analysis methods obtained on this base are illustrated by considering some EEG data and some rock surface data.

Chaotic systems share with stochastic processes several properties that make them almost undistinguishable. In this communication we introduce a representation space, to be called the complexity-entropy causality plane. Its horizontal and vertical axis are suitable functionals of the pertinent probability distribution, namely, the entropy of the system and an appropriate statistical complexity measure, respectively. These two functionals are evaluated using the Bandt-Pompe recipe to assign a probability distribution function to the time series generated by the system. Several well-known model-generated time series, usually regarded as being of either stochastic or chaotic nature, are analyzed so as to illustrate the approach. The main achievement of this communication is the possibility of clearly distinguishing between them in our representation space, something that is rather difficult otherwise.