Temporal organization of magnetospheric fluctuations unveiled by recurrence patterns in the Dst index.

Magnetic storms constitute the most remarkable large-scale phenomena of nonlinear magnetospheric dynamics. Studying the dynamical organization of macroscopic variability in terms of geomagnetic activity index data by means of complexity measures provides a promising approach for identifying the underlying processes and associated time scales. Here, we apply a suite of characteristics from recurrence quantification analysis (RQA) and recurrence network analysis (RNA) in order to unveil some key nonlinear features of the hourly Disturbance storm-time (Dst) index during periods with magnetic storms and such of normal variability. Our results demonstrate that recurrence-based measures can serve as excellent tracers for changes in the dynamical complexity along non-stationary records of geomagnetic activity. In particular, trapping time (characterizing the typical length of "laminar phases" in the observed dynamics) and recurrence network transitivity (associated with the number of the system's effective dynamical degrees of freedom) allow for a very good discrimination between magnetic storm and quiescence phases. In general, some RQA and RNA characteristics distinguish between storm and non-storm times equally well or even better than other previously considered nonlinear characteristics like Hurst exponent or symbolic dynamics based entropy concepts. Our results point to future potentials of recurrence characteristics for unveiling temporal changes in the dynamical complexity of the magnetosphere.

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title: Complex network approach for recurrence analysis of time series
abstract: We propose a novel approach for analysing time series using complex network theory. We identify the recurrence matrix (calculated from time series) with the adjacency matrix of a complex network and apply measures for the characterisation of complex networks to this recurrence matrix. By using the logistic map, we illustrate the potential of these complex network measures for the detection of dynamical transitions. Finally, we apply the proposed approach to a marine palaeo-climate record and identify the subtle changes to the climate regime.

title: Recurrence networks—a novel paradigm for nonlinear time series analysis
abstract: This paper presents a new approach for analysing the structural properties of time series from complex systems. Starting from the concept of recurrences in phase space, the recurrence matrix of a time series is interpreted as the adjacency matrix of an associated complex network, which links different points in time if the considered states are closely neighboured in phase space. In comparison with similar network-based techniques the new approach has important conceptual advantages, and can be considered as a unifying framework for transforming time series into complex networks that also includes other existing methods as special cases. It has been demonstrated here that there are fundamental relationships between many topological properties of recurrence networks and different nontrivial statistical properties of the phase space density of the underlying dynamical system. Hence, this novel interpretation of the recurrence matrix yields new quantitative characteristics (such as average path length, clustering coefficient, or centrality measures of the recurrence network) related to the dynamical complexity of a time series, most of which are not yet provided by other existing methods of nonlinear time series analysis.

title: Testing time series irreversibility using complex network methods
abstract: The absence of time-reversal symmetry is a fundamental property of many nonlinear time series. Here, we propose a new set of statistical tests for time series irreversibility based on standard and horizontal visibility graphs. Specifically, we statistically compare the distributions of time-directed variants of the common complex network measures degree and local clustering coefficient. Our approach does not involve surrogate data and is applicable to relatively short time series. We demonstrate its performance for paradigmatic model systems with known time-reversal properties as well as for picking up signatures of nonlinearity in neuro-physiological data.

title: Multifractal analysis of solar flare indices and their horizontal visibility graphs
abstract: The multifractal properties of the daily solar X-ray brightness, X l and X s , during the period from 1 January 1986 to 31 December 2007 which includes two solar cycles are examined using the universal multifractal approach and multifractal detrended fluctuation analysis. Then we convert these time series into networks using the horizontal visibility graph technique. Multifractal analysis of the resulting networks is performed using an algorithm proposed by us. The results from the multifractal analysis show that multifractality exists in both raw daily time series of X-ray brightness and their horizontal visibility graphs. It is also found that the empirical K ( q ) curves of raw time series can be fitted by the universal multifractal model. The numerical results on the raw data show that the Solar Cycle 23 is weaker than the Solar Cycle 22 in multifractality. The values of h (2) from multifractal detrended fluctuation analysis for these time series indicate that they are stationary and persistent, and the correlations in the time series of Solar Cycle 23 are stronger than those for Solar Cycle 22. Furthermore, the multifractal scaling for the networks of the time series can reflect some properties which cannot be picked up by using the same analysis on the original time series. This suggests a potentially useful method to explore geophysical data.

title: Long-term changes in the north-south asymmetry of solar activity: a nonlinear dynamics characterization using visibility graphs
abstract: Solar activity is characterized by complex dynamics superimposed onto an almost periodic, approximately 11-year cycle. One of its main features is the presence of a marked, time-varying hemispheric asymmetry, the deeper reasons for which have not yet been completely uncovered. Traditionally, this asymmetry has been studied by considering amplitude and phase differences. Here, we use visibility graphs, a novel tool of nonlinear time series analysis, to obtain complementary information on hemispheric asymmetries in dynamical properties. Our analysis provides deep insights into the potential and limitations of this method, revealing a complex interplay between factors relating to statistical and dynamical properties, i.e., effects due to the probability distribution and the regularity of observed fluctuations. We demonstrate that temporal changes in the hemispheric predominance of the graph properties lag those directly associated with the total hemispheric sunspot areas. Our findings open a new dynamical perspective on studying the north–south sunspot asymmetry, which is to be further explored in future work.

title: The geometry of chaotic dynamics — a complex network perspective
abstract: Recently, several complex network approaches to time series analysis have been developed
and applied to study a wide range of model systems as well as real-world data, e.g.,
geophysical or financial time series. Among these techniques, recurrence-based concepts
and prominently ε-recurrence networks, most faithfully represent the
geometrical fine structure of the attractors underlying chaotic (and less interestingly
non-chaotic) time series. In this paper we demonstrate that the well known graph
theoretical properties local clustering coefficient and global (network) transitivity can
meaningfully be exploited to define two new local and two new global measures of dimension
in phase space: local upper and lower clustering dimension as well as global upper and
lower transitivity dimension. Rigorous analytical as well as numerical results for
self-similar sets and simple chaotic model systems suggest that these measures are
well-behaved in most non-pathological situations and that they can be estimated reasonably
well using ε-recurrence networks constructed from relatively short time
series. Moreover, we study the relationship between clustering and transitivity dimensions
on the one hand, and traditional measures like pointwise dimension or local Lyapunov
dimension on the other hand. We also provide further evidence that the local clustering
coefficients, or equivalently the local clustering dimensions, are useful for identifying
unstable periodic orbits and other dynamically invariant objects from time series. Our
results demonstrate that ε-recurrence networks exhibit an important link
between dynamical systems and graph theory.

title: Analytical framework for recurrence network analysis of time series.
abstract: Recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems. While there are already many successful applications ranging from medicine to paleoclimatology, a solid theoretical foundation of the method has still been missing so far. Here, we interpret an ɛ-recurrence network as a discrete subnetwork of a "continuous" graph with uncountably many vertices and edges corresponding to the system's attractor. This step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by ɛ. In particular, we introduce local measures such as the ɛ-clustering coefficient, mesoscopic measures such as ɛ-motif density, path-based measures such as ɛ-betweennesses, and global measures such as ɛ-efficiency. This new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of ɛ via a percolation threshold, and it shows that estimation can be improved by so-called node splitting invariant versions of the measures. We finally illustrate the framework for a number of archetypical chaotic attractors such as those of the Bernoulli and logistic maps, periodic and two-dimensional quasiperiodic motions, and for hyperballs and hypercubes by deriving analytical expressions for the novel measures and comparing them with data from numerical experiments. More generally, the theoretical framework put forward in this work describes random geometric graphs and other networks with spatial constraints, which appear frequently in disciplines ranging from biology to climate science.

title: Identifying shrimps in continuous dynamical systems using recurrence-based methods
abstract: In this work, we demonstrate that nonlinear measures based on recurrence plots obtained from individual trajectories provide a practicable alternative for automatically distinguishing periodic and chaotic dynamics and, hence, numerically detecting shrimps. Traditional diagonal line-based measures of recurrence quantification analysis as well as measures from complex network theory are shown to allow an excellent classification of periodic and chaotic behavior in parameter space. Average path length and clustering coefficient of the resulting recurrence networks are found to be particularly powerful discriminatory statistics for the identification of shrimps. Our results are illustrated in detail for the Roessler system. Implications for detecting bifurcations between regular and chaotic dynamics in other models of geophysical phenomena (such as the Lorenz-84 system) are discussed.

title: Non-linear regime shifts in Holocene Asian monsoon variability: potential impacts on cultural change and migratory patterns
abstract: The Asian monsoon system is an important tipping element in Earth's climate with a large impact on human societies in the past and present. In light of the potentially severe impacts of present and future anthropogenic climate change on Asian hydrology, it is vital to understand the forcing mechanisms of past climatic regime shifts in the Asian monsoon domain. Here we use novel recurrence network analysis techniques for detecting episodes with pronounced non-linear changes in Holocene Asian monsoon dynamics recorded in speleothems from caves distributed throughout the major branches of the Asian monsoon system. A newly developed multi-proxy methodology explicitly considers dating uncertainties with the COPRA (COnstructing Proxy Records from Age models) approach and allows for detection of continental-scale regime shifts in the complexity of monsoon dynamics. Several epochs are characterised by non-linear regime shifts in Asian monsoon variability, including the periods around 8.5–7.9, 5.7–5.0, 4.1–3.7, and 3.0–2.4 ka BP. The timing of these regime shifts is consistent with known episodes of Holocene rapid climate change (RCC) and high-latitude Bond events. Additionally, we observe a previously rarely reported non-linear regime shift around 7.3 ka BP, a timing that matches the typical 1.0–1.5 ky return intervals of Bond events. A detailed review of previously suggested links between Holocene climatic changes in the Asian monsoon domain and the archaeological record indicates that, in addition to previously considered longer-term changes in mean monsoon intensity and other climatic parameters, regime shifts in monsoon complexity might have played an important role as drivers of migration, pronounced cultural changes, and the collapse of ancient human societies.

title: Spurious Structures in Recurrence Plots Induced by Embedding
abstract: In this paper we show that delay embedding produces spurious structures in a recurrence plot (RP) that are not present in the real attractor. We analyze typical sets of simulated data, such as white noise and data from the chaotic Rössler system to show the relevance of this effect.In the second part of the paper we show that the second order Rényi entropy and the correlation dimension are dynamical invariants that can be estimated from Recurrence Plots with arbitrary embedding dimension and delay.

title: Statistics A Guide To The Use Of Statistical Methods In The Physical Sciences
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title: Individual Comparisons by Ranking Methods
abstract: The comparison of two treatments generally falls into one of the following two categories: (a) we may have a number of replications for each of the two treatments, which are unpaired, or (b) we may have a number of paired comparisons leading to a series of differences, some of which may be positive and some negative. The appropriate methods for testing the significance of the differences of the means in these two cases are described in most of the textbooks on statistical methods.

title: Detection of Dynamical Complexity Changes in Dst Time Series Using Entropy Concepts and Rescaled Range Analysis
abstract: Using an array of diagnostic tools including entropy concepts and rescaled range analysis, we establish that the Dst index time series exhibits long-range correlations, and that the underlying stochastic process can be modeled as fractional Brownian motion. We show the emergence of two distinct patterns in the geomagnetic variability of the terrestrial magnetosphere: (1) a pattern associated with intense magnetic storms, which is characterized by a higher degree of organization (i.e., lower complexity or higher predictability for the system) and persistent behavior, and (2) a pattern associated with normal periods, which is characterized by a lower degree of organization (i.e., higher complexity or lower predictability for the system) and anti-persistent behavior.

title: Approximate entropy as a measure of system complexity.
abstract: Techniques to determine changing system complexity from data are evaluated. Convergence of a frequently used correlation dimension algorithm to a finite value does not necessarily imply an underlying deterministic model or chaos. Analysis of a recently developed family of formulas and statistics, approximate entropy (ApEn), suggests that ApEn can classify complex systems, given at least 1000 data values in diverse settings that include both deterministic chaotic and stochastic processes. The capability to discern changing complexity from such a relatively small amount of data holds promise for applications of ApEn in a variety of contexts.

title: Physiological time-series analysis using approximate entropy and sample entropy.
abstract: Entropy, as it relates to dynamical systems, is the rate of information production. Methods for estimation of the entropy of a system represented by a time series are not, however, well suited to analysis of the short and noisy data sets encountered in cardiovascular and other biological studies. Pincus introduced approximate entropy (ApEn), a set of measures of system complexity closely related to entropy, which is easily applied to clinical cardiovascular and other time series. ApEn statistics, however, lead to inconsistent results. We have developed a new and related complexity measure, sample entropy (SampEn), and have compared ApEn and SampEn by using them to analyze sets of random numbers with known probabilistic character. We have also evaluated cross-ApEn and cross-SampEn, which use cardiovascular data sets to measure the similarity of two distinct time series. SampEn agreed with theory much more closely than ApEn over a broad range of conditions. The improved accuracy of SampEn statistics should make them useful in the study of experimental clinical cardiovascular and other biological time series.

title: Characterization of Surface EMG Signal Based on Fuzzy Entropy
abstract: Fuzzy entropy (FuzzyEn), a new measure of time series regularity, was proposed and applied to the characterization of surface electromyography (EMG) signals. Similar to the two existing related measures ApEn and SampEn, FuzzyEn is the negative natural logarithm of the conditional probability that two vectors similar for m points remain similar for the next m+1 points. Importing the concept of fuzzy sets, vectors' similarity is fuzzily defined in FuzzyEn on the basis of exponential function and their shapes. Besides possessing the good properties of SampEn superior to ApEn, FuzzyEn also succeeds in giving the entropy definition in the case of small parameters. Its performance on characterizing surface EMG signals, as well as independent, identically distributed (i.i.d.) random numbers and periodical sinusoidal signals, shows that FuzzyEn can more efficiently measure the regularity of time series. The method introduced here can also be applied to other noisy physiological signals with relatively short datasets.

title: Universality in solar flare, magnetic storm and earthquake dynamics using Tsallis statistical mechanics
abstract: The universal character of the dynamics of various extreme phenomena is an outstanding scientific challenge. We show that X-ray flux and Dst time series during powerful solar flares and intense magnetic storms, respectively, obey a nonextensive energy distribution function for earthquake dynamics with similar values for the Tsallis entropic index q. Thus, evidence for universality in solar flares, magnetic storms and earthquakes arise naturally in the framework of Tsallis statistical mechanics. The observed similarity suggests a common approach to the interpretation of these diverse phenomena in terms of driving physical mechanisms that have the same character.

title: Distinguishing dynamics using recurrence-time statistics.
abstract: The probability densities of the mean recurrence time, which is the average time needed for a system to recur to a previously visited neighborhood, are investigated in various dynamical regimes and are found to be in agreement with those of the finite-time Lyapunov exponents. The important advantages of the former ones are that they are easy to estimate and that comparable short time series are sufficient. Asymmetric distributions with exponential tails are observed for intermittency and crisis-induced intermittency, while for typical chaos, the distribution has a Gaussian shape. Further, the shape of the distribution distinguishes intermittent strange nonchaotic attractors from those appearing through fractalization and tori collision mechanisms. Furthermore, statistics performed on the peaks in the frequency distribution of recurrence times unveil scaling behavior in agreement with that obtained from the spectral distribution function defined as the number of peaks in the Fourier spectrum greater than a predefined value. The results of the present recurrence statistics are of relevance in classifying different dynamics and providing important insights into the dynamics of a system when only one realization of this system is available. The practical use of this approach for experimental data is shown on experimental electrochemical time series.