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The recurrence plot and the recurrence quantification analysis (RQA) are well-established methods for the analysis of data from complex systems. They provide important insights into the nature of the dynamics, periodicity, regime changes, and many more. These methods are used in different fields of research, such as finance, engineering, life, and earth science. To use them, the data have usually to be uniformly sampled, posing difficulties in investigations that provide non-uniformly sampled data, as typical in medical data (e.g., heart-beat based measurements), paleoclimate archives (such as sediment cores or stalagmites), or astrophysics (supernova or pulsar observations). One frequently used solution is interpolation to generate uniform time series. However, this preprocessing step can introduce bias to the RQA measures, particularly those that rely on the diagonal or vertical line structure in the recurrence plot. Using prototypical model systems, we systematically analyze differences in the RQA measure average diagonal line length for data with different sampling and interpolation. For real data, we show that the course of this measure strongly depends on the choice of the sampling rate for interpolation. Furthermore, we suggest a correction scheme, which is capable of correcting the bias introduced by the prepossessing step if the interpolation ratio is an integer. [ABSTRACT FROM AUTHOR]

The identification of cycles in periodic signals is a ubiquitous problem in time series analysis. Many real-world datasets only record a signal as a series of discrete events or symbols. In some cases, only a sequence of (non-equidistant) times can be assessed. Many of these signals are furthermore corrupted by noise and offer a limited number of samples, e.g., cardiac signals, astronomical light curves, stock market data, or extreme weather events. We propose a novel method that provides a power spectral estimate for discrete data. The edit distance is a distance measure that allows us to quantify similarities between non-equidistant event sequences of unequal lengths. However, its potential to quantify the frequency content of discrete signals has so far remained unexplored. We define a measure of serial dependence based on the edit distance, which can be transformed into a power spectral estimate (EDSPEC), analogous to the Wiener–Khinchin theorem for continuous signals. The proposed method is applied to a variety of discrete paradigmatic signals representing random, correlated, chaotic, and periodic occurrences of events. It is effective at detecting periodic cycles even in the presence of noise and for short event series. Finally, we apply the EDSPEC method to a novel catalog of European atmospheric rivers (ARs). ARs are narrow filaments of extensive water vapor transport in the lower troposphere and can cause hazardous extreme precipitation events. Using the EDSPEC method, we conduct the first spectral analysis of European ARs, uncovering seasonal and multi-annual cycles along different spatial domains. The proposed method opens new research avenues in studying of periodic discrete signals in complex real-world systems. [ABSTRACT FROM AUTHOR]

Couplings in complex real-world systems are often nonlinear and scale dependent. In many cases, it is crucial to consider a multitude of interlinked variables and the strengths of their correlations to adequately fathom the dynamics of a high-dimensional nonlinear system. We propose a recurrence-based dependence measure that quantifies the relationship between multiple time series based on the predictability of their joint evolution. The statistical analysis of recurrence plots (RPs) is a powerful framework in nonlinear time series analysis that has proven to be effective in addressing many fundamental problems, e.g., regime shift detection and identification of couplings. The recurrence flow through an RP exploits artifacts in the formation of diagonal lines, a structure in RPs that reflects periods of predictable dynamics. Using time-delayed variables of a deterministic uni-/multivariate system, lagged dependencies with potentially many time scales can be captured by the recurrence flow measure. Given an RP, no parameters are required for its computation. We showcase the scope of the method for quantifying lagged nonlinear correlations and put a focus on the delay selection problem in time-delay embedding which is often used for attractor reconstruction. The recurrence flow measure of dependence helps to identify non-uniform delays and appears as a promising foundation for a recurrence-based state space reconstruction algorithm. [ABSTRACT FROM AUTHOR]

An editorial is presented on the development in the field of recurrence plots (RPs), recurrence quantification analysis (RQA), and recurrence networks. Topics include suggesting a pseudo-basis approach based on the recurrence matrix for better understanding and improving the conversion between time series and recurrence matrix; and cardiological states related to specific problems of hemodialysis and the relationship between heart rate and cognitive performance.

The quantification of spatial propagation of extreme precipitation events is vital in water resources planning and disaster mitigation. However, quantifying these extreme events has always been challenging as many traditional methods are insufficient to capture the nonlinear interrelationships between extreme event time series. Therefore, it is crucial to develop suitable methods for analyzing the dynamics of extreme events over a river basin with a diverse climate and complicated topography. Over the last decade, complex network analysis emerged as a powerful tool to study the intricate spatiotemporal relationship between many variables in a compact way. In this study, we employ two nonlinear concepts of event synchronization and edit distance to investigate the extreme precipitation pattern in the Ganga river basin. We use the network degree to understand the spatial synchronization pattern of extreme rainfall and identify essential sites in the river basin with respect to potential prediction skills. The study also attempts to quantify the influence of precipitation seasonality and topography on extreme events. The findings of the study reveal that (1) the network degree is decreased in the southwest to northwest direction, (2) the timing of 50th percentile precipitation within a year influences the spatial distribution of degree, (3) the timing is inversely related to elevation, and (4) the lower elevation greatly influences connectivity of the sites. The study highlights that edit distance could be a promising alternative to analyze event-like data by incorporating event time and amplitude and constructing complex networks of climate extremes. [ABSTRACT FROM AUTHOR]

The identification of recurrences at various timescales in extreme event-like time series is challenging because of the rare occurrence of events which are separated by large temporal gaps. Most of the existing time series analysis techniques cannot be used to analyze an extreme event-like time series in its unaltered form. The study of the system dynamics by reconstruction of the phase space using the standard delay embedding method is not directly applicable to event-like time series as it assumes a Euclidean notion of distance between states in the phase space. The edit distance method is a novel approach that uses the point-process nature of events. We propose a modification of edit distance to analyze the dynamics of extreme event-like time series by incorporating a nonlinear function which takes into account the sparse distribution of extreme events and utilizes the physical significance of their temporal pattern. We apply the modified edit distance method to event-like data generated from point process as well as flood event series constructed from discharge data of the Mississippi River in the USA and compute their recurrence plots. From the recurrence analysis, we are able to quantify the deterministic properties of extreme event-like data. We also show that there is a significant serial dependency in the flood time series by using the random shuffle surrogate method. [ABSTRACT FROM AUTHOR]

In this paper, we present the new frequency spectrum recurrence analysis technique by means of electro-encephalon signals (EES) analyses. The technique is suitable for time series analysis with noise and disturbances. EES were collected, and alpha waves of the occipital region were analysed by comparing the signals from participants in two states, eyes open and eyes closed. Firstly, EES were characterized and analysed by means of techniques already known to compare with the results of the innovative technique that we present here. We verified that, standard recurrence quantification analysis by means of EES time series cannot statistically distinguish the two states. However, the new frequency spectrum recurrence quantification exhibit quantitatively whether the participants have their eyes open or closed. In sequence, new quantifiers are created for analysing the recurrence concentration on frequency bands. These analyses show that EES with similar frequency spectrum have different recurrence levels revealing different behaviours of the nervous system. The technique can be used to deepen the study on depression, stress, concentration level and other neurological issues and also can be used in any complex system. [ABSTRACT FROM AUTHOR]

The identification of recurrences at various time scales in extreme event-like time series is challenging because of the rare occurrence of events which are separated by large temporal gaps. Most of the existing time series analysis techniques cannot be used to analyse extreme event-like time series in its unaltered form. The study of the system dynamics by reconstruction of the phase space using the standard delay embedding method is not directly applicable to event-like time series as it assumes a Euclidean notion of distance between states in the phase space. The edit distance method is a novel approach that uses the point-process nature of events. We propose a modification of edit distance to analyze the dynamics of extreme event-like time series by incorporating a nonlinear function which takes into account the sparse distribution of extreme events and utilizes the physical significance of their temporal pattern. We apply the modified edit distance method to event-like data generated from point process as well as flood event series constructed from discharge data of the Mississippi River in USA, and compute their recurrence plots. From the recurrence analysis, we are able to quantify the deterministic properties of extreme event-like data. We also show that there is a significant serial dependency in the flood time series by using the random shuffle surrogate method. [ABSTRACT FROM AUTHOR]

Intrinsic predictability is imperative to quantify inherent information contained in a time series and assists in evaluating the performance of different forecasting methods to get the best possible prediction. Model forecasting performance is the measure of the probability of success. Nevertheless, model performance or the model does not provide understanding for improvement in prediction. Intuitively, intrinsic predictability delivers the highest level of predictability for a time series and informative in unfolding whether the system is unpredictable or the chosen model is a poor choice. We introduce a novel measure, the Wavelet Entropy Energy Measure (WEEM), based on wavelet transformation and information entropy for quantification of intrinsic predictability of time series. To investigate the efficiency and reliability of the proposed measure, model forecast performance was evaluated via a wavelet networks approach. The proposed measure uses the wavelet energy distribution of a time series at different scales and compares it with the wavelet energy distribution of white noise to quantify a time series as deterministic or random. We test the WEEM using a wide variety of time series ranging from deterministic, non-stationary, and ones contaminated with white noise with different noise-signal ratios. Furthermore, a relationship is developed between the WEEM and Nash–Sutcliffe Efficiency, one of the widely known measures of forecast performance. The reliability of WEEM is demonstrated by exploring the relationship to logistic map and real-world data. [ABSTRACT FROM AUTHOR]

This paper introduces a novel measure to assess similarity between event hydrographs. It is based on cross recurrence plots (CRP) and recurrence quantification analysis (RQA), which have recently gained attention in a range of disciplines when dealing with complex systems. The method attempts to quantify the event runoff dynamics and is based on the time delay embedded phase space representation of discharge hydrographs. A phase space trajectory is reconstructed from the event hydrograph, and pairs of hydrographs are compared to each other based on the distance of their phase space trajectories. Time delay embedding allows considering the multidimensional relationships between different points in time within the event. Hence, the temporal succession of discharge values is taken into account, such as the impact of the initial conditions on the runoff event. We provide an introduction to cross recurrence plots and discuss their parameterization. An application example based on flood time series demonstrates how the method can be used to measure the similarity or dissimilarity of events, and how it can be used to detect events with rare runoff dynamics. It is argued that this methods provides a more comprehensive approach to quantify hydrograph similarity compared to conventional hydrological signatures. Key Points: This manuscript presents the first application of recurrence plots in catchment hydrologyCRP is used as a novel method for quantifying comprehensive hydrograph similarity based on runoff dynamicsTime delay embedded phase space trajectory allows to consider relationships between magnitudes of different point in time [ABSTRACT FROM AUTHOR]

One main challenge in constructing a reliable recurrence plot (RP) and, hence, its quantification [recurrence quantification analysis (RQA)] of a continuous dynamical system is the induced noise that is commonly found in observation time series. This induced noise is known to cause disrupted and deviated diagonal lines despite the known deterministic features and, hence, biases the diagonal line based RQA measures and can lead to misleading conclusions. Although discontinuous lines can be further connected by increasing the recurrence threshold, such an approach triggers thick lines in the plot. However, thick lines also influence the RQA measures by artificially increasing the number of diagonals and the length of vertical lines [e.g., Determinism (D E T) and Laminarity (L A M) become artificially higher]. To take on this challenge, an extended RQA approach for accounting disrupted and deviated diagonal lines is proposed. The approach uses the concept of a sliding diagonal window with minimal window size that tolerates the mentioned deviated lines and also considers a specified minimal lag between points as connected. This is meant to derive a similar determinism indicator for noisy signal where conventional RQA fails to capture. Additionally, an extended local minima approach to construct RP is also proposed to further reduce artificial block structures and vertical lines that potentially increase the associated RQA like LAM. The methodology and applicability of the extended local minima approach and D E T equivalent measure are presented and discussed, respectively. [ABSTRACT FROM AUTHOR]

The appropriate selection of recurrence thresholds is a key problem in applications of recurrence quantification analysis and related methods across disciplines. Here, we discuss the distribution of pairwise distances between state vectors in the studied system’s state space reconstructed by means of time-delay embedding as the key characteristic that should guide the corresponding choice for obtaining an adequate resolution of a recurrence plot. Specifically, we present an empirical description of the distance distribution, focusing on characteristic changes of its shape with increasing embedding dimension. Our results suggest that selecting the recurrence threshold according to a fixed percentile of this distribution reduces the dependence of recurrence characteristics on the embedding dimension in comparison with other commonly used threshold selection methods. Numerical investigations on some paradigmatic model systems with time-dependent parameters support these empirical findings. [ABSTRACT FROM AUTHOR]

Identifying abrupt transitions is a key question in various disciplines. Existing transition detection methods, however, do not rigorously account for time series uncertainties, often neglecting them altogether or assuming them to be independent and qualitatively similar. Here, we introduce a novel approach suited to handle uncertainties by representing the time series as a time-ordered sequence of probability density functions. We show how to detect abrupt transitions in such a sequence using the community structure of networks representing probabilities of recurrence. Using our approach, we detect transitions in global stock indices related to well-known periods of politico-economic volatility. We further uncover transitions in the El Niño-Southern Oscillation which coincide with periods of phase locking with the Pacific Decadal Oscillation. Finally, we provide for the first time an ‘uncertainty-aware’ framework which validates the hypothesis that ice-rafting events in the North Atlantic during the Holocene were synchronous with a weakened Asian summer monsoon. [ABSTRACT FROM AUTHOR]

The temporal dynamics of climate processes are spread across different timescales and, as such, the study of these processes at only one selected timescale might not reveal the complete mechanisms and interactions within and between the (sub-)processes. To capture the non-linear interactions between climatic events, the method of event synchronization has found increasing attention recently. The main drawback with the present estimation of event synchronization is its restriction to analysing the time series at one reference timescale only. The study of event synchronization at multiple scales would be of great interest to comprehend the dynamics of the investigated climate processes. In this paper, the wavelet-based multi-scale event synchronization (MSES) method is proposed by combining the wavelet transform and event synchronization. Wavelets are used extensively to comprehend multi-scale processes and the dynamics of processes across various timescales. The proposed method allows the study of spatio-temporal patterns across different timescales. The method is tested on synthetic and real-world time series in order to check its replicability and applicability. The results indicate that MSES is able to capture relationships that exist between processes at different timescales. [ABSTRACT FROM AUTHOR]

Recurrence is an ubiquitous feature which provides deep insights into the dynamics of real dynamical systems. A suitable tool for investigating recurrences is recurrence quantification analysis (RQA). It allows, e.g., the detection of regime transitions with respect to varying control parameters. We investigate the complexity of different coexisting nonlinear dynamical regimes of the plasma floating potential fluctuations at different magnetic fields and discharge voltages by using recurrence quantification variables, in particular, DET, Lmax, and Entropy. The recurrence analysis reveals that the predictability of the system strongly depends on discharge voltage. Furthermore, the persistent behaviour of the plasma time series is characterized by the Detrended fluctuation analysis technique to explore the complexity in terms of long range correlation. The enhancement of the discharge voltage at constant magnetic field increases the nonlinear correlations; hence, the complexity of the system decreases, which corroborates the RQA analysis. [ABSTRACT FROM AUTHOR]

Synchronous occurrences of heavy rainfall events and the study of their relation in time and space are of large socio-economical relevance, for instance for the agricultural and insurance sectors, but also for the general well-being of the population. In this study, the spatial synchronization structure is analyzed as a regional climate network constructed from precipitation event series. The similarity between event series is determined by the number of synchronous occurrences. We propose a novel standardization of this number that results in synchronization scores which are not biased by the number of events in the respective time series. Additionally, we introduce a new version of the network measure directionality that measures the spatial directionality of weighted links by also taking account of the effects of the spatial embedding of the network. This measure provides an estimate of heavy precipitation isochrones by pointing out directions along which rainfall events synchronize. We propose a climatological interpretation of this measure in terms of propagating fronts or event traces and confirm it for Germany by comparing our results to known atmospheric circulation patterns. [ABSTRACT FROM AUTHOR]

Recurrence plot based measures of complexity are capable tools for characterizing complex dynamics. In this letter we show the potential of selected recurrence plot measures for the investigation of even high-dimensional dynamics. We apply this method on spatially extended chaos, such as derived from the Lorenz96 model and show that the recurrence plot based measures can qualitatively characterize typical dynamical properties such as chaotic or periodic dynamics. Moreover, we demonstrate its power by analysing satellite image time series of vegetation cover with contrasting dynamics as a spatially extended and potentially high-dimensional example from the real world. [ABSTRACT FROM AUTHOR]

A method to identify distinct dynamical regimes and transitions between those regimes in a short univariate time series was recently introduced [N. Malik etal., Europhys. Lett. 97,40009 (2012)], employing the computation of fluctuations in a measure of nonlinear similarity based on local recurrence properties. In this work, we describe the details of the analytical relationships between this newly introduced measure and the well-known concepts of attractor dimensions and Lyapunov exponents. We show that the new measure has linear dependence on the effective dimension of the attractor and it measures the variations in the sum of the Lyapunov spectrum. To illustrate the practical usefulness of the method, we identify various types of dynamical transitions in different nonlinear models. We present testbed examples for the new method's robustness against noise and missing values in the time series. We also use this method to analyze time series of social dynamics, specifically an analysis of the US crime record time series from 1975 to 1993. Using this method, we find that dynamical complexity in robberies was influenced by the unemployment rate until the late 1980s. We have also observed a dynamical transition in homicide and robbery rates in the late 1980s and early 1990s, leading to increase in the dynamical complexity of these rates. [ABSTRACT FROM AUTHOR]

Predicting a transition point in behavioral data should take into account the complexity of the signal being influenced by contextual factors. In this paper, we propose to analyze changes in the embedding dimension as contextual information indicating a proceeding transitive point, called OPtimal Embedding tRANsition Detection (OPERAND). Three texts were processed and translated to time-series of emotional polarity. It was found that changes in the embedding dimension proceeded transition points in the data. These preliminary results encourage further research into changes in the embedding dimension as generic markers of an approaching transition point. [ABSTRACT FROM AUTHOR]

Complex network approaches have recently been applied to continuous spatial dynamical systems, like climate, successfully uncovering the system's interaction structure. However the relationship between the underlying atmospheric or oceanic flow's dynamics and the estimated network measures have remained largely unclear. We bridge this crucial gap in a bottom-up approach and define a continuous analytical analogue of Pearson correlation networks for advection-diffusion dynamics on a background flow. Analysing complex networks of prototypical flows and from time series data of the equatorial Pacific, we find that our analytical model reproduces the most salient features of these networks and thus provides a general foundation of climate networks. The relationships we obtain between velocity field and network measures show that line-like structures of high betweenness mark transition zones in the flow rather than, as previously thought, the propagation of dynamical information. [ABSTRACT FROM AUTHOR]

Potential paleoclimatic driving mechanisms acting on human evolution present an open problem of cross-disciplinary scientific interest. The analysis of paleoclimate archives encoding the environmental variability in East Africa during the past 5 Ma has triggered an ongoing debate about possible candidate processes and evolutionary mechanisms. In this work, we apply a nonlinear statistical technique, recurrence network analysis, to three distinct marine records of terrigenous dust flux. Our method enables us to identify three epochs with transitions between qualitatively different types of environmental variability in North and East Africa during the (i) Middle Pliocene (3.35-3.15 Ma B.P.), (ii) Early Pleistocene (2.25-1.6 Ma B.P.), and (iii) Middle Pleistocene (1.1-0.7 Ma B.P.). A deeper examination of these transition periods reveals potential climatic drivers, including (i) large-scale changes in ocean currents due to a spatial shift of the Indonesian throughflow in combination with an intensification of Northern Hemisphere glaciation, (ii) a global reorganization of the atmospheric Walker circulation induced in the tropical Pacific and Indian Ocean, and (iii) shifts in the dominating temporal variability pattern of glacial activity during the Middle Pleistocene, respectively. A reexamination of the available fossil record demonstrates statistically significant coincidences between the detected transition periods and major steps in hominin evolution. This result suggests that the observed shifts between more regular and more erratic environmental variability may have acted as a trigger for rapid change in the development of humankind in Africa. [ABSTRACT FROM AUTHOR]

Recurrence plots and recurrence quantification analysis have become popular in the last two decades. Recurrence based methods have on the one hand a deep foundation in the theory of dynamical systems and are on the other hand powerful tools for the investigation of a variety of problems. The increasing interest encompasses the growing risk of misuse and uncritical application of these methods. Therefore, we point out potential problems and pitfalls related to different aspects of the application of recurrence plots and recurrence quantification analysis. [ABSTRACT FROM AUTHOR]

Complex networks are an important paradigm of modern complex systems sciences which allows quantitatively assessing the structural properties of systems composed of different interacting entities. During the last years, intensive efforts have been spent on applying network-based concepts also for the analysis of dynamically relevant higher-order statistical properties of time series. Notably, many corresponding approaches are closely related to the concept of recurrence in phase space. In this paper, we review recent methodological advances in time series analysis based on complex networks, with a special emphasis on methods founded on recurrence plots. The potentials and limitations of the individual methods are discussed and illustrated for paradigmatic examples of dynamical systems as well as for real-world time series. Complex network measures are shown to provide information about structural features of dynamical systems that are complementary to those characterized by other methods of time series analysis and, hence, substantially enrich the knowledge gathered from other existing (linear as well as nonlinear) approaches. [ABSTRACT FROM AUTHOR]

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. [Copyright &y& Elsevier]

Identifying and characterising dynamical regime shifts, critical transitions or potential tipping points in palaeoclimate time series is relevant for improving the understanding of often highly nonlinear Earth system dynamics. Beyond linear changes in time series properties such as mean, variance, or trend, these nonlinear regime shifts can manifest as changes in signal predictability, regularity, complexity, or higher-order stochastic properties such as multi-stability. In recent years, several classes of methods have been put forward to study these critical transitions in time series data that are based on concepts from nonlinear dynamics, complex systems science, information theory, and stochastic analysis. These include approaches such as phase space-based recurrence plots and recurrence networks, visibility graphs, order pattern-based entropies, and stochastic modelling. Here, we review and compare in detail several prominent methods from these fields by applying them to the same set of marine palaeoclimate proxy records of African climate variations during the past 5 million years. Applying these methods, we observe notable nonlinear transitions in palaeoclimate dynamics in these marine proxy records and discuss them in the context of important climate events and regimes such as phases of intensified Walker circulation, marine isotope stage M2, the onset of northern hemisphere glaciation and the mid-Pleistocene transition. We find that the studied approaches complement each other by allowing us to point out distinct aspects of dynamical regime shifts in palaeoclimate time series. We also detect significant correlations of these nonlinear regime shift indicators with variations of Earth's orbit, suggesting the latter as potential triggers of nonlinear transitions in palaeoclimate. Overall, the presented study underlines the potentials of nonlinear time series analysis approaches to provide complementary information on dynamical regime shifts in palaeoclimate and their driving processes that cannot be revealed by linear statistics or eyeball inspection of the data alone. [ABSTRACT FROM AUTHOR]

Dynamical processes in Earth sciences are often considered to be of complex nature. The term complexity is often used for processes that are either unpredictable (e.g. nonlinear dynamics), consist of many different components, or exhibit regime transitions (e.g. tipping points). To measure complexity, the Shannon entropy is often used. Here we present various entropy measures that have been defined on the base of the recurrence plot. Because of the different features that are used, these entropy measures represent different aspects of the analysed system and, thus, behave differently. In the past, this fact has lead to difficulties in interpreting and understanding those measures. We summarize the definitions, the motivation and interpretation of these entropy measures, compare their differences and discuss some of the pitfalls when using them. Finally, we illustrate their potential in an application on palaeoclimate time series. Using entropy measures, changes and transitions in the climate dynamics in the past can be identified and interpreted. [ABSTRACT FROM AUTHOR]

We have introduced a multiplex recurrence network approach by combining recurrence networks with the multiplex network approach in order to investigate multivariate time series. The potential use of this approach is demonstrated on coupled map lattices and a typical example from palaeobotany research. In both examples, topological changes in the multiplex recurrence networks allow for the detection of regime changes in their dynamics. The method goes beyond classical interpretation of pollen records by considering the vegetation as a whole and using the intrinsic similarity in the dynamics of the different regional vegetation elements. We find that the different vegetation types behave more similarly when one environmental factor acts as the dominant driving force. [ABSTRACT FROM AUTHOR]

Describing a time series parsimoniously is the first step to study the underlying dynamics. For a time-discrete system, a generating partition provides a compact description such that a time series and a symbolic sequence are one-to-one. But, for a time-continuous system, such a compact description does not have a solid basis. Here, we propose to describe a time-continuous time series using a local cross section and the times when the orbit crosses the local cross section. We show that if such a series of crossing times and some past observations are given, we can predict the system's dynamics with fine accuracy. This reconstructability neither depends strongly on the size nor the placement of the local cross section if we have a sufficiently long database. We demonstrate the proposed method using the Lorenz model as well as the actual measurement of wind speed. [ABSTRACT FROM AUTHOR]

Abstract In the last decade, there has been a growing body of literature addressing the utilization of complex network methods for the characterization of dynamical systems based on time series. While both nonlinear time series analysis and complex network theory are widely considered to be established fields of complex systems sciences with strong links to nonlinear dynamics and statistical physics, the thorough combination of both approaches has become an active field of nonlinear time series analysis, which has allowed addressing fundamental questions regarding the structural organization of nonlinear dynamics as well as the successful treatment of a variety of applications from a broad range of disciplines. In this report, we provide an in-depth review of existing approaches of time series networks, covering their methodological foundations, interpretation and practical considerations with an emphasis on recent developments. After a brief outline of the state-of-the-art of nonlinear time series analysis and the theory of complex networks, we focus on three main network approaches, namely, phase space based recurrence networks, visibility graphs and Markov chain based transition networks, all of which have made their way from abstract concepts to widely used methodologies. These three concepts, as well as several variants thereof will be discussed in great detail regarding their specific properties, potentials and limitations. More importantly, we emphasize which fundamental new insights complex network approaches bring into the field of nonlinear time series analysis. In addition, we summarize examples from the wide range of recent applications of these methods, covering rather diverse fields like climatology, fluid dynamics, neurophysiology, engineering and economics, and demonstrating the great potentials of time series networks for tackling real-world contemporary scientific problems. The overall aim of this report is to provide the readers with the knowledge how the complex network approaches can be applied to their own field of real-world time series analysis. [ABSTRACT FROM AUTHOR]

The asymmetry of coupling between complex systems can be studied by conditional probabilities of recurrence, which can be estimated by joint recurrence plots. This approach is applied for the first time on experimental data: time series of the human cardiorespiratory system in order to investigate the couplings between heart rate, mean arterial blood pressure and respiration. We find that the respiratory system couples towards the heart rate, and the heart rate towards the mean arterial blood pressure. However, our analysis could not detect a clear coupling direction between the mean arterial blood pressure and respiration. [ABSTRACT FROM AUTHOR]

We analyse cardiovascular time series with the aim of performing early prediction of preeclampsia (PE), a pregnancy-specific disorder causing maternal and foetal morbidity and mortality. The analysis is made using a novel approach, namely the ε-recurrence networks applied to a phase space constructed by means of the time series of the variabilities of the heart rate and the blood pressure (systolic and diastolic). All the possible coupling structures among these variables are considered for the analysis. Network measures such as average path length, mean coreness, global clustering coefficient and scale-local transitivity dimension are computed and constitute the parameters for the subsequent quadratic discriminant analysis. This allows us to predict PE with a sensitivity of 91.7 per cent and a specificity of 68.1 per cent, thus validating the use of this method for classifying healthy and preeclamptic patients. [ABSTRACT FROM AUTHOR]

PyRQA is a software package that efficiently conducts recurrence quantification analysis (RQA) on time series consisting of more than one million data points. RQA is a method from non-linear time series analysis that quantifies the recurrent behaviour of systems. Existing implementations to RQA are not capable of analysing such very long time series at all or require large amounts of time to calculate the quantitative measures. PyRQA overcomes their limitations by conducting the RQA computations in a highly parallel manner. Building on the OpenCL framework, PyRQA leverages the computing capabilities of a variety of parallel hardware architectures, such as GPUs. The underlying computing approach partitions the RQA computations and enables to employ multiple compute devices at the same time. The goal of this publication is to demonstrate the features and the runtime efficiency of PyRQA . For this purpose we employ a real-world example, comparing the dynamics of two climatological time series, and a synthetic example, reducing the runtime regarding the analysis of a series consisting of over one million data points from almost eight hours using state-of-the-art RQA software to roughly 69 s using PyRQA . [ABSTRACT FROM AUTHOR]

While it is an important problem to identify the existence of causal associations between two components of a multivariate time series, a topic addressed in Runge, Heitzig, Petoukhov, and Kurths [Phys. Rev. Lett. 108, 258701 (2012)], it is even more important to assess the strength of their association in a meaningful way. In the present article we focus on the problem of defining a meaningful coupling strength using information-theoretic measures and demonstrate the shortcomings of the well-known mutual information and transfer entropy. Instead, we propose a certain time-delayed conditional mutual information, the momentary information transfer (MIT), as a lag-specific measure of association that is general, causal, reflects a well interpretable notion of coupling strength, and is practically computable. Rooted in information theory, MIT is general in that it does not assume a certain model class underlying the process that generates the time series. As discussed in a previous paper [Runge, Heitzig, Petoukhov, and Kurths, Phys. Rev. Lett. 108, 258701 (2012)], the general framework of graphical models makes MIT causal in that it gives a nonzero value only to lagged components that are not independent conditional on the remaining process. Further, graphical models admit a low-dimensional formulation of conditions, which is important for a reliable estimation of conditional mutual information and, thus, makes MIT practically computable. MIT is based on the fundamental concept of source entropy, which we utilize to yield a notion of coupling strength that is, compared to mutual information and transfer entropy, well interpretable in that, for many cases, it solely depends on the interaction of the two components at a certain lag. In particular, MIT is, thus, in many cases able to exclude the misleading influence of autodependency within a process in an information-theoretic way. We formalize and prove this idea analytically and numerically for a general class of nonlinear stochastic processes and illustrate the potential of MIT on climatological data. [ABSTRACT FROM AUTHOR]

We suggest an approach for detecting and identifying ignition failure on a internal combustion engine used in aviation through the analysis of vibration time series. The research is carried out at the experimental stage, where time series of vibrations are collected from sensors installed in various parts of the facility at various rotational speeds and various operating conditions (no failure/failure of a selected piston). The time series were decomposed into periodic components centered around dominant frequencies. Data with greater dimensionality was statistically described using linear and non-linear indicators in short time windows, and labeled accordingly. Instead of examining the statistical significance of the characteristics of individual groups, machine learning classification methods were used, which allowed to distinguish the operating state of the engine (damaged/undamaged), and also to identify a specific unfired cylinder. The use of non-linear indicators allowed us to obtain 100% classification accuracy with a small number of samples. • The article discusses the detection and identification of cylinder misfire in small aircraft engines in different operating conditions. • The detection is based on the linear and non-linear properties of vibrational signals. • The complexity and path of diagnostic signals require finding changes in frequency components. • The location of sensors does not affect the monitoring of the engine's operating status. • Machine learning classified each of the damaged cylinder, and the use of non-linear statistics increased the accuracy by 23% with 100% classification. [ABSTRACT FROM AUTHOR]

Irregular sampling of data sets is one of the challenges often encountered in time-series analysis, since traditional methods cannot be applied and the frequently used interpolation approach can corrupt the data and bias the subsequence analysis. Here we present the TrAnsformation-Cost Time-Series (TACTS) method, which allows us to analyze irregularly sampled data sets without degenerating the quality of the data set. Instead of using interpolation we consider time-series segments and determine how close they are to each other by determining the cost needed to transform one segment into the following one. Using a limited set of operations--with associated costs--to transform the time series segments, we determine a new time series, that is our transformation-cost time series. This cost time series is regularly sampled and can be analyzed using standard methods. While our main interest is the analysis of paleoclimate data, we develop our method using numerical examples like the logistic map and the Rössler oscillator. The numerical data allows us to test the stability of our method against noise and for different irregular samplings. In addition we provide guidance on how to choose the associated costs based on the time series at hand. The usefulness of the TACTS method is demonstrated using speleothem data from the Secret Cave in Borneo that is a good proxy for paleoclimatic variability in the monsoon activity around the maritime continent. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a geometric method for identifying the coupling direction between two dynamical systems based on a bivariate extension of recurrence network analysis. Global characteristics of the resulting inter-system recurrence networks provide a correct discrimination for weakly coupled Rössler oscillators not yet displaying generalised synchronisation. Investigating two real-world palaeoclimate time series representing the variability of the Asian monsoon over the last 10,000 years, we observe indications for a considerable influence of the Indian summer monsoon on climate in Eastern China rather than vice versa. The proposed approach can be directly extended to studying coupled subsystems. [Copyright &y& Elsevier]

Deep-time paleoclimatic records document large-scale shifts and perturbations in Earth's climate; during the Cenozoic in particular transitions have been recorded on time scales of 10 thousand to 1 million years. Bifurcations in the leading dynamical modes could be a key element driving these events. Such bifurcation-induced critical transitions are typically preceded by characteristic early-warning signals, for example in terms of rising standard deviation and lag-one autocorrelation. These early-warning signals are generated by a widening of the underlying basin of attraction when approaching the bifurcation, a phenomenon dubbed critical slowing down. The associated dynamical transitions should therefore be preceded by characteristic signals that can be detected by statistical methods. Here, we reveal the presence of significant early-warning signals prior to several climate events within a paleoclimate record spanning the last 66 million years - the Cenozoic Era. We computed standard deviation and lag-one autocorrelation of the CENOzoic Global Reference benthic foraminifer carbon and oxygen Isotope Dataset (CENOGRID), comprising two time series of deep sea carbonate isotope variations of 18O and 13C. We find significant early-warning signals for five out of nine previously identified Cenozoic paleoclimatic events in at least one of the two records, which can be considered as viable candidates for bifurcation-induced transitions to be analysed in follow-up studies. Our results suggest that some of the major climate events of the last 66 Ma were triggered by bifurcations in leading modes of variability, indicating bifurcations could be a key component of Earth's climate system deep-time evolution. • Analysis of abrupt climate changes in paleoclimate time series for statistical early-warning signals of critical transitions. • Evidence for critical slowing down in 5 out of 9 examined climate events during Cenozoic. • Synchronized occurrence of early-warning signals in 13C and 18O isotope records. [ABSTRACT FROM AUTHOR]

The Chew Bahir Drilling Project (CBDP) aims to test hypothesized linkages between climate and human evolution, dispersal and technological innovation by the acquisition and analysis of long (~280 m) sediment cores that have recorded environmental change in the Chew Bahir basin, southern Ethiopia. In this time-series analysis project, we consider the Chew Bahir palaeolake to be a dynamical system consisting of interactions between its different components, such as the waterbody, the sediment beneath lake, and the organisms living within and around the lake, and humans within the lake catchment. Recurrence is a common feature of such dynamical systems, with recurring patterns in the state of the system reflecting typical influences. Identifying and defining these influences contributes significantly to our understanding of the dynamics of the system.We use methods of linear and nonlinear time series analysis, such as change point detection, semblance analysis and recurrence plots, to identify and classify recurring types of variability and transitions on the time scales of human life spans. For example, we investigate the rapidness of transitions, possible precursor events, and tipping points in our palaeoenvironmental data and discuss their possible impact on the living conditions of humans in the region. First results of the analysis show that we indeed find, as an example, recurring threshold-type transitions, when the Chew Bahir system switched from one stable mode to another, such as from stable wet to dry conditions. Such a rapid change of climate in response to a relatively modest change in forcing appears to be typical of tipping points in complex systems such as the Chew Bahir. If this is the case then the 14 dry events idenfified at the end of the African Humid Period (15–5 kyr BP) could represent precursors of an imminent tipping point that, if properly interpreted, would allow predictions to be made of future climate change in the Chew Bahir basin. [ABSTRACT FROM AUTHOR]

Irregularly sampled time series usually require data preprocessing before a desired time-series analysis can be applied. We propose an approach for distance measuring of pairs of data points which is directly applicable to irregularly sampled time series. In order to apply recurrence plot analysis to irregularly sampled time series, we use this approach and detect regime transitions in prototypical models and for an application from palaeoclimatatology. This approach might be useful for any method that is based on distance measuring, e.g., correlation sum or Lyapunov exponent estimation. [ABSTRACT FROM AUTHOR]