Multiscale atmospheric dynamics: cross-frequency phase-amplitude coupling in the air temperature

Causes of extreme events revealed by Rényi information transfer

Information-theoretic generalization of Granger causality principle, based on evaluation of conditional mutual information, also known as transfer entropy (CMI/TE), is redefined in the framework of Rényi entropy (RCMI/RTE). Using numerically generated data with a defined causal structure and examples of real data from the climate system, it is demonstrated that RCMI/RTE is able to identify the cause variable responsible for the occurrence of extreme values in an effect variable. In the presented example, the Siberian High was identified as the cause responsible for the increased probability of cold extremes in the winter and spring surface air temperature in Europe, while the North Atlantic Oscillation and blocking events can induce shifts of the whole temperature probability distribution.

Granger causality for compressively sensed sparse signals

Compressed sensing is a scheme that allows for sparse signals to be acquired, transmitted, and stored using far fewer measurements than done by conventional means employing the Nyquist sampling theorem. Since many naturally occurring signals are sparse (in some domain), compressed sensing has rapidly seen popularity in a number of applied physics and engineering applications, particularly in designing signal and image acquisition strategies, e.g., magnetic resonance imaging, quantum state tomography, scanning tunneling microscopy, and analog to digital conversion technologies. Contemporaneously, causal inference has become an important tool for the analysis and understanding of processes and their interactions in many disciplines of science, especially those dealing with complex systems. Direct causal analysis for compressively sensed data is required to avoid the task of reconstructing the compressed data. Also, for some sparse signals, such as for sparse temporal data, it may be difficult to discover causal relations directly using available data-driven or model-free causality estimation techniques. In this work, we provide a mathematical proof that structured compressed sensing matrices, specifically circulant and Toeplitz, preserve causal relationships in the compressed signal domain, as measured by Granger causality (GC). We then verify this theorem on a number of bivariate and multivariate coupled sparse signal simulations which are compressed using these matrices. We also demonstrate a real world application of network causal connectivity estimation from sparse neural spike train recordings from rat prefrontal cortex. In addition to demonstrating the effectiveness of structured matrices for GC estimation from sparse signals, we also show a computational time advantage of the proposed strategy for causal inference from compressed signals of both sparse and regular autoregressive processes as compared to standard GC estimation from original signals.

Compression complexity with ordinal patterns for robust causal inference in irregularly sampled time series

Distinguishing cause from effect is a scientific challenge resisting solutions from mathematics, statistics, information theory and computer science. Compression-Complexity Causality (CCC) is a recently proposed interventional measure of causality, inspired by Wiener–Granger’s idea. It estimates causality based on change in dynamical compression-complexity (or compressibility) of the effect variable, given the cause variable. CCC works with minimal assumptions on given data and is robust to irregular-sampling, missing-data and finite-length effects. However, it only works for one-dimensional time series. We propose an ordinal pattern symbolization scheme to encode multidimensional patterns into one-dimensional symbolic sequences, and thus introduce the Permutation CCC (PCCC). We demonstrate that PCCC retains all advantages of the original CCC and can be applied to data from multidimensional systems with potentially unobserved variables which can be reconstructed using the embedding theorem. PCCC is tested on numerical simulations and applied to paleoclimate data characterized by irregular and uncertain sampling and limited numbers of samples.

Causal Inference in Time Series in Terms of Rényi Transfer Entropy

Uncovering causal interdependencies from observational data is one of the great challenges of a nonlinear time series analysis. In this paper, we discuss this topic with the help of an information-theoretic concept known as Rényi’s information measure. In particular, we tackle the directional information flow between bivariate time series in terms of Rényi’s transfer entropy. We show that by choosing Rényi’s parameter α, we can appropriately control information that is transferred only between selected parts of the underlying distributions. This, in turn, is a particularly potent tool for quantifying causal interdependencies in time series, where the knowledge of “black swan” events, such as spikes or sudden jumps, are of key importance. In this connection, we first prove that for Gaussian variables, Granger causality and Rényi transfer entropy are entirely equivalent. Moreover, we also partially extend these results to heavy-tailed α-Gaussian variables. These results allow establishing a connection between autoregressive and Rényi entropy-based information-theoretic approaches to data-driven causal inference. To aid our intuition, we employed the Leonenko et al. entropy estimator and analyzed Rényi’s information flow between bivariate time series generated from two unidirectionally coupled Rössler systems. Notably, we find that Rényi’s transfer entropy not only allows us to detect a threshold of synchronization but it also provides non-trivial insight into the structure of a transient regime that exists between the region of chaotic correlations and synchronization threshold. In addition, from Rényi’s transfer entropy, we could reliably infer the direction of coupling and, hence, causality, only for coupling strengths smaller than the onset value of the transient regime, i.e., when two Rössler systems are coupled but have not yet entered synchronization.

Contributions to surface air temperature trends estimated from climate time series: Medium-term causalities

Contributions of various natural and anthropogenic factors to trends of surface air temperatures at different latitudes of the Northern and Southern hemispheres on various temporal horizons are estimated from climate data since the 19th century in empirical autoregressive models. Along with anthropogenic forcing, we assess the impact of several natural climate modes including Atlantic Multidecadal Oscillation, El-Nino/Southern Oscillation, Interdecadal Pacific Oscillation, Pacific Decadal Oscillation, and Antarctic Oscillation. On relatively short intervals of the length of two or three decades, contributions of climate variability modes are considerable and comparable to the contributions of greenhouse gases and even exceed the latter. On longer intervals of about half a century and greater, the contributions of greenhouse gases dominate at all latitudinal belts including polar, middle, and tropical ones.

Generative formalism of causality quantifiers for processes

The concept of dynamical causal effect (DCE) is generalized and equipped with a formalism which allows one to formulate in a unified manner and interrelate a variety of causality quantifiers used in time series analysis. An elementary DCE from a subsystem Y to a subsystem X is defined within the stochastic dynamical systems framework as a response of a future X state to an appropriate variation of an initial (X,Y)-state distribution or a certain parameter of Y or of the coupling element Y→X; this response is quantified in a probabilistic sense via a certain distinction functional; elementary DCEs are assembled over a set of initial variations via an assemblage functional. To include all those aspects, a "triple brackets formula" for the general DCE is suggested and serves as a first principle to produce specific causality quantifiers as realizations of the general DCE. As an application, transfer entropy and Liang-Kleeman information flow are related surprisingly as opposite limit cases in a family of DCEs; it is shown that their "nats per time unit" may differ drastically. The suggested DCE viewpoint links any formal causality quantifier to "intervention-effect" experiments, i.e., future responses to initial variations, and so provides its dynamical interpretation, opening a way to its further physical interpretations in studies of physical systems.

Phase-dynamic causalities within dynamical effects framework

This work investigates numerics of several widely known phase-dynamic quantifiers of directional (causal) couplings between oscillatory systems: transfer entropy (TE), differential quantifier, and squared-coefficients quantifier based on an evolution map. The study is performed on the system of two stochastic Kuramoto oscillators within the framework of dynamical causal effects. The quantifiers are related to each other and to an asymptotic effect of the coupling on phase diffusion. Several novel findings are listed as follows: (i) for a non-synchronous regime and high enough noise levels, the TE rate multiplied by a certain characteristic time (called here reduced TE) equals twice an asymptotic effect of a directional coupling on phase diffusion; (ii) "information flow" expressed by the TE rate unboundedly rises with the coupling coefficient even in the domain of effective synchronization; (iii) in any effective synchronization regime, the reduced TE is equal to 1/8 n.u. in each direction for equal coupling coefficients and equal noise intensities, and it is in general a simple function of the ratio of noise intensities and the ratio of coupling coefficients.

Trends of temperature variability: Which variability and what health implications?

A large majority of climate change studies carried out to date are on changes in mean climate, which have comparatively downplayed variability. In terms of trend analysis or forecast, the scientific output and common knowledge for global warming are much more robust than for changes in temperature variability. Quantification of temperature variability adds another dimension of temporal scale, requiring immense labor and presenting great uncertainty. Regardless, this endeavor is necessary since changes in ambient temperature variabilities could also contribute to current and future human health burden besides changes in mean quantities. Here, we review the current literature on trends of surface air temperature variability defined at a range of timescales, aiming to tease out the welter of evidence and thus improving the scientific recognition of changes in air temperature variability in the context of climate change. The findings of reviewed studies from numerous regions differ substantially over various temporal scales. In general, the ambient temperature variability on short time scales (e.g., diurnal or inter-day) shows a downward trend, while it is increasing on longer time scales (e.g., inter-annual). We then move beyond the review and deliver an extended discussion of potential implications for future research related to ambient temperature variability. We highlight the need to consider the methodological choices, especially timescales of interest, in the trend analysis as well as health impact studies. Continued research focusing on temperature variability at multiple timescales, with concerted efforts from scientists of all relevant stripes, is meaningful in synthesizing knowledge and reducing uncertainties surrounding air temperature variability.

Cross-Scale Causality and Information Transfer in Simulated Epileptic Seizures

An information-theoretic approach for detecting causality and information transfer was applied to phases and amplitudes of oscillatory components related to different time scales and obtained using the wavelet transform from a time series generated by the Epileptor model. Three main time scales and their causal interactions were identified in the simulated epileptic seizures, in agreement with the interactions of the model variables. An approach consisting of wavelet transform, conditional mutual information estimation, and surrogate data testing applied to a single time series generated by the model was demonstrated to be successful in the identification of all directional (causal) interactions between the three different time scales described in the model. Thus, the methodology was prepared for the identification of causal cross-frequency phase–phase and phase–amplitude interactions in experimental and clinical neural data.

Fast and effective pseudo transfer entropy for bivariate data-driven causal inference

Identifying, from time series analysis, reliable indicators of causal relationships is essential for many disciplines. Main challenges are distinguishing correlation from causality and discriminating between direct and indirect interactions. Over the years many methods for data-driven causal inference have been proposed; however, their success largely depends on the characteristics of the system under investigation. Often, their data requirements, computational cost or number of parameters limit their applicability. Here we propose a computationally efficient measure for causality testing, which we refer to as pseudo transfer entropy (pTE), that we derive from the standard definition of transfer entropy (TE) by using a Gaussian approximation. We demonstrate the power of the pTE measure on simulated and on real-world data. In all cases we find that pTE returns results that are very similar to those returned by Granger causality (GC). Importantly, for short time series, pTE combined with time-shifted (T-S) surrogates for significance testing strongly reduces the computational cost with respect to the widely used iterative amplitude adjusted Fourier transform (IAAFT) surrogate testing. For example, for time series of 100 data points, pTE and T-S reduce the computational time by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$82\%$$\end{document}82% with respect to GC and IAAFT. We also show that pTE is robust against observational noise. Therefore, we argue that the causal inference approach proposed here will be extremely valuable when causality networks need to be inferred from the analysis of a large number of short time series.

Causality and Information Transfer Between the Solar Wind and the Magnetosphere-Ionosphere System

An information-theoretic approach for detecting causality and information transfer is used to identify interactions of solar activity and interplanetary medium conditions with the Earth's magnetosphere-ionosphere systems. A causal information transfer from the solar wind parameters to geomagnetic indices is detected. The vertical component of the interplanetary magnetic field (Bz) influences the auroral electrojet (AE) index with an information transfer delay of 10 min and the geomagnetic disturbances at mid-latitudes measured by the symmetric field in the H component (SYM-H) index with a delay of about 30 min. Using a properly conditioned causality measure, no causal link between AE and SYM-H, or between magnetospheric substorms and magnetic storms can be detected. The observed causal relations can be described as linear time-delayed information transfer.

Transfer entropies within dynamical effects framework

Transfer entropy (TE) is widely used in time-series analysis to detect causal couplings between temporally evolving objects. As a coupling strength quantifier, the TE alone often seems insufficient, raising the question of its further interpretations. Here the TE is related to dynamical causal effects (DCEs) which quantify long-term responses of a coupling recipient to variations in a coupling source or in a coupling itself: Detailed relationships are established for a paradigmatic stochastic dynamical system of bidirectionally coupled linear overdamped oscillators, their practical applications and possible extensions are discussed. It is shown that two widely used versions of the TE (original and infinite-history) can become qualitatively distinct, diverging to different long-term DCEs.

Multiscale measures of phase-space trajectories

Characterizing the multiscale nature of fluctuations from nonlinear and nonstationary time series is one of the most intensively studied contemporary problems in nonlinear sciences. In this work, we address this problem by combining two established concepts-empirical mode decomposition (EMD) and generalized fractal dimensions-into a unified analysis framework. Specifically, we demonstrate that the intrinsic mode functions derived by EMD can be used as a source of local (in terms of scales) information about the properties of the phase-space trajectory of the system under study, allowing us to derive multiscale measures when looking at the behavior of the generalized fractal dimensions at different scales. This formalism is applied to three well-known low-dimensional deterministic dynamical systems (the Hénon map, the Lorenz '63 system, and the standard map), three realizations of fractional Brownian motion with different Hurst exponents, and two somewhat higher-dimensional deterministic dynamical systems (the Lorenz '96 model and the on-off intermittency model). These examples allow us to assess the performance of our formalism with respect to practically relevant aspects like additive noise, different initial conditions, the length of the time series under study, low- vs high-dimensional dynamics, and bursting effects. Finally, by taking advantage of two real-world systems whose multiscale features have been widely investigated (a marine stack record providing a proxy of the global ice volume variability of the past 5×10⁶ years and the SYM-H geomagnetic index), we also illustrate the applicability of this formalism to real-world time series.

Analysis of 20th century surface air temperature using linear dynamical modes

A Bayesian Linear Dynamical Mode (LDM) decomposition method is applied to isolate robust modes of climate variability in the observed surface air temperature (SAT) field. This decomposition finds the optimal number of internal modes characterized by their own time scales, which enter the cost function through a specific choice of prior probabilities. The forced climate response, with time dependence estimated from state-of-the-art climate-model simulations, is also incorporated in the present LDM decomposition and shown to increase its optimality from a Bayesian standpoint. On top of the forced signal, the decomposition identifies five distinct LDMs of internal climate variability. The first three modes exhibit multidecadal scales, while the remaining two modes are attributable to interannual-to-decadal variability associated with El Niño-Southern oscillation; all of these modes contribute to the secular climate signal-the so-called global stadium wave-missing in the climate-model simulations. One of the multidecadal LDMs is associated with Atlantic multidecadal oscillation. The two remaining slow modes have secular time scales and patterns exhibiting regional-to-global similarities to the forced-signal pattern. These patterns have a global scale and contribute significantly to SAT variability over the Southern and Pacific Oceans. In combination with low-frequency modulation of the fast LDMs, they explain the vast majority of the variability associated with interdecadal Pacific oscillation. The global teleconnectivity of the secular climate modes and their possible crucial role in shaping the forced climate response are the two key dynamical questions brought about by the present analysis.

Sparse identification of slow timescale dynamics

Multiscale phenomena that evolve on multiple distinct timescales are prevalent throughout the sciences. It is often the case that the governing equations of the persistent and approximately periodic fast scales are prescribed, while the emergent slow scale evolution is unknown. Yet the course-grained, slow scale dynamics is often of greatest interest in practice. In this work we present an accurate and efficient method for extracting the slow timescale dynamics from signals exhibiting multiple timescales that are amenable to averaging. The method relies on tracking the signal at evenly spaced intervals with length given by the period of the fast timescale, which is discovered by using clustering techniques in conjunction with the dynamic mode decomposition. Sparse regression techniques are then used to discover a mapping which describes iterations from one data point to the next. We show that, for sufficiently disparate timescales, this discovered mapping can be used to discover the continuous-time slow dynamics, thus providing a novel tool for extracting dynamics on multiple timescales.

Coupling in complex systems as information transfer across time scales

Complex systems such as the human brain or the Earth's climate consist of many subsystems interacting in intricate, nonlinear ways. Moreover, variability of such systems extends over broad ranges of spatial and temporal scales and dynamical phenomena on different scales also influence each other. In order to explain how to detect cross-scale causal interactions, we review information-theoretic formulation of the Granger causality in combination with computational statistics (surrogate data method) and demonstrate how this method can be used to infer driver-response relations from amplitudes and phases of coupled nonlinear dynamical systems. Considering complex systems evolving on multiple time scales, the reviewed methodology starts with a wavelet decomposition of a multi-scale signal into quasi-oscillatory modes of a limited bandwidth, described using their instantaneous phases and amplitudes. Then statistical associations, in particular, causality relations between phases or between phases and amplitudes on different time scales are tested using the conditional mutual information. As an application, we present the analysis of cross-scale interactions and information transfer in the dynamics of the El Niño Southern Oscillation. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Bifurcation structure determines different phase-amplitude coupling patterns in the activity of biologically plausible neural networks

Phase-amplitude cross frequency coupling (PAC) is a rather ubiquitous phenomenon that has been observed in a variety of physical domains; however, the mechanisms underlying the emergence of PAC and its functional significance in the context of neural processes are open issues under debate. In this work we analytically demonstrate that PAC phenomenon naturally emerges in mean-field models of biologically plausible networks, as a signature of specific bifurcation structures. The proposed analysis, based on bifurcation theory, allows the identification of the mechanisms underlying oscillatory dynamics that are essentially different in the context of PAC. Specifically, we found that two PAC classes can coexist in the complex dynamics of the analyzed networks: 1) harmonic PAC which is an epiphenomenon of the nonsinusoidal waveform shape characterized by the linear superposition of harmonically related spectral components, and 2) nonharmonic PAC associated with "true" coupled oscillatory dynamics with independent frequencies elicited by a secondary Hopf bifurcation and mechanisms involving periodic excitation/inhibition (PEI) of a network population. Importantly, these two PAC types have been experimentally observed in a variety of neural architectures confounding traditional parametric and nonparametric PAC metrics, like those based on linear filtering or the waveform shape analysis, due to the fact that these methods operate on a single one-dimensional projection of an intrinsically multidimensional system dynamics. We exploit the proposed tools to study the functional significance of the PAC phenomenon in the context of Parkinson's disease (PD). Our results show that pathological slow oscillations (e.g. β band) and nonharmonic PAC patterns emerge from dissimilar underlying mechanisms (bifurcations) and are associated to the competition of different BG-thalamocortical loops. Thus, this study provides theoretical arguments that demonstrate that nonharmonic PAC is not an epiphenomenon related to the pathological β band oscillations, thus supporting the experimental evidence about the relevance of PAC as a potential biomarker of PD.

Disentangling the multi-scale effects of sea-surface temperatures on global precipitation: A coupled networks approach

The oceans and atmosphere interact via a multiplicity of feedback mechanisms, shaping to a large extent the global climate and its variability. To deepen our knowledge of the global climate system, characterizing and investigating this interdependence is an important task of contemporary research. However, our present understanding of the underlying large-scale processes is greatly limited due to the manifold interactions between essential climatic variables at different temporal scales. To address this problem, we here propose to extend the application of complex network techniques to capture the interdependence between global fields of sea-surface temperature (SST) and precipitation (P) at multiple temporal scales. For this purpose, we combine time-scale decomposition by means of a discrete wavelet transform with the concept of coupled climate network analysis. Our results demonstrate the potential of the proposed approach to unravel the scale-specific interdependences between atmosphere and ocean and, thus, shed light on the emerging multiscale processes inherent to the climate system, which traditionally remain undiscovered when investigating the system only at the native resolution of existing climate data sets. Moreover, we show how the relevant spatial interdependence structures between SST and P evolve across time-scales. Most notably, the strongest mutual correlations between SST and P at annual scale (8-16 months) concentrate mainly over the Pacific Ocean, while the corresponding spatial patterns progressively disappear when moving toward longer time-scales.

Transient and equilibrium causal effects in coupled oscillators

Two quite different types of causal effects are given by (i) changes in near future states of a driven system under changes in a current state of a driving system and (ii) changes in statistical characteristics of a driven system dynamics under changes in coupling parameters, e.g., under switching the coupling off. The former can be called transient causal effects and can be estimated from a time series within the well established framework of the Wiener-Granger causality, while the latter represent equilibrium (or stationary) causal effects which are often most interesting but generally inaccessible to estimation from an observed time series recorded at fixed coupling parameters. In this work, relationships between the two kinds of causal effects are found for unidirectionally coupled stochastic linear oscillators depending on their frequencies and damping factors. Approximate closed-form expressions for these relationships are derived. Their limitations and possible extensions are discussed, and their practical applicability to extracting equilibrium causal effects from time series is argued.

Causality, dynamical systems and the arrow of time

Using several methods for detection of causality in time series, we show in a numerical study that coupled chaotic dynamical systems violate the first principle of Granger causality that the cause precedes the effect. While such a violation can be observed in formal applications of time series analysis methods, it cannot occur in nature, due to the relation between entropy production and temporal irreversibility. The obtained knowledge, however, can help to understand the type of causal relations observed in experimental data, namely, it can help to distinguish linear transfer of time-delayed signals from nonlinear interactions. We illustrate these findings in causality detected in experimental time series from the climate system and mammalian cardio-respiratory interactions.

Cross-Frequency Transfer Entropy Characterize Coupling of Interacting Nonlinear Oscillators in Complex Systems

The purpose of this study is to introduce a method in quantifying cross-frequency information transfer to characterize directional couplers between irregular oscillations in complex systems. Importantly, the method should be able to reflect the intrinsic mechanism of interacting oscillations faithfully. Six types of interacting oscillators, including phase-amplitude, amplitude-amplitude, and component-amplitude cross-frequency transfer entropy as well as their inverse transfer entropies, are within our scope in untangling the brain connectivity. Challenges with nonlinear and nonstationary patterns are designed to validate the robustness of the proposed method. We suggest this approach could be effective in identifying driving and responding elements of interacting oscillators across different time scales. Meanwhile, an atlas of interacting oscillators in sleep is constructed. High-frequency amplitude can inversely drive low-frequency phase stronger than the standard phase-amplitude coupling, and the low-frequency amplitude can be the driving force to the high-frequency amplitude in addition to the low-frequency phase. Unlike the standard phase-amplitude coupling, the proposed cross-frequency transfer entropy is applicable to quantify the interactions across phases, amplitudes, or even the components without methodological adjustments. Meanwhile, the exploration of causal relationship enables the identification of the driving force of information flow.

Detection of coupling delay: A problem not yet solved

Nonparametric detection of coupling delay in unidirectionally and bidirectionally coupled nonlinear dynamical systems is examined. Both continuous and discrete-time systems are considered. Two methods of detection are assessed-the method based on conditional mutual information-the CMI method (also known as the transfer entropy method) and the method of convergent cross mapping-the CCM method. Computer simulations show that neither method is generally reliable in the detection of coupling delays. For continuous-time chaotic systems, the CMI method appears to be more sensitive and applicable in a broader range of coupling parameters than the CCM method. In the case of tested discrete-time dynamical systems, the CCM method has been found to be more sensitive, while the CMI method required much stronger coupling strength in order to bring correct results. However, when studied systems contain a strong oscillatory component in their dynamics, results of both methods become ambiguous. The presented study suggests that results of the tested algorithms should be interpreted with utmost care and the nonparametric detection of coupling delay, in general, is a problem not yet solved.

Identification of the driving forces of climate change using the longest instrumental temperature record

The identification of causal effects is a fundamental problem in climate change research. Here, a new perspective on climate change causality is presented using the central England temperature (CET) dataset, the longest instrumental temperature record, and a combination of slow feature analysis and wavelet analysis. The driving forces of climate change were investigated and the results showed two independent degrees of freedom -a 3.36-year cycle and a 22.6-year cycle, which seem to be connected to the El Niño-Southern Oscillation cycle and the Hale sunspot cycle, respectively. Moreover, these driving forces were modulated in amplitude by signals with millennial timescales.

Smooth information flow in temperature climate network reflects mass transport

A directed climate network is constructed by Granger causality analysis of air temperature time series from a regular grid covering the whole Earth. Using winner-takes-all network thresholding approach, a structure of a smooth information flow is revealed, hidden to previous studies. The relevance of this observation is confirmed by comparison with the air mass transfer defined by the wind field. Their close relation illustrates that although the information transferred due to the causal influence is not a physical quantity, the information transfer is tied to the transfer of mass and energy.

Coupling strength versus coupling impact in nonidentical bidirectionally coupled dynamics

The understanding of interacting dynamics is important for the characterization of real-world networks. In general, real-world networks are heterogeneous in the sense that each node of the network is a dynamics with different properties. For coupled nonidentical dynamics symmetric interactions are not straightforwardly defined from the coupling strength values. Thus, a challenging issue is whether we can define a symmetric interaction in this asymmetric setting. To address this problem we introduce the notion of the coupling impact. The coupling impact considers not only the coupling strength but also the energy of the individual dynamics, which is conveyed via the coupling. To illustrate this concept, we follow a data-driven approach by analyzing signals from pairs of coupled model dynamics using two different connectivity measures. We find that the coupling impact, but not the coupling strength, correctly detects a symmetric interaction between pairs of coupled dynamics regardless of their degree of asymmetry. Therefore, this approach allows us to reveal the real impact that one dynamics has on the other and hence to define symmetric interactions in pairs of nonidentical dynamics.

Discrimination of coupling structures using causality networks from multivariate time series

Measures of Granger causality on multivariate time series have been used to form the so-called causality networks. A causality network represents the interdependence structure of the underlying dynamical system or coupled dynamical systems, and its properties are quantified by network indices. In this work, it is investigated whether network indices on networks generated by an appropriate Granger causality measure can discriminate different coupling structures. The information based Granger causality measure of partial mutual information from mixed embedding (PMIME) is used to form causality networks, and a large number of network indices are ranked according to their ability to discriminate the different coupling structures. The evaluation of the network indices is done with a simulation study based on two dynamical systems, the coupled Mackey-Glass delay differential equations and the neural mass model, both of 25 variables, and three prototypes of coupling structures, i.e., random, small-world, and scale-free. It is concluded that the setting of PMIME combined with a network index attains high level of discrimination of the coupling structures solely on the basis of the observed multivariate time series. This approach is demonstrated to identify epileptic seizures emerging during electroencephalogram recordings.

On the temporal variability of the surface solar radiation by means of spectral representations

Abstract. This work deals with the temporal variability of daily means of the global broadband surface solar irradiance (SSI) impinging on a horizontal plane by studying a decennial time-series of high-quality measurements recorded at a BSRN ground station. Since the data have a non-linear and non-stationary character, two time-frequency-energy representations of signal processing are compared in their ability to resolve the temporal variability of the pyranometric signal. First, the continuous wavelet transform is used to construct the wavelet power spectrum of the data. Second, the adaptive, noise-assisted empirical mode decomposition is employed to extract the intrinsic mode functions of the signal, followed by Hilbert spectral analysis. In both spectral representations, the temporal variability of the SSI is portrayed having clearly distinguishable features: a plateau between scales of two days and two-three months that has decreasing power with increasing scale, a large spectral peak corresponding to the annual variability cycle, and a low power regime in between the previous two. It is shown that the data-driven, noise-assisted method yields a somewhat more sparse representation and that it is a suitable tool for inspecting the temporal variability of SSI measurements.

Identifying causal gateways and mediators in complex spatio-temporal systems

Identifying regions important for spreading and mediating perturbations is crucial to assess the susceptibilities of spatio-temporal complex systems such as the Earth's climate to volcanic eruptions, extreme events or geoengineering. Here a data-driven approach is introduced based on a dimension reduction, causal reconstruction, and novel network measures based on causal effect theory that go beyond standard complex network tools by distinguishing direct from indirect pathways. Applied to a data set of atmospheric dynamics, the method identifies several strongly uplifting regions acting as major gateways of perturbations spreading in the atmosphere. Additionally, the method provides a stricter statistical approach to pathways of atmospheric teleconnections, yielding insights into the Pacific-Indian Ocean interaction relevant for monsoonal dynamics. Also for neuroscience or power grids, the novel causal interaction perspective provides a complementary approach to simulations or experiments for understanding the functioning of complex spatio-temporal systems with potential applications in increasing their resilience to shocks or extreme events.

Relating Granger causality to long-term causal effects

In estimation of causal couplings between observed processes, it is important to characterize coupling roles at various time scales. The widely used Granger causality reflects short-term effects: it shows how strongly perturbations of a current state of one process affect near future states of another process, and it quantifies that via prediction improvement (PI) in autoregressive models. However, it is often more important to evaluate the effects of coupling on long-term statistics, e.g., to find out how strongly the presence of coupling changes the variance of a driven process as compared to an uncoupled case. No general relationships between Granger causality and such long-term effects are known. Here, we pose the problem of relating these two types of coupling characteristics, and we solve it for a class of stochastic systems. Namely, for overdamped linear oscillators, we rigorously derive that the above long-term effect is proportional to the short-term effects, with the proportionality coefficient depending on the prediction interval and relaxation times. We reveal that this coefficient is typically considerably greater than unity so that small normalized PI values may well correspond to quite large long-term effects of coupling. The applicability of the derived relationship to wider classes of systems, its limitations, and its value for further research are discussed. To give a real-world example, we analyze couplings between large-scale climatic processes related to sea surface temperature variations in equatorial Pacific and North Atlantic regions.

Assessing the direction of climate interactions by means of complex networks and information theoretic tools

An estimate of the net direction of climate interactions in different geographical regions is made by constructing a directed climate network from a regular latitude-longitude grid of nodes, using a directionality index (DI) based on conditional mutual information (CMI). Two datasets of surface air temperature anomalies-one monthly averaged and another daily averaged-are analyzed and compared. The network links are interpreted in terms of known atmospheric tropical and extra-tropical variability patterns. Specific and relevant geographical regions are selected, the net direction of propagation of the atmospheric patterns is analyzed, and the direction of the inferred links is validated by recovering some well-known climate variability structures. These patterns are found to be acting at various time-scales, such as atmospheric waves in the extratropics or longer range events in the tropics. This analysis demonstrates the capability of the DI measure to infer the net direction of climate interactions and may contribute to improve the present understanding of climate phenomena and climate predictability. The work presented here also stands out as an application of advanced tools to the analysis of empirical, real-world data.

Quantification of causal couplings via dynamical effects: a unifying perspective

Quantitative characterization of causal couplings from time series is crucial in studies of complex systems of different origin. Various statistical tools for that exist and new ones are still being developed with a tendency to creating a single, universal, model-free quantifier of coupling strength. However, a clear and generally applicable way of interpreting such universal characteristics is lacking. This work suggests a general conceptual framework for causal coupling quantification, which is based on state space models and extends the concepts of virtual interventions and dynamical causal effects. Namely, two basic kinds of interventions (state space and parametric) and effects (orbital or transient and stationary or limit) are introduced, giving four families of coupling characteristics. The framework provides a unifying view of apparently different well-established measures and allows us to introduce new characteristics, always with a definite "intervention-effect" interpretation. It is shown that diverse characteristics cannot be reduced to any single coupling strength quantifier and their interpretation is inevitably model based. The proposed set of dynamical causal effect measures quantifies different aspects of "how the coupling manifests itself in the dynamics," reformulating the very question about the "causal coupling strength."