Nonuniform state-space reconstruction and coupling detection

A robust time-delay selection criterion applied to convergent cross mapping

This work presents a heuristic for the selection of a time delay based on optimizing the global maximum of mutual information in orthonormal coordinates for embedding a dynamical system. This criterion is demonstrated to be more robust compared to methods that utilize a local minimum, as the global maximum is guaranteed to exist in the proposed coordinate system for any dynamical system. By contrast, methods using local minima can be ill-posed as a local minimum can be difficult to identify in the presence of noise or may simply not exist. The performance of the global maximum and local minimum methods are compared in the context of causality detection using convergent cross mapping using both a noisy Lorenz system and experimental data from an oscillating plasma source. The proposed heuristic for time lag selection is shown to be more consistent in the presence of noise and closer to an optimal uniform time lag selection.

A rigorous and versatile statistical test for correlations between stationary time series

In disciplines from biology to climate science, a routine task is to compute a correlation between a pair of time series and determine whether the correlation is statistically significant (i.e., unlikely under the null hypothesis that the time series are independent). This problem is challenging because time series typically exhibit autocorrelation and thus cannot be properly analyzed with the standard iid-oriented statistical tests. Although there are well-known parametric tests for time series, these are designed for linear correlation statistics and thus not suitable for the increasingly popular nonlinear correlation statistics. There are also nonparametric tests that can be used with any correlation statistic, but for these, the conditions that guarantee correct false positive rates are either restrictive or unclear. Here, we describe the truncated time-shift (TTS) test, a nonparametric procedure to test for dependence between 2 time series. We prove that this test correctly controls the false positive rate as long as one of the time series is stationary, a minimally restrictive requirement among current tests. The TTS test is versatile because it can be used with any correlation statistic. Using synthetic data, we demonstrate that this test performs correctly even while other tests suffer high false positive rates. In simulation examples, simple guidelines for parameter choices allow high statistical power to be achieved with sufficient data. We apply the test to datasets from climatology, animal behavior, and microbiome science, verifying previously discovered dependence relationships and detecting additional relationships.

Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems

Real-world non-autonomous systems are open, out-of-equilibrium systems that evolve in and are driven by temporally varying environments. Such systems can show multiple timescale and transient dynamics together with transitions to very different and, at times, even disastrous dynamical regimes. Since such critical transitions disrupt the systems' intended or desired functionality, it is crucial to understand the underlying mechanisms, to identify precursors of such transitions, and to reliably detect them in time series of suitable system observables to enable forecasts. This review critically assesses the various steps of investigation involved in time-series-analysis-based detection of critical transitions in real-world non-autonomous systems: from the data recording to evaluating the reliability of offline and online detections. It will highlight pros and cons to stimulate further developments, which would be necessary to advance understanding and forecasting nonlinear behavior such as critical transitions in complex systems.

The causality measure of partial mutual information from mixed embedding (PMIME) revisited

The measure of partial mutual information from mixed embedding (PMIME) is an information theory-based measure to accurately identify the direct and directional coupling, termed Granger causality or simply causality, between the observed variables or subsystems of a high-dimensional dynamical and complex system, without any a priori assumptions about the nature of the coupling relationship. In its core, it is a forward selection procedure that aims to iteratively identify the lag-dependence structure of a given observed variable (response) to all the other observed variables (candidate drivers). This model-free approach is capable of detecting nonlinear interactions, abundantly present in real-world complex systems, and it was shown to perform well on multivariate time series of moderately high dimension. However, the PMIME presents some inefficiencies in its performance mainly when applied on strongly stochastic (linear or nonlinear) systems as it may falsely detect non-existent relationships. Moreover, and by construction, the measure cannot extract purely synergetic relationships present in a system. In the current work, the issue of false detections is addressed by introducing an improved resampling significance test and a procedure of rechecking the identified drivers (backward revision). Regarding the inability to detect synergetic relationships, the PMIME is further enhanced by checking pairs as candidate drivers for the response variable after having considered all drivers individually. The effects of these modifications are investigated in a systematic simulation study on properly designed systems involving strong stochasticity, regressor terms with synergetic effects, and a system dimension ranging from 3 to 30. The overall results of the simulations indicate that these modifications indeed improve the performance of PMIME and alleviate to a significant degree the issues of the original algorithm. Guidelines for balancing between accuracy and computational efficiency are also given, particularly relevant for real-world applications. Finally, the measure performance is investigated in the study of futures of various government bonds and stock market indices in the period around COVID-19 pandemic.

Information transfers and flows in Markov chains as dynamical causal effects

A logical sequence of information-theoretic quantifiers of directional (causal) couplings in Markov chains is generated within the framework of dynamical causal effects (DCEs), starting from the simplest DCEs (in terms of localization of their functional elements) and proceeding step-by-step to more complex ones. Thereby, a system of 11 quantifiers is readily obtained, some of them coinciding with previously known causality measures widely used in time series analysis and often called "information transfers" or "flows" (transfer entropy, Ay-Polani information flow, Liang-Kleeman information flow, information response, etc.,) By construction, this step-by-step generation reveals logical relationships between all these quantifiers as specific DCEs. As a further concretization, diverse quantitative relationships between the transfer entropy and the Liang-Kleeman information flow are found both rigorously and numerically for coupled two-state Markov chains.

Non-Asymptotic Guarantees for Reliable Identification of Granger Causality via the LASSO

Granger causality is among the widely used data-driven approaches for causal analysis of time series data with applications in various areas including economics, molecular biology, and neuroscience. Two of the main challenges of this methodology are: 1) over-fitting as a result of limited data duration, and 2) correlated process noise as a confounding factor, both leading to errors in identifying the causal influences. Sparse estimation via the LASSO has successfully addressed these challenges for parameter estimation. However, the classical statistical tests for Granger causality resort to asymptotic analysis of ordinary least squares, which require long data duration to be useful and are not immune to confounding effects. In this work, we address this disconnect by introducing a LASSO-based statistic and studying its non-asymptotic properties under the assumption that the true models admit sparse autoregressive representations. We establish fundamental limits for reliable identification of Granger causal influences using the proposed LASSO-based statistic. We further characterize the false positive error probability and test power of a simple thresholding rule for identifying Granger causal effects and provide two methods to set the threshold in a data-driven fashion. We present simulation studies and application to real data to compare the performance of our proposed method to ordinary least squares and existing LASSO-based methods in detecting Granger causal influences, which corroborate our theoretical results.

Recurrence Quantification Analysis of Crowd Sound Dynamics

When multiple individuals interact in a conversation or as part of a large crowd, emergent structures and dynamics arise that are behavioral properties of the interacting group rather than of any individual member of that group. Recent work using traditional signal processing techniques and machine learning has demonstrated that global acoustic data recorded from a crowd at a basketball game can be used to classify emergent crowd behavior in terms of the crowd's purported emotional state. We propose that the description of crowd behavior from such global acoustic data could benefit from nonlinear analysis methods derived from dynamical systems theory. Such methods have been used in recent research applying nonlinear methods to audio data extracted from music and group musical interactions. In this work, we used nonlinear analyses to extract features that are relevant to the behavioral interactions that underlie acoustic signals produced by a crowd attending a sporting event. We propose that recurrence dynamics measured from these audio signals via recurrence quantification analysis (RQA) reflect information about the behavioral dynamics of the crowd itself. We analyze these dynamics from acoustic signals recorded from crowds attending basketball games, and that were manually labeled according to the crowds' emotional state across six categories: angry noise, applause, cheer, distraction noise, positive chant, and negative chant. We show that RQA measures are useful to differentiate the emergent acoustic behavioral dynamics between these categories, and can provide insight into the recurrence patterns that underlie crowd interactions.

Information theory-based direct causality measure to assess cardiac fibrillation dynamics

Understanding the mechanism sustaining cardiac fibrillation can facilitate the personalization of treatment. Granger causality analysis can be used to determine the existence of a hierarchical fibrillation mechanism that is more amenable to ablation treatment in cardiac time-series data. Conventional Granger causality based on linear predictability may fail if the assumption is not met or given sparsely sampled, high-dimensional data. More recently developed information theory-based causality measures could potentially provide a more accurate estimate of the nonlinear coupling. However, despite their successful application to linear and nonlinear physical systems, their use is not known in the clinical field. Partial mutual information from mixed embedding (PMIME) was implemented to identify the direct coupling of cardiac electrophysiology signals. We show that PMIME requires less data and is more robust to extrinsic confounding factors. The algorithms were then extended for efficient characterization of fibrillation organization and hierarchy using clinical high-dimensional data. We show that PMIME network measures correlate well with the spatio-temporal organization of fibrillation and demonstrated that hierarchical type of fibrillation and drivers could be identified in a subset of ventricular fibrillation patients, such that regions of high hierarchy are associated with high dominant frequency.

On the Validity of Single Regression Strategy for Granger Causality Assessment in Cardiovascular and Cardiorespiratory Control Studies

Granger causality (GC) analysis is based on the comparison between prediction error variances computed over the full and restricted models after identifying the coefficients of appropriate vector regressions. GC markers can be computed via a double regression (DR) approach identifying two separate, independent models and a single regression (SR) strategy optimizing the description of the dynamics of the target over the full model and, then, reusing some parts of it in the restricted model. The present study compares the SR and DR strategies over heart period (HP), systolic arterial pressure (SAP) and respiration (R) beat-to-beat series collected during a graded orthostatic challenge induced by head-up tilt in 17 healthy individuals (age: 21-36 yrs; median: 29 yrs; 9 females and 8 males). We found that the DR approach was more powerful than the SR one in detecting the expected stronger involvement of the baroreflex during the challenge, while the expected weaker cardiorespiratory coupling was identified by both SR and DR strategies. The less powerful ability of the SR approach was the result of the greater variance of GC markers compared to the DR strategy. We conclude that, contrary to the suggestions present in literature, the SR approach is not necessarily associated with a smaller dispersion of GC markers. Moreover, we suggest that additional factors, such as the strategy utilized to build embedding spaces and metric utilized to compare prediction error variances, might play an important role in differentiating SR and DR approaches.

Conditional cross-map-based technique: From pairwise dynamical causality to causal network reconstruction

Causality detection methods based on mutual cross mapping have been fruitfully developed and applied to data originating from nonlinear dynamical systems, where the causes and effects are non-separable. However, these pairwise methods still have shortcomings in discriminating typical network structures, including common drivers, indirect dependencies, and facing the curse of dimensionality, when they are stepping to causal network reconstruction. A few endeavors have been devoted to conquer these shortcomings. Here, we propose a novel method that could be regarded as one of these endeavors. Our method, named conditional cross-map-based technique, can eliminate third-party information and successfully detect direct dynamical causality, where the detection results can exactly be categorized into four standard normal forms by the designed criterion. To demonstrate the practical usefulness of our model-free, data-driven method, data generated from different representative models covering all kinds of network motifs and measured from real-world systems are investigated. Because correct identification of the direct causal links is essential to successful modeling, predicting, and controlling the underlying complex systems, our method does shed light on uncovering the inner working mechanisms of real-world systems only using the data experimentally obtained in a variety of disciplines.

Comparison of discretization strategies for the model-free information-theoretic assessment of short-term physiological interactions

This work presents a comparison between different approaches for the model-free estimation of information-theoretic measures of the dynamic coupling between short realizations of random processes. The measures considered are the mutual information rate (MIR) between two random processes X and Y and the terms of its decomposition evidencing either the individual entropy rates of X and Y and their joint entropy rate, or the transfer entropies from X to Y and from Y to X and the instantaneous information shared by X and Y. All measures are estimated through discretization of the random variables forming the processes, performed either via uniform quantization (binning approach) or rank ordering (permutation approach). The binning and permutation approaches are compared on simulations of two coupled non-identical Hènon systems and on three datasets, including short realizations of cardiorespiratory (CR, heart period and respiration flow), cardiovascular (CV, heart period and systolic arterial pressure), and cerebrovascular (CB, mean arterial pressure and cerebral blood flow velocity) measured in different physiological conditions, i.e., spontaneous vs paced breathing or supine vs upright positions. Our results show that, with careful selection of the estimation parameters (i.e., the embedding dimension and the number of quantization levels for the binning approach), meaningful patterns of the MIR and of its components can be achieved in the analyzed systems. On physiological time series, we found that paced breathing at slow breathing rates induces less complex and more coupled CR dynamics, while postural stress leads to unbalancing of CV interactions with prevalent baroreflex coupling and to less complex pressure dynamics with preserved CB interactions. These results are better highlighted by the permutation approach, thanks to its more parsimonious representation of the discretized dynamic patterns, which allows one to explore interactions with longer memory while limiting the curse of dimensionality.

Detecting Nonlinear Interactions in Complex Systems: Application in Financial Markets

Emerging or diminishing nonlinear interactions in the evolution of a complex system may signal a possible structural change in its underlying mechanism. This type of structural break may exist in many applications, such as in climate and finance, and standard methods for change-point detection may not be sensitive to it. In this article, we present a novel scheme for detecting structural breaks through the occurrence or vanishing of nonlinear causal relationships in a complex system. A significance resampling test was developed for the null hypothesis (H₀) of no nonlinear causal relationships using (a) an appropriate Gaussian instantaneous transform and vector autoregressive (VAR) process to generate the resampled multivariate time series consistent with H₀; (b) the modelfree Granger causality measure of partial mutual information from mixed embedding (PMIME) to estimate all causal relationships; and (c) a characteristic of the network formed by PMIME as test statistic. The significance test was applied to sliding windows on the observed multivariate time series, and the change from rejection to no-rejection of H₀, or the opposite, signaled a non-trivial change of the underlying dynamics of the observed complex system. Different network indices that capture different characteristics of the PMIME networks were used as test statistics. The test was evaluated on multiple synthetic complex and chaotic systems, as well as on linear and nonlinear stochastic systems, demonstrating that the proposed methodology is capable of detecting nonlinear causality. Furthermore, the scheme was applied to different records of financial indices regarding the global financial crisis of 2008, the two commodity crises of 2014 and 2020, the Brexit referendum of 2016, and the outbreak of COVID-19, accurately identifying the structural breaks at the identified times.

Adaptation of Partial Mutual Information from Mixed Embedding to Discrete-Valued Time Series

A causality analysis aims at estimating the interactions of the observed variables and subsequently the connectivity structure of the observed dynamical system or stochastic process. The partial mutual information from mixed embedding (PMIME) is found appropriate for the causality analysis of continuous-valued time series, even of high dimension, as it applies a dimension reduction by selecting the most relevant lag variables of all the observed variables to the response, using conditional mutual information (CMI). The presence of lag components of the driving variable in this vector implies a direct causal (driving-response) effect. In this study, the PMIME is appropriately adapted to discrete-valued multivariate time series, called the discrete PMIME (DPMIME). An appropriate estimation of the discrete probability distributions and CMI for discrete variables is implemented in the DPMIME. Further, the asymptotic distribution of the estimated CMI is derived, allowing for a parametric significance test for the CMI in the DPMIME, whereas for the PMIME, there is no parametric test for the CMI and the test is performed using resampling. Monte Carlo simulations are performed using different generating systems of discrete-valued time series. The simulation suggests that the parametric significance test for the CMI in the progressive algorithm of the DPMIME is compared favorably to the corresponding resampling significance test, and the accuracy of the DPMIME in the estimation of direct causality converges with the time-series length to the accuracy of the PMIME. Further, the DPMIME is used to investigate whether the global financial crisis has an effect on the causality network of the financial world market.

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.

Phase-based causality analysis with partial mutual information from mixed embedding

Instantaneous phases extracted from multivariate time series can retain information about the relationships between the underlying mechanisms that generate the series. Although phases have been widely used in the study of nondirectional coupling and connectivity, they have not found similar appeal in the study of causality. Herein, we present a new method for phase-based causality analysis, which combines ideas from the mixed embedding technique and the information-theoretic approach to causality in coupled oscillatory systems. We then use the introduced method to investigate causality in simulated datasets of bivariate, unidirectionally paired systems from combinations of Rössler, Lorenz, van der Pol, and Mackey-Glass equations. We observe that causality analysis using the phases can capture the true causal relation for coupling strength smaller than the analysis based on the amplitudes can capture. On the other hand, the causality estimation based on the phases tends to have larger variability, which is attributed more to the phase extraction process than the actual phase-based causality method. In addition, an application on real electroencephalographic data from an experiment on elicited human emotional states reinforces the usefulness of phases in causality identification.

Coordination dynamics of multi-agent interaction in a musical ensemble

Humans interact with other humans at a variety of timescales and in a variety of social contexts. We exhibit patterns of coordination that may differ depending on whether we are genuinely interacting as part of a coordinated group of individuals vs merely co-existing within the same physical space. Moreover, the local coordination dynamics of an interacting pair of individuals in an otherwise non-interacting group may spread, propagating change in the global coordination dynamics and interaction of an entire crowd. Dynamical systems analyses, such as Recurrence Quantification Analysis (RQA), can shed light on some of the underlying coordination dynamics of multi-agent human interaction. We used RQA to examine the coordination dynamics of a performance of "Welcome to the Imagination World", composed for wind orchestra. This performance enacts a real-life simulation of the transition from uncoordinated, non-interacting individuals to a coordinated, interacting multi-agent group. Unlike previous studies of social interaction in musical performance which rely on different aspects of video and/or acoustic data recorded from each individual, this project analyzes group-level coordination patterns solely from the group-level acoustic data of an audio recording of the performance. Recurrence and stability measures extracted from the audio recording increased when musicians coordinated as an interacting group. Variability in these measures also increased, indicating that the interacting ensemble of musicians were able to explore a greater variety of behavior than when they performed as non-interacting individuals. As an orchestrated (non-emergent) example of coordination, we believe these analyses provide an indication of approximate expected distributions for recurrence patterns that may be measurable before and after truly emergent coordination.

Detecting the relationships among multivariate time series using reduced auto-regressive modeling

An information theoretic reduction of auto-regressive modeling called the Reduced Auto-Regressive (RAR) modeling is applied to several multivariate time series as a method to detect the relationships among the components in the time series. The results are compared with the results of the transfer entropy, one of the common techniques for detecting causal relationships. These common techniques are pairwise by definition and could be inappropriate in detecting the relationships in highly complicated dynamical systems. When the relationships between the dynamics of the components are linear and the time scales in the fluctuations of each component are in the same order of magnitude, the results of the RAR model and the transfer entropy are consistent. When the time series contain components that have large differences in the amplitude and the time scales of fluctuation, however, the transfer entropy fails to detect the correct relationships between the components, while the results of the RAR modeling are still correct. For a highly complicated dynamics such as human brain activity observed by electroencephalography measurements, the results of the transfer entropy are drastically different from those of the RAR modeling.

Variational Embedding Multiscale Sample Entropy: A Tool for Complexity Analysis of Multichannel Systems

Entropy-based methods have received considerable attention in the quantification of structural complexity of real-world systems. Among numerous empirical entropy algorithms, conditional entropy-based methods such as sample entropy, which are associated with amplitude distance calculation, are quite intuitive to interpret but require excessive data lengths for meaningful evaluation at large scales. To address this issue, we propose the variational embedding multiscale sample entropy (veMSE) method and conclusively demonstrate its ability to operate robustly, even with several times shorter data than the existing conditional entropy-based methods. The analysis reveals that veMSE also exhibits other desirable properties, such as the robustness to the variation in embedding dimension and noise resilience. For rigor, unlike the existing multivariate methods, the proposed veMSE assigns a different embedding dimension to every data channel, which makes its operation independent of channel permutation. The veMSE is tested on both stimulated and real world signals, and its performance is evaluated against the existing multivariate multiscale sample entropy methods. The proposed veMSE is also shown to exhibit computational advantages over the existing amplitude distance-based entropy methods.

Connectivity Analysis for Multivariate Time Series: Correlation vs. Causality

The study of the interdependence relationships of the variables of an examined system is of great importance and remains a challenging task. There are two distinct cases of interdependence. In the first case, the variables evolve in synchrony, connections are undirected and the connectivity is examined based on symmetric measures, such as correlation. In the second case, a variable drives another one and they are connected with a causal relationship. Therefore, directed connections entail the determination of the interrelationships based on causality measures. The main open question that arises is the following: can symmetric correlation measures or directional causality measures be applied to infer the connectivity network of an examined system? Using simulations, we demonstrate the performance of different connectivity measures in case of contemporaneous or/and temporal dependencies. Results suggest the sensitivity of correlation measures when temporal dependencies exist in the data. On the other hand, causality measures do not spuriously indicate causal effects when data present only contemporaneous dependencies. Finally, the necessity of introducing effective instantaneous causality measures is highlighted since they are able to handle both contemporaneous and causal effects at the same time. Results based on instantaneous causality measures are promising; however, further investigation is required in order to achieve an overall satisfactory performance.

A modified phase transfer entropy for cross-frequency directed coupling estimation in brain network

Cross-frequency coupling of neural oscillation is widespread during the complex cognitive process. Therefore, identifying cross-frequency information flow is essential for revealing neural dynamics mechanisms in the brain network. A current method based on the information theory, phase transfer entropy (PTE), has been proved its effectiveness in estimating directional coupling in several recent studies. However, there remains some limits in PTE: (1)lack of multivariable effect, (2) poor robustness, (3)curse of dimensionality in the high dimensional system. This study introduced a novel multivariate phase transfer entropy method named "MPTE_NUE" to solve the above issues. In MPTE_NUE, it considered the influence of remaining confounding variables, which guaranteed its applicability in a multivariable system. Meanwhile, a nonuniform embedding (NUE) approach for state reconstruction was adopted to eliminate the dimensional curse problem. We performed a series of numerical simulations based on the typical Hénon map model. The results proved that the MPTE_NUE achieved better noise robustness and effectively avoided the curse of dimension; meanwhile, the accuracy and sensitivity can reach 96.9% and 99.2%, respectively.

Causality indices for bivariate time series data: A comparative review of performance

Inferring nonlinear and asymmetric causal relationships between multivariate longitudinal data is a challenging task with wide-ranging application areas including clinical medicine, mathematical biology, economics, and environmental research. A number of methods for inferring causal relationships within complex dynamic and stochastic systems have been proposed, but there is not a unified consistent definition of causality in the context of time series data. We evaluate the performance of ten prominent causality indices for bivariate time series across four simulated model systems that have different coupling schemes and characteristics. Pairwise correlations between different methods, averaged across all simulations, show that there is generally strong agreement between methods, with minimum, median, and maximum Pearson correlations between any pair (excluding two similarity indices) of 0.298, 0.719, and 0.955, respectively. In further experiments, we show that these methods are not always invariant to real-world relevant transformations (data availability, standardization and scaling, rounding errors, missing data, and noisy data). We recommend transfer entropy and nonlinear Granger causality as particularly strong approaches for estimating bivariate causal relationships in real-world applications. Both successfully identify causal relationships and a lack thereof across multiple simulations, while remaining robust to rounding errors, at least 20% missing data and small variance Gaussian noise. Finally, we provide flexible open-access Python code for computation of these methods and for the model simulations.

Benchmarking Transfer Entropy Methods for the Study of Linear and Nonlinear Cardio-Respiratory Interactions

Transfer entropy (TE) has been used to identify and quantify interactions between physiological systems. Different methods exist to estimate TE, but there is no consensus about which one performs best in specific applications. In this study, five methods (linear, k-nearest neighbors, fixed-binning with ranking, kernel density estimation and adaptive partitioning) were compared. The comparison was made on three simulation models (linear, nonlinear and linear + nonlinear dynamics). From the simulations, it was found that the best method to quantify the different interactions was adaptive partitioning. This method was then applied on data from a polysomnography study, specifically on the ECG and the respiratory signals (nasal airflow and respiratory effort around the thorax). The hypothesis that the linear and nonlinear components of cardio-respiratory interactions during light and deep sleep change with the sleep stage, was tested. Significant differences, after performing surrogate analysis, indicate an increased TE during deep sleep. However, these differences were found to be dependent on the type of respiratory signal and sampling frequency. These results highlight the importance of selecting the appropriate signals, estimation method and surrogate analysis for the study of linear and nonlinear cardio-respiratory interactions.

Risk spillover networks in financial system based on information theory

Since the financial system has illustrated an increasingly prominent characteristic of inextricable connections, information theory is gradually utilized to study the financial system. By collecting the daily data of industry index (2005-2020) and region index (2012-2020) listed in China as samples, this paper applies an innovative measure named partial mutual information on mixed embedding to generate directed networks. Based on the analysis of nonlinear relationships among sectors, this paper realizes the accurate construction of "time-varying" financial network from the perspective of risk spillover. The results are presented as follow: (1) interactions can be better understood through the nonlinear networks among distinct sectors, and sectors in the networks could be classified into different types according to their topological properties connected to risk spillover; (2) in the rising stage, information is transmitted rapidly in the network, so the risk is fast diffused and absorbed; (3) in the declining stage, the network topology is more complex and panic sentiments have long term impact leading to more connections; (4) The US market, Japan market and Hongkong market have significant affect on China's market. The results suggest that this nonlinear measure is an effective approach to develop financial networks and explore the mechanism of risk spillover.

Multiscale Transfer Spectral Entropy for Quantifying Corticomuscular Interaction

Corticomuscular coupling reflects nonlinear interactions and multi-layer neural information transmission between the motor cortex and effector muscle in the sensorimotor system. Transfer spectral entropy (TSE) method has been used to describe corticomuscular coupling within single scale. As an extension of TSE, multiscale transfer spectral entropy (MSTSE) is proposed in this paper to depict multi-layer of neural information transfer between two coupling signals. The reliability and effectiveness of MSTSE were verified on data generated by nonlinear numerical models and those of a force tracking task. Compared with TSE, MSTSE is more robust to the embedding dimension and performs optimally in the detection of the coupling properties. Further analysis of the physiological signals showed that the MSTSE provided more detailed band characteristics than the single scale TSE measurement. MSTSE indicates significant coupling scattered in alpha, beta and low gamma bands during the force tracking task. Besides, the coupling strength in the descending direction of the beta band was significantly higher than that in the ascending direction. This study constructs multi-scale coupling information to provide a new perspective for exploring corticomuscular interaction.

Multiscale multivariate transfer entropy and application to functional corticocortical coupling

Objective. Complex biological systems consist of multi-level mechanism in terms of within- and cross-subsystems correlations, and they are primarily manifested in terms of connectivity, multiscale properties, and nonlinearity. Existing studies have each only explored one aspect of the functional corticocortical coupling (FCCC), which has some limitations in portraying the complexity of multivariable systems. The present study investigated the direct interactions of brain networks at multiple time scales.Approach. We extended the multivariate transfer entropy (MuTE) method and proposed a novel method, named multiscale multivariate transfer entropy (MSMVTE), to explore the direct interactions of brain networks across multiple time scale. To verify this aim, we introduced three simulation models and compared them with multiscale transfer entropy (MSTE) and MuTE methods. We then applied MSMVTE method to analyze FCCC during a unilateral right-hand steady-state force task.Main results. Simulation results showed that the MSMVTE method, compared with MSTE and MuTE methods, better detected direct interactions and avoid the spurious effects of indirect relationships. Further analysis of experimental data showed that the connectivity from left premotor/sensorimotor cortex to right premotor/sensorimotor cortex was significantly higher than that of opposite directionality. Furthermore, the connectivities from central motor areas to both sides of premotor/sensorimotor areas were higher than those of opposite directionalities. Additionally, the maximum coupling strength was found to occur at a specific scale (3-10).Significance. Simulation results confirmed the effectiveness of the MSMVTE method to describe direct relationships and multiscale characteristics in complex systems. The enhancement of FCCC reflects the interaction of more extended activation in cortical motor regions. Additionally, the neurodynamic process of brain depends not only on emergent behavior at small scales, but also on the constraining effects of the activity at large scales. Taken together, our findings provide a basis for better understanding dynamics in brain networks.

Identifying bidirectional total and non-linear information flow in functional corticomuscular coupling during a dorsiflexion task: a pilot study

Background

The key challenge to constructing functional corticomuscular coupling (FCMC) is to accurately identify the direction and strength of the information flow between scalp electroencephalography (EEG) and surface electromyography (SEMG). Traditional TE and TDMI methods have difficulty in identifying the information interaction for short time series as they tend to rely on long and stable data, so we propose a time-delayed maximal information coefficient (TDMIC) method. With this method, we aim to investigate the directional specificity of bidirectional total and nonlinear information flow on FCMC, and to explore the neural mechanisms underlying motor dysfunction in stroke patients.

Methods

We introduced a time-delayed parameter in the maximal information coefficient to capture the direction of information interaction between two time series. We employed the linear and non-linear system model based on short data to verify the validity of our algorithm. We then used the TDMIC method to study the characteristics of total and nonlinear information flow in FCMC during a dorsiflexion task for healthy controls and stroke patients.

Results

The simulation results showed that the TDMIC method can better detect the direction of information interaction compared with TE and TDMI methods. For healthy controls, the beta band (14–30 Hz) had higher information flow in FCMC than the gamma band (31–45 Hz). Furthermore, the beta-band total and nonlinear information flow in the descending direction (EEG to EMG) was significantly higher than that in the ascending direction (EMG to EEG), whereas in the gamma band the ascending direction had significantly higher information flow than the descending direction. Additionally, we found that the strong bidirectional information flow mainly acted on Cz, C3, CP3, P3 and CPz. Compared to controls, both the beta-and gamma-band bidirectional total and nonlinear information flows of the stroke group were significantly weaker. There is no significant difference in the direction of beta- and gamma-band information flow in stroke group.

Conclusions

The proposed method could effectively identify the information interaction between short time series. According to our experiment, the beta band mainly passes downward motor control information while the gamma band features upward sensory feedback information delivery. Our observation demonstrate that the center and contralateral sensorimotor cortex play a major role in lower limb motor control. The study further demonstrates that brain damage caused by stroke disrupts the bidirectional information interaction between cortex and effector muscles in the sensorimotor system, leading to motor dysfunction.

Inferring network properties from time series using transfer entropy and mutual information: Validation of multivariate versus bivariate approaches

Functional and effective networks inferred from time series are at the core of network neuroscience. Interpreting properties of these networks requires inferred network models to reflect key underlying structural features. However, even a few spurious links can severely distort network measures, posing a challenge for functional connectomes. We study the extent to which micro- and macroscopic properties of underlying networks can be inferred by algorithms based on mutual information and bivariate/multivariate transfer entropy. The validation is performed on two macaque connectomes and on synthetic networks with various topologies (regular lattice, small-world, random, scale-free, modular). Simulations are based on a neural mass model and on autoregressive dynamics (employing Gaussian estimators for direct comparison to functional connectivity and Granger causality). We find that multivariate transfer entropy captures key properties of all network structures for longer time series. Bivariate methods can achieve higher recall (sensitivity) for shorter time series but are unable to control false positives (lower specificity) as available data increases. This leads to overestimated clustering, small-world, and rich-club coefficients, underestimated shortest path lengths and hub centrality, and fattened degree distribution tails. Caution should therefore be used when interpreting network properties of functional connectomes obtained via correlation or pairwise statistical dependence measures, rather than more holistic (yet data-hungry) multivariate models.

We compare bivariate and multivariate methods for inferring networks from time series, which are generated using a neural mass model and autoregressive dynamics. We assess their ability to reproduce key properties of the underlying structural network. Validation is performed on two macaque connectomes and on synthetic networks with various topologies (regular lattice, small-world, random, scale-free, modular). Even a few spurious links can severely bias key network properties. Multivariate transfer entropy performs best on all topologies for longer time series.

A protein structural study based on the centrality analysis of protein sequence feature networks

In this paper, we use network approaches to analyze the relations between protein sequence features for the top hierarchical classes of CATH and SCOP. We use fundamental connectivity measures such as correlation (CR), normalized mutual information rate (nMIR), and transfer entropy (TE) to analyze the pairwise-relationships between the protein sequence features, and use centrality measures to analyze weighted networks constructed from the relationship matrices. In the centrality analysis, we find both commonalities and differences between the different protein 3D structural classes. Results show that all top hierarchical classes of CATH and SCOP present strong non-deterministic interactions for the composition and arrangement features of Cystine (C), Methionine (M), Tryptophan (W), and also for the arrangement features of Histidine (H). The different protein 3D structural classes present different preferences in terms of their centrality distributions and significant features.

Noise-assisted Multivariate Empirical Mode Decomposition based Causal Decomposition for brain-physiological network in bivariate and multiscale time series

Objective.Noise-assisted Multivariate Empirical Mode Decomposition (NA-MEMD) based Causal Decomposition depicts a cause and effect relationship that is not based on the term of prediction, but rather on the phase dependence of time series. Here, we present the NA-MEMD based Causal Decomposition approach according to the covariation and power views traced to Hume and Kant: a priori cause-effect interaction is first acquired, and the presence of a candidate cause and of the effect is then computed from the sensory input somehow.Approach.Based on the definition of NA-MEMD based Causal Decomposition, we show such causal relation is a phase relation where the candidate causes are not merely followed by effects, but rather produce effects.Main results.The predominant methods used in neuroscience (Granger causality, EMD-based Causal Decomposition) are validated, showing the applicability of NA-MEMD based Causal Decomposition, particular to brain physiological processes in bivariate and multiscale time series.Significance.We point to the potential use in the causality inference analysis in a complex dynamic process.

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.

Refined nonuniform embedding for coupling detection in multivariate time series

State-space reconstruction is essential to analyze the dynamics and internal interactions of complex systems. However, it is difficult to estimate high-dimensional conditional mutual information and select the optimal time delays in most existing nonuniform state-space reconstruction methods. Therefore, we propose a nonuniform embedding method framed in information theory for state-space reconstruction. We use a low-dimensional approximation of conditional mutual information criterion for time delay selection, which is effectively solved by the particle swarm optimization algorithm. The obtained embedded vector has relatively strong independence and low redundancy, which better characterizes multivariable complex systems and detects coupling within complex systems. In addition, the proposed nonuniform embedding method exhibits good performance in coupling detection of linear stochastic, nonlinear stochastic, chaotic systems. In the actual application, the importance of small airports that cause delay propagation has been demonstrated by constructing the delay propagation network.

Non-Uniform Embedding Scheme and Low-Dimensional Approximation Methods for Causality Detection

Information causality measures have proven to be very effective in uncovering the connectivity patterns of multivariate systems. The non-uniform embedding (NUE) scheme has been developed to address the “curse of dimensionality”, since the estimation relies on high-dimensional conditional mutual information (CMI) terms. Although the NUE scheme is a dimension reduction technique, the estimation of high-dimensional CMIs is still required. A possible solution is the utilization of low-dimensional approximation (LA) methods for the computation of CMIs. In this study, we aim to provide useful insights regarding the effectiveness of causality measures that rely on NUE and/or on LA methods. In a comparative study, three causality detection methods are evaluated, namely partial transfer entropy (PTE) defined using uniform embedding, PTE using the NUE scheme (PTENUE), and PTE utilizing both NUE and an LA method (LATE). Results from simulations on well known coupled systems suggest the superiority of PTENUE over the other two measures in identifying the true causal effects, having also the least computational cost. The effectiveness of PTENUE is also demonstrated in a real application, where insights are presented regarding the leading forces in financial data.

Construction of functional brain connectivity networks from fMRI data with driving and modulatory inputs: an extended conditional Granger causality approach

We propose a numerical-based approach extending the conditional MVAR Granger causality (MVGC) analysis for the construction of directed connectivity networks in the presence of both exogenous/stimuli and modulatory inputs. The performance of the proposed scheme is validated using both synthetic stochastic data considering also the influence of haemodynamics latencies and a benchmark fMRI dataset related to the role of attention in the perception of visual motion. The particular fMRI dataset has been used in many studies to evaluate alternative model hypotheses using the Dynamic Causal Modelling (DCM) approach. Based on the use of the Bayes factor, we show that the obtained GC connectivity network compares well to a reference model that has been selected through DCM analysis among other candidate models. Thus, our findings suggest that the proposed scheme can be successfully used as a stand-alone or complementary to DCM approach to find directed causal connectivity patterns in task-related fMRI studies.

Risk Evaluation for a Manufacturing Process Based on a Directed Weighted Network

The quality of a manufacturing process can be represented by the complex coupling relationship between quality characteristics, which is defined by the directed weighted network to evaluate the risk of the manufacturing process. A multistage manufacturing process model is established to extract the quality information, and the quality characteristics of each process are mapped to nodes of the network. The mixed embedded partial conditional mutual information (PMIME) is used to analyze the causal effect between quality characteristics, wherein the causal relationships are mapped as the directed edges, while the magnitudes of the causal effects are defined as the weight of edges. The node centrality is measured based on information entropy theory, and the influence of a node is divided into two parts, which are local and indirect effects. Moreover, the entropy value of the directed weighted network is determined according to the weighted average of the centrality of the nodes, and this value is defined as the risk of the manufacturing process. Finally, the method is verified through a public dataset.

Forecasting high-dimensional dynamics exploiting suboptimal embeddings

Delay embedding—a method for reconstructing dynamical systems by delay coordinates—is widely used to forecast nonlinear time series as a model-free approach. When multivariate time series are observed, several existing frameworks can be applied to yield a single forecast combining multiple forecasts derived from various embeddings. However, the performance of these frameworks is not always satisfactory because they randomly select embeddings or use brute force and do not consider the diversity of the embeddings to combine. Herein, we develop a forecasting framework that overcomes these existing problems. The framework exploits various “suboptimal embeddings” obtained by minimizing the in-sample error via combinatorial optimization. The framework achieves the best results among existing frameworks for sample toy datasets and a real-world flood dataset. We show that the framework is applicable to a wide range of data lengths and dimensions. Therefore, the framework can be applied to various fields such as neuroscience, ecology, finance, fluid dynamics, weather, and disaster prevention.

Detecting Causality in Multivariate Time Series via Non-Uniform Embedding

Causal analysis based on non-uniform embedding schemes is an important way to detect the underlying interactions between dynamic systems. However, there are still some obstacles to estimating high-dimensional conditional mutual information and forming optimal mixed embedding vector in traditional non-uniform embedding schemes. In this study, we present a new non-uniform embedding method framed in information theory to detect causality for multivariate time series, named LM-PMIME, which integrates the low-dimensional approximation of conditional mutual information and the mixed search strategy for the construction of the mixed embedding vector. We apply the proposed method to simulations of linear stochastic, nonlinear stochastic, and chaotic systems, demonstrating its superiority over partial conditional mutual information from mixed embedding (PMIME) method. Moreover, the proposed method works well for multivariate time series with weak coupling strengths, especially for chaotic systems. In the actual application, we show its applicability to epilepsy multichannel electrocorticographic recordings.

Evaluation of Granger Causality Measures for Constructing Networks from Multivariate Time Series

Granger causality and variants of this concept allow the study of complex dynamical systems as networks constructed from multivariate time series. In this work, a large number of Granger causality measures used to form causality networks from multivariate time series are assessed. These measures are in the time domain, such as model-based and information measures, the frequency domain, and the phase domain. The study aims also to compare bivariate and multivariate measures, linear and nonlinear measures, as well as the use of dimension reduction in linear model-based measures and information measures. The latter is particular relevant in the study of high-dimensional time series. For the performance of the multivariate causality measures, low and high dimensional coupled dynamical systems are considered in discrete and continuous time, as well as deterministic and stochastic. The measures are evaluated and ranked according to their ability to provide causality networks that match the original coupling structure. The simulation study concludes that the Granger causality measures using dimension reduction are superior and should be preferred particularly in studies involving many observed variables, such as multi-channel electroencephalograms and financial markets.

Large-scale directed network inference with multivariate transfer entropy and hierarchical statistical testing

Network inference algorithms are valuable tools for the study of large-scale neuroimaging datasets. Multivariate transfer entropy is well suited for this task, being a model-free measure that captures nonlinear and lagged dependencies between time series to infer a minimal directed network model. Greedy algorithms have been proposed to efficiently deal with high-dimensional datasets while avoiding redundant inferences and capturing synergistic effects. However, multiple statistical comparisons may inflate the false positive rate and are computationally demanding, which limited the size of previous validation studies. The algorithm we present-as implemented in the IDTxl open-source software-addresses these challenges by employing hierarchical statistical tests to control the family-wise error rate and to allow for efficient parallelization. The method was validated on synthetic datasets involving random networks of increasing size (up to 100 nodes), for both linear and nonlinear dynamics. The performance increased with the length of the time series, reaching consistently high precision, recall, and specificity (>98% on average) for 10,000 time samples. Varying the statistical significance threshold showed a more favorable precision-recall trade-off for longer time series. Both the network size and the sample size are one order of magnitude larger than previously demonstrated, showing feasibility for typical EEG and magnetoencephalography experiments.

Nonuniform State Space Reconstruction for Multivariate Chaotic Time Series

State space reconstruction is the foundation of chaotic system modeling. Selection of reconstructed variables is essential to the analysis and prediction of multivariate chaotic time series. As most existing state space reconstruction theorems deal with univariate time series, we have presented a novel nonuniform state space reconstruction method using information criterion for multivariate chaotic time series. We derived a new criterion based on low dimensional approximation of joint mutual information for time delay selection, which can be solved efficiently through the use of an intelligent optimization algorithm with low computation complexity. The embedding dimension is determined by conditional entropy, after which the reconstructed variables have relatively strong independence and low redundancy. The scheme, which integrates nonuniform embedding and feature selection, results in better reconstructions for multivariate chaotic systems. Moreover, the proposed nonuniform state space reconstruction method shows good performance in forecasting benchmark and actual multivariate chaotic time series.

Inferring symmetric and asymmetric interactions between animals and groups from positional data

Interactions between domestic and wild species has become a global problem of growing interest. Global Position Systems (GPS) allow collection of vast records of time series of animal spatial movement, but there is need for developing analytical methods to efficiently use this information to unravel species interactions. This study assesses different methods to infer interactions and their symmetry between individual animals, social groups or species. We used two data sets, (i) a simulated one of the movement of two grazing species under different interaction scenarios by-species and by-individual, and (ii) a real time series of GPS data on the movements of sheep and deer grazing a large moorland plot. Different time series transformations were applied to capture the behaviour of the data (convex hull area, k^th nearest neighbour distance, distance to centre of mass, Voronoi tessellation area, distance to past position) to assess their efficiency in inferring the interactions using different techniques (cross correlation, Granger causality, network properties). The results indicate that the methods are more efficient assessing by-group interaction than by-individual interaction, and different transformations produce different outputs of the nature of the interaction. Both species maintained a consistent by-species grouping structure. The results do not provide clear evidence of inter-species interaction based on the traditional framework of niche partitioning in the guild of large herbivores. In view of the transformation-dependent results, it seems that in our experimental framework both species co-exist showing complex interactions. We provide guidelines for the use of the different transformations with respect to study aims and data quality. The study attempts to provide behavioural ecologists with tools to infer animal interactions and their symmetry based on positional data recorded by visual observation, conventional telemetry or GPS technology.

A study for multiscale information transfer measures based on conditional mutual information

As the big data science develops, efficient methods are demanded for various data analysis. Granger causality provides the prime model for quantifying causal interactions. However, this theoretic model does not meet the requirement for real-world data analysis, because real-world time series are diverse whose models are usually unknown. Therefore, model-free measures such as information transfer measures are strongly desired. Here, we propose the multi-scale extension of conditional mutual information measures using MORLET wavelet, which are named the WM and WPM. The proposed measures are computational efficient and interpret information transfer by multi-scales. We use both synthetic data and real-world examples to demonstrate the efficiency of the new methods. The results of the new methods are robust and reliable. Via the simulation studies, we found the new methods outperform the wavelet extension of transfer entropy (WTE) in both computational efficiency and accuracy. The features and properties of the proposed measures are also discussed.

Optimization of relative parameters in transfer entropy estimation and application to corticomuscular coupling in humans

BACKGROUND

As a non-modeled information theoretical measure, the transfer entropy (TE) could be applied to quantitatively analyze the linear and nonlinear coupling characteristics between two observations. However, the parameters selection of TE (the parameters used in state space reconstruction and estimating Shannon entropy) has a serious influence on the accuracy of its results.

NEW METHOD

In this study, the hybrid particle swarm optimization (HPSO) was applied to improve the accuracy of TE by optimizing its parameters. In HPSO, the TE calculation and significant analysis were integrated into the fitness function, and the optimal parameters group within the parameter space could be automatically found through an iteration process.

RESULTS

The TE results computed under the parameters optimized by HPSO (HPSO-TE), was assessed with a numerical non-linear model, the neural mass model and the recorded electroencephalogram (EEG) and electromyogram (EMG) signals. Compared with TE, HPSO-TE could reduce the 'false positive' in non-linear model, and 'spurious coupling', i.e. two nonzero TEs for unidirectionally coupled systems, especially when coupling strength was weak. The robustness against noise and long time-delay was improved. Moreover, the experimental data analysis showed HPSO-TE revealed the dominant direction (EEG → EMG) in corticomuscular coupling, and had higher values than TE which showed the same dominant direction.

COMPARISON WITH EXISTING METHOD

The implication of HPSO improved the accuracy of TE in estimating the coupling strength and direction.

CONCLUSIONS

The efficiency of TE could be improved by HPSO for estimating coupling relationships, especially for weakly coupled, strong noisy and long time-delay series.

Causal decomposition in the mutual causation system

Inference of causality in time series has been principally based on the prediction paradigm. Nonetheless, the predictive causality approach may underestimate the simultaneous and reciprocal nature of causal interactions observed in real-world phenomena. Here, we present a causal-decomposition approach that is not based on prediction, but based on the covariation of cause and effect: cause is that which put, the effect follows; and removed, the effect is removed. Using empirical mode decomposition, we show that causal interaction is encoded in instantaneous phase dependency at a specific time scale, and this phase dependency is diminished when the causal-related intrinsic component is removed from the effect. Furthermore, we demonstrate the generic applicability of our method to both stochastic and deterministic systems, and show the consistency of causal-decomposition method compared to existing methods, and finally uncover the key mode of causal interactions in both modelled and actual predator–prey systems.

Causality inference in time series analysis based on temporal precedence principle between cause and effect fails to detect mutual causal interactions. Here, Yang et al. introduce a causal decomposition approach based on the covariation principle of cause and effect that overcomes this limitation.

Detecting directional couplings from multivariate flows by the joint distance distribution

The identification of directional couplings (or drive-response relationships) in the analysis of interacting nonlinear systems is an important piece of information to understand their dynamics. This task is especially challenging when the analyst's knowledge of the systems reduces virtually to time series of observations. Spurred by the success of Granger causality in econometrics, the study of cause-effect relationships (not to be confounded with statistical correlations) was extended to other fields, thus favoring the introduction of further tools such as transfer entropy. Currently, the research on old and new causality tools along with their pitfalls and applications in ever more general situations is going through a time of much activity. In this paper, we re-examine the method of the joint distance distribution to detect directional couplings between two multivariate flows. This method is based on the forced Takens theorem, and, more specifically, it exploits the existence of a continuous mapping from the reconstructed attractor of the response system to the reconstructed attractor of the driving system, an approach that is increasingly drawing the attention of the data analysts. The numerical results with Lorenz and Rössler oscillators in three different interaction networks (including hidden common drivers) are quite satisfactory, except when phase synchronization sets in. They also show that the method of the joint distance distribution outperforms the lowest dimensional transfer entropy in the cases considered. The robustness of the results to the sampling interval, time series length, observational noise, and metric is analyzed too.

Space-time nature of causality

In a causal world the direction of the time arrow dictates how past causal events in a variable X produce future effects in Y. X is said to cause an effect in Y, if the predictability (uncertainty) about the future states of Y increases (decreases) as its own past and the past of X are taken into consideration. Causality is thus intrinsic dependent on the observation of the past events of both variables involved, to the prediction (or uncertainty reduction) of future event of the other variable. We will show that this temporal notion of causality leads to another natural spatiotemporal definition for it, and that can be exploited to detect the arrow of influence from X to Y, either by considering shorter time-series of X and longer time-series of Y (an approach that explores the time nature of causality) or lower precision measured time-series in X and higher precision measured time-series in Y (an approach that explores the spatial nature of causality). Causality has thus space and time signatures, causing a break of symmetry in the topology of the probabilistic space, or causing a break of symmetry in the length of the measured time-series, a consequence of the fact that information flows from X to Y.

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.