Comparison of six methods for the detection of causality in a bivariate time series

Decomposing causality into its synergistic, unique, and redundant components

Causality lies at the heart of scientific inquiry, serving as the fundamental basis for understanding interactions among variables in physical systems. Despite its central role, current methods for causal inference face significant challenges due to nonlinear dependencies, stochastic interactions, self-causation, collider effects, and influences from exogenous factors, among others. While existing methods can effectively address some of these challenges, no single approach has successfully integrated all these aspects. Here, we address these challenges with SURD: Synergistic-Unique-Redundant Decomposition of causality. SURD quantifies causality as the increments of redundant, unique, and synergistic information gained about future events from past observations. The formulation is non-intrusive and applicable to both computational and experimental investigations, even when samples are scarce. We benchmark SURD in scenarios that pose significant challenges for causal inference and demonstrate that it offers a more reliable quantification of causality compared to previous methods.

On formal limitations of causal ecological networks

Causal multivariate time-series analysis, combined with network theory, provide a powerful tool for studying complex ecological interactions. However, these methods have limitations often underestimated when used in graphical modelling of ecological systems. In this opinion article, I examine the relationship between formal logic methods used to describe causal networks and their inherent statistical and epistemological limitations. I argue that while these methods offer valuable insights, they are restricted by axiomatic assumptions, statistical constraints and the incompleteness of our knowledge. To prove that, I first consider causal networks as formal systems, define causality and formalize their axioms in terms of modal logic and use ecological counterexamples to question the axioms. I also highlight the statistical limitations when using multivariate time-series analysis and Granger causality to develop ecological networks, including the potential for spurious correlations among other data characteristics. Finally, I draw upon Gödel's incompleteness theorems to highlight the inherent limits of fully understanding complex networks as formal systems and conclude that causal ecological networks are subject to initial rules and data characteristics and, as any formal system, will never fully capture the intricate complexities of the systems they represent. This article is part of the theme issue 'Connected interactions: enriching food web research by spatial and social interactions'.

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.

Using empirical dynamic modeling to identify the impact of meteorological factors on hemorrhagic fever with renal syndrome in Weifang, Northeastern China, from 2011 to 2020

BACKGROUND

Hemorrhagic Fever with Renal Syndrome (HFRS) continues to pose a significant public health threat to the well-being of the population. Given that the spread of HFRS is susceptible to meteorological factors, we aim to probe into the meteorological drivers of HFRS. Thus, novel techniques that can discern time-delayed non-linear relationships from nonlinear dynamical systems are compulsory.

METHODS

We analyze the epidemiological features of HFRS in Weifang City, 2011-2020, via the employment of the Empirical Dynamic Modeling (EDM) method. Our analysis delves into the intricate web of time-delayed non-linear associations between meteorological factors and HFRS. Additionally, we investigate the repercussions of minor perturbations in meteorological variables on future HFRS incidence.

RESULTS

A total of 2515 HFRS cases were reported in Weifang from 2011 to 2020. The number of cases per week was 4.81, and the average weekly incidence was 0.52 per 1,000,000. The propagation of HFRS is significantly impacted by the mean weekly temperature, relative humidity, cumulative rainfall, and wind speed, and the ρCCM converges to 0.55,0.48,0.38 and 0.39, respectively. The graphical representation of the relationship between temperature (lagged by 2 weeks) and the incidence of HFRS exhibits an inverted U-shaped curve, whereby the incidence of HFRS culminates as the temperature reaches 10 °C. Moreover, temperature, relative humidity, cumulative rainfall, and wind speed exhibit a positive correlation with HFRS incidence, with a time lag of 4-6 months.

CONCLUSIONS

Our discoveries suggest that meteorological factors can drive the transmission of HFRS both at a macroscopic and microscopic scale. Prospective alterations in meteorological conditions, for instance, elevations in temperature, relative humidity, and precipitation will instigate an upsurge in the incidence of HFRS after 4-6 months, and thus, timely public health measures should be taken to mitigate these changes.

Testing dynamic correlations and nonlinearity in bivariate time series through information measures and surrogate data analysis

The increasing availability of time series data depicting the evolution of physical system properties has prompted the development of methods focused on extracting insights into the system behavior over time, discerning whether it stems from deterministic or stochastic dynamical systems. Surrogate data testing plays a crucial role in this process by facilitating robust statistical assessments. This ensures that the observed results are not mere occurrences by chance, but genuinely reflect the inherent characteristics of the underlying system. The initial process involves formulating a null hypothesis, which is tested using surrogate data in cases where assumptions about the underlying distributions are absent. A discriminating statistic is then computed for both the original data and each surrogate data set. Significantly deviating values between the original data and the surrogate data ensemble lead to the rejection of the null hypothesis. In this work, we present various surrogate methods designed to assess specific statistical properties in random processes. Specifically, we introduce methods for evaluating the presence of autodependencies and nonlinear dynamics within individual processes, using Information Storage as a discriminating statistic. Additionally, methods are introduced for detecting coupling and nonlinearities in bivariate processes, employing the Mutual Information Rate for this purpose. The surrogate methods introduced are first tested through simulations involving univariate and bivariate processes exhibiting both linear and nonlinear dynamics. Then, they are applied to physiological time series of Heart Period (RR intervals) and respiratory flow (RESP) variability measured during spontaneous and paced breathing. Simulations demonstrated that the proposed methods effectively identify essential dynamical features of stochastic systems. The real data application showed that paced breathing, at low breathing rate, increases the predictability of the individual dynamics of RR and RESP and dampens nonlinearity in their coupled dynamics.

Robust inference of causality in high-dimensional dynamical processes from the Information Imbalance of distance ranks

We introduce an approach which allows detecting causal relationships between variables for which the time evolution is available. Causality is assessed by a variational scheme based on the Information Imbalance of distance ranks, a statistical test capable of inferring the relative information content of different distance measures. We test whether the predictability of a putative driven system Y can be improved by incorporating information from a potential driver system X, without explicitly modeling the underlying dynamics and without the need to compute probability densities of the dynamic variables. This framework makes causality detection possible even between high-dimensional systems where only few of the variables are known or measured. Benchmark tests on coupled chaotic dynamical systems demonstrate that our approach outperforms other model-free causality detection methods, successfully handling both unidirectional and bidirectional couplings. We also show that the method can be used to robustly detect causality in human electroencephalography data.

The effective connectivity analysis of fMRI based on asymmetric detection of transfer brain entropy

It is important to explore causal relationships in functional magnetic resonance imaging study. However, the traditional effective connectivity analysis method is easy to produce false causality, and the detection accuracy needs to be improved. In this paper, we introduce a novel functional magnetic resonance imaging effective connectivity method based on the asymmetry detection of transfer entropy, which quantifies the disparity in predictive information between forward and backward time, subsequently normalizing this disparity to establish a more precise criterion for detecting causal relationships while concurrently reducing computational complexity. Then, we evaluate the effectiveness of this method on the simulated data with different level of nonlinearity, and the results demonstrated that the proposed method outperforms others methods on the detection of both linear and nonlinear causal relationships, including Granger Causality, Partial Granger Causality, Kernel Granger Causality, Copula Granger Causality, and traditional transfer entropy. Furthermore, we applied it to study the effective connectivity of brain functional activities in seafarers. The results showed that there are significantly different causal relationships between different brain regions in seafarers compared with non-seafarers, such as Temporal lobe related to sound and auditory information processing, Hippocampus related to spatial navigation, Precuneus related to emotion processing as well as Supp_Motor_Area associated with motor control and coordination, which reflects the occupational specificity of brain function of seafarers.

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.

Quantifying the diversity of multiple time series with an ordinal symbolic approach

The main motivation of this paper is to introduce the ordinal diversity, a symbolic tool able to quantify the degree of diversity of multiple time series. Analytical, numerical, and experimental analyses illustrate the utility of this measure to quantify how diverse, from an ordinal perspective, a set of many time series is. We have shown that ordinal diversity is able to characterize dynamical richness and dynamical transitions in stochastic processes and deterministic systems, including chaotic regimes. This ordinal tool also serves to identify optimal operating conditions in the machine learning approach of reservoir computing. These results allow us to envision potential applications for the handling and characterization of large amounts of data, paving the way for addressing some of the most pressing issues facing the current big data paradigm.

Inferring connectivity of an oscillatory network via the phase dynamics reconstruction

We review an approach for reconstructing oscillatory networks' undirected and directed connectivity from data. The technique relies on inferring the phase dynamics model. The central assumption is that we observe the outputs of all network nodes. We distinguish between two cases. In the first one, the observed signals represent smooth oscillations, while in the second one, the data are pulse-like and can be viewed as point processes. For the first case, we discuss estimating the true phase from a scalar signal, exploiting the protophase-to-phase transformation. With the phases at hand, pairwise and triplet synchronization indices can characterize the undirected connectivity. Next, we demonstrate how to infer the general form of the coupling functions for two or three oscillators and how to use these functions to quantify the directional links. We proceed with a different treatment of networks with more than three nodes. We discuss the difference between the structural and effective phase connectivity that emerges due to high-order terms in the coupling functions. For the second case of point-process data, we use the instants of spikes to infer the phase dynamics model in the Winfree form directly. This way, we obtain the network's coupling matrix in the first approximation in the coupling strength.

Causal inference from cross-sectional earth system data with geographical convergent cross mapping

Causal inference in complex systems has been largely promoted by the proposal of some advanced temporal causation models. However, temporal models have serious limitations when time series data are not available or present insignificant variations, which causes a common challenge for earth system science. Meanwhile, there are few spatial causation models for fully exploring the rich spatial cross-sectional data in Earth systems. The generalized embedding theorem proves that observations can be combined together to construct the state space of the dynamic system, and if two variables are from the same dynamic system, they are causally linked. Inspired by this, here we show a Geographical Convergent Cross Mapping (GCCM) model for spatial causal inference with spatial cross-sectional data-based cross-mapping prediction in reconstructed state space. Three typical cases, where clearly existing causations cannot be measured through temporal models, demonstrate that GCCM could detect weak-moderate causations when the correlation is not significant. When the coupling between two variables is significant and strong, GCCM is advantageous in identifying the primary causation direction and better revealing the bidirectional asymmetric causation, overcoming the mirroring effect.

Usefulness and limitations of convergent cross sorting and continuity scaling methods for their application in simulated and real-world time series

Causality detection methods are valuable tools for detecting causal links in complex systems. The efficiency of continuity scaling (CS) and the convergent cross sorting (CSS) methods to detect causality was analysed. Usefulness and limitations of both methods in their application to simulated and real-world time series was explored under different scenarios. We find that CS is more robust and efficient than the CSS method for all simulated systems, even when increasing noise levels were considered. Both methods were not able to infer causality when time series with a marked difference in their main frequencies were analysed. Minimum time-series length required for the detection of a causal link depends on intrinsic system dynamics and on the method selected to detect it. Using simulated time series, only the CS method was capable to detect bidirectional causality. Causality detection, using the CS method, should at least include: (i) causality strength convergence analysis, (ii) statistical tests of significance, (iii) time-series standardization, and (iv) causality strength ratios as a strength indicator of relative causality between systems. Causality cannot be detected by either method in simulated time series that exhibit generalized synchronization.

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.

Relationship between Schreiber's transfer entropy and Liang-Kleeman information flow from the perspective of stochastic thermodynamics

Schreiber's transfer entropy is an important index for investigating the causal relationship between random variables. The Liang-Kleeman information flow is another analysis to demonstrate the causality within dynamical systems. Horowitz's information flow is introduced through multicomponent stochastic thermodynamics. In this study, I elucidate the relationship between Schreiber's transfer entropy and the Liang-Kleeman information flow through Horowitz's information flow. I consider the case in which the system changes according to the stochastic differential equation. I find that the Liang-Kleeman and Horowitz information flows differ by a term derived from the stochastic fluctuation. I also show that Schreiber's transfer entropy is not less than Horowitz's information flow. This study helps unify various indexes that determine the causal relationship between variables.

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.

Optimal time lags from causal prediction model help stratify and forecast nervous system pathology

Traditional clinical approaches diagnose disorders of the nervous system using standardized observational criteria. Although aiming for homogeneity of symptoms, this method often results in highly heterogeneous disorders. A standing question thus is how to automatically stratify a given random cohort of the population, such that treatment can be better tailored to each cluster's symptoms, and severity of any given group forecasted to provide neuroprotective therapies. In this work we introduce new methods to automatically stratify a random cohort of the population composed of healthy controls of different ages and patients with different disorders of the nervous systems. Using a simple walking task and measuring micro-fluctuations in their biorhythmic motions, we combine non-linear causal network connectivity analyses in the temporal and frequency domains with stochastic mapping. The methods define a new type of internal motor timings. These are amenable to create personalized clinical interventions tailored to self-emerging clusters signaling fundamentally different types of gait pathologies. We frame our results using the principle of reafference and operationalize them using causal prediction, thus renovating the theory of internal models for the study of neuromotor control.

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.

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.

The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

Inferring causal networks of dynamical systems through transient dynamics and perturbation

Inferring causal relations from time series measurements is an ill-posed mathematical problem, where typically an infinite number of potential solutions can reproduce the given data. We explore in depth a strategy to disambiguate between possible underlying causal networks by perturbing the network, where the forcings are either targeted or applied at random. The resulting transient dynamics provide the critical information necessary to infer causality. Two methods are shown to provide accurate causal reconstructions: Granger causality (GC) with perturbations, and our proposed perturbation cascade inference (PCI). Perturbed GC is capable of inferring smaller networks under low coupling strength regimes. Our proposed PCI method demonstrated consistently strong performance in inferring causal relations for small (2-5 node) and large (10-20 node) networks, with both linear and nonlinear dynamics. Thus, the ability to apply a large and diverse set of perturbations to the network is critical for successfully and accurately determining causal relations and disambiguating between various viable networks.

Implementation of two causal methods based on predictions in reconstructed state spaces

If deterministic dynamics is dominant in the data, then methods based on predictions in reconstructed state spaces can serve to detect causal relationships between and within the systems. Here we introduce two algorithms for such causal analysis. They are designed to detect causality from two time series but are potentially also applicable in a multivariate context. The first method is based on cross-predictions, and the second one on the so-called mixed predictions. In terms of performance, the cross-prediction method is considerably faster and less prone to false negatives. The predictability improvement method is slower, but in addition to causal detection, in a multivariate scenario, it also reveals which specific observables can help the most if we want to improve prediction. The study also highlights cases where our methods and state-space approaches generally seem to lose reliability. We propose a new perspective on these situations, namely that the variables under investigation have weak observability due to the complex nonlinear information flow in the system. Thus, in such cases, the failure of causality detection cannot be attributed to the methods themselves but to the use of data that do not allow reliable reconstruction of the underlying dynamics.

A Refinement of Recurrence Analysis to Determine the Time Delay of Causality in Presence of External Perturbations

This article describes a refinement of recurrence analysis to determine the delay in the causal influence between a driver and a target, in the presence of additional perturbations affecting the time series of the response observable. The methodology is based on the definition of a new type of recurrence plots, the Conditional Joint Recurrence plot. The potential of the proposed approach resides in the great flexibility of recurrence plots themselves, which allows extending the technique to more than three quantities. Autoregressive time series, both linear and nonlinear, with different couplings and percentage of additive Gaussian noise have been investigated in detail, with and without outliers. The approach has also been applied to the case of synthetic periodic signals, representing realistic situations of synchronization experiments in thermonuclear fusion. The results obtained have been very positive; the proposed Conditional Joint Recurrence plots have always managed to identify the right interval of the causal influences and are very competitive with alternative techniques such as the Conditional Transfer Entropy.

Image-Based Methods to Investigate Synchronization between Time Series Relevant for Plasma Fusion Diagnostics

Advanced time series analysis and causality detection techniques have been successfully applied to the assessment of synchronization experiments in tokamaks, such as Edge Localized Modes (ELMs) and sawtooth pacing. Lag synchronization is a typical strategy for fusion plasma instability control by pace-making techniques. The major difficulty, in evaluating the efficiency of the pacing methods, is the coexistence of the causal effects with the periodic or quasi-periodic nature of the plasma instabilities. In the present work, a set of methods based on the image representation of time series, are investigated as tools for evaluating the efficiency of the pace-making techniques. The main options rely on the Gramian Angular Field (GAF), the Markov Transition Field (MTF), previously used for time series classification, and the Chaos Game Representation (CGR), employed for the visualization of large collections of long time series. The paper proposes an original variation of the Markov Transition Matrix, defined for a couple of time series. Additionally, a recently proposed method, based on the mapping of time series as cross-visibility networks and their representation as images, is included in this study. The performances of the method are evaluated on synthetic data and applied to JET measurements.

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.

Detecting causality from time series in a machine learning framework

Detecting causality from observational data is a challenging problem. Here, we propose a machine learning based causality approach, Reservoir Computing Causality (RCC), in order to systematically identify causal relationships between variables. We demonstrate that RCC is able to identify the causal direction, coupling delay, and causal chain relations from time series. Compared to a well-known phase space reconstruction based causality method, Extended Convergent Cross Mapping, RCC does not require the estimation of the embedding dimension and delay time. Moreover, RCC has three additional advantages: (i) robustness to noisy time series; (ii) computational efficiency; and (iii) seamless causal inference from high-dimensional data. We also illustrate the power of RCC in identifying remote causal interactions of high-dimensional systems and demonstrate its usability on a real-world example using atmospheric circulation data. Our results suggest that RCC can accurately detect causal relationships in complex systems.

On the Potential of Time Delay Neural Networks to Detect Indirect Coupling between Time Series

Determining the coupling between systems remains a topic of active research in the field of complex science. Identifying the proper causal influences in time series can already be very challenging in the trivariate case, particularly when the interactions are non-linear. In this paper, the coupling between three Lorenz systems is investigated with the help of specifically designed artificial neural networks, called time delay neural networks (TDNNs). TDNNs can learn from their previous inputs and are therefore well suited to extract the causal relationship between time series. The performances of the TDNNs tested have always been very positive, showing an excellent capability to identify the correct causal relationships in absence of significant noise. The first tests on the time localization of the mutual influences and the effects of Gaussian noise have also provided very encouraging results. Even if further assessments are necessary, the networks of the proposed architecture have the potential to be a good complement to the other techniques available in the market for the investigation of mutual influences between time series.

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.

Correlation Dimension Detects Causal Links in Coupled Dynamical Systems

It is becoming increasingly clear that causal analysis of dynamical systems requires different approaches than, for example, causal analysis of interconnected autoregressive processes. In this study, a correlation dimension estimated in reconstructed state spaces is used to detect causality. If deterministic dynamics plays a dominant role in data then the method based on the correlation dimension can serve as a fast and reliable way to reveal causal relationships between and within the systems. This study demonstrates that the method, unlike most other causal approaches, detects causality well, even for very weak links. It can also identify cases of uncoupled systems that are causally affected by a hidden common driver.

Causality Detection Methods Applied to the Investigation of Malaria Epidemics

Malaria, a disease with major health and socio-economic impacts, is driven by multiple factors, including a complex interaction with various climatic variables. In this paper, five methods developed for inferring causal relations between dynamic processes based on the information encapsulated in time series are applied on cases previously studied in literature by means of statistical methods. The causality detection techniques investigated in the paper are: a version of the kernel Granger causality, transfer entropy, recurrence plot, causal decomposition and complex networks. The methods provide coherent results giving a quite good confidence in the conclusions.

Transfer entropy computation using the Perron-Frobenius operator

We propose a method for computing the transfer entropy between time series using Ulam's approximation of the Perron-Frobenius (transfer) operator associated with the map generating the dynamics. Our method differs from standard transfer entropy estimators in that the invariant measure is estimated not directly from the data points, but from the invariant distribution of the transfer operator approximated from the data points. For sparse time series and low embedding dimension, the transfer operator is approximated using a triangulation of the attractor, whereas for data-rich time series or higher embedding dimension, we use a faster grid approach. We compare the performance of our methods with existing estimators such as the k nearest neighbors method and kernel density estimation method, using coupled instances of well known chaotic systems: coupled logistic maps and a coupled Rössler-Lorenz system. We find that our estimators are robust against moderate levels of noise. For sparse time series with less than 100 observations and low embedding dimension, our triangulation estimator shows improved ability to detect coupling directionality, relative to standard transfer entropy estimators.

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.