Causality, dynamical systems and the arrow of time

Linear and nonlinear causality in financial markets

Identifying and quantifying co-dependence between financial instruments is a key challenge for researchers and practitioners in the financial industry. Linear measures such as the Pearson correlation are still widely used today, although their limited explanatory power is well known. In this paper, we present a much more general framework for assessing co-dependencies by identifying linear and nonlinear causalities in the complex system of financial markets. To do so, we use two different causal inference methods, transfer entropy and convergent cross-mapping, and employ Fourier transform surrogates to separate their linear and nonlinear contributions. We find that stock indices in Germany and the U.S. exhibit a significant degree of nonlinear causality and that correlation, while a very good proxy for linear causality, disregards nonlinear effects and hence underestimates causality itself. The presented framework enables the measurement of nonlinear causality, the correlation-causality fallacy, and motivates how causality can be used for inferring market signals, pair trading, and risk management of portfolios. Our results suggest that linear and nonlinear causality can be used as early warning indicators of abnormal market behavior, allowing for better trading strategies and risk management.

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.

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.

Understanding Training Load as Exposure and Dose

Various terms used in sport and exercise science, and medicine, are derived from other fields such as epidemiology, pharmacology and causal inference. Conceptual and nomological frameworks have described training load as a multidimensional construct manifested by two causally related subdimensions: external and internal training load. In this article, we explain how the concepts of training load and its subdimensions can be aligned to classifications used in occupational medicine and epidemiology, where exposure can also be differentiated into external and internal dose. The meanings of terms used in epidemiology such as exposure, external dose, internal dose and dose-response are therefore explored from a causal perspective and their underlying concepts are contextualised to the physical training process. We also explain how these concepts can assist in the validation process of training load measures. Specifically, to optimise training (i.e. within a causal context), a measure of exposure should be reflective of the mediating mechanisms of the primary outcome. Additionally, understanding the difference between intermediate and surrogate outcomes allows for the correct investigation of the effects of exposure measures and their interpretation in research and applied settings. Finally, whilst the dose-response relationship can provide evidence of the validity of a measure, conceptual and computational differentiation between causal (explanatory) and non-causal (descriptive and predictive) dose-response relationships is needed. Regardless of how sophisticated or "advanced" a training load measure (and metric) appears, in a causal context, if it cannot be connected to a plausible mediator of a relevant response (outcome), it is likely of little use in practice to support and optimise the training process.

Causal Inference in Time Series in Terms of Rényi Transfer Entropy

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

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.

Generative formalism of causality quantifiers for processes

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

Embedding entropy: a nonlinear measure of dynamical causality

Research on concepts and computational methods of causality has a long history, and there are various concepts of causality as well as corresponding computing algorithms based on measured data. Here, by considering causes and effects from a dynamical perspective, we present a unified mathematical framework for the so-called dynamical causality (DC), which can detect causal interactions over time; notably, this framework covers Granger causality, transfer entropy, embedding causality and their conditional versions. Based on this framework, we further propose a causality criterion called embedding entropy (EE) to measure the DC between two variables. Moreover, its conditional version, conditional embedding entropy (cEE), is also derived for detecting conditional/direct causality. The significant advantages of EE and cEE are that they can be employed for solving not only nonlinear causal inference but also the non-separability problem, and they can reduce the scale bias in numerical calculation. The performance and robustness of EE and cEE were demonstrated through numerical simulations, and causal inference on various real-world datasets validated their effectiveness.

Time-irreversibility test for random-length time series: The matching-time approach applied to DNA

In this work, we implement the so-called matching-time estimators for estimating the entropy rate as well as the entropy production rate for symbolic sequences. These estimators are based on recurrence properties of the system, which have been shown to be appropriate for testing irreversibility, especially when the sequences have large correlations or memory. Based on limit theorems for matching times, we derive a maximum likelihood estimator for the entropy rate by assuming that we have a set of moderately short symbolic time series of finite random duration. We show that the proposed estimator has several properties that make it adequate for estimating the entropy rate and entropy production rate (or for testing the irreversibility) when the sample sequences have different lengths, such as the coding sequences of DNA. We test our approach with controlled examples of Markov chains, non-linear chaotic maps, and linear and non-linear autoregressive processes. We also implement our estimators for genomic sequences to show that the degree of irreversibility of coding sequences in human DNA is significantly larger than that for the corresponding non-coding sequences.

Localizing category-related information in speech with multi-scale analyses

Measurements of the physical outputs of speech-vocal tract geometry and acoustic energy-are high-dimensional, but linguistic theories posit a low-dimensional set of categories such as phonemes and phrase types. How can it be determined when and where in high-dimensional articulatory and acoustic signals there is information related to theoretical categories? For a variety of reasons, it is problematic to directly quantify mutual information between hypothesized categories and signals. To address this issue, a multi-scale analysis method is proposed for localizing category-related information in an ensemble of speech signals using machine learning algorithms. By analyzing how classification accuracy on unseen data varies as the temporal extent of training input is systematically restricted, inferences can be drawn regarding the temporal distribution of category-related information. The method can also be used to investigate redundancy between subsets of signal dimensions. Two types of theoretical categories are examined in this paper: phonemic/gestural categories and syntactic relative clause categories. Moreover, two different machine learning algorithms were examined: linear discriminant analysis and neural networks with long short-term memory units. Both algorithms detected category-related information earlier and later in signals than would be expected given standard theoretical assumptions about when linguistic categories should influence speech. The neural network algorithm was able to identify category-related information to a greater extent than the discriminant analyses.

Measuring the importance of individual units in producing the collective behavior of a complex network

A quantitative evaluation of the contribution of individual units in producing the collective behavior of a complex network can allow us to understand the potential damage to the structure integrity due to the failure of local nodes. Given a time series for each unit, a natural way to do this is to find the information flowing from the unit of concern to the rest of the network. In this study, we show that this flow can be rigorously derived in the setting of a continuous-time dynamical system. With a linear assumption, a maximum likelihood estimator can be obtained, allowing us to estimate it in an easy way. As expected, this "cumulative information flow" does not equal the sum of the information flows to other individual units, reflecting the collective phenomenon that a group is not the addition of individual members. For the purpose of demonstration and validation, we have examined a network made of Stuart-Landau oscillators. Depending on the topology, the computed information flow may differ. In some situations, the most crucial nodes for the network are not the hubs, i.e., nodes with high degrees; they may have low degrees and, if depressed or attacked, will cause the failure of the entire network. This study can help diagnose neural network problems, control epidemic diseases, trace city traffic bottlenecks, identify the potential cause of power grid failure (e.g., the 2003 great power outage that darkened much of North America), build robust computer networks, and so forth.

Causality in Reversed Time Series: Reversed or Conserved?

The inference of causal relations between observable phenomena is paramount across scientific disciplines; however, the means for such enterprise without experimental manipulation are limited. A commonly applied principle is that of the cause preceding and predicting the effect, taking into account other circumstances. Intuitively, when the temporal order of events is reverted, one would expect the cause and effect to apparently switch roles. This was previously demonstrated in bivariate linear systems and used in design of improved causal inference scores, while such behaviour in linear systems has been put in contrast with nonlinear chaotic systems where the inferred causal direction appears unchanged under time reversal. The presented work explores the conditions under which the causal reversal happens—either perfectly, approximately, or not at all—using theoretical analysis, low-dimensional examples, and network simulations, focusing on the simplified yet illustrative linear vector autoregressive process of order one. We start with a theoretical analysis that demonstrates that a perfect coupling reversal under time reversal occurs only under very specific conditions, followed up by constructing low-dimensional examples where indeed the dominant causal direction is even conserved rather than reversed. Finally, simulations of random as well as realistically motivated network coupling patterns from brain and climate show that level of coupling reversal and conservation can be well predicted by asymmetry and anormality indices introduced based on the theoretical analysis of the problem. The consequences for causal inference are discussed.

Normalized Multivariate Time Series Causality Analysis and Causal Graph Reconstruction

Causality analysis is an important problem lying at the heart of science, and is of particular importance in data science and machine learning. An endeavor during the past 16 years viewing causality as a real physical notion so as to formulate it from first principles, however, seems to have gone unnoticed. This study introduces to the community this line of work, with a long-due generalization of the information flow-based bivariate time series causal inference to multivariate series, based on the recent advance in theoretical development. The resulting formula is transparent, and can be implemented as a computationally very efficient algorithm for application. It can be normalized and tested for statistical significance. Different from the previous work along this line where only information flows are estimated, here an algorithm is also implemented to quantify the influence of a unit to itself. While this forms a challenge in some causal inferences, here it comes naturally, and hence the identification of self-loops in a causal graph is fulfilled automatically as the causalities along edges are inferred. To demonstrate the power of the approach, presented here are two applications in extreme situations. The first is a network of multivariate processes buried in heavy noises (with the noise-to-signal ratio exceeding 100), and the second a network with nearly synchronized chaotic oscillators. In both graphs, confounding processes exist. While it seems to be a challenge to reconstruct from given series these causal graphs, an easy application of the algorithm immediately reveals the desideratum. Particularly, the confounding processes have been accurately differentiated. Considering the surge of interest in the community, this study is very timely.

Cross-Scale Causality and Information Transfer in Simulated Epileptic Seizures

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

Granger Causality on forward and Reversed Time Series

In this study, the information flow time arrow is investigated for stochastic data defined by vector autoregressive models. The time series are analyzed forward and backward by different Granger causality detection methods. Besides the normal distribution, which is usually required for the validity of Granger causality analysis, several other distributions of predictive errors are considered. A clear effect of a change in the order of cause and effect on the time-reversed series of unidirectionally connected variables was detected with standard Granger causality test (GC), when the product of the connection strength and the ratio of the predictive errors of the driver and the recipient was below a certain level, otherwise bidirectional causal connection was detected. On the other hand, opposite causal link was detected unconditionally by the methods based on the time reversal testing, but they were not able to detect correct bidirectional connection. The usefulness of the backward analysis is manifested in cases where falsely detected unidirectional connections can be rejected by applying the result obtained after the time reversal, and in cases of uncorrelated causally independent variables, where the absence of a causal link detected by GC on the original series should be confirmed on the time-reversed series.

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

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

Time-Reversibility, Causality and Compression-Complexity

Detection of the temporal reversibility of a given process is an interesting time series analysis scheme that enables the useful characterisation of processes and offers an insight into the underlying processes generating the time series. Reversibility detection measures have been widely employed in the study of ecological, epidemiological and physiological time series. Further, the time reversal of given data provides a promising tool for analysis of causality measures as well as studying the causal properties of processes. In this work, the recently proposed Compression-Complexity Causality (CCC) measure (by the authors) is shown to be free of the assumption that the "cause precedes the effect", making it a promising tool for causal analysis of reversible processes. CCC is a data-driven interventional measure of causality (second rung on the Ladder of Causation) that is based on Effort-to-Compress (ETC), a well-established robust method to characterize the complexity of time series for analysis and classification. For the detection of the temporal reversibility of processes, we propose a novel measure called the Compressive Potential based Asymmetry Measure. This asymmetry measure compares the probability of the occurrence of patterns at different scales between the forward-time and time-reversed process using ETC. We test the performance of the measure on a number of simulated processes and demonstrate its effectiveness in determining the asymmetry of real-world time series of sunspot numbers, digits of the transcedental number π and heart interbeat interval variability.

Transfer entropies within dynamical effects framework

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

Multiscale measures of phase-space trajectories

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

Explicit Granger causality in kernel Hilbert spaces

Granger causality (GC) is undoubtedly the most widely used method to infer cause-effect relations from observational time series. Several nonlinear alternatives to GC have been proposed based on kernel methods. We generalize kernel Granger causality by considering the variables' cross-relations explicitly in Hilbert spaces. The framework is shown to generalize the linear and kernel GC methods and comes with tighter bounds of performance based on Rademacher complexity. We successfully evaluate its performance in standard dynamical systems, as well as to identify the arrow of time in coupled Rössler systems, and it is exploited to disclose the El Niño-Southern Oscillation phenomenon footprints on soil moisture globally.

Atmospheric Temperature and CO2: Hen-Or-Egg Causality?

It is common knowledge that increasing CO2 concentration plays a major role in enhancement of the greenhouse effect and contributes to global warming. The purpose of this study is to complement the conventional and established theory, that increased CO2 concentration due to human emissions causes an increase in temperature, by considering the reverse causality. Since increased temperature causes an increase in CO2 concentration, the relationship of atmospheric CO2 and temperature may qualify as belonging to the category of “hen-or-egg” problems, where it is not always clear which of two interrelated events is the cause and which the effect. We examine the relationship of global temperature and atmospheric carbon dioxide concentration in monthly time steps, covering the time interval 1980–2019 during which reliable instrumental measurements are available. While both causality directions exist, the results of our study support the hypothesis that the dominant direction is T → CO2. Changes in CO2 follow changes in T by about six months on a monthly scale, or about one year on an annual scale. We attempt to interpret this mechanism by involving biochemical reactions as at higher temperatures, soil respiration and, hence, CO2 emissions, are increasing.

Atmospheric Temperature and CO2: Hen-or-Egg Causality?

It is common knowledge that increasing CO2 concentration plays a major role in enhancement of the greenhouse effect and contributes to global warming. The purpose of this study is to complement the conventional and established theory that increased CO2 concentration due to human emissions causes an increase of temperature, by considering the reverse causality. Since increased temperature causes an increase in CO2 concentration, the relationship of atmospheric CO2 and temperature may qualify as belonging to the category of “hen-or-egg” problems, where it is not always clear which of two interrelated events is the cause and which the effect. We examine the relationship of global temperature and atmospheric carbon dioxide concentration at the monthly time step, covering the time interval 1980–2019, in which reliable instrumental measurements are available. While both causality directions exist, the results of our study support the hypothesis that the dominant direction is T → CO2. Changes in CO2 follow changes in T by about six months on a monthly scale, or about one year on an annual scale. We attempt to interpret this mechanism by involving biochemical reactions, as at higher temperatures soil respiration, and hence CO2 emission, are increasing.

Atmospheric Temperature and CO2: Hen-or-Egg Causality?

It is common knowledge that increasing CO2 concentration plays a major role in enhancement of the greenhouse effect and contributes to global warming. The purpose of this study is to complement the conventional and established theory that increased CO2 concentration due to human emissions causes an increase of temperature, by considering the reverse causality. Since increased temperature causes an increase in CO2 concentration, the relationship of atmospheric CO2 and temperature may qualify as belonging to the category of “hen-or-egg” problems, where it is not always clear which of two interrelated events is the cause and which the effect. We examine the relationship of global temperature and atmospheric carbon dioxide concentration at the monthly time step, covering the time interval 1980–2019, in which reliable instrumental measurements are available. While both causality directions exist, the results of our study support the hypothesis that the dominant direction is T → CO2. Changes in CO2 follow changes in T by about six months on a monthly scale, or about one year on an annual scale. We attempt to interpret this mechanism by involving biochemical reactions, as at higher temperatures soil respiration, and hence CO2 emission, are increasing.

Atmospheric Temperature and CO2: Hen-or-Egg Causality?

It is common knowledge that increasing CO2 concentration plays a major role in enhancement of the greenhouse effect and contributes to global warming. The purpose of this study is to complement the conventional and established theory that increased CO2 concentration due to human emissions causes an increase of temperature, by considering the reverse causality. Since increased temperature causes an increase in CO2 concentration, the relationship of atmospheric CO2 and temperature may qualify as belonging to the category of “hen-or-egg” problems, where it is not always clear which of two interrelated events is the cause and which the effect. We examine the relationship of global temperature and atmospheric carbon dioxide concentration at the monthly time step, covering the time interval 1980–2019, in which reliable instrumental measurements are available. While both causality directions exist, the results of our study support the hypothesis that the dominant direction is T → CO2. Changes in CO2 follow changes in T by about six months on a monthly scale, or about one year on an annual scale. We attempt to interpret this mechanism by involving biochemical reactions, as at higher temperatures soil respiration, and hence CO2 emission, are increasing.

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.

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.

Measuring Information Coupling between the Solar Wind and the Magnetosphere-Ionosphere System

The interaction between the solar wind and the Earth’s magnetosphere–ionosphere system is very complex, being essentially the result of the interplay between an external driver, the solar wind, and internal processes to the magnetosphere–ionosphere system. In this framework, modelling the Earth’s magnetosphere–ionosphere response to the changes of the solar wind conditions requires a correct identification of the causality relations between the different parameters/quantities used to monitor this coupling. Nowadays, in the framework of complex dynamical systems, both linear statistical tools and Granger causality models drastically fail to detect causal relationships between time series. Conversely, information theory-based concepts can provide powerful model-free statistical quantities capable of disentangling the complex nature of the causal relationships. In this work, we discuss how to deal with the problem of measuring causal information in the solar wind–magnetosphere–ionosphere system. We show that a time delay of about 30–60 min is found between solar wind and magnetospheric and ionospheric overall dynamics as monitored by geomagnetic indices, with a great information transfer observed between the z component of the interplanetary magnetic field and geomagnetic indices, while a lower transfer is found when other solar wind parameters are considered. This suggests that the best candidate for modelling the geomagnetic response to solar wind changes is the interplanetary magnetic field component Bz. A discussion of the relevance of our results in the framework of Space Weather is also provided.

The complex neurobiology of resilient functioning after childhood maltreatment

BACKGROUND

Childhood maltreatment has been associated with significant impairment in social, emotional and behavioural functioning later in life. Nevertheless, some individuals who have experienced childhood maltreatment function better than expected given their circumstances.

MAIN BODY

Here, we provide an integrated understanding of the complex, interrelated mechanisms that facilitate such individual resilient functioning after childhood maltreatment. We aim to show that resilient functioning is not facilitated by any single 'resilience biomarker'. Rather, resilient functioning after childhood maltreatment is a product of complex processes and influences across multiple levels, ranging from 'bottom-up' polygenetic influences, to 'top-down' supportive social influences. We highlight the complex nature of resilient functioning and suggest how future studies could embrace a complexity theory approach and investigate multiple levels of biological organisation and their temporal dynamics in a longitudinal or prospective manner. This would involve using methods and tools that allow the characterisation of resilient functioning trajectories, attractor states and multidimensional/multilevel assessments of functioning. Such an approach necessitates large, longitudinal studies on the neurobiological mechanisms of resilient functioning after childhood maltreatment that cut across and integrate multiple levels of explanation (i.e. genetics, endocrine and immune systems, brain structure and function, cognition and environmental factors) and their temporal interconnections.

CONCLUSION

We conclude that a turn towards complexity is likely to foster collaboration and integration across fields. It is a promising avenue which may guide future studies aimed to promote resilience in those who have experienced childhood maltreatment.

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.

Introduction to Focus Issue: Causation inference and information flow in dynamical systems: Theory and applications

Questions of causation are foundational across science and often relate further to problems of control, policy decisions, and forecasts. In nonlinear dynamics and complex systems science, causation inference and information flow are closely related concepts, whereby "information" or knowledge of certain states can be thought of as coupling influence onto the future states of other processes in a complex system. While causation inference and information flow are by now classical topics, incorporating methods from statistics and time series analysis, information theory, dynamical systems, and statistical mechanics, to name a few, there remain important advancements in continuing to strengthen the theory, and pushing the context of applications, especially with the ever-increasing abundance of data collected across many fields and systems. This Focus Issue considers different aspects of these questions, both in terms of founding theory and several topical applications.