Detecting directional couplings from multivariate flows by the joint distance distribution

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.

Ordinal methods for a characterization of evolving functional brain networks

Ordinal time series analysis is based on the idea to map time series to ordinal patterns, i.e., order relations between the values of a time series and not the values themselves, as introduced in 2002 by C. Bandt and B. Pompe. Despite a resulting loss of information, this approach captures meaningful information about the temporal structure of the underlying system dynamics as well as about properties of interactions between coupled systems. This-together with its conceptual simplicity and robustness against measurement noise-makes ordinal time series analysis well suited to improve characterization of the still poorly understood spatiotemporal dynamics of the human brain. This minireview briefly summarizes the state-of-the-art of uni- and bivariate ordinal time-series-analysis techniques together with applications in the neurosciences. It will highlight current limitations to stimulate further developments, which would be necessary to advance characterization of evolving functional brain networks.

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.

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.