# Quantifying coupling and causality in dynamic bivariate systems: a unified framework for time-domain, spectral, and information-theoretic analysis

## TL;DR

This paper introduces a unified framework to analyze interactions in dynamic systems using time, frequency, and information-theoretic methods, supported by a software toolbox.

## Contribution

A unified framework connecting causal, symmetric, and spectral measures with information-theoretic approaches for bivariate system analysis.

## Key findings

- The framework enables frequency-specific representation of information-theoretic metrics.
- Both linear and non-linear estimation approaches are described for robust analysis.
- Applications to cardiovascular and climate data demonstrate the framework's practical utility.

## Abstract

Understanding the underlying dynamics of complex real-world systems, such as neurophysiological and climate systems, requires quantifying the functional interactions between the system units under different scenarios. This tutorial paper offers a comprehensive description to time, frequency and information-theoretic domain measures for assessing the interdependence between pairs of time series describing the dynamical activities of physical systems, supporting flexible and robust analyses of statistical dependencies and directional relationships. Classical time and frequency domain correlation-based measures, as well as directional approaches derived from the notion of Granger causality, are introduced and discussed, along with information-theoretic measures of symmetrical and directional coupling. Both linear model-based and non-linear model-free estimation approaches are thoroughly described, the latter including binning, permutation, and nearest-neighbour estimators. Special emphasis is placed on the description of a unified framework that establishes a connection between causal and symmetric, as well as spectral and information-theoretic measures. This framework enables the frequency-specific representation of information-theoretic metrics, allowing for a detailed investigation of oscillatory components in bivariate systems. The practical computation of the interaction measures is favoured by presenting a software toolbox and two exemplary applications to cardiovascular and climate data. By bridging theoretical concepts with practical tools, this work enables researchers to effectively investigate a wide range of dynamical behaviours in various real-world scenarios in Network Physiology and beyond.