Kernel-based Joint Independence Tests for Multivariate Stationary and Non-stationary Time Series

Time series that record temporal changes in sets of system variables are ubiquitous across many scientific disciplines [1], from physics and engineering [2] to biomedicine [3, 4], climate science [5, 6], economics [7, 8] or online human behaviour [9, 10]. Many real-world systems are thus described as multivariate time series of (possibly) interlinked processes tracking the temporal evolution (deterministic or random) of groups of observables of interest. The relationships between the measured variables are often complex, in many cases displaying inter-dependencies among each other. For example, the spreading of Covid-19 in Indonesia was dependent on weather conditions [11]; the Sustainable Development Goals have extensive interlinkages [12]; there are strong interconnections between foreign exchange and cryptocurrencies [13]; and the brain displays multiple spatial and temporal scales of functional connectivity [14]. Driven by technological advances (e.g., imaging techniques in the brain sciences [15], or the increased connectivity of personal devices via the Internet of Things [16]), there is a rapid expansion in the collection and storage of multivariate time series data sets, which underlines the need for mathematical tools to analyze the interdependencies within complex high-dimensional time series data.