Multivariate time series data that capture the temporal evolution of interconnected systems are ubiquitous in diverse areas. Understanding the complex relationships and potential dependencies among co-observed variables is crucial for the accurate statistical modelling and analysis of such systems. Here, we introduce kernel-based statistical tests of joint independence in multivariate time series by extending the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) to encompass both stationary and non-stationary processes, thus allowing broader real-world applications. By leveraging resampling techniques tailored for both single- and multiple-realisation time series, we show how the method robustly uncovers significant higher-order dependencies in synthetic examples, including frequency mixing data and logic gates, as well as real-world climate and socioeconomic data. Our method adds to the mathematical toolbox for the analysis of multivariate time series and can aid in uncovering high-order interactions in data.

翻译：捕捉互联系统时间演化特征的多变量时间序列数据广泛存在于各个领域。理解共同观测变量间的复杂关系和潜在依赖关系对于此类系统的精确统计建模与分析至关重要。本文通过将d变量希尔伯特-施密特独立性准则(dHSIC)扩展至平稳与非平稳过程，引入基于核的多变量时间序列联合独立性统计检验方法，从而拓展其现实应用范围。通过利用针对单实现与多实现时间序列定制的重采样技术，我们展示了该方法如何在合成实例（包括频率混叠数据与逻辑门）以及真实气候与社会经济数据中稳健地揭示显著高阶依赖关系。本方法丰富了多变量时间序列分析的数学工具库，有助于揭示数据中的高阶相互作用。