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 nonstationary random processes, thus allowing broader real-world applications. By leveraging resampling techniques tailored for both single- and multiple-realization time series, we show how the method robustly uncovers significant higher-order dependencies in synthetic examples, including frequency mixing data, as well as real-world climate and socioeconomic data. Our method adds to the mathematical toolbox for the analysis of complex high-dimensional time-series datasets.

翻译：捕捉互联系统时间演化的多元时间序列数据在各个领域无处不在。理解共同观测变量之间的复杂关系和潜在依赖性对于准确统计建模和分析此类系统至关重要。本文通过将d变量希尔伯特-施密特独立性准则（dHSIC）扩展至涵盖平稳和非平稳随机过程，引入基于核的多元时间序列联合独立性统计检验，从而拓宽了实际应用范围。通过利用针对单实现和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示例（包括频率混合数据）以及真实气候和社会经济数据中稳健地揭示显著的高阶依赖性。该方法为分析复杂高维时间序列数据集增添了数学工具。