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

arXiv: https://arxiv.org/abs/2305.08529 · HTML: https://mlsanity.com/paper/2305.08529/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 59 references that are in the map.
- First posted on arXiv 15 May 2023.

## Builds on (most cited first, 15 of 15)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [The physics of higher-order interactions in complex systems](https://mlsanity.com/paper/2110.06023/) (2021)
- [Kernel-based Tests for Joint Independence](https://mlsanity.com/paper/1603.00285/) (2016)
- [Forecasting Economics and Financial Time Series: ARIMA vs. LSTM](https://mlsanity.com/paper/1803.06386/) (2018)
- [Quantifying High-order Interdependencies via Multivariate Extensions of the Mutual Information](https://mlsanity.com/paper/1902.11239/) (2019)
- [A Wild Bootstrap for Degenerate Kernel Tests](https://mlsanity.com/paper/1408.5404/) (2014)
- [A Kernel Independence Test for Random Processes](https://mlsanity.com/paper/1402.4501/) (2014)
- [A Kernel Test for Three-Variable Interactions](https://mlsanity.com/paper/1306.2281/) (2013)
- [Data-driven unsupervised clustering of online learner behaviour](https://mlsanity.com/paper/1902.04047/) (2019)
- [Kernel Two-Sample and Independence Tests for Non-Stationary Random Processes](https://mlsanity.com/paper/2010.00271/) (2020)
- [A Kernel Test for Three-Variable Interactions with Random Processes](https://mlsanity.com/paper/1603.00929/) (2016)
- [Discovering and Deciphering Relationships Across Disparate Data Modalities](https://mlsanity.com/paper/1609.05148/) (2016)
- [Independence Testing for Temporal Data](https://mlsanity.com/paper/1908.06486/) (2019)
- [Similarity measure for sparse time course data based on Gaussian processes](https://mlsanity.com/paper/2102.12342/) (2021)

## Built on by (most cited first, 3 of 3)

- [Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions](https://mlsanity.com/paper/2306.00904/) (2023)
- [Information-Theoretic Measures on Lattices for High-Order Interactions](https://mlsanity.com/paper/2408.07533/) (2024)
- [Permutation-Free High-Order Interaction Tests](https://mlsanity.com/paper/2506.05963/) (2025)

## Abstract (arXiv)

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, neuroscience, 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.