# Normalized multivariate time series causality analysis and causal graph reconstruction

arXiv: https://arxiv.org/abs/2104.11360 · HTML: https://mlsanity.com/paper/2104.11360/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 39 references that are in the map.
- First posted on arXiv 23 Apr 2021.

## Builds on (most cited first, 1 of 1)

- [From Ordinary Differential Equations to Structural Causal Models: the deterministic case](https://mlsanity.com/paper/1408.2063/) (2014)

## Built on by (most cited first, 7 of 7)

- [Information Flow Rate for Cross-Correlated Stochastic Processes](https://mlsanity.com/paper/2401.04950/) (2024)
- [An overview of the quantitative causality analysis and causal graph reconstruction based on a rigorous formalism of information flow](https://mlsanity.com/paper/2112.14839/) (2021)
- [Quantitative causality, causality-guided scientific discovery, and causal machine learning](https://mlsanity.com/paper/2402.13427/) (2024)
- [System States Forecasting of Microservices with Dynamic Spatio-Temporal Data](https://mlsanity.com/paper/2408.07894/) (2024)
- [Unsupervised Anomaly Prediction with N-BEATS and Graph Neural Network in Multi-variate Semiconductor Process Time Series](https://mlsanity.com/paper/2510.20718/) (2025)
- [FairMed-XGB: A Bayesian-Optimised Multi-Metric Framework with Explainability for Demographic Equity in Critical Healthcare Data](https://mlsanity.com/paper/2603.14947/) (2026)
- [GL-LFGNN:A Global-Local Dual-branch Causal Graph Neural Network Based on Liang-Kleeman Information Flow for EEG Emotion Recognition](https://mlsanity.com/paper/2605.25061/) (2026)

## Abstract (arXiv)

Causality analysis is an important problem lying at the heart of science, and is of particular importance in data science and machine learning. An endeavor during the past 16 years viewing causality as real physical notion so as to formulate it from first principles, however, seems to go unnoticed. This study introduces to the community this line of work, with a long-due generalization of the information flow-based bivariate time series causal inference to multivariate series, based on the recent advance in theoretical development. The resulting formula is transparent, and can be implemented as a computationally very efficient algorithm for application. It can be normalized, and tested for statistical significance. Different from the previous work along this line where only information flows are estimated, here an algorithm is also implemented to quantify the influence of a unit to itself. While this forms a challenge in some causal inferences, here it comes naturally, and hence the identification of self-loops in a causal graph is fulfilled automatically as the causalities along edges are inferred. To demonstrate the power of the approach, presented here are two applications in extreme situations. The first is a network of multivariate processes buried in heavy noises (with the noise-to-signal ratio exceeding 100), and the second a network with nearly synchronized chaotic oscillators. In both graphs, confounding processes exist. While it seems to be a huge challenge to reconstruct from given series these causal graphs, an easy application of the algorithm immediately reveals the desideratum. Particularly, the confounding processes have been accurately differentiated. Considering the surge of interest in the community, this study is very timely.