# Estimation of Information Flow-Based Causality with Coarsely Sampled Time Series

## TL;DR

This paper introduces a new method for analyzing causality in time series data sampled at low frequencies, using Lie groups instead of Lie algebras.

## Contribution

A novel approach using Lie groups to improve causality estimation in coarsely sampled nonlinear systems.

## Key findings

- The new method works well for linear systems and reduces bias in nonlinear systems with low sampling rates.
- The approach was successfully tested on coupled Rössler oscillators, even when they were nearly synchronized.
- The method relies on sample covariances and avoids complex differential equations.

## Abstract

The past decade has seen growing applications of the information flow-based causality analysis, particularly with the concise formula of its maximum likelihood estimator. At present, the algorithm for its estimation is based on differential dynamical systems, which, however, may raise an issue for coarsely sampled time series. Here, we show that, for linear systems, this is suitable at least qualitatively, but, for highly nonlinear systems, the bias increases significantly as the sampling frequency is reduced. This study provides a partial solution to this problem, showing how causality analysis can be made faithful with coarsely sampled series, provided that the statistics are sufficient. The key point here is that, instead of working with a Lie algebra, we turn to work with its corresponding Lie group. An explicit and concise formula is obtained, with only sample covariances involved. It is successfully applied to a system comprising a pair of coupled Rössler oscillators. Particularly remarkable is the success when the two oscillators are nearly synchronized. As more often than not observations may be scarce, this solution, albeit partial, is very timely.