# Multiscale Granger causality

## TL;DR

This paper extends Granger causality to analyze directed interactions across multiple temporal scales in complex systems, using state space models to improve estimation accuracy and reveal multiscale information transfer patterns.

## Contribution

It introduces a theoretical framework for multiscale Granger causality using state space models, addressing practical issues like filtering and downsampling.

## Key findings

- State space approach yields higher estimation accuracy than traditional AR models.
- Multiscale GC reveals meaningful information transfer patterns in climate data.
- Method effectively captures dynamics across multiple time scales.

## Abstract

In the study of complex physical and biological systems represented by multivariate stochastic processes, an issue of great relevance is the description of the system dynamics spanning multiple temporal scales. While methods to assess the dynamic complexity of individual processes at different time scales are well-established, multiscale analysis of directed interactions has never been formalized theoretically, and empirical evaluations are complicated by practical issues such as filtering and downsampling. Here we extend the very popular measure of Granger causality (GC), a prominent tool for assessing directed lagged interactions between joint processes, to quantify information transfer across multiple time scales. We show that the multiscale processing of a vector autoregressive (AR) process introduces a moving average (MA) component, and describe how to represent the resulting ARMA process using state space (SS) models and to combine the SS model parameters for computing exact GC values at arbitrarily large time scales. We exploit the theoretical formulation to identify peculiar features of multiscale GC in basic AR processes, and demonstrate with numerical simulations the much larger estimation accuracy of the SS approach compared with pure AR modeling of filtered and downsampled data. The improved computational reliability is exploited to disclose meaningful multiscale patterns of information transfer between global temperature and carbon dioxide concentration time series, both in paleoclimate and in recent years.