# On the computation of the extremal index for time series

## TL;DR

This paper reviews the extremal index in dynamical systems, introduces a new estimator, and explores its applications to noisy systems and climate data, enhancing understanding of local and global system properties.

## Contribution

It provides a comprehensive review of the extremal index in dynamical systems and introduces a novel estimator linked to existing statistical methods.

## Key findings

- New estimator of the extremal index introduced
- Relation established between the new and existing estimators
- Applications demonstrated on climate data

## Abstract

The extremal index is a quantity introduced in extreme value theory to measure the presence of clusters of exceedances. In the dynamical systems framework, it provides important information about the dynamics of the underlying systems. In this paper we provide a review of the meaning of the extremal index in dynamical systems. Depending on the observables used, this quantity can inform on local properties of attractors such as periodicity, stability and persistence in phase space, or on global properties such as the Lyapunov exponents. We also introduce a new estimator of the extremal index and shows its relation with those previously introduced in the statistical literature. We reserve a particular focus to the systems perturbed with noise as they are a good paradigm of many natural phenomena. Different kind of noises are investigated in the annealed and quenched situations. Applications to climate data are also presented.