Asymptotically consistent prediction of extremes in chaotic systems:1 stationary case

TL;DR

This paper develops a theoretical framework for predicting extreme events in stationary chaotic systems, demonstrating asymptotic consistency especially when the system's attractor dimension is known, with extensions to higher dimensions.

Contribution

It introduces a new theoretical approach for predicting extremes in stationary chaotic systems, including empirical results and extensions to higher-dimensional cases.

Findings

Asymptotic consistency achieved for known attractor dimensions

Empirical results support the theoretical framework

Extensions to higher-dimensional systems discussed

Abstract

In many real world chaotic systems, the interest is typically in determining when the system will behave in an extreme manner. Flooding and drought, extreme heatwaves, large earthquakes, and large drops in the stock market are examples of the extreme behaviors of interest. For clarity, in this paper we confine ourselves to the case where the chaotic system to be predicted is stationary so theory for asymptotic consistency can be easily illuminated. We will start with a simple case, where the attractor of the chaotic system is of known dimension so the answer is clear from prior work. Some extension will be made to stationary chaotic system with higher dimension where a number of empirical results will be described and a theoretical framework proposed to help explain them.