Extreme value laws and mean squared error growth in dynamical systems

Background. Extreme value theory for chaotic deterministic dynamical systems is a rapidly expanding area of research. Given a system and a scalar observable defined on its state space, extreme value theory studies the asymptotic probability distributions of large values attained by the observable along evolutions of the system. Objective. The aim of this paper is to study the relation between the statistics and predictability of extremes. Methods. Predictability is measured by the mean squared error (MSE), which is estimated from the difference of pairs of forecasts conditional on one of the forecasts exceeding a threshold. Results. Under the assumption that pairs of forecast variables satisfy a linear regression model, we show that the MSE can be decomposed into the sum of three terms: a threshold-independent constant, a mean term that always increases with threshold, and a variance term that can either increase, decrease, or stay constant with threshold. Using the generalised Pareto distribution to model excesses over a threshold, we show that the MSE always increases with threshold at sufficiently high threshold. However, when the forecasts have a negative tail index the MSE can be a decreasing function of threshold at lower thresholds. Conclusions. Our method is illustrated by means of four examples: the tent map, the cusp map, and two low-order models for atmospheric regime transitions and the El Ni\~{n}o-Southern Oscillation phenomenon. These examples clearly show that predictability depends on the observable and the invariant measure of the system.