Exceedance and force of centrality for functional data

Exceedance refers to instances where a dynamic process surpasses given thresholds, e.g., the occurrence of a heat wave. We propose a novel exceedance framework for functional data, where each observed random trajectory is transformed into an exceedance function, which quantifies exceedance durations as a function of threshold levels. An inherent relationship between exceedance functions and probability distributions makes it possible to draw on distributional data analysis techniques such as Fr'echet regression to study the dependence of exceedances on Euclidean predictors, e.g., calendar year when the exceedances are observed. We use local linear estimators to obtain exceedance functions from discretely observed functional data with noise and study the convergence of the proposed estimators. New concepts of interest include the force of centrality that quantifies the propensity of a system to revert to lower levels when a given threshold has been exceeded, conditional exceedance functions when conditioning on Euclidean covariates, and threshold exceedance functions, which characterize the size of exceedance sets in dependence on covariates for any fixed threshold. We establish consistent estimation with rates of convergence for these targets. The practical merits of the proposed methodology are illustrated through simulations and applications for annual temperature curves and medfly activity profiles.