Functional Peaks-Over-Threshold Analysis

Abstract

Peaks‐over‐threshold analysis using the generalised Pareto distribution is widely applied in modelling tails of univariate random variables, but much information may be lost when complex extreme events are studied using univariate results. In this paper, we extend peaks‐over‐threshold analysis to extremes of functional data. Threshold exceedances defined using a functional r are modelled by the generalised r‐Pareto process, a functional generalisation of the generalised Pareto distribution that covers the three classical regimes for the decay of tail probabilities, and that is the only possible continuous limit for r‐exceedances of a properly rescaled process. We give construction rules, simulation algorithms and inference procedures for generalised r‐Pareto processes, discuss model validation and apply the new methodology to extreme European windstorms and heavy spatial rainfall.

Space-Time Trend Detection and Dependence Modeling in Extreme Event Approaches by Functional Peaks-Over-Thresholds: Application to Precipitation in Burkina Faso

“…After removing the nonstationarity, thanks to the (2), the modeling is then focused on the evaluation of the extreme spatial dependence structure in Z using functional POT [24][25][26]. In the multivariate and spatial framework a threshold exceedances for a random function Z � Z(s), s ∈ S { } is defined by [26] to be an event of the form ℓ(Z) > u { } for some u ≥ 0, where ℓ: C(S) ⟶ R + is a continuous and homogeneous nonnegative risk function, i.e., there exists α > 0 such that ℓ(λy) � λ α ℓ(y) when y ∈ C + (S) and λ > 0. e risk function ℓ determines the type of extreme events of interest.…”

“…where a > 0 and b are continuous functions on S, respectively, scale and location functions and Y ℓ is a stochastic process whose probability measure is completely determined by the limit measure Λ. More details on generalized ℓ-Pareto processes can be found in [24,25].…”

“…Let Z(s), s ∈ S { } be a latent spatial stochastic process indexed by s ∈ S ⊂ R d with sample paths in C(S) of continuous marginal distribution F Z and common right endpoint z F . As mentioned in [25,27], we assume that the spatial process Z is a general functional regular variation (GRV), i.e., that there exists suitable sequences of continuous functions a n :…”

“…Although attractive, these models are expensive in computing time when tending toward large dimensions and are not adapted for modeling threshold exceedances. e threshold exceedance approach introduced by [21,22] was first extended to multivariate environments [23] before being generalized to functional data [24][25][26][27] to give birth to the family of generalized ℓ-Pareto processes. is family of processes are good candidates for modeling the spatial dependence structure of exceedance data.…”

“…First temporal and spatial nonstationarities are controlled via a tail trend function in marginal distributions [5][6][7]. Subsequently, the spatial dependence structure is taken into account by a hidden auxiliary spatial process using generalized ℓ-Pareto processes [25,26]. Finally, we use the model to simulate and predict future extreme events.…”