A latent trawl process model for extreme values

Abstract

This paper presents a new model for characterising temporal dependence in exceedances above a threshold. The model is based on the class of trawl processes, which are stationary, infinitely divisible stochastic processes. The model for extreme values is constructed by embedding a trawl process in a hierarchical framework, which ensures that the marginal distribution is generalised Pareto, as expected from classical extreme value theory. We also consider a modified version of this model that works with a wider class of generalised Pareto distributions, and has the advantage of separating marginal and temporal dependence properties. The model is illustrated by applications to environmental time series, and it is shown that the model offers considerable flexibility in capturing the dependence structure of extreme value data.

Ground-level ozone: Evidence of increasing serial dependence in the extremes

“…In the second paper, the authors assume that extreme events of ozone levels, or of ozone levels given the covariates, are temporally independent, while in the third the author works with slightly less correlated weekly maxima under an independence assumption. Noven et al (2015) are the first to consider the temporal dependence of a time series of extreme ozone levels.…”

“…Following Bortot and Gaetan (2014), we use a decomposition of the GP distribution to construct a hierarchical model for exceedances that preserves the GP distribution marginally. Rather than using a Markov chain as the latent process as in Bortot and Gaetan (2014), we proceed as in Noven et al (2017) and use a trawl process as the latent process. Trawl processes are stationary, infinitely divisible stochastic processes that have been recently introduced by Barndorff-Nielsen et al (2011) and that have already shown wide applicability, see e.g.…”

“…Applying a subsequent marginal transformation model as suggested by Bortot and Gaetan (2014) and explored in Noven et al (2017), we manage to separate the marginal parameters and the dependence parameter. A change-point test for the extremal dependence can then be established testing the stability of the dependence parameter over time.…”

Hierarchical Space-Time Modeling of Asymptotically Independent Exceedances With an Application to Precipitation Data

“…Strong motivation also comes from Bortot and Gaetan (2014) by developing a generalization of their latent temporal process. Alternatively, our latent process could be viewed as a space-time version of the temporal trawl processes introduced by Barndorff-Nielsen et al (2014) and exploited for extreme values by Noven et al (2015), with spatial extensions recently proposed by Opitz (2017). Our approach is based on representing a generalized Pareto distribution as a Gamma mixture of an exponential distribution, enabling us to keep easily tractable marginal distributions which remain coherent with univariate extreme value theory.…”

Modelling, Simulation and Inference for Multivariate Time Series of Counts Using Trawl Processes

“…In the univariate case, trawl processes -not necessarily restricted to the integer-valued case -have been introduced by Barndorff-Nielsen (2011). Also, Noven et al (2015) used such processes in an hierarchical model in the context of extreme value theory. The univariate integer-valued case has been developed in detail in Barndorff-Nielsen et al (2014).…”