Abstract
The peaks-over-threshold (POT) method has a long tradition in modelling extremes in environmental variables. However, it has originally been introduced under the assumption of independently and identically distributed (iid) data. Since environmental data often exhibits a time series structure, this assumption is likely to be violated due to short- and long-term dependencies in practical settings, leading to clustering of high-threshold exceedances. In this paper, we first review popular approaches that either focus on modelling short- or long-range dynamics explicitly. In particular, we consider conditional POT variants and the Mittag–Leffler distribution modelling waiting times between exceedances. Further, we propose a new two-step approach capturing both short- and long-range correlations simultaneously. We suggest the autoregressive fractionally integrated moving average peaks-over-threshold (ARFIMA-POT) approach, which in a first step fits an ARFIMA model to the original series and then in a second step utilises a classical POT model for the residuals. Applying these models to an oceanographic time series of significant wave heights measured on the Sefton coast (UK), we find that neither solely modelling short- nor long-range dependencies satisfactorily explains the clustering of extremes. The ARFIMA-POT approach, however, provides a significant improvement in terms of model fit, underlining the need for models that jointly incorporate short- and long-range dependence to address extremal clustering, and their theoretical justification.
Highlights
Extreme value theory (EVT) has emerged as an invaluable toolkit for a broad range of sciences in recent decades
We begin by presenting the inference results for the Hawkes-POT and SEPOT models as reviewed in Section 3.1 with both epidemic-type aftershock sequence (ETAS) (16) and exponential (17) decay functions, which aim at accommodating extremal clustering by explicitly modelling short-range dependencies
autoregressive fractionally integrated moving average (ARFIMA)-POT procedure as described in detail with the application to significant wave heights in Section 4.3: we first estimate the ARFIMA model with the AR and MA orders being selected according to the Bayesian information criterion (BIC), and calculate the ARFIMA residuals, to which we apply the classical POT model
Summary
Extreme value theory (EVT) has emerged as an invaluable toolkit for a broad range of sciences in recent decades. Hees et al [43] provide a parametric approach to modelling a setting in which the event occurrences follow a fractional Poisson process which was proved to be long-range dependent by Biard and Saussereau [44] They focus on characterising the inter-event times via the Mittag–Leffler distribution, which generalises the exponential distribution [45,46]. We consider strongly dependent significant wave height data for which high level exceedances appear in an extremely clustered manner As this setting poses a particular challenge for traditional POT approaches, we analyse whether popular models dealing with short- or long-range dependence convey enough information about the dependence structure in the data.
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