TY - JOUR

T1 - Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights

AU - Dissanayake, Pushpa

AU - Flock, Teresa

AU - Meier, Johanna

AU - Sibbertsen, Philipp

N1 - Funding Information: Acknowledgments: The authors gratefully acknowledge Jennifer Brown from National Oceanographic Centre, Liverpool, and Andrew Martin from Sefton Metropolitan Borough Council for providing access to the wave data. This study is part of the MoDECS (Modification of Dune Erosion by adjacent Coastal Systems) project funded by the German Research Foundation (DFG: DI 2139/2-1). The authors thank two anonymous referees for providing helpful and detailed feedback that led to an improved manuscript.

PY - 2021/11/5

Y1 - 2021/11/5

N2 - 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.

AB - 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.

KW - ARFIMA models

KW - Extremal clustering

KW - Extreme value theory

KW - Long-range dependence

KW - Peaks-over-threshold

KW - Sefton coast

KW - Significant wave heights

UR - http://www.scopus.com/inward/record.url?scp=85119132899&partnerID=8YFLogxK

U2 - 10.3390/math9212817

DO - 10.3390/math9212817

M3 - Article

AN - SCOPUS:85119132899

VL - 9

JO - Mathematics

JF - Mathematics

IS - 21

M1 - 2817

ER -