Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights
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...
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| Format: | Article |
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MDPI AG
2021-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/21/2817 |
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| author | Pushpa Dissanayake Teresa Flock Johanna Meier Philipp Sibbertsen |
| author_facet | Pushpa Dissanayake Teresa Flock Johanna Meier Philipp Sibbertsen |
| author_sort | Pushpa Dissanayake |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T05:56:42Z |
| format | Article |
| id | doaj.art-42ff30a1664345f2a491018eda4ee6dd |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T05:56:42Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-42ff30a1664345f2a491018eda4ee6dd2023-11-22T21:19:16ZengMDPI AGMathematics2227-73902021-11-01921281710.3390/math9212817Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave HeightsPushpa Dissanayake0Teresa Flock1Johanna Meier2Philipp Sibbertsen3Coastal Geology and Sedimentology, Institute of Geosciences, Kiel University, 24118 Kiel, GermanyFaculty of Economics and Management, Institute of Statistics, Leibniz University Hannover, 30167 Hannover, GermanyFaculty of Economics and Management, Institute of Statistics, Leibniz University Hannover, 30167 Hannover, GermanyFaculty of Economics and Management, Institute of Statistics, Leibniz University Hannover, 30167 Hannover, GermanyThe 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.https://www.mdpi.com/2227-7390/9/21/2817peaks-over-thresholdextremal clusteringlong-range dependenceARFIMA modelsextreme value theorysignificant wave heights |
| spellingShingle | Pushpa Dissanayake Teresa Flock Johanna Meier Philipp Sibbertsen Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights Mathematics peaks-over-threshold extremal clustering long-range dependence ARFIMA models extreme value theory significant wave heights |
| title | Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights |
| title_full | Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights |
| title_fullStr | Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights |
| title_full_unstemmed | Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights |
| title_short | Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights |
| title_sort | modelling short and long term dependencies of clustered high threshold exceedances in significant wave heights |
| topic | peaks-over-threshold extremal clustering long-range dependence ARFIMA models extreme value theory significant wave heights |
| url | https://www.mdpi.com/2227-7390/9/21/2817 |
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