##
**Simulating flood event sets using extremal principal components.**
*(English)*
Zbl 07692385

Summary: Hazard event sets, a collection of synthetic extreme events over a given period, are important for catastrophe modelling. This paper addresses the issue of generating event sets of extreme river flow for northern England and southern Scotland, a region which has been particularly affected by severe flooding over the past 20 years. We start by analysing historical extreme river flow across 45 gauges, using methods from extreme value analysis, including the concept of extremal principal components. Our analysis reveals interesting connections between the extremal dependence structure and the region’s topography/climate. We then introduce a framework which is based on modelling the distribution of the extremal principal components in order to generate synthetic events of extreme river flow. The generative framework is dimension-reducing in that it distinctly handles the principal components based on their contribution to describing the nature of extreme river flow across the study region. We also detail a data-driven approach to select the optimal dimension. Synthetic flood events are subsequently generated efficiently by sampling from the fitted distribution. Our results indicate good agreement between the observed and simulated extreme river flow dynamics and, therefore, illustrate the usefulness of our approach to practitioners. For the considered application, we also find that our approach outperforms existing statistical approaches for generating hazard event sets.

### MSC:

62Pxx | Applications of statistics |

### Keywords:

multivariate extreme value theory; nonparametric bootstrapping; principal component analysis; spatial flood risk analysis### Software:

ismev
PDFBibTeX
XMLCite

\textit{C. Rohrbeck} and \textit{D. Cooley}, Ann. Appl. Stat. 17, No. 2, 1333--1352 (2023; Zbl 07692385)

### References:

[1] | Asadi, P., Davison, A. C. and Engelke, S. (2015). Extremes on river networks. Ann. Appl. Stat. 9 2023-2050. · Zbl 1397.62482 |

[2] | BALLANI, F. and SCHLATHER, M. (2011). A construction principle for multivariate extreme value distributions. Biometrika 98 633-645. · Zbl 1230.62073 |

[3] | BARLOW, A. M., SHERLOCK, C. and TAWN, J. (2020). Inference for extreme values under threshold-based stopping rules. J. R. Stat. Soc. Ser. C. Appl. Stat. 69 765-789. |

[4] | BEIRLANT, J., GOEGEBEUR, Y., SEGERS, J. and TEUGELS, J. (2004). Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics. Wiley, Chichester. |

[5] | BHATIA, S., JAIN, A. and HOOI, B. (2021). ExGAN: Adversarial generation of extreme samples. In Proceedings of the AAAI Conference on Artificial Intelligence 35 6750-6758. |

[6] | BOLDI, M.-O. and DAVISON, A. C. (2007). A mixture model for multivariate extremes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 69 217-229. · Zbl 1120.62030 |

[7] | BOULAGUIEM, Y., ZSCHEISCHLER, J., VIGNOTTO, E., VAN DER WIEL, K. and ENGELKE, S. (2022). Modeling and simulating spatial extremes by combining extreme value theory with generative adversarial networks. Environ. Data Sci. 1 e5. |

[8] | BRACKEN, C., RAJAGOPALAN, B., CHENG, L., KLEIBER, W. and GANGOPADHYAY, S. (2016). Spatial Bayesian hierarchical modeling of precipitation extremes over a large domain. Water Resour. Res. 52 6643-6655. |

[9] | CAMICI, S., BROCCA, L., MELONE, F. and MORAMARCO, T. (2014). Impact of climate change on flood frequency using different climate models and downscaling approaches. J. Hydrol. Eng. 19 04014002. |

[10] | Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics. Springer London, Ltd., London. |

[11] | COLES, S., HEFFERNAN, J. E. and TAWN, J. A. (1999). Dependence measures for extreme value analyses. Extremes 2 339-365. · Zbl 0972.62030 |

[12] | COLES, S. G. and TAWN, J. A. (1991). Modelling extreme multivariate events. J. Roy. Statist. Soc. Ser. B 53 377-392. · Zbl 0800.60020 |

[13] | COOLEY, D., DAVIS, R. A. and NAVEAU, P. (2010). The pairwise beta distribution: A flexible parametric multivariate model for extremes. J. Multivariate Anal. 101 2103-2117. · Zbl 1203.62104 |

[14] | COOLEY, D., NYCHKA, D. and NAVEAU, P. (2007). Bayesian spatial modeling of extreme precipitation return levels. J. Amer. Statist. Assoc. 102 824-840. · Zbl 1469.62389 |

[15] | COOLEY, D. and THIBAUD, E. (2019). Decompositions of dependence for high-dimensional extremes. Biometrika 106 587-604. · Zbl 1464.62278 |

[16] | Davison, A. C., Padoan, S. A. and Ribatet, M. (2012). Statistical modeling of spatial extremes. Statist. Sci. 27 161-186. · Zbl 1330.86021 |

[17] | DAVISON, A. C. and SMITH, R. L. (1990). Models for exceedances over high thresholds. J. Roy. Statist. Soc. Ser. B 52 393-425. · Zbl 0706.62039 |

[18] | DE CARVALHO, M. and DAVISON, A. C. (2014). Spectral density ratio models for multivariate extremes. J. Amer. Statist. Assoc. 109 764-776. · Zbl 1367.62270 |

[19] | DREES, H. and SABOURIN, A. (2021). Principal component analysis for multivariate extremes. Electron. J. Stat. 15 908-943. |

[20] | DREVETON, C. and GUILLOU, Y. (2004). Use of a principal components analysis for the generation of daily time series. J. Appl. Meteorol. 43 984-996. |

[21] | EASTOE, E. F. (2019). Nonstationarity in peaks-over-threshold river flows: A regional random effects model. Environmetrics 30 e2560, 18 pp. |

[22] | EASTOE, E. F. and TAWN, J. A. (2009). Modelling non-stationary extremes with application to surface level ozone. J. R. Stat. Soc. Ser. C. Appl. Stat. 58 25-45. |

[23] | Engelke, S. and Hitz, A. S. (2020). Graphical models for extremes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 82 871-932. · Zbl 07554779 |

[24] | Engelke, S. and Ivanovs, J. (2021). Sparse structures for multivariate extremes. Annu. Rev. Stat. Appl. 8 241-270. |

[25] | ENVIRONMENT AGENCY (2018). Estimating the economic costs of the winter floods 2015 to 2016. Ref: LIT 10736. Available at https://www.gov.uk/government/publications/floods-of-winter-2015-to-2016-estimating-the-costs. |

[26] | GROSSI, P. and KUNREUTHER, H. (2005). Catastrophe Modeling: A New Approach to Managing Risk. Springer, New York. |

[27] | HALL, P., WATSON, G. S. and CABRERA, J. (1987). Kernel density estimation with spherical data. Biometrika 74 751-762. · Zbl 0632.62033 |

[28] | HEFFERNAN, J. E. and TAWN, J. A. (2004). A conditional approach for multivariate extreme values. J. R. Stat. Soc. Ser. B. Stat. Methodol. 66 497-546. · Zbl 1046.62051 |

[29] | HUSER, R. and WADSWORTH, J. L. (2022). Advances in statistical modeling of spatial extremes. Wiley Interdiscip. Rev.: Comput. Stat. 14 Paper No. e1537, 28 pp. |

[30] | Hüsler, J. and Reiss, R.-D. (1989). Maxima of normal random vectors: Between independence and complete dependence. Statist. Probab. Lett. 7 283-286. · Zbl 0679.62038 |

[31] | INSTITUTE OF HYDROLOGY (GREAT BRITAIN) (1975). Flood Studies Report. Natural Environment Research Council, London. |

[32] | KEEF, C., TAWN, J. A. and LAMB, R. (2013). Estimating the probability of widespread flood events. Environmetrics 24 13-21. |

[33] | LARSSON, M. and RESNICK, S. I. (2012). Extremal dependence measure and extremogram: The regularly varying case. Extremes 15 231-256. · Zbl 1329.60153 |

[34] | NORTHROP, P. J., ATTALIDES, N. and JONATHAN, P. (2017). Cross-validatory extreme value threshold selection and uncertainty with application to ocean storm severity. J. R. Stat. Soc. Ser. C. Appl. Stat. 66 93-120. |

[35] | Pickands, J. III (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119-131. · Zbl 0312.62038 |

[36] | QUINN, N., BATES, P. D., NEAL, J., SMITH, A., WING, O., SAMPSON, C., SMITH, J. and HEFFERNAN, J. E. (2019). The spatial dependence of flood hazard and risk in the United States. Water Resour. Res. 55 1890-1911. |

[37] | ROHRBECK, C. and COOLEY, D. (2023). Supplement to “Simulating flood event sets using extremal principal components.” https://doi.org/10.1214/22-AOAS1672SUPPA, https://doi.org/10.1214/22-AOAS1672SUPPB |

[38] | ROHRBECK, C. and TAWN, J. A. (2021). Bayesian spatial clustering of extremal behavior for hydrological variables. J. Comput. Graph. Statist. 30 91-105. · Zbl 07499884 |

[39] | Tawn, J. A. (1988). Bivariate extreme value theory: Models and estimation. Biometrika 75 397-415. · Zbl 0653.62045 |

[40] | WADSWORTH, J. L. (2016). Exploiting structure of maximum likelihood estimators for extreme value threshold selection. Technometrics 58 116-126 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.