EVA and Multivariate Analysis

The area of multivariate extremes values is more complicated and less developed than univariate extreme modelling. Multivariate extreme value theory is a generalization of the univariate case where multi-dimensional data are involved (Liu and Tawn, 2014). This research follows the method described by Wyncoll and Gouldby (2013) that applied the Heffernan and Tawn (2004) approach. Heffernan and Tawn (2004) developed a methodology for modelling the distribution of a d-dimensional variable when at least one of its components is extreme.

The traditional extreme value approaches have, in the past, tended to have been restricted in terms of the number of variables that have been considered and the spatial extents that are covered (Wyncoll and Gouldby, 2013). Recent developments in multivariate extreme value methods have removed some of these constraints and opened opportunities for improving flood risk analysis methods (e.g. Keef et al., 2009a; Keef et al., 2009b; Liu and Tawn, 2014; Southworth and Heffernan, 2015). However validation of system-based probabilistic models remains a challenge (Wyncoll and Gouldby, 2013); ongoing research on the development of benchmark tests for probabilistic models is likely to afford greater opportunity for rigorous validation in the future.

An issue that quickly arises is how to define multivariate extreme observations. Dependences between variables in the data do not necessarily say anything at all about extremal dependences. If an observation has to be extreme in all components simultaneously, the amount of data to model is reduced drastically. The number of multivariate exceedances is too small to do anything meaningful with.

The measure of extremal dependences is a dominant issue in the multivariate extreme analysis (Coles et al., 1999; Schlather and Tawn, 2002). Dependence occurs, for example, when different processes under study have a stochastic behaviour being driven by common meteorological conditions (Liu and Tawn, 2014). Dependence may also arise when a single process (i.e. storm surge) is studied at different spatial locations as in this research. Coastal

flooding occurs at any particular location when the combined sea level and waves exceed a critical level. Concurrently short-term coastal defences are likely to be sustainable; persistent high sea levels may cause severe damage. Furthermore, strong spatial dependence will lead to a widespread regional extreme event when the conditions occurring simultaneously along the entire coast, such as the same storm surge (i.e. major extreme events occurred in 1953, 2007 and 2013/14).

2.6 Chapter Summary

This chapter provided an overview of the key features and the necessary background information that will be exemplified on the subsequent chapters of the thesis. As the aim of this research is to obtain a storm surge flooding model for the East Coast of the UK, the challenge is to critically identify issues in the multivariate extreme value theory, such as the estimation of thresholds and coping with time series missing values to determine the likelihood of future significant coastal storm surge flooding.

For that reason, knowledge about coastal flooding is key to understand and predict extreme events. The number of extreme events by hydrological natural hazards such coastal flooding are increasing, maybe due to the impact of climate change. Flooding is a widespread natural hazard with a high economic and human cost. In view of that, UNISDR implemented the Sendai Framework for Disaster Risk Reduction 2015-2030 to promote a strategic and systematic approach to reducing vulnerabilities and risks to hazards. Flood risk is a significant and rising problem for the UK, where 2.4 million properties (households and non-residential properties) are at risk from river and coastal flooding, caused by the climate change and new developments planned on existing floodplains. Three big events have occurred on the East Coast of the UK in 1953, 2007 and 2013/2014 with catastrophic damages.

The study of skew surge (research data) will help to understand one of the sources of coastal flooding. Skew surge is the difference between the elevation of predicted astronomical high tide and nearest (in time) experienced high water. Although the sea level is subject to timescales fluctuations (climate change or glacial isostatic adjustment) or

anthropogenic activities, the main benefits of skew surge data are that they remove all timing differences and they are independent and identically distributed (i.i.d.) events.

There are a range of mathematical models to predict and simulated coastal flooding: physical-based models, catastrophe modelling or GIS. An effective solution to assess the risk of widespread and concurrent extreme events is the extreme value analysis (EVA). EVA aim to predict the probability of the next extreme flood event.

However, one of the key challenges in the extreme value analysis is to determine the threshold above which the events are extreme. In the literature, the traditional approach relies on the subjective graphical technique. Recently, analytical threshold selection methods have been proposed, but they are complex to replicate and not producing a decisive verdict to choose a suitable threshold. Moreover, one of the limitations in the multivariate extreme analysis (e.g. method of Heffernan and Tawn, 2004) for modelling dependences is to handle missing data and temporal dependences. Keef et al. (2009a) and Keef et al. (2009b) undertake a dependence model that takes into account missing value gaps between variables at only three locations with 9 years of concurrent data and time series span of 19-47 year. Subsequently, Keef et al. (2011) improve the Heffernan and Tawn method by a multivariate normal copula model.

The historical and methodological aspects of EVA were reviewed in this chapter. For a more general information on the theory of the extreme value statistics see these three references: Coles (2001); Heffernan and Tawn (2004), Wyncoll and Gouldby (2003), amongst others.

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