Paul Thompson


The study of the behaviour of the extreme values of a variable such as wave height is very important in engineering applications such as flood risk assessment and coastal design. Storm wave modelling usually adopts a univariate extreme value theory approach, essentially identifying the extreme observations of one variable and fitting a standard extreme value distribution to these values. Often it is of interest to understand how extremes of a variable such as wave height depend on a covariate such as wave direction. An important associated concept is that of return level, a value that is expected to be exceeded once in a certain time period. The main areas of research discussed in this thesis involve making improvements to the way that extreme observations are identified and to the use of quantile regression as an alternative methodology for understanding the dependence of extreme values on a covariate. Both areas of research provide developments to existing return level methodology so enhancing the accuracy of predicted future storm wave events. We illustrate the methodology that we have developed using both coastal and offshore wave data sets. In particular, we present an automated and computationally inexpensive method to select the threshold used to identify observations for extreme value modelling. Our method is based on the distribution of model parameter estimates across a range of thresholds. We also assess the effect of the uncertainty associated with threshold selection on return level estimation by using a bootstrap procedure. Furthermore, we extend our approach so that the selection of the threshold can also depends on the value of a covariate such as wave direction. As a biproduct of our methodological development we have improved existing techniques for estimating and making inference about the parameters of a standard extreme value distribution. We also present a new technique that extends existing Bayesian quantile regression methodology by modelling the dependence of a quantile of one variable on the values of another using a natural cubic spline. Inference is based on the posterior density of the spline and an associated smoothing parameter and is performed by means of a specially tuned Markov chain Monte Carlo algorithm. We show that our nonparametric methodology provides more flexible modelling than the current polynomial based approach for a range of examples.

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