Development of Bivariate Extreme Value Distributions for Applications in Marine Technology

Abstract

The extreme value theory for applications in such a responsible branch of industry as offshore and maritime engineering requires a robust, straightforward and reliable method for estimating the statistics of extremes. A method must be able to extract as much statistical information as possible from a recorded time series of data. In addition, a method must be capable to utilize the information regarding the temporal dependence structure of the process, as well as spatial dependence characteristics of the given time series in the bivariate case. In this thesis, a newly developed method for the purpose of predicting extremes associated with the observed process is studied thoroughly and improved. The method is referred to as the average conditional exceedance rate (ACER) method. It avoids the problem of having to decluster the data to ensure independence, which is a requisite component in the application of, for example, the standard peaks-over-threshold (POT) method. Moreover, the ACER method is specifically designed to account for statistical dependence between the sampled data points in a precise manner. The proposed method also targets the use of sub-asymptotic data to improve prediction accuracy. The research shows that the ACER method, if properly implemented, is able to provide a statistical representation with error bounds of the exact extreme value distribution given by the data. In the first part of the thesis, the method is demonstrated in detail by application to both synthetic and real environmental data. From a practical point of view, it appears to perform better than the POT and block maxima methods, and, with an appropriate modification, it is directly applicable to non-stationary time series. In the second part of the thesis, the ACER method for estimation of extreme value statistics is extended in a natural way to also cover the case of bivariate time series. This is achieved by introducing a cascade of conditioning approximations to the exact bivariate extreme value distribution. The results show that when the cascade converges, an accurate empirical estimate of the extreme value distribution can be obtained. It is also revealed that the possible functional representation of the empirically estimated bivariate ACER surface can be derived from the properties of the extreme-value copula. In this thesis, application of the bivariate ACER method is substantially studied for bivariate synthetic data. Finally, performance of the method is demonstrated for measured coupled wind speed and wave height data as well as simultaneous wind speed measurements from two separate locations.