This note summarizes our work in the framework of the Neptune project.
The EUR analysed the joint statistics of extreme sea state variables (offshore) within the framework of multivariate extreme value theory. The basic assumption in our work is the validity of the multivariate extreme value condition. This condition allows multivariate extreme value behavior to be analysed in terms of marginal extreme value behavior and the dependence between the marginal extremes. Moreover it implies the extreme value condition for the marginals, thus allowing extrapolation outside the sample. Marginal extreme value behavior may be described using well known univariate techniques. The dependence between extremes may be described in terms of the tail dependence function or the spectral measure.
Our work aimed at developing and using techniques for the analysis of the dependence structure of marginal extremes, and estimation methods for probabilities of extreme events or alternatively for design parameters corresponding to extreme probabilities.
In our work we emphasize the use of non- or semi-parametric estimation methods, thereby minimizing assumptions made about the underlying joint distribution of the variables.
We have applied our methods to the problem of estimating failure probabilities of a particular sea wall: the ``Pettemer zeewering''. RWS supplied data on offshore sea state variables (wave height, wave period, direction, still water level) and supplied an initial definition of failure: a linear combination of wave height and water level exceeding a certain value. Later in the project DH supplied a function describing the the transformation of offshore variables to onshore variables and a reliability function translating onshore variables into a load on the sea wall.
In practical applications, e.g. estimating failure probabilities or design values (quantiles), estimation methods depend on the (in)dependence of the marginal extremes. Therefore analysis starts with analysis of the dependence structure of the joint distribution of marginal extremes. We developed and demonstrated several ways of representing dependence of extremes: estimating the tail dependence function and studying its behavior using contourplots and estimating and graphing the spectral measure.
In the case of asymptotic dependence, failure probabilities of defence structures may be estimated using a simple semi parametric estimator. We have shown asymptotic dependence to be a reasonable assumption for relevant offshore sea state variables, and have applied the method both in a simple bivariate setting and in a more complex and realistic trivariate setting.
Our final effort was directed at computing confidence intervals for the estimated failure probabilities. We developed and applied theory for a confidence interval for our semi parametric estimator. Moreover we estimated failure probability with confidence intervals in a maximum likelihood setting, using a fully parametric model.
Our findings are presented in the following summaries of our reports. Apart from these reports we contributed to a simulation study carried out by the Lancaster University [1]. In this study our methods correctly identified the dependence structure in three sets of randomly generated data. The reports (in zipped PostScript format) are available from our ftp-site.
An introduction to concepts and methods in bivariate extreme value theory. The theory is applied to the question of the reliability of a sea-wall (the ``Pettemer zeewering''). Failure of the sea wall is defined by a simple failure region: linear combinations of wave height and sea level exceeding a design value. The dependence of extremes is represented by contour plots of the tail dependence function and by the spectral measure. These show that asymptotic dependende is a reasonable assumption. A simple semi parametric estimator is used to estimate the failure probability.
This notes discusses topics raised over report EUR--04, at the Neptune workshop in Norwich 1995. They include:
Application of multivariate extreme value theory to a 3-dimensional problem. The theory is applied to the question of the reliability of a sea-wall (the ``Pettemer zeewering''). Failure of the sea-wall is defined in terms of offshore wave height and period and sea level. The report presents the trivariate variants of concepts and graphs defined in EUR--04 (contourplanes of the tail dependence function, scatterplots to represent the spectral measure). Failure of the sea wall is defined by a transformation function that transforms offshore values to nearshore values, and a reliability function that translates nearshore values into the load on the sea wall. Our estimation procedure treats these functions as black box functions defining a failure region in terms of offshore sea state variables.
Application of univariate extreme value theory to a 3- dimensional problem (see EUR--07). The trivariate problem is reduced to a univariate problem, by calculating the load on the sea wall caused by extreme combinations of wave height, period and sea level, and applying univariate methods to the load variable. The estimated failure probability differs by a factor 100 from the the trivariate estimate.
This note compares five estimators of the extreme value index. A new estimator is shown to approximate the efficiency of the moment estimator, while maintaining the `shift invariance' of the Pickands estimator. The estimators are applied to North Sea wave height and sea level data. (This note was written in collaboration with P. Deheuvels (University of Paris) and M. T. Themido Pereira (University of Lisbon).
In this report we use a fully parametric model for estimating failure probabilities with confidence intervals. We present a numerical integration procedure for estimating failure probability. Model parameters are estimated by maximum likelihood estimation. The integration procedure is used to obtain a point estimate and a confidence interval for the failure probability.
It is applied to the problem of estimating the failure probability of the Pettemer zeewering used in earlier reports. Good agreement is shown between this method and a simple semi parametric method for estimating failure probabilities used in earlier reports. But jointly estimating model parameters by maximum likelihood surprisingly leads to widely different parameter estimates for the marginals.
We use a bivariate extreme value threshold model to analyse the extreme behavior of wave height and sea level and obtain the probability of failure, i.e. to have an observation in a specific failure region.
We use a combination of non-parametric and parametric methods for estimating the model parameters: first the dependence structure is estimated non-parametrically and then marginal parameters are estimated by maximum likelihood estimation.
Our final report shows how to construct a confidence interval for the semi parametric estimator of an extreme failure probability. The procedure is applied to the case of the Pettemer zeewering.
In terms of the original project definition we made a definite contribution to the following tasks:
There is still a long way to go. The estimates we produced turned out to be highly dependent on the estimation method used, on the portion of the sample actually used for estimation etc. We were only able to show how probabilities of extreme events may be estimated, but for reliable estimates much more data will be needed.