Extreme value predictions based on nonstationary time series of wave data

A new method for calculating return periods of various level values from nonstationary time series data is presented. The key idea of the method is a new definition of the return period, based on the MEan Number of Upcrossings of the level x* (MENU method). In the present article, the case of Gaussian periodically correlated time series is studied in detail. The whole procedure is numerically implemented and applied to synthetic wave data in order to test the stability of the method. Results obtained by using several variants of traditional methods (Gumbel's approach and the POT method) are also presented for comparison purposes. The results of the MENU method showed an extraordinary stability, in contrast to the wide variability of the traditional methods. The predictions obtained by means of the MENU method are lower than the traditional predictions. This is in accordance with the results of other methods that also take into account the dependence structure of the examined time series. Copyright © 2005 John Wiley & Sons, Ltd.

[1]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[2]  Jonathan A. Tawn,et al.  An extreme-value theory model for dependent observations , 1988 .

[3]  Christos Stefanakos,et al.  A nonstationary stochastic model for long-term time series of significant wave height , 1995 .

[4]  Enrique Castillo,et al.  Engineering analysis of extreme value data : selection of models , 1992 .

[5]  C. Guedes Soares,et al.  An Application of the Peaks Over Threshold Method to Predict Extremes of Significant Wave Height , 1998 .

[6]  Keith R. Thompson,et al.  Return periods of extreme sea levels from short records , 1986 .

[7]  J. Labeyrie,et al.  Time scales and statistical uncertainties in the prediction of extreme environmental conditions , 1991 .

[8]  Arvid Naess,et al.  Crossing rate statistics of quadratic transformations of Gaussian processes , 2001 .

[9]  C. Guedes Soares,et al.  Application of the r largest-order statistics for long-term predictions of significant wave height , 2004 .

[10]  C. Guedes Soares,et al.  On the choice of data transformation for modelling time series of significant wave height , 1999 .

[11]  Irving I. Gringorten,et al.  A plotting rule for extreme probability paper , 1963 .

[12]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[13]  D.J.T. Carter,et al.  Methods of fitting the Fisher-Tippett type 1 extreme value distribution , 1983 .

[14]  A. H. El-Shaarawi,et al.  ON THE CALCULATION OF EXTREME WAVE HEIGHTS: A REVIEW , 1986 .

[15]  J. Pickands.  Statistical Inference Using Extreme Order Statistics , 1975 .

[16]  E. J. Gumbel,et al.  Statistics of Extremes. , 1960 .

[17]  Julia Abrahams,et al.  A Survey of Recent Progress on Level-Crossing Problems for Random Processes , 1986 .

[18]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[19]  Michel K. Ochi,et al.  Stochastic Analysis and Probabilistic Prediction of Random Seas , 1982 .

[20]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[21]  S. Rice.  Mathematical analysis of random noise , 1944 .

[22]  Edward F. Thompson,et al.  Case Studies of Extreme Wave Analysis: A Comparative Analysis , 1994 .

[23]  Konstandinos A. Belibassakis,et al.  Nonstationary Stochastic Modelling of Multivariate Long-Term Wind and Wave Data , 2005 .

[24]  C. D. Beaumont,et al.  Probabilistic Models in Engineering Sciences Vol. 1: Random Variables and Stochastic Processes , 1980 .

[25]  C. Guedes Soares,et al.  Representation of non-stationary time series of significant wave height with autoregressive models , 1996 .

[26]  J. Hosking,et al.  Parameter and quantile estimation for the generalized pareto distribution , 1987 .

[27]  Arvid Naess.  Statistical Extrapolation of Extreme Value Data Based on the Peaks Over Threshold Method , 1998 .

[28]  Christos Stefanakos,et al.  A unified methodology for the analysis, completion and simulation of nonstationary time series with missing values, with application to wave data , 2001 .

[29]  S. Grimshaw.  Computing Maximum Likelihood Estimates for the Generalized Pareto Distribution , 1993 .

[30]  Gerassimos A. Athanassoulis,et al.  Bivariate Stochastic Simulation Based On Nonstationary Time Series Modelling , 2003 .

[31]  R. J. Sobey,et al.  Triple annual maximum series in wave climate analyses , 1995 .

[32]  A. Walden,et al.  Maximum likelihood estimation of the parameters of the generalized extreme-value distribution , 1980 .

[33]  Enrique Castillo.  Extreme value theory in engineering , 1988 .