Estimation of Wave Height Return Periods Using a Nonstationary Time Series Modelling
01 Jan 2006-pp 337-345
TL;DR: In this article, a new method for calculating return periods of various level values from nonstationary time series data is presented, based on the Mean Number of Upcrossings of the level x* (MENU method).
Abstract: 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). The whole procedure is numerically implemented and applied to long-term measured time series of significant wave height. The method is compared with other more classical approaches that take into acount the time dependance for time series of significant wave height. Estimates of the extremal index are given and for each method bootstrap confidence intervals are computed. The predictions obtained by means of MENU method are lower than the traditional predictions. This is in accordance with the results of other methods that take also into account the dependence structure of the examined time series.Copyright © 2006 by ASME
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TL;DR: It was found that limited effort has been put on developing statistical models for waves incorporating spatial and long-term temporal variability and it is suggested that model improvements could be achieved by adopting approaches from other application areas.
Abstract: This paper presents a literature survey on time-dependent statistical modelling of extreme waves and sea states. The focus is twofold: on statistical modelling of extreme waves and space- and time-dependent statistical modelling. The first part will consist of a literature review of statistical modelling of extreme waves and wave parameters, most notably on the modelling of extreme significant wave height. The second part will focus on statistical modelling of time- and space-dependent variables in a more general sense, and will focus on the methodology and models used also in other relevant application areas. It was found that limited effort has been put on developing statistical models for waves incorporating spatial and long-term temporal variability and it is suggested that model improvements could be achieved by adopting approaches from other application areas. In particular, Bayesian hierarchical space–time models were identified as promising tools for spatio-temporal modelling of extreme waves. Finally, a review of projections of future extreme wave climate is presented.
64 citations
Cites methods from "Estimation of Wave Height Return Pe..."
...A method for calculating return periods of various levels from long-term nonstationary time series data of significant wave height based on a new definition of the return period is presented in Stefanakos and Monbet (2006) and Stefanakos and Athanassoulis (2006)....
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Journal Article•
TL;DR: In this paper, the authors discuss some of the existing statistical models for the analysis of extreme value data in the case of independence, pointing out their excellence and possible sources of error.
Abstract: This paper discusses some of the existing statistical models for the analysis of extreme value data in the case of independence, pointing out their excellence and possible sources of error. Initially, the concept of order statistics is introduced, and the joint distribution of any set of order statistics is given. As simple examples, the distribution of the maximum, the minimum, any single order statistic, or any pair of order statistics are derived. Then, the problem of limit distribution is raised and carefully analyzed making a clear distinction between maxima and minima. It is shown that all models can be grouped in the Von Mises‐Jenkinson families, which include the three classical families. Several methods for selecting an adequate limit distribution based on data, including probability papers, least‐squares methods, and the curvature method are described. To clarify concepts, several illustrative examples of applications are included. Finally, a practical method for determining the limit distributi...
36 citations
TL;DR: In this paper, a composite stochastic model is proposed to resolve the state-by-state, seasonal and interannual variabilities of HS, which is a combination of two cyclostationary random processes modeling the variability of mean monthly values and mean monthly standard deviations.
Abstract: [1] In the present paper, a composite stochastic model is formulated and validated, resolving the state-by-state, seasonal and interannual variabilities of HS. The model is a combination of two cyclostationary random processes modeling the variability of mean monthly values and mean monthly standard deviations, respectively, and of a stationary random process modeling the residual, state-by-state, variability. In this way, the time series of significant wave height is given the structure of a multiple-scale composite stochastic process. The present model is a generalization of the nonstationary stochastic modeling introduced by the authors in previous works.
14 citations
01 Mar 2008
TL;DR: In this article, the initial non-stationary series is decomposed into a seasonal (periodic) mean value m(t) and a residual time series W(t).
Abstract: It is well known that long‐term time series of wind and wave data are modelled as non‐stationary stochastic processes with a yearly periodic mean value and standard deviation (periodically correlated or cyclostationary stochastic processes). Using this model, the initial non‐stationary series are decomposed into a seasonal (periodic) mean value m(t) and a residual time series W(t) multiplied by a seasonal (periodic) standard deviation s(t), of the form Y(t) = m(t)+s(t)W(t). The periodic components m(t) and s(t) are estimated using mean monthly values, and the residual time series W(t) is examined for stationarity. For this purpose, spectral densities of W(t) are obtained from different seasonal segments, calculated, and compared with each other. It is shown that W(t) can indeed be considered stationary, and thus Y(t) can be considered periodically correlated. This analysis has been applied to model wind and wave data from several locations in the Mediterranean Sea. It turns out that the spectrum of W(t) i...
5 citations
01 Jan 2013
TL;DR: This chapter aims at providing a comprehensive, up-to-date review of statistical models proposed for modeling long-term variability in extreme waves and sea states as well as a review of alternative approaches from other areas of application.
Abstract: This chapter aims at providing a comprehensive, up-to-date review of statistical models proposed for modeling long-term variability in extreme waves and sea states as well as a review of alternative approaches from other areas of application. A review of wave climate projections is also included. Efforts have been made to include all relevant and important work to make this literature survey as complete as possible, which has resulted in a rather voluminous list of references at the end of the chapter. Notwithstanding, due to the enormous amount of literature in this field some important works might inevitably have been omitted. This is unintended and it should be noted that important contributions to the discussion herein might exist of which I have not been aware. Nevertheless, it is believed that this literature study contains a fair review of relevant literature and as such that it gives a good indication of state of the art within the field and may serve as a basis for further research on stochastic modeling of extreme waves and sea states.
2 citations
References
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TL;DR: In this paper, the authors used the representations of the noise currents given in Section 2.8 to derive some statistical properties of I(t) and its zeros and maxima.
Abstract: In this section we use the representations of the noise currents given in section 2.8 to derive some statistical properties of I(t). The first six sections are concerned with the probability distribution of I(t) and of its zeros and maxima. Sections 3.7 and 3.8 are concerned with the statistical properties of the envelope of I(t). Fluctuations of integrals involving I2(t) are discussed in section 3.9. The probability distribution of a sine wave plus a noise current is given in 3.10 and in 3.11 an alternative method of deriving the results of Part III is mentioned. Prof. Uhlenbeck has pointed out that much of the material in this Part is closely connected with the theory of Markoff processes. Also S. Chandrasekhar has written a review of a class of physical problems which is related, in a general way, to the present subject.22
5,806 citations
"Estimation of Wave Height Return Pe..." refers background in this paper
...As it is well known [12], an upcrossing of the level x∗ by the process X(τ;β) occurs when...
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..., a stationary stochastic process and calculate its extremes based on the theory of these processes [12, 13]....
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...Equation (7) was first derived by Rice [12]....
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TL;DR: In this paper, the authors analyze the recent development of the theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i).
Abstract: Abstract. Let X j denote the life length of the j th component of a machine. In reliability theory, one is interested in the life length Z n of the machine where n signifies its number of components. Evidently, Z n = min (X j : 1 ≤ j ≤ n). Another important problem, which is extensively discussed in the literature, is the service time W n of a machine with n components. If Y j is the time period required for servicing the j th component, then W n = max (Y j : 1 ≤ j ≤ n). In the early investigations, it was usually assumed that the X's or Y's are stochastically independent and identically distributed random variables. If n is large, then asymptotic theory is used for describing Z n or W n . Classical theory thus gives that the (asymptotic) distribution of these extremes (Z n or W n ) is of Weibull type. While the independence assumptions are practically never satisfied, data usually fits well the assumed Weibull distribution. This contradictory situation leads to the following mathematical problems: (i) What type of dependence property of the X's (or the Y's) will result in a Weibull distribution as the asymptotic law of Z n (or W n )? (ii) given the dependence structure of the X's (or Y's), what type of new asymptotic laws can be obtained for Z n (or W n )? The aim of the present paper is to analyze the recent development of the (mathematical) theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i). In regard to (ii), the following result holds: the class of limit laws of extremes for exchangeable variables is identical to the class of limit laws of extremes for arbitrary random variables. One can therefore limit attention to exchangeable variables. The basic references to this paper are the author's recent papers in Duke Math. J. 40 (1973), 581–586, J. Appl. Probability 10 (1973, 122–129 and 11 (1974), 219–222 and Zeitschrift fur Wahrscheinlichkeitstheorie 32 (1975), 197–207. For multivariate extensions see H. A. David and the author, J. Appl. Probability 11 (1974), 762–770 and the author's paper in J. Amer. Statist. Assoc. 70 (1975), 674–680. Finally, we shall point out the difficulty of distinguishing between several distributions based on data. Hence, only a combination of theoretical results and experimentations can be used as conclusive evidence on the laws governing the behavior of extremes.
1,964 citations
"Estimation of Wave Height Return Pe..." refers background in this paper
...In this approach, when it is based on annual maxima, the distribution of the population of maxima, denoted by G(x), is known as n→∞ [26]....
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Book•
01 Jan 1977TL;DR: In this article, the authors analyze the recent development of the theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i).
Abstract: . Let X j denote the life length of the j th component of a machine. In reliability theory, one is interested in the life length Z n of the machine where n signifies its number of components. Evidently, Z n = min (X j : 1 ≤ j ≤ n). Another important problem, which is extensively discussed in the literature, is the service time W n of a machine with n components. If Y j is the time period required for servicing the j th component, then W n = max (Y j : 1 ≤ j ≤ n). In the early investigations, it was usually assumed that the X's or Y's are stochastically independent and identically distributed random variables. If n is large, then asymptotic theory is used for describing Z n or W n . Classical theory thus gives that the (asymptotic) distribution of these extremes (Z n or W n ) is of Weibull type. While the independence assumptions are practically never satisfied, data usually fits well the assumed Weibull distribution. This contradictory situation leads to the following mathematical problems: (i) What type of dependence property of the X's (or the Y's) will result in a Weibull distribution as the asymptotic law of Z n (or W n )? (ii) given the dependence structure of the X's (or Y's), what type of new asymptotic laws can be obtained for Z n (or W n )? The aim of the present paper is to analyze the recent development of the (mathematical) theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i). In regard to (ii), the following result holds: the class of limit laws of extremes for exchangeable variables is identical to the class of limit laws of extremes for arbitrary random variables. One can therefore limit attention to exchangeable variables. The basic references to this paper are the author's recent papers in Duke Math. J. 40 (1973), 581–586, J. Appl. Probability 10 (1973, 122–129 and 11 (1974), 219–222 and Zeitschrift fur Wahrscheinlichkeitstheorie 32 (1975), 197–207. For multivariate extensions see H. A. David and the author, J. Appl. Probability 11 (1974), 762–770 and the author's paper in J. Amer. Statist. Assoc. 70 (1975), 674–680. Finally, we shall point out the difficulty of distinguishing between several distributions based on data. Hence, only a combination of theoretical results and experimentations can be used as conclusive evidence on the laws governing the behavior of extremes.
1,953 citations
996 citations