The effect of the generalized extreme value distribution parameter estimation methods in extreme wind speed prediction

Abstract

The modeling and prediction of extreme values of geophysical variables, such as wind, ocean surface waves, sea level, temperature and river flow, has always been a field of main concern for engineers and scientists. The analysis of extreme wind speed particularly plays an important role in natural disasters’ preparedness, prevention, mitigation and management and in various ocean, environmental and civil engineering applications, such as the design of offshore platforms and coastal marine structures, coastal management, wind climate analysis and structural safety. The block maxima (BM) approach is fundamental for extreme value analysis. BM method is closely related to the generalized extreme value (GEV) distribution, which unifies the three asymptotic extreme value distributions into a single one. The most common methods used for the estimation of the GEV parameters are maximum likelihood (ML) and probability weighted moments methods. In this work, several very common as well some less known estimation methods are firstly assessed through a simulation analysis. The results of the analysis showed that the maximum product of spacings (MPS), the elemental percentile (EP), the ordinary entropy method and, in a lesser degree, the ML methods seem to be, in general, superior to the other examined methods with respect to bias, mean squared error and variance of the estimated parameters. The effects of the estimation methods have been also assessed with respect to the n-year design values of real wind speed measurements. The obtained results suggest that the MPS and EP methods, which are rather unknown to the engineering community, describe adequately well the extreme quantiles of the wind speed data samples.

Appendix

1.1 The concept of return period and design value

In relevant wind and coastal engineering applications, the concept of return period and design value is widely used. The formal definition of the return period implies that the design value is expected to be exceeded on average once during the next n years. The period of n years is called return period RP, associated with the design value.

Specifically, let F(x) = Pr [X ≤ x] be the cdf of the block (annual) maximum of the random variable X. The return period is associated with the exceedance event X > x _p , that has probability of occurrence Pr [X > x _p ] = 1 − Pr [X ≤ x _p ] = 1 − F(x _p ). Therefrom, the return period is defined as follows:

$${\text{RP}}\left( {x_{p} } \right) = \frac{1}{{1 - F\left( {x_{p} } \right)}},$$

(90)

where x _p is the design value associated with the return period RP, or else the RP-year design value. Some authors use, instead of F(x _p ), the expression 1 − p, where p = Pr [X > x _p ]. In this case, x _p is the pth-quantile.

1.2 Plotting position formulae

The plotting position problem has been discussed by many authors, see, for example, Cunnane (1978), Makkonen (2006), Makkonen (2008), Kim et al. (2012), Gringorten (1963), Arnell et al. (1986), In-na and Nyuyen (1989) and Goel and De (1993). By 1960, many new formulae had appeared, but there was no criterion by which a single formula could be chosen to give unique results over all distributions. In order to choose the best plotting position formula, the estimated quantile should be free of bias and have minimum variance among graphical estimates. See also the relevant discussion in Makkonen (2006), Makkonen (2008) about plotting position formulae in EVA. Since quantiles are an important component in plotting position and return period calculations, there is a clear connection between them: First, the data (e.g., annual maxima) are ranked in increasing order of magnitude and a cumulative probability is associated with each point. Then, a best-fit line is fitted to the ranked values by some fitting procedure. An extrapolation of this line provides long-return periods of the extreme value of interest.

Let x _min = x _1:n  ≤ x _2:n  ≤ ···x _n:n  = x _max be an ordered random sequence. The most well-known plotting position formula is probably the Weibull formula, i.e.,

$$F\left( {x_{i:n} } \right) = \Pr \left[ {X < x_{i:n} } \right] = p_{i:n} = \frac{i}{n + 1},\quad i = 1, 2, \ldots , n.$$

(91)

In the above relation, as well as for all plotting positions formulae, F(x _i:n ) is the empirical estimate of the non-exceedance probability of the ith smallest member in an ordered sample. It is noted in Makkonen (2006), Makkonen (2008) that the plotting position provided by relation (91) is the only one justified for return period calculations.

In Cunnane (1978), the following general representation for the plotting position formulae is proposed:

$$p_{i:n} = \left({i - a} \right)/n, {\text{or}}p_{i:n} = \left({i - a} \right)/\left({n + 1 - 2a} \right),\quad {\text{for}}0 < a < 1.$$

(92)

The value of a (plotting position parameter) in the above relation yields approximately unbiased plotting positions for a variety of different distributions and determines the efficiency of the plotting position as regards the fit of a given theoretical distribution. For example, a = 0 is valid for all distributions (Weibull formula), a = 0.44 for the GEV and exponential distributions (Gringorten’s formula), a = 0.5 for the GEV distribution (Hazen’s formula) and a = 3/8 for the normal distribution.

Recently, in Kim et al. (2012), the authors using a genetic optimization method proposed the following plotting position formula for the GEV distribution:

$$p_{j:n} = \frac{j - 0.32}{{n + 0.0149g^{2} - 0.1364g + 0.3225}}, \, \quad j = 1,2, \ldots ,n,$$

(93)

where g denotes the skewness coefficient. The authors compared also the proposed formula with the formulas provided in Cunnane (1978), Gringorten (1963), Arnell et al. (1986), In-na and Nyuyen (1989) and Goel and De (1993).