**10. Maximum likelihood method**

such cases. An extreme event *xT* with return period *T* can occur more than one time in a year

*<sup>T</sup>* (2)

*<sup>T</sup>* (3)

*<sup>T</sup>* ) (4)

<sup>∞</sup>*x <sup>r</sup> f* (*x*)*dx* (5)

*<sup>P</sup> <sup>X</sup>* <sup>&</sup>gt; *xT* <sup>=</sup> <sup>1</sup>

Thus, the above mentioned non-exceedance probability can be presented as follows:

*<sup>F</sup>* (*xT* ) <sup>=</sup>*<sup>P</sup> <sup>X</sup>* <sup>≤</sup> *xT* =1 - *<sup>P</sup> <sup>X</sup>* <sup>&</sup>gt; *xT* =1 - <sup>1</sup>

*xp* <sup>=</sup> *<sup>F</sup>* -1(*p*)= *<sup>F</sup>* -1(1 - <sup>1</sup>

which gives the value of *xp* corresponding to any particular value of *p* or *T* .

Equation (3) is presenting the magnitude of an extreme event correspond to a return period *T* . For a preselected value of return period and corresponding non-exceedance probability

*<sup>T</sup>* ), it is possible to determine the quantile *xp* using the inverse function of *F* :

The probability distribution is a function for describing the probability of occurrence of a random event. Large amount of statistical information will be summarized in the distribution and its parameters by fitting a probability distribution on a set of hydrologic data. The most important and widely used methods for estimation of distribution parameters from data samples are method of moments, linear moments, and maximum likelihood which will be

The method of moments was first introduced by Pearson (1902). He found that the appropriate estimations of the parameters of a probability distribution are those which their moments match with corresponding sample moments in the best way. In this method, general formula

The method of moments is describing the relation between moments and distribution param‐ eters. The most important moments around the mean are the mean, variance, skewness, and

for calculation of moments of order *r* of the distribution *f* (*x*) around the mean is:

*μ <sup>r</sup>* =*∫* -∞

kurtosis, which are the one to four order moments, respectively.

and its exceedance probability can be expressed by:

220 New Developments in Renewable Energy

**8. Selection of probability distribution**

described more in next paragraph.

**9. Method of moments**

(*<sup>p</sup>* =1 - <sup>1</sup>

Maximum-likelihood estimation was recommended, analyzed and vastly popularized by R. A. Fisher between 1912 and 1922 (Aldrich, John 1997). He argued that the best value of a parameter of a probability distribution should be one, which maximizes the likelihood or joint probability of occurrence of the sample. Assume that the sample space is divided into parts with length *dx* and *xi* is selected from independent and identically distributed observations *x*1, *x*2, …, *xn*. The probability density for *xi* can be denoted by *f* (*xi* ). The probability that a random event occur in a distance consisting *xi* would be equal to *f* (*xi* ) *dx*. As the samples are independent, the joint density function for all observations can be calculated by:

$$f\left(\mathbf{x}\_1\right)d\mathbf{x}.f\left(\mathbf{x}\_1\right)d\mathbf{x}\dots f\left(\mathbf{x}\_n\right)d\mathbf{x} = \left(\prod\_{i=1}^n f\left(\mathbf{x}\_i\right)\right)(d\mathbf{x})^n\tag{6}$$

As the distance *dx* is constant, the maximization of above joint density function is equivalent to maximization of likelihood function defined as:

$$L\left(\mathbf{x}\_{1}, \mathbf{x}\_{2}, \dots, \mathbf{x}\_{n}\right) = f\left(\mathbf{x}\_{1}, \mathbf{x}\_{2}, \dots, \mathbf{x}\_{n}\right) = \prod\_{i=1}^{n} f\left(\mathbf{x}\_{i}\right) \tag{7}$$

The maximum likelihood method is theoretically the most accurate method in estimation of the parameters of probability distributions. In fact, it is estimating those parameters with minimum average error with respect to correct parameters.

#### **11. L-moments method**

Probability weighted moments (PWMs) are defined by Greenwood et al. (1979) as

$$\beta\_r = E\left(\mathbf{x}\{F(\mathbf{x})\}'\right) \tag{8}$$

Which can be rewritten as:

$$\beta\_r = \underset{0}{\text{fx}} \{ F \} F \quad \text{'} \quad \text{'} \qquad r = 0, \text{ 1, } \quad \text{....} \quad \text{'s} \tag{9}$$

Where *F* = *F* (*x*) is CDF of *x* and *x*(*F* ) is its inverse. In the case of *r* =0, *β<sup>r</sup>* is equal to the mean of the distribution *μ* =*E*(*x*). PWMs are precursors of L-moments and developed more by the works of Hosking (1986, 1990) and Hosking and Wallis (1991, 1993, 1997). Hosking defined Lmoments as:

$$
\lambda\_{r+1} = \sum\_{k=0}^{r} \beta\_r (-1)^{r+k} \binom{r}{k} \binom{r+k}{k} \tag{10}
$$

The first four order L-moments can be calculated as bellow:

$$
\lambda\_1 = \beta\_0 \tag{11}
$$

ones divided by the sample size. Among theoretical probability distributions, some of them are selected by researchers for describing wind speed, which are presented in Table 2.

**Year Distribution function Scientists** 1940 to 1945 Pearson Type III Putnum\* 1951 Pearson Type III Sherlock\*

1970s Isotropic Gaussian Model of McWilliams et al. Justus and Koeppl \*

1978 Three-parameter Weibull Stewart and Essenvanger \*

1976 to 1977 Square-root Normal Model Winger \*

 Three-Parameter Generalized Gamma Auwera \* Inverse Gaussian Bardsley \* Pearson Type I (Beta) Lavagnini et al. \* Weibull Stelios Pasardes \* Log-Normal Bogardi and Matyasovski

2009 Beta Carta et al. 2009 Two-Component Weibull Akdog et al. \* 2010 Gumbel and Weibull Deepthi and Deo

One of the main problems in frequency analysis is the lack of adequate long time data in locations under study, which beside insufficient accuracy in data recording in at-site estimates, has caused regional frequency analysis to be more applicable in such studies. Steps of regional

For the screening of similar sites, the discordancy measure, in terms of the sample L-moment ratios (L-CV, L-skew, and L-kurt.) of the gauging sites' observed data is suggested by Hosking

1974, 1976, 1977 Three-Parameter Log-normal

**Table 2.** Probability distribution function of wind speed in the literature

**13. Regional frequency analysis**

frequency analysis are presented below.

**14. Discordancy test**

Bivariate Distributions of Two Components Essenwanger \*

Two-Parameter Normal Crutcher and Bear \*

Wind Speed Regionalization Under Climate Change Conditions

Luna and Church \* Kaminsky, Justus et al.

http://dx.doi.org/10.5772/55985

223

1950 to 1970

\*

by Carta et al. 2009

$$
\lambda\_2 = 2\beta\_1 \cdot \beta\_0 \tag{12}
$$

$$
\lambda\_3 = 6\beta\_2 - 6\beta\_1 + \beta\_0 \tag{13}
$$

$$
\lambda\_4 = 20\beta\_3 - 30\beta\_2 + 12\beta\_1 - \beta\_0 \tag{14}
$$

L-moment ratios, which are analogous to conventional moment ratios, are defined by Hosking (1990) as:

$$
\pi = \lambda\_2 / \lambda\_1 \tag{15}
$$

$$
\pi\_r = \lambda\_r / \lambda\_2 \quad , \quad r \ge 3 \tag{16}
$$

where *λ*1 is a measure of location, *τ* is a measure of scale and dispersion (L-CV), *τ*3 is a measure of skewness (L-skew), and *τ*4 is a measure of kurtosis (L-kurt.). Analogous to conventional (product) moments, the L-moments of order one to four characterize location, scale, skewness and kurtosis, respectively (Karvanen, 2006). In order to estimate the distribution parameters in the method of L-moments, similar to other methods, sample L-moment ratios are calculated by replacing distribution L-moments *λr* by their sample estimates.

However, L-moments have significant advantages over PWM's, specially their ability to summarize a statistical distribution in a more meaningful way. Since L-moment estimators are linear functions of the ordered data values, they are virtually unbiased and less influenced by outliers. Also they have relatively small sampling variance and the bias of their small sample estimates remains quite small. L-moments have become popular tools for solving various problems related to parameter estimation, distribution identification, and regionalization in different fields such as hydrology, water resources, and especially in regional analysis of rainfall and floods.
