**7. Probability distribution function and frequency formula**

To describe the probability distribution of a random variable *X* , a cumulative distribution function (CDF) is used. The value of this function *F* (*x*) is simply the probability *P* of the event that the random variable takes on value equal to or less than the argument:

$$F(\mathbf{x}) = P[X \le \mathbf{x}] \tag{1}$$

This is the probability of the random variable *X* , it will not exceed *x* and is shown by the nonexceedance probability *F* (*x*).

The occurrence of extreme events is not according to a constant regime or with a fixed magnitude and the time interval between two such events is variable. Thus, the return period defined as the average inter-arrival time between two extreme events is an applicable tool in such cases. An extreme event *xT* with return period *T* can occur more than one time in a year and its exceedance probability can be expressed by:

$$\mathbb{E}\left[\mathbf{X} \geq \mathbf{x}\_{T}\right] = \frac{1}{T} \tag{2}$$

**10. Maximum likelihood method**

*x*1, *x*2, …, *xn*. The probability density for *xi*

to maximization of likelihood function defined as:

minimum average error with respect to correct parameters.

*β<sup>r</sup>* = *∫* 0 1 *x*(*F* )*F <sup>r</sup>*

> *λr*+1 = ∑ *k*=0 *r βr*

random event occur in a distance consisting *xi* would be equal to *f* (*xi*

independent, the joint density function for all observations can be calculated by:

*L* (*x*1, *x*2, …, *xn*) = *f* (*x*1, *x*2, …, *xn*) =∏

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

*f* (*x*1) *dx*. *f* (*x*1) *dx* … *f* (*xn*) *dx* =(∏

with length *dx* and *xi*

**11. L-moments method**

Which can be rewritten as:

moments as:

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

is selected from independent and identically distributed observations

Wind Speed Regionalization Under Climate Change Conditions

*i*=1 *n f* (*xi*

*<sup>β</sup><sup>r</sup>* <sup>=</sup>*E*(*x*{*<sup>F</sup>* (*x*)}*r*) (8)

*dF* , *r* =0, 1, …, s (9)

*<sup>k</sup>* ) (10)

). The probability that a

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

221

) *dx*. As the samples are

))(*dx*)*<sup>n</sup>* (6)

) (7)

can be denoted by *f* (*xi*

*i*=1 *n f* (*xi*

As the distance *dx* is constant, the maximization of above joint density function is equivalent

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

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 L-

(-1)*<sup>r</sup>*-*<sup>k</sup>* ( *<sup>r</sup>*

*<sup>k</sup>* )( *<sup>r</sup>* <sup>+</sup> *<sup>k</sup>*

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

$$P\left(\mathbf{x}\_{T}\right) = P\left[X \le \mathbf{x}\_{T}\right] = 1 - P\left[X > \mathbf{x}\_{T}\right] = 1 - \frac{1}{T} \tag{3}$$

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>p</sup>* =1 - <sup>1</sup> *<sup>T</sup>* ), it is possible to determine the quantile *xp* using the inverse function of *F* :

$$\propto\_p = F^{-1}(p) = F^{-1}\left(1 - \frac{1}{T}\right) \tag{4}$$

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