**14. Discordancy test**

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

222 New Developments in Renewable Energy

(1990) as:

rainfall and floods.

**12. At-station goodness of fit tests**

*λ*<sup>1</sup> =*β*<sup>0</sup> (11)

*λ*<sup>2</sup> =2*β*<sup>1</sup> - *β*<sup>0</sup> (12)

*λ*<sup>3</sup> =6*β*<sup>2</sup> - 6*β*<sup>1</sup> + *β*<sup>0</sup> (13)

*τ* =*λ*<sup>2</sup> / *λ*<sup>1</sup> (15)

*τ<sup>r</sup>* =*λ<sup>r</sup>* / *λ*<sup>2</sup> , *r* ≥3 (16)

λ<sup>4</sup> =20β<sup>3</sup> - 30β<sup>2</sup> + 12β<sup>1</sup> - β<sup>0</sup> (14)

L-moment ratios, which are analogous to conventional moment ratios, are defined by Hosking

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

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

After estimation of the parameter of the prescribed distributions, the usual question is the selection of the best fitted distribution to the observation sample. For this aim, the goodnessof-fit tests are used to compare fitted theoretical distributions and observations. Two very common goodness-of-fit tests with wide application in the literature are Chi-square and Kolmogorov-Smirnov tests. Another extensively used test is root mean square error (RMSE) which estimates the root of the square differences between observed values and calculated

by replacing distribution L-moments *λr* by their sample estimates.

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 and Wallis (1997). The aim of the data screening performed using L-moments based on a discordancy measure *Di* is to identify data that are grossly discordant with the group as a whole for the regional flood frequency analysis. Hosking and Wallis (1997) defined the discordancy measure (*Di* ) considering that there are *N* sites in the sample. Let *ui* =(*<sup>t</sup>* (*i*) , *t*<sup>3</sup> (*i*) , *t*<sup>4</sup> (*i*) ) be a vector containing the sample L-moment ratios for site *i*(*i* =1, 2, …, *N* ) (Hosking and Wallis, 1993, 1997). The classical discordancy measure for any gauging site *i* is calculated as follows (Hosking and Wallis 1997):

$$D\_i = \frac{1}{3} \begin{pmatrix} \mathbf{u}\_i \ \mathbf{u} \end{pmatrix}^T \mathbf{S} \cdot \mathbf{1} \begin{pmatrix} \mathbf{u}\_i \ \mathbf{u} \end{pmatrix} \tag{17}$$

$$\stackrel{\cdots}{\mu} = \frac{1}{N} \sum\_{i=1}^{N} \mu\_i \tag{18}$$

*V*<sup>1</sup> ={∑ *i*=1 *N ni* (*τ* (*i*)

*H* =

(*Vi* - *μV <sup>i</sup>* ) *σV i*

*τ <sup>R</sup>* = ∑ *i*=1 *N ni τ* (*i*) / ∑ *i*=1 *N*

*τ*3 *<sup>R</sup>* = ∑ *i*=1 *N ni τ*3 (*i*) / ∑ *i*=1 *N*

*τ*4 *<sup>R</sup>* = ∑ *i*=1 *N ni τ*4 (*i*) / ∑ *i*=1 *N*

heterogeneous" if 1≤*H* <2, and "definitely heterogeneous" if *H* ≥2.

may need to resolve *H* values very close to 1 or 2.

*V*<sup>2</sup> = ∑ *i*=1 *N ni* {(*τ* (*i*)

*V*<sup>3</sup> = ∑ *i*=1 *N ni* {(*τ*<sup>3</sup> (*i*) - *τ*<sup>3</sup>

The heterogeneity measure is then defined as:

where *μVi*

and *σVi*

where N is the number of sites, ni

correlation or serial correlation.


> (*i*) - *τ*<sup>3</sup>

respectively. The regional averages L-moment ratios are determined by following equations:

*<sup>R</sup>*)2}0.5 / <sup>∑</sup> *i*=1 *N*

are the mean and the standard deviation of the simulated value of the V,

is the record length at site i, and *τ* (*i*)

sample L-moment ratios at site *i*. After fitting a Kappa distribution to the regional average Lmoment ratios, a large number of realizations *Nsim* of a region with *N* sites, each having the Kappa distribution is simulated. The simulated regions are homogenous and have no cross-

Declare the region to be heterogeneous if *H* is sufficiently large. Hosking and Wallis (1993) suggested that the region be regarded as "acceptably homogeneous" if *H* <1, "possibly

In order to achieve reliable estimates *μV* and *σ<sup>V</sup>* , Hosking and Wallis (1993) recognized that the value of *Nsim* =500 should usually be adequate. Furthermore, they judged that larger values

Hosking and Wallis (1993) revealed that *H* statistics based on *V*2 and *V*<sup>3</sup> lacks a power to distinguish between homogeneous and heterogeneous regions and the *H* statistics based on *V*<sup>1</sup> has much better discriminatory power and thus, it is suggested as heterogeneity measure.


*<sup>R</sup>*)<sup>2</sup> <sup>+</sup> (*τ*<sup>4</sup> (*i*) - *τ*<sup>4</sup> *<sup>R</sup>*)<sup>2</sup> }0.5 / <sup>∑</sup> *i*=1 *N*

(20)

225

*ni* (21)

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

*ni* (22)

; *i* =1, 2, 3 (23)

Wind Speed Regionalization Under Climate Change Conditions

*ni* (24)

*ni* (25)

*ni* (26)

, *τ*<sup>3</sup> (*i*)

, and *τ*<sup>4</sup> (*i*) are the

$$\mathbf{S} = \frac{1}{N \cdot 1} \sum\_{i=1}^{N} \begin{pmatrix} \mathbf{u} \\ \boldsymbol{\mu}\_{i} - \boldsymbol{\mu} \end{pmatrix} \begin{pmatrix} \mathbf{u} \\ \boldsymbol{\mu}\_{i} - \boldsymbol{\mu} \end{pmatrix}^{T} \tag{19}$$

Where N is the number of stations, *ui* is the vector of L-moments, *u* and *S* are the sample mean and covariance matrix respectively, and *T* denotes the transposition of a vector or matrix. Large values of *Di* indicate the sites that are the most discordant from the group as a whole and are the most suitable for investigating the existence of data errors (Hosking and Wallis 1993). Generally, if a site's *D* statistic exceeds three when the number of sites in one region is greater than 15, its data are considered to be discordant from the rest of the regional data (Hosking and Wallis 1997). Large values of *Di* indicate that cautious investigation of the *i*th site should be carried out to detect the presence of data errors.

#### **15. Heterogeneity test**

L-moment heterogeneity tests allow assessing whether a group of sites may reasonably be treated as a homogeneous region. The heterogeneity measure compares the between-site variations in sample L-moments for a group of sites with what would be expected for a homogeneous region (Hosking and Wallis, 1993).

The homogeneity test used in this study is the test that is proposed by Hosking and Wallis (1993) and is based on various orders of sample L-moment ratios. It is particularly based on the variability of three different levels of tests: a test based on the L-CV only; a test based on the L-CV and L-skew; and a test based on the L-skew and L-kurt. These tests are called the Vstatistics and are respectively defined as:

#### Wind Speed Regionalization Under Climate Change Conditions http://dx.doi.org/10.5772/55985 225

$$V\_1 = \left| \sum\_{i=1}^{N} n\_i \{ \pi^{(i)} \text{ - } \pi^{(R)} \} ^2 \left| \sum\_{i=1}^{N} n\_i \right| ^{0.5} \tag{20}$$

$$\ln V\_{2} = \sum\_{i=1}^{N} n\_{i} \left| \left\{ \left( \pi^{\binom{i}{2}} - \pi^{\cdot R} \right)^{2} + \left( \pi\_{3}^{\binom{i}{3}} - \pi\_{3}^{\cdot R} \right)^{2} \right\}^{0.5} \right| \sum\_{i=1}^{N} n\_{i} \tag{21}$$

$$\Delta V\_{\mathcal{B}} = \sum\_{i=1}^{N} n\_i \left| \left\{ \pi\_3^{(j)} \text{ -- } \pi\_3^{R} \right\}^2 + \left\{ \pi\_4^{(j)} \text{ -- } \pi\_4^{R} \right\}^2 \right| \stackrel{0.5}{\sum\_{i=1}^{N} n\_i}{\sum\_{i=1}^{N}} \tag{22}$$

The heterogeneity measure is then defined as:

and Wallis (1997). The aim of the data screening performed using L-moments based on a

whole for the regional flood frequency analysis. Hosking and Wallis (1997) defined the

be a vector containing the sample L-moment ratios for site *i*(*i* =1, 2, …, *N* ) (Hosking and Wallis, 1993, 1997). The classical discordancy measure for any gauging site *i* is calculated as

> *<sup>S</sup>* -1(*ui* - *<sup>u</sup>* -

is the vector of L-moments, *u*

and covariance matrix respectively, and *T* denotes the transposition of a vector or matrix. Large values of *Di* indicate the sites that are the most discordant from the group as a whole and are the most suitable for investigating the existence of data errors (Hosking and Wallis 1993). Generally, if a site's *D* statistic exceeds three when the number of sites in one region is greater than 15, its data are considered to be discordant from the rest of the regional data (Hosking and Wallis 1997). Large values of *Di* indicate that cautious investigation of the *i*th site should

L-moment heterogeneity tests allow assessing whether a group of sites may reasonably be treated as a homogeneous region. The heterogeneity measure compares the between-site variations in sample L-moments for a group of sites with what would be expected for a

The homogeneity test used in this study is the test that is proposed by Hosking and Wallis (1993) and is based on various orders of sample L-moment ratios. It is particularly based on the variability of three different levels of tests: a test based on the L-CV only; a test based on the L-CV and L-skew; and a test based on the L-skew and L-kurt. These tests are called the V-

*Di* <sup>=</sup> <sup>1</sup> 3 (*ui* - *<sup>u</sup>* - ) *T*

*<sup>S</sup>* <sup>=</sup> <sup>1</sup> *<sup>N</sup>* - <sup>1</sup> ∑ *i*=1 *<sup>N</sup>* (*ui* - *<sup>u</sup>* - )(*ui* - *<sup>u</sup>* - ) *T*

*u* - = 1 *<sup>N</sup>* ∑ *i*=1 *N*

is to identify data that are grossly discordant with the group as a

, *t*<sup>3</sup> (*i*) , *t*<sup>4</sup> (*i*) )

(19)

) (17)

and *S* are the sample mean

*ui* (18)


) considering that there are *N* sites in the sample. Let *ui* =(*<sup>t</sup>* (*i*)

discordancy measure *Di*

224 New Developments in Renewable Energy

discordancy measure (*Di*

follows (Hosking and Wallis 1997):

Where N is the number of stations, *ui*

**15. Heterogeneity test**

be carried out to detect the presence of data errors.

homogeneous region (Hosking and Wallis, 1993).

statistics and are respectively defined as:

$$H = \frac{\{V\_i \cdot \mu\_{V\_i}\}}{\sigma\_{V\_i}} \quad ; \quad i = 1, \ 2, \ 3 \tag{23}$$

where *μVi* and *σVi* are the mean and the standard deviation of the simulated value of the V, respectively. The regional averages L-moment ratios are determined by following equations:

$$\pi \,\, \pi \,\, ^R = \sum\_{i=1}^N \pi\_i \,\, \pi \,\, ^{(j)} \Big/ \sum\_{i=1}^N \pi\_i \,\, \tag{24}$$

$$\pi\_3^R = \sum\_{i=1}^N \pi\_i \pi\_3^{(i)} \Big/ \sum\_{i=1}^N \pi\_i \tag{25}$$

$$\pi\_{\mathbf{4}}\pi\_{\mathbf{4}}^{R} = \sum\_{i=1}^{N} \pi\_{i}\pi\_{\mathbf{4}}^{\{i\}} \Big/ \sum\_{i=1}^{N} \pi\_{i} \tag{26}$$

where N is the number of sites, ni is the record length at site i, and *τ* (*i*) , *τ*<sup>3</sup> (*i*) , and *τ*<sup>4</sup> (*i*) are the sample L-moment ratios at site *i*. After fitting a Kappa distribution to the regional average Lmoment ratios, a large number of realizations *Nsim* of a region with *N* sites, each having the Kappa distribution is simulated. The simulated regions are homogenous and have no crosscorrelation or serial correlation.

Declare the region to be heterogeneous if *H* is sufficiently large. Hosking and Wallis (1993) suggested that the region be regarded as "acceptably homogeneous" if *H* <1, "possibly heterogeneous" if 1≤*H* <2, and "definitely heterogeneous" if *H* ≥2.

In order to achieve reliable estimates *μV* and *σ<sup>V</sup>* , Hosking and Wallis (1993) recognized that the value of *Nsim* =500 should usually be adequate. Furthermore, they judged that larger values may need to resolve *H* values very close to 1 or 2.

Hosking and Wallis (1993) revealed that *H* statistics based on *V*2 and *V*<sup>3</sup> lacks a power to distinguish between homogeneous and heterogeneous regions and the *H* statistics based on *V*<sup>1</sup> has much better discriminatory power and thus, it is suggested as heterogeneity measure.
