• Akdağ, S. A., , H. S. Bagiorgas, , and G. Mihalakakou, 2010: Use of two-component Weibull mixtures in the analysis of wind speed in the eastern Mediterranean. Appl. Energy, 87, 25662573.

    • Search Google Scholar
    • Export Citation
  • Byrd, R. H., , J. C. Gilbert, , and J. Nocedal, 2000: A trust region method based on interior point techniques for nonlinear programming. Math. Program., 89, 149185.

    • Search Google Scholar
    • Export Citation
  • Carta, J. A., , and P. Ramírez, 2007a: Analysis of two-component mixture Weibull statistics for estimation of wind speed distributions. Renew. Energy, 32, 518531.

    • Search Google Scholar
    • Export Citation
  • Carta, J. A., , and P. Ramírez, 2007b: Use of finite mixture distribution models in the analysis of wind energy in the Canarian Archipelago. Energy Convers. Manage., 48, 281291.

    • Search Google Scholar
    • Export Citation
  • Carta, J. A., , P. Ramírez, , and S. Velázquez, 2009: A review of wind speed probability distributions used in wind energy analysis: Case studies in the Canary Islands. Renew. Sustainable Energy Rev., 13, 933955.

    • Search Google Scholar
    • Export Citation
  • Chang, T. P., 2010: Wind speed and power density analyses based on mixture Weibull and maximum entropy distributions. Int. J. Appl. Sci. Eng., 8, 3946.

    • Search Google Scholar
    • Export Citation
  • Chang, T. P., 2011: Estimation of wind energy potential using different probability density functions. Appl. Energy, 88, 18481856.

  • Dorvlo, A. S. S., 2002: Estimating wind speed distribution. Energy Convers. Manage., 43, 23112318.

  • Garcia, A., , J. L. Torres, , E. Prietoa, , and A. de Francisco, 1998: Fitting wind speed distribution: A case study. Sol. Energy, 2, 139144.

    • Search Google Scholar
    • Export Citation
  • Jaramillo, O. A., , and M. A. Borja, 2004: Wind speed analysis in La Ventosa, Meixco: A bimodal probability distribution case. Renew. Energy, 29, 16131630.

    • Search Google Scholar
    • Export Citation
  • Massey, F. J., Jr., 1951: The Kolmogorov-Smirnov test for goodness of fit. J. Amer. Stat. Assoc., 46, 6878.

  • Meng, Q., , and M. Tang, 1997: On asymptotic distributions of yearly maximum values of surface temperature and wind speed over Chengdu and the estimation of their parameters. J. Chengdu Inst. Meteor., 12, 284291.

    • Search Google Scholar
    • Export Citation
  • Morgen, E. C., , M. Lackner, , R. M. Vogela, , and L. G. Baise, 2011: Probability distributions for offshore wind speeds. Energy Convers. Manage., 52, 1526.

    • Search Google Scholar
    • Export Citation
  • Murthy, D. N. P., , M. Xie, , and R. Jiang, 2004: Weibull Models. John Wiley and Sons, 383 pp.

  • Waltz, R. A., , J. L. Morales, , J. Nocedal, , and D. Orban, 2006: An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math. Program., 107, 391408.

    • Search Google Scholar
    • Export Citation
  • Zaharim, A., , A. M. Razali, , R. Z. Abidin, , and K. Sopian, 2009: Fitting of statistical distributions to wind speed data in Malaysia. Eur. J. Sci. Res., 26, 612.

    • Search Google Scholar
    • Export Citation
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Two Improved Mixture Weibull Models for the Analysis of Wind Speed Data

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  • 1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China
  • | 2 Faculty of Science, Xi’an Jiaotong University, Xi’an, China
  • | 3 Key Laboratory of Regional Climate-Environment Research for Temperature East Asia, Institute of Atmospheric Physics, Chinese Academy of Science, Beijing, China
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Abstract

In this paper, the authors propose two improved mixture Weibull distribution models by adding one or two location parameters to the existing two-component mixture two-parameter Weibull distribution [MWbl(2, 2)] model. One improved model is the mixture two-parameter Weibull and three-parameter Weibull distribution [MWbl(2, 3)] model. The other improved model is the two-component mixture three-parameter Weibull distribution [MWbl(3, 3)] model. In contrast to existing literature, which has focused on the MWbl(2, 2) and the typical Weibull distribution models, the authors apply the MWbl(2, 3) model and MWbl(3, 3) model to fit the distribution of wind speed data with nearly zero percentages of null wind speed. The parameters of the two improved models are estimated by the maximum likelihood method in which the maximization problem is regarded as a nonlinear programming problem with only inequality constraints and is solved numerically by the interior-point method. The experimental results show that the mixture Weibull models proposed in this paper are more flexible than the existing models for the analysis of wind speed data in practice.

Corresponding author address: Xu Qin, School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China. E-mail: jiayouxuxu@gmail.com

Abstract

In this paper, the authors propose two improved mixture Weibull distribution models by adding one or two location parameters to the existing two-component mixture two-parameter Weibull distribution [MWbl(2, 2)] model. One improved model is the mixture two-parameter Weibull and three-parameter Weibull distribution [MWbl(2, 3)] model. The other improved model is the two-component mixture three-parameter Weibull distribution [MWbl(3, 3)] model. In contrast to existing literature, which has focused on the MWbl(2, 2) and the typical Weibull distribution models, the authors apply the MWbl(2, 3) model and MWbl(3, 3) model to fit the distribution of wind speed data with nearly zero percentages of null wind speed. The parameters of the two improved models are estimated by the maximum likelihood method in which the maximization problem is regarded as a nonlinear programming problem with only inequality constraints and is solved numerically by the interior-point method. The experimental results show that the mixture Weibull models proposed in this paper are more flexible than the existing models for the analysis of wind speed data in practice.

Corresponding author address: Xu Qin, School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China. E-mail: jiayouxuxu@gmail.com

1. Introduction

In this paper we analyze the application of Weibull distribution models on wind speed data. The typical Weibull distribution (named after the Swedish physicist W. Weibull, who applied it when studying material strength in tension and fatigue in the 1930s) provides a close approximation to the probability laws of many natural phenomena (Murthy et al. 2004). In recent years most attention has been focused on this distribution for wind energy applications not only because of its greater flexibility and simplicity but also because it can give a good fit to experiment data.

During the past decades, the typical two-parameter Weibull has been a flexible distribution that is useful for describing unimodal frequency distributions of wind speeds at many sites. For instance, Dorvlo (2002) used the two-parameter Weibull distribution to model wind speeds at four locations that have the highest long-term average wind speeds in Oman; in Garcia et al. (1998) the Weibull and lognormal models were used for fitting the potential shape of wind speed data. That study dealt with the estimation of the annual Weibull and lognormal parameters from 20 locations in Navarre. Both distribution models gave a good fit, giving better results for the Weibull distribution; Zaharim et al. (2009) adopted two-parameter Weibull distribution to fit the wind data recorded in Faculty of Engineering, University Kebangsaan, Malaysia. The scale and shape parameters were estimated by using maximum likelihood method; Meng and Tang (1997) obtained that asymptotic distribution of the yearly maximum wind speed in Chengdu, China, had relations with the three-parameter Weibull distribution.

However, in some of regions of the world, the use of the typical Weibull distributions leads to incorrect results. Just as Carta and Ramírez (2007a) stated, a two-component mixture Weibull distribution was even more useful than the typical two-parameter Weibull because the former was additionally able to represent heterogeneous wind regimes in which there was evidence of bimodality or bitangentiality or, simply, unimodality. In that study, the authors used the most frequently used methods, that is, the moments, likelihood maximum, and least squares methods, to estimate the five parameters of the two-component mixture two-parameter Weibull distribution [MWbl(2, 2)]. Recently, authors mostly tended to use the MWbl(2, 2) models that showed good results for the analysis of wind speed data (Carta and Ramírez 2007b; Jaramillo and Borja 2004; Carta et al. 2009; Chang 2011, 2010; Akdağ et al. 2010; Morgen et al. 2011).

In practice, we may encounter wind speed data with nearly zero percentages of null wind speed, which means that the minimum of wind speed series is always higher than some certain limit. In the three-parameter Weibull model [Wbl(3)], there is a location parameter that is associated with the minimum of wind speed, which inspires us to introduce location parameters in the MWbl(2, 2) model. So in this paper we try to extend the MWbl(2, 2) model with five parameters to the MWbl(2, 3) model with six parameters and the MWbl(3, 3) model with seven parameters. The newly added parameters are location parameters. The parameters of the two extended models are estimated by the maximum likelihood (ML) method where the maximization problem is regarded as a nonlinear programming problem with only inequality constraints and is then resolved by the classical interior-point algorithm (Byrd et al. 2000; Waltz et al. 2006).

The rest of the paper is organized as follows. In section 2, we simply introduce the existing Weibull distribution models for the analysis of wind speed data and the improved mixture Weibull distribution models proposed in this paper, including the typical two-parameter Weibull [Wbl(2)], the typical Wbl(3), the MWbl(2, 2), the MWbl(2, 3), and the MWbl(3, 3) models. The method of estimating parameters is introduced in section 3. In section 4, we apply these models to fit the distribution of several real daily wind speed sequences. Then we compare the fitting effect using the sum of squares due to error (SSE) and Akaike information criterion (AIC). A few concluding remarks are given in section 5.

2. The models

In this section, we mainly introduce the Weibull distribution models for the analysis of the wind speed.

a. The existing models

The Weibull distribution function that is a three-parameter function, but for wind speed, can be expressed mathematically as
eq1
and the corresponding cumulative distribution is
eq2
where υ > 0 is the wind speed (m s−1), α > 0 is a shape parameter, β > 0 is a scale parameter (m s−1), and γ is a location parameter with 0 ≤ γ < υ. If the location parameter γ is equal to zero, the three-parameter model [Wbl(3)] becomes the two-parameter model [Wbl(2)]. Then, we have
eq3
and the corresponding cumulative distribution is
eq4

In statistics, a mixture density is a probability density function that is a convex linear combination of other probability density functions.

Suppose that Vi (i = 1, 2) are independently distributed as two-parameter Weibull f(υ, αi, βi). Then a random variable V that is distributed as Vi with mixing parameters ωi (ω1 + ω2 = 1) is said to have a two-component mixture Weibull distribution, that is, MWbl(2, 2).

The density function of MWbl(2, 2), which depends on five parameters (α1, β1, α2, β2, ω), is given by
eq5
where for υ > 0, α1, β1, α2, β2 > 0, 0 ≤ ω ≤ 1 and ff2,2[υ; (α1, β1, α2, β2, ω)] = 0, for υ ≤ 0. The cumulative distribution function is given by
eq6

b. The improved model

In this paper, by introducing one or two location parameters we extend the MWbl(2, 2) model to the MWbl(2, 3) and the MWbl(3, 3) model.

The density function of MWbl(2, 3), which depends on six parameters (α1, β1, α2, β2, ω, γ0), is given by
e1
where
eq7
and
eq8
The corresponding cumulative distribution function is given by
e2
The density function of MWbl(3, 3), which depends on seven parameters (α1, β1, α2, β2, ω, γ1, γ2), is given by
e3
where
eq9
and
eq10
The corresponding cumulative distribution function is given by
e4

3. Methodology of analysis

In statistics, moments (M), ML, and least squares (LS) methods are three common methods to estimate the parameters of a distribution model. In this paper, we mainly apply the ML method to estimate the parameters of the MWbl(2, 3) and MWbl(3, 3) distributions where the maximization problem is regarded as the nonlinear programming and is solved by the interior-point algorithm.

The interior-point method is one classical method of solving the nonlinear programming with only inequality constraints. The basic idea of the interior methods is to convert nonlinear programming problems with only inequality constraints into linear programming problems by means of barrier functions. And the role of these functions is to set barrier at the border of the feasible region and make the solution process always carried out in the interior of the feasible region. So, the interior-point method is also called interior penalty or barrier method.

In this paper, the parameters of the MWbl(2, 3) model are estimated by the ML method, which can be regarded as one nonlinear programming problem as follows:
e5
where . The interior of the feasible region
eq11

To solve the nonlinear programming with only inequality constraints, we introduce a penalty function with logarithmic form, G(θ, μ):

eq12
where μ is a very small positive number.
Then, the original problem Eq. (5) is converted to a nonlinear programming problem with no constraints:
e6
To solve the above programming, we adopt sequential unconstrained minimization technique (SUMT) where we take a strictly monotone decreasing positive sequence {μk}, which tends to 0.

Summarize the interior-point algorithm in the above mentioned programming problem as follows:

  1. Choose the initial parameter values. θ0 ∈ int S, μ1 = 0.1, β = 0.1, ϵ = 1 × e−6, k = 1.
  2. Solve the unconstrained problem. Set θk−1 as the initial value and compute Eq. (6). Let the optimal solution of Eq. (6) be θk.
  3. Check the stopping criterion. If , then the iteration is stopped and θk is the approximately optimal solution of Eq. (5). Else, let μk+1 = βμk, k = k + 1 and return to step 2.

In the above-mentioned algorithm, the choice of the initial parameter values plays an important role in fitted effect. In view of the six parameters in MWbl(2, 3), the initial values of the four parameters (α1, β1, α2, β2) are chosen according to Carta and Ramírez (2007a), that is,
e7
where m and s2 are the mean and variance of the wind speed sequence {υi, i = 1, 2, … , n}, respectively. To select the initial values of ω and γ0, we restrict them to varying within [0, 1] and [0, min(υ)) where the lengths of sampling interval are 0.5 and 0.1 (m s−1), respectively. Then we use these different possible initial parameter values to solve Eq. (5) and obtain a series of parameters. Then we compare the corresponding Kolmogorov–Smirnov (K–S) test values (Massey 1951) of different models and finally choose the corresponding to the maximal K–S test value to be the initial values of ω and γ0.

For MWbl(3, 3) model, the process of parameter estimation is similar to that of MWbl(2, 2) model. The parameters of MWbl(3, 3) are α1, α2, β1, β2, ω, γ1, and γ2. The initial values of the first four parameters are taken as Eq. (7). For the initial values of the last three parameters, we restrict them to varying with [0, 1], [0, min(υ)) and [0, min(υ)) where the lengths of sampling interval of them are 0.5, 0.1 (m s−1) and 0.1 (m s−1), respectively.

4. Numerical experiments

To make a comparison between our models and the existing models [MWbl(2, 2), Wbl(2), Wbl(3)] in fitting the distribution of wind speed data, six daily wind speed series are picked as the examples that are shown in Fig. 1. The first three series are extracted from the NCEP reanalysis dataset (http://cdc.cma.gov.cn). The corresponding grid points are (30°N, 112.5°E), (42.5°N, 125°E), and (32.5°N, 117.5°E), whose locations are shown in Fig. 2. We choose the daily wind speed in 2005 at 850 hPa; the last three series are extracted from A Regional, Integrated Hydrological Monitoring System for the Pan-Arctic Landmass (ArcticRIMS; http://rims.unh.edu) dataset. The countries chosen are the United States, Denmark, and Norway. The six data series are all from 1 January to 31 December 2005. The sample lengths are all 365. We analyze the descriptive statistics of the six series. Table 1 shows the corresponding results that including the maximum, the minimum, and the mean value of wind speed series. We can find that the wind speed values of grid points are higher than ones of countries as a whole. For the grid point (30°N, 112.5°E), the minimum value is 0.1 m s−1. That is to say that there exists nearly null wind speed for the series.

Fig. 1.
Fig. 1.

The series of wind speed. (a)–(c) Grid points; (d)–(f) country stations.

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 2.
Fig. 2.

The location of the grid points.

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Table 1.

The analysis results of descriptive statistics for wind speed series.

Table 1.

In this paper, the parameters of the five models are all estimated by ML method. The parameter estimation values are shown in Table 2. With different wind speed sequences, the results of estimation are different. When the MWbl(3, 3) and MWbl(2, 3) models are used to fit the distribution for grid point (30°N, 112.5°E), the corresponding location parameters are nearly zero. For the United States, the mixture parameter estimation of MWbl(3, 3) is 1. That is to say, we may just use the typical Wbl(3) to fit the wind speed distribution of the country.

Table 2.

Parameter estimators of the models.

Table 2.

Figures 38 display the histograms and fitting results of the five models for the distribution analysis of the five wind speed sequences, where the width of bins are all taken as 1 (m s−1). We find that the density curves of the mixture models always display the fluctuation of the histograms. That is to say, the effect of the mixture models is better than the effect of the two typical models. From the results for Denmark, we also find that the MWbl(2, 2) models do not always provide a good fit for wind speed.

Fig. 3.
Fig. 3.

The (a) histogram and (b) probability graph of wind speed series for grid point (30°N, 112°E).

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for grid point (42.5°N, 125°E).

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 5.
Fig. 5.

As in Fig. 3, but for grid point (32.5°N, 117.5°E).

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 6.
Fig. 6.

As in Fig. 3, but for the United States.

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 7.
Fig. 7.

As in Fig. 3, but for Denmark.

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 8.
Fig. 8.

As in Fig. 3, but for Norway.

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

For further comparison of the fitting results, we consider the SSE and AIC values corresponding to different models. The computation of SSE is as follows:
eq13
where Pi are the values of the empirical cumulative relative frequencies and the FFis are the forecast values using the theoretical distribution functions. The lower the SSE is, the better is the fit. The computation of AIC is as follows:
eq14
where K is the number of parameters and L is the likelihood function. Figures 914 give the comparison results of the six wind speed series fitting with different distribution models.
Fig. 9.
Fig. 9.

The comparison of fitting results with different models for grid point (30°N, 112.5°E). (a) Corresponding AIC values; (b) corresponding SSE values.

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for grid point (42.5°N, 125°E).

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 11.
Fig. 11.

As in Fig. 9, but for grid point (32.5°N, 117.5°E).

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 12.
Fig. 12.

As in Fig. 9, but for the United States.

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 13.
Fig. 13.

As in Fig. 9, but for Denmark.

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

Fig. 14.
Fig. 14.

As in Fig. 9, but for Norway.

Citation: Journal of Applied Meteorology and Climatology 51, 7; 10.1175/JAMC-D-11-0231.1

For grid point (30°N, 112.5°E) with nearly null wind speed, the fitting effect of improved models are not better than that of MWbl(2, 2). The corresponding parameter estimators also illustrate this point. For grid point (42.5°N, 125°E) and Norway, the fitting effect of MWbl(3, 3) is good. For grid point (32.5°N, 117.5°E), the MWbl(2, 3) and MWbl(3, 3) can both provide good fit. For the United States, the typical Wbl(3) has the best fitting effect, which can be also seen from the corresponding parameter estimation results. For Denmark, the two extended models are both satisfactory for fitting the corresponding wind speed. From the parameter estimation result, we find the γ1 of MWbl(3, 3) is nearly equal to 0. This point illustrates that we just choose the MWbl(2, 3) with a smaller number of parameters to fit the corresponding series.

5. Conclusions and discussion

This paper investigates two improved mixture Weibull distribution models for the analysis of the wind speed. The models mix one two-parameter (three component) and one three-parameter Weibull distribution. The parameters are estimated by the ML method in which the maximization problem is regarded as a nonlinear programming with only inequality constraints and is solved by the interior-point method. We apply the MWbl(2, 3), MWbl(3, 3), and the other three Weibull distribution models—MWbl(2, 2), Wbl(2), and Wbl(3)—to six wind speed sequences.

From the results of experiments, we can obtain the following conclusions: 1) There does not exist a general model that can fit all the real wind speed series under study. 2) In practice, we can first use the MWbl(3, 3) model to fit a given series whose modality is unknown. Then we estimate the parameters of the MWbl(3, 3) model. We consider the features of γ1, γ2, and ω, which can give the basis for reducing properly the form of models. Consequently, we can choose a suitable model for fitting the given wind speed series. 3) With a number of experiments, we find that although we add one or two new parameters in the existing MWbl(2, 2) model, the computation time does not change much.

In this paper, the estimation of parameters in the MWbl(2, 3) model is numerical and the choice of the initial parameter values is very important. We do a number of experiments to choose best initial values according to the goodness of fit, which will be further discussed in future research. In addition, we mainly use the ML method to estimate the parameters of models. To develop other estimation methods will also be one of the issues we address.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant 61075006) and the National Basic Research Program of China (973 Program) (Grant 2007CB311002). The authors are grateful to the editors and anonymous referees for their valuable comments, which led to improvements in this article.

REFERENCES

  • Akdağ, S. A., , H. S. Bagiorgas, , and G. Mihalakakou, 2010: Use of two-component Weibull mixtures in the analysis of wind speed in the eastern Mediterranean. Appl. Energy, 87, 25662573.

    • Search Google Scholar
    • Export Citation
  • Byrd, R. H., , J. C. Gilbert, , and J. Nocedal, 2000: A trust region method based on interior point techniques for nonlinear programming. Math. Program., 89, 149185.

    • Search Google Scholar
    • Export Citation
  • Carta, J. A., , and P. Ramírez, 2007a: Analysis of two-component mixture Weibull statistics for estimation of wind speed distributions. Renew. Energy, 32, 518531.

    • Search Google Scholar
    • Export Citation
  • Carta, J. A., , and P. Ramírez, 2007b: Use of finite mixture distribution models in the analysis of wind energy in the Canarian Archipelago. Energy Convers. Manage., 48, 281291.

    • Search Google Scholar
    • Export Citation
  • Carta, J. A., , P. Ramírez, , and S. Velázquez, 2009: A review of wind speed probability distributions used in wind energy analysis: Case studies in the Canary Islands. Renew. Sustainable Energy Rev., 13, 933955.

    • Search Google Scholar
    • Export Citation
  • Chang, T. P., 2010: Wind speed and power density analyses based on mixture Weibull and maximum entropy distributions. Int. J. Appl. Sci. Eng., 8, 3946.

    • Search Google Scholar
    • Export Citation
  • Chang, T. P., 2011: Estimation of wind energy potential using different probability density functions. Appl. Energy, 88, 18481856.

  • Dorvlo, A. S. S., 2002: Estimating wind speed distribution. Energy Convers. Manage., 43, 23112318.

  • Garcia, A., , J. L. Torres, , E. Prietoa, , and A. de Francisco, 1998: Fitting wind speed distribution: A case study. Sol. Energy, 2, 139144.

    • Search Google Scholar
    • Export Citation
  • Jaramillo, O. A., , and M. A. Borja, 2004: Wind speed analysis in La Ventosa, Meixco: A bimodal probability distribution case. Renew. Energy, 29, 16131630.

    • Search Google Scholar
    • Export Citation
  • Massey, F. J., Jr., 1951: The Kolmogorov-Smirnov test for goodness of fit. J. Amer. Stat. Assoc., 46, 6878.

  • Meng, Q., , and M. Tang, 1997: On asymptotic distributions of yearly maximum values of surface temperature and wind speed over Chengdu and the estimation of their parameters. J. Chengdu Inst. Meteor., 12, 284291.

    • Search Google Scholar
    • Export Citation
  • Morgen, E. C., , M. Lackner, , R. M. Vogela, , and L. G. Baise, 2011: Probability distributions for offshore wind speeds. Energy Convers. Manage., 52, 1526.

    • Search Google Scholar
    • Export Citation
  • Murthy, D. N. P., , M. Xie, , and R. Jiang, 2004: Weibull Models. John Wiley and Sons, 383 pp.

  • Waltz, R. A., , J. L. Morales, , J. Nocedal, , and D. Orban, 2006: An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math. Program., 107, 391408.

    • Search Google Scholar
    • Export Citation
  • Zaharim, A., , A. M. Razali, , R. Z. Abidin, , and K. Sopian, 2009: Fitting of statistical distributions to wind speed data in Malaysia. Eur. J. Sci. Res., 26, 612.

    • Search Google Scholar
    • Export Citation
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