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The goal of this paper is to show how to derive the multivariate Weibull probability density function from the multivariate Standard Normal one and to show its applications. Having Weibull distribution parameters and a correlation matrix as input data, the proposal is to obtain a precise multivariate Weibull distribution that can be applied in the analysis and simulation of wind speeds and wind powers at different locations. The main advantage of the distribution obtained, over those generally used, is that it is defined by the classical parameters of the univariate Weibull distributions and the correlation coefficients and all of them can be easily estimated. As a special case, attention has been paid to the bivariate Weibull distribution, where the hypothesis test of the correlation coefficient is defined.

The Weibull distribution is a continuous probability distribution that was described by Waloddi Weibull in 1951 [

When more than one Weibull-described variable is being considered and the dependence among them has certain relevance, the so called multivariate distribution [

So far, most of the multivariate Weibull expressions for Cumulative Distribution Function (CDF) or Probability Distribution Function (PDF) are based on models [

In this paper, a model is proposed for the multivariate Weibull PDF, based on the classic parameters used in the definition of a univariate Weibull model and on the correlation coefficients among the marginal distributions. It develops the change of variables from Normal to Weibull used in [

The structure of the paper is as follows:

The proposed multivariate Weibull distribution is obtained by means of a Normal to Weibull change of variables. The procedure to derive it is as follows:

The key point of the procedure is to define a change of variable from a Standard Normal to a Weibull distributed one. This transformation must be differentiable and the inverse function has to exist. The Standard Normal distributed variable is created in order to use its known features and transfer them to the Weibull one;

Establish as many changes as the number of variables, n, considering the different Weibull parameters for each case;

Obtain the multivariate Weibull PDF from the multivariate Standard Normal PDF applying the change of variable.

Finally, the relationship between the correlation coefficients in the multivariate Standard Normal PDF and in the Weibull one has to be checked in order to establish a multivariate Weibull PDF that depends on marginal Weibull parameters and the correlation coefficients between pairs of Weibull variables.

In order to define a change of variables from a Normal distributed variable to a Weibull one, the Probability Integral Transform is applied [

The CDF of a univariate Weibull distribution [_{u}(

On the other hand, the CDF of the univariate Normal distribution [_{x}(

As mentioned earlier, the Probability Integral Transform states that variables from any given continuous distribution can be converted into variables having a uniform distribution, _{u} = F_{u}(_{x} = F_{x}(

Both variables y_{u} and y_{x} can be matched in order to establish a relationship between a Weibull and a Normal distributed variable. The whole process can be understood in two steps: First, the conversion of a Normal distributed variable into a Uniform one and then the conversion of this one into a Weibull one. Therefore, if both CDFs are matched, then the Weibull distributed variable, u, can be expressed as a function of the Normal one

Thus, from a Normal distributed variable with given parameters (µ, σ), a Weibull one can be derived, with the desired parameters (C, k). Notice that this transformation can be applied to other types of variables to obtain an equation that relates two variables with different distributions.

In order to derive further results, the transformation given in Equation (3) will be referred to as ntw(x;C,k) where x is a Standard Normal variable (µ = 0 and σ = 1), the Weibull parameters are C and k. For a single variable, x_{i}_{i}_{i}

The inverse transformation is also needed and denoted as ntw^{−1}(u;C,k) where u is the Weibull variable. For a single variable, u_{i}^{−1}( ) is the inverse of the error function [^{−1}(u;C,k) is shown in

Standardized Normal variable as a function of a Weibull one, with various values of the scale parameter C.

The derivative of ntw( ) is denoted as ntwʹ(x;C,k) and expressed in Equation (6), also for a single variable.

In Equation (6) the derivative of the error function is used [

By using Equations (4–6), the change of variables can be extended to multiple variables.

In order to broaden the transformation to several variables, the multivariate Standard Normal distribution has to be considered. Its PDF for when the covariance matrix is positive definite is shown in Equation (8):
_{ij}_{ii}_{ij}_{ij}_{ij}_{i}_{j}^{−1} means inversion of matrix ^{t} means transposition of vector

By using Equation (8), the PDF corresponding to the multivariate Weibull distribution is obtained through Equation (9):

Thus, the Jacobian matrix has non-zero elements in its diagonal. The determinant of this matrix is obtained through the equation expressed in Equation (11):

The multivariate Weibull PDF of a group of variables is shown in Equation (12) as a function of the multivariate Standard Normal PDF and ntw( ).

Equation (12) can be expressed as in Equation (13) if Equation (8) is taken into account.

Notice that Equation (13) depends on two parameters per variable (C_{i}_{i}_{ij}_{i}_{j}

Therefore, in many cases the parameters used in Equation (13) are defined by the behavior of the group of Weibull variables.

Even though Equation (13) seems a trifle complex, it should be emphasized that, when introducing it in a software application, its complexity does not depend on the number of variables,

According to [_{i}_{i}_{i}_{1},…,v_{n}), with Weibull parameters C_{i}_{i}_{v} referred to the pairs of Normally distributed variables, is shown in Equation (15):
^{−1}( ) and ntwʹ( ) are defined in Equations (5) and (6) respectively.

On the other hand, in most cases the Weibull parameters of the wind speed distributions lie in the intervals expressed in Equation (14), so, as it is explained in _{v} can also represent the correlation matrix of the v_{i}

The relationship between wind speed and wind power [_{i}_{i}_{i}

Therefore, applying a change of variables, P_{i}_{i}_{i}_{i}_{i}_{i}_{i}

And in Equation (19), as a function of the parameters of the wind speed distribution.

In both cases, Equations (18) and (19), the matrix R_{P} contains the correlation coefficients corresponding to the pairs of normally distributed variables. In most cases the parameters of the Weibull distributions defined in Equation (17) lie outside the intervals expressed in Equation (14), so, as it is explained in _{P} does not represent the correlation matrix of the P_{i}

In many cases the bivariate Weibull distribution is sought in order to describe the wind speed or wind power behavior in a pair of locations. Due to its importance, we have considered it interesting to develop here as a particular case.

Equation (9) specifically for n = 2 is shown in Equation (20):

And the bivariate Standard Normal PDF is expressed in Equation (21):

By using Equations (20) and (21) the bivariate Weibull PDF Equation (22) is obtained as a function of C_{1}, k_{1}, C_{2}, k_{2} and ρ, which stands for the correlation coefficient between _{1} and _{2} but, as has been stated above, can be considered as the correlation coefficient between v_{1} and v_{2}.

In order to simplify Equation (22), as it depends on x_{1} and x_{2}, their relationships with v_{1} and v_{2} are shown in Equation (23):

As stated, in most cases the Weibull parameters to define the wind speed behavior lie in the intervals given in Equation (14), so Normal and wind speed correlation coefficients can be considered equal. The bivariate wind speed PDF for several values of ρ is shown in _{1} = 8, k_{1} = 2, C_{2} = 8, k_{2} = 2).

Bivariate wind speed Probability Distribution Function (PDF) for ρ=0.0.

Bivariate wind speed PDF for ρ=0.5.

Bivariate wind speed PDF for ρ=1.0.

In order to perform the correlation coefficient inference between two variables (u_{1}, u_{2}), the sample correlation coefficient has to be tested with a hypothesis [_{1}, y_{2}).
_{ij}_{i}_{i}_{i}

The hypotheses to be checked in this case are the following:
_{0}: _{1}: _{0} is a known value that corresponds to a bivariate Normal distribution, which needs to be tested. In order to do so, a new variable

According to [

Depending on the significance level, α, a confidence interval, CI = [zmin, zmax], is established, in which its limits are expressed in an implicit way in Equation (27):
_{z}( ) is the CDF of the variable

So, if _{0} can be accepted with significance level α, and if not, it cannot be accepted.

If H_{0} is accepted, the correlation coefficient corresponding to the bivariate Weibull distribution has to be obtained.

According to previous sections, the correlation coefficients corresponding to the bivariate Normal distribution and to the Weibull one, when it represents a pair of wind speed variables, are approximately the same, so if H_{0} is accepted, it can be said that ρ_{0} can be taken as an estimation of the correlation coefficient in a bivariate Weibull distribution with significance level of α.

According to Equation (19), the bivariate wind power PDF is expressed in Equation (28):
_{1} and x_{2} are shown in Equation (29):

Relationship between Standard Normal and wind power correlation coefficients.

The bivariate wind power PDF for several values of ρ is shown in _{1} = 8, k_{1} = 2, A_{1} = 7853 m^{2}, d_{1} = 1.225 kg/m^{3}, C_{2} = 8, k_{2} = 2, A_{2} = 7853 m^{2}, d_{2} = 1.225 kg/m^{3}).

Bivariate wind power PDF for ρ = 0.0.

Bivariate wind power PDF for ρ = 0.5.

Bivariate wind power PDF for ρ = 1.0.

As a case study, it can be assessed if a certain model of correlation coefficient between a pair of locations can be accepted in order to estimate its value as a function of the distance between them.

The main features of the behavior of the wind in Galicia, in the Northwest of Spain, are that during winter, the winds blow from the Southwest and are very constant and powerful and during the summer, the winds normally blow softly from the Northeast. There are a great number of meteorological stations spread throughout Galicia [

In order to estimate the correlation coefficient for locations with a low number of simultaneous sample values of wind speed measures, the relationship between the correlation coefficient and the distance can be analyzed [_{ij}_{E} is the Earth’s radius; lat_{i}_{i}

Therefore, including all the possible pairs (distance, correlation coefficient) in Galicia and by means of the least square method, the relationship Equation (31) is derived.
^{−1} and b = 0.6589.

So, if the correlation coefficient between a location not included in the previous analysis (Coto Muiño), and another one that is included (Melide), needs to be estimated, all that has to be done is to apply Equation (30) between both locations, and then Equation (31), after which a value of ρ_{0} = 0.6012 is obtained.

Moreover, by utilizing simultaneous sample data (n = 1000) from both locations, the sample correlation coefficient can be obtained through Equation (24), r = 0.6202, and the change suggested in Equation (25) applied, to obtain z = 0.7253.

Considering α = 0.05, Z ~ N(0.6951, 0.0317), the CI obtained is CI = [0.6330, 0.7571]. Therefore, as explained in the previous sections, as z_{min}< _{max}, ρ_{0} = 0.6012 can be accepted as the correlation coefficient of the bivariate Weibull distributions corresponding to the wind speed data of those locations. The results are shown in

Results of the case study.

Variable | Obtained value | Minimum value | Maximum value |
---|---|---|---|

r | 0.6202 | 0.5601 | 0.6394 |

z | 0.7253 | 0.6330 | 0.7571 |

In this paper, a Normal to Weibull change of variables has been defined. It should be noticed that the process can be applied to any type of variables, even inversely. The multivariate Weibull PDF has been obtained and justified, depending on the classic parameters of a single variable and the correlation coefficients between pairs of them. It upgrades former approaches that mainly consist of models based on parameters that have no direct relationship with the univariate parameters and the usual dependence measurement. The function proposed can be easily implemented in a software application regardless of the number of variables. The bivariate case seems a bit complex, compared to other models, but it uses the correlation coefficient between both variables. From the point of view of n variables, each defined by a Weibull distribution with correlation coefficients between pairs given, the PDF proposed is not an approximation, it provides exact results. Additionally, the application of the multivariate Weibull PDF to wind speed and wind power has been explained and derived. The bivariate case for both has also been specified due to its relevance, and some figures are given for clarification purposes. Moreover, the inference of the correlation coefficient in the bivariate wind speed distribution is explained and applied to a particular case.

The decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose is called the Cholesky decomposition. A Hermitian matrix is a square matrix with complex entries that is equal to its conjugate-transpose.

Therefore, given Ω, Hermitian and positive-definite, the Cholesky decomposition consists of obtaining L, fulfilling Equation (32):
^{*} means the conjugate transpose of the matrix L.

The Cholesky decomposition is mainly used for the numerical solution of linear equations, linear least squares problems, non-linear optimization or in Kalman filters. Here it is utilized in another application: the Monte Carlo [

Given X, a matrix of uncorrelated series of samples, and Ω, the desired correlation matrix for these series, Equation (33) is applied in order to obtain Y, a matrix of correlated series of samples according to Ω, where L is the result of the Cholesky decomposition of Ω.

Moreover, if X is a matrix of series of samples where each of these series follows a Normal distribution, the resulting matrix in Equation (33), Y, is also a matrix of series of samples that follows a Normal distribution, which can easily be demonstrated. Equation (33) has the shape shown in Equation (34).

The

If x_{li} is distributed according to a N(µ_{l},σ_{l}) distribution, then y_{ji}_{1} = µ_{2} =…= µ and σ_{1} = σ_{2} =…= σ, then y_{ji}_{ji}

On the other hand, as the condition that L fulfills is Equation (32), it is always true that _{1} = µ_{2} =…= 0, σ_{1} = σ_{2}=…= 1, y_{ji}

The authors declare no conflict of interest.