Weibull Distribution for Estimating the Parameters

2.1. Linear Least Square Method (LLSM)

Least square method is used to calculate the parameter(s) in a formula when modeling an experiment of a phenomenon and it can give an estimation of the parameters. When using least square method, the sum of the squares of the deviations S which is defined as below, should be minimized.

In the equation, xi is the wind speed, yi is the probability of the wind speed rank, so (xi, yi) mean the data plot, wi is a weight value of the plot and n is a number of the data plot. The estimation technique we shall discuss is known as the Linear Least Square Method (LLSM), which is a computational approach to fitting a mathematical or statistical model to data. It is so commonly applied in engineering and mathematics problem that is often not thought of as an estimation problem. The linear least square method (LLSM) is a special case for the least square method with a formula which consists of some linear functions and it is easy to use. And in the more special case that the formula is line, the linear least square method is much easier. The Weibull distribution function is a non-linear function, which is

i.e.

i.e.

But the cumulative Weibull distribution function is transformed to a linear function like below:

Again

Equation (15) can be written asY=bX+a

where

Y=lnln{11−F(v)},X=lnv,a=−klnc,b=k

By Linear regression formula

2.2. Maximum Likelihood Estimator(MLE)

The method of maximum likelihood (Harter and Moore (1965a), Harter and Moore (1965b), and Cohen (1965)) is a commonly used procedure because it has very desirable properties.

Letx1,x2,........................xnbe a random sample of size n drawn from a probability density functionf(x,θ)where θ is an unknown parameter. The likelihood function of this random sample is the joint density of the n random variables and is a function of the unknown parameter. Thus

is the Likelihood function. The Maximum Likelihood Estimator (MLE) of θ, sayθ¯, is the value of θ, that maximizes L or, equivalently, the logarithm of L. Often, but not always, the MLE of q is a solution of

Now, we apply the MLE to estimate the Weibull parameters, namely the shape parameter and the scale parameters. Consider the Weibull probability density function (pdf) given in (2), then likelihood function will be

On taking the logarithms of (20), differentiating with respect to k and c in turn and equating to zero, we obtain the estimating equations

On eliminating c between these two above equations and simplifying, we get

which may be solved to get the estimate of k. This can be accomplished by Newton-Raphson method. Which can be written in the form

Where

And

Once k is determined, c can be estimated using equation (22) as

2.3. Some results

When a location has c=6 the pdf under various values of k are shown in Fig. 1. A higher value of k such as 2.5 or 4 indicates that the variation of Mean Wind speed is small. A lower value of k such as 1.5 or 2 indicates a greater deviation away from Mean Wind speed.

When a location has k=3 the pdf under various valus of c are shown in Fig.2. A higher value of c such as 12 indicates a greater deviation away from Mean Wind speed.

Fig. 3 represents the characteristic curve ofΓ(1+1k).versus shape parameter k. The values ofΓ(1+1k).varies around.889 when k is between 1.9 to 2.6.

Fig.4 represents the characteristic curve ofcv¯versus shape parameter k.Normally the wind speed data collected at a specified location are used to calculate Mean Wind speed. A good estimate for parameter c can be obtained from Fig.4 asc=1.128v¯where k ranges from 1.6 to 4. If the parameter k is less than unity, the ratiocv¯decrease rapidly. Hence c is directly proportional to Mean Wind speed for1.6≤k≤4and Mean Wind speed is mainly affected by c. The most good wind farms have k in this specified range and estimation of c in terms ofv¯may have wide applications.

Example: Consider the following example wherexirepresents the Average Monthly Wind Speed (m/s) at kolkata (from 1^st March, 2009 to 31^st March, 2009)

Also letF(xi)=in+1and using equations (16) and (17) we get k= 1.013658 and c=29.9931.

But if we apply maximum Likelihood Method we get k = 1.912128 and c=1.335916. There is a huge difference in value of c by the above two methods. This is due to the mean rank ofF(xi)and k value is tends to unity.