DIFFERENT METHODS OF WEIBULL PARAMETER ESTIMATION ... Where; β is the shape parameter, α the scale parameter and γ is the location parameter.
IJIRST –International Journal for Innovative Research in Science & Technology| Volume 2 | Issue 11 | April 2016 ISSN (online): 2349-6010
A Comparative Study of Different Methods to Estimate the Parameters of Weibull Distribution Subodh Kumar Assistant Professor Department of Production Engineering B.I.T. Sindri, Dhanbad
Jamshed Anwar U.G. Student Department of Production Engineering B.I.T. Sindri, Dhanbad Manoj Kumar Mahto U.G. Student Department of Production Engineering B.I.T. Sindri, Dhanbad
Akash Kumar U.G. Student Department of Production Engineering B.I.T. Sindri, Dhanbad
Abstract Weibull reliability is found to be very beneficial to characterize the equipment time to failure, and to design their appropriate maintenance strategy by knowing the expected number of failure to occur in a given period of time. Through the use of fitted Weibull model as predictive tool, the operation management can take appropriate decision in advance to avoid operational upset. The time to failure or time between failure data will be utilized to perform the analysis. In this paper, some methods to estimate Weibull parameters, namely, shape parameters (β) and scale parameter (α) are presented. The methods can be grouped into two main categories: graphical and analytical methods. Results computed by different methods are compared. Keywords: Analytical Method, Graphical method, Least Square Method, Parameter estimation, Weibull distribution _______________________________________________________________________________________________________ I. INTRODUCTION Results of mechanical tests on any simple to complex mechanism are often summarized by a distribution function fit to the data. This fitted distribution can then be used in calculating properties based on percentiles of the data, in reliability- based design calculations, or in simulations of the performance of that particular mechanism The Weibull distribution (named after Waloddi Weibull, a Swedish physicist who used it in 1939 to describe the breaking strength of material) is playing an increasingly important role in this type of research and has become a part of several American Society of Testing and Materials (ASTM) standards. Its popularity with researchers is due in part to one of the parameters the shape parameter which allows it to look like a variety of other distributions, such as the normal, lognormal, and exponential distributions. II. DIFFERENT METHODS OF WEIBULL PARAMETER ESTIMATION Graphical method Usually the graphical methods are used because of their simplicity and speed. However, the involve a great probability of error. Two main graphical methods are: Weibull Probability Plotting The Weibull distribution density function (Mann et. al, 1974) is given by:
f ( x)
x
(
)
1
(
e
x
)
; β > 0, α > 0
(1)
The cumulative Weibull distribution function is given by: (
x
)
F ( x) 1 e (2) Where; β is the shape parameter, α the scale parameter and γ is the location parameter. In order to obtain the relation between the CDF and the two parameters (β, α), we take double logarithmic transformation of the CDF and letting γ = 0 from (2) F ( x) 1 e
(
x
)
ln{ 1 F ( x )} e
ln[ln{
1 1 F ( x)
(
x
)
}] ln x ln
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532
A Comparative Study of Different Methods to Estimate the Parameters of Weibull Distribution (IJIRST/ Volume 2 / Issue 10 / 093)
Where F(xi) can be estimated by different methods Table – 1: Method F(xi)
i
Mean Rank
n 1 i 0 .3
Median Rank
n 0 .4
Hazard Probability Plotting The hazard plotting technique is an estimation procedure for the Weibull parameters. This is done by plotting cumulative hazard function H(x) against failure times on a hazard paper or a simple log-log paper. x 1 h( x) ( ) The cumu7lative hazard function is given by: x (3) H ( x) h( x) ( ) We can transform (3) by taking the logarithm as follows:
ln H ( x ) {ln x ln } or
ln x
1
ln H ( x ) ln
(4)
From (4), we can plot ln H(x) vs ln x, and upon completing the estimated parameters will be x 1 ; at H=1, α=x y slope Analytical Method Due to the high probability of error in using graphical methods, we prefer to use the analytical methods. Some analytical methods are: Maximum Likelihood Estimator (MLE) The method of maximum likelihood is a commonly used procedure because it has very desirable properties. Let x1 , x 2 , x3 ,..... x n be a random sample of size n drawn from a probability density function f x ( x ; ) 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 n
L
f x ( xi , )
(5)
i 1
^
is the likelihood function. The maximum likelihood estimator (MLE) of θ, say , is the value of θ that maximizes L. often, but not always, the MLE of θ is a solution of d log L (6) 0 d where solution that are not the functions of the sample values x1 , x 2 , x3 ,..... x n are not admissible, nor are solutions hich are not in the parameter space. Consider the Weibull pdf given in (1), then likelihood function will be n
L( x1 , x 2 , x3 ,..... x n ,β,α) = ( i 1
)(
xi
)
1
(
e
xi
)
(7)
taking log and differentiating with respect to β and α and equating to zero, we obtain the estimating equations
ln L ln L
n
ln xi i 1
n
1
2
n
x
1
x
i
ln x i 0
(8)
i 1
i
n
0
(9)
i 1
On eliminating α between these two equations and simplifying, we have
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533
A Comparative Study of Different Methods to Estimate the Parameters of Weibull Distribution (IJIRST/ Volume 2 / Issue 10 / 093) n
x
i
ln x i
i 1
n
x
1
1
n
n
ln x
i
0
(10)
i 1
i
i 1
which may be solved to get the estimate of β. Once β is determined, α can be estimated using equation(9). Least Square Method (LSM) For the estimation of Weibull parameters, the least square method (LSM)is extensively used in engineering and mathematics problems. We can get a linear relationship between the two parameters by taking the double logarithm as 1 (11) ln{ln[ ]} ln x ln 1 F ( x) Equation (11) is a linear equation, and we can write _
x
n
1
ln{ln[ n i 1
_
y
1 n
^
1 F ( xi )
]}
(12)
n
ln x
(13)
i
i 1
n
1
{ n (ln x i )(ln{ln[ i 1
n
1 1 F ( x)
]})} { ln(ln[ i 1 n
n
1 1 F ( x)
{ n (ln x i ) } { (ln x i )} 2
i 1
^
e
_
_
n
]) ln x i i 1
(14)
2
i 1
^
( y x/ )
(15) III. ILLUSTRATIVE EXAMPLE
The objective of our study is to compare the above two analytical method namely, MLE and LSM. We have generated random samples with known parameters with the help of software “Easy Fit 5.0 Professional”. For each data set we have varied the sample size from 10 to 50. After generating the random samples, we estimate their parameters with “Weibull – itool” Table - 1 Data set 1: β=1, α = 15 Sample size 10 20 30 40
50
Sample size 10 20 30
Sample Data 0.76, 3.8, 4.18, 4.21, 8.48, 8.6, 15.42, 16.14, 55.19, 66.75 1.42, 3.38, 4.44, 6.62, 6.76, 7.06, 7.83, 8.05, 8.53, 8.66, 9.11, 9.78, 10.18, 13. 8, 18.39, 22.5, 28.17, 41.22, 54.33 0.17, 0.52, 0.9, 0.96, 0.98, 2.06, 208, 5.24, 5.4, 6.8, 7.8, 10.53, 11, 12.14, 12.21, 12.9, 13.03, 13.32, 15.03, 15.58, 17.19, 19.98, 22.08, 22.93, 23.75, 25.2, 34.68, 57.03, 75.56 0.08, 0.68, 1.44, 2.24, 2.38, 2.41, 2.44, 2.56, 3.25, 3.81, 4.12, 5.84, 6.69, 7.78, 8.17, 8.25, 8.26, 8.28, 9.21, 9.86, 10.34, 10.72, 10.81, 12.99, 15.31, 15.39, 17.47, 18.09, 19.45, 21.38, 24.61, 24.81, 26.47, 39.46, 44.28, 44.49, 44.82, 47.13, 69.43,88.69 0.16, 0.3, 0.55, 0.56, 0.87, 1.88, 2.98, 3.25, 3.64, 3.67, 3.83, 3.95, 4.01, 4.02, 4.48, 4.58, 4.82, 5.2, 6.74, 7.24, 7.81, 7.93, 8.04, 8.06, 8.08, 8.9, 9.31, 9.39, 9.8, 11.55, 12.37, 14.01, 15.47, 19.51, 19.58, 21.14, 21.26, 23.81, 24.07, 24.72, 24.98, 27.32, 28.97, 30.89, 34.4, 37.95, 38.91, 44.85,60.04,62.94 Table - 2 Data set 2: β=1.5, α = 125
MLE β α 0.86 16.8
LSM β Α 0.81 17.13
1.21
14.86
1.32
14.78
0.92
14.59
0.83
15.11
0.93
17.26
0.88
17.67
0.97
14.65
0.91
14.88
MLE
Sample Data 15.53, 54.03, 68.53, 77.32, 84.64, 104.28, 160.31, 226.47, 274.42, 281.97 11.09, 23.69, 30.34, 35.28, 37.6, 55.28, 58.8, 64.19, 70.16, 89.82, 100.36, 138.59, 144.23, 154.41, 171.86, 206.15, 231.34, 256.19, 276.35, 351.25 46.8, 50.34, 51.77, 63.07, 66.28, 68.24, 70.32, 72.32, 73.81, 81.63, 83.35, 85.91, 96.05, 96.51, 106.27,108.63, 119.97, 125.26,132.29, 146.3, 176.78, 179.72, 180.67, 184.51, 208.37, 223.81, 258.2, 266.06, 292.28, 344.59
LSM
β 1.5
α 149.32
β 1.23
α 154.37
1.36
138.19
1.28
138.76
1.87
153.57
2.02
152.36
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534
A Comparative Study of Different Methods to Estimate the Parameters of Weibull Distribution (IJIRST/ Volume 2 / Issue 10 / 093)
40
50
Sample size 10 20 30
40
50
4.26, 11.24, 14.37, 32.62, 33.08, 35.98, 36.2, 40.42, 43.57,53.94, 65.57, 68.2, 69.65, 75.72, 76.15, 86.17, 87.68, 91.04, 95.6, 98.97, 99.74, 112.83, 113.17,120.08, 122.29, 124.76, 126.12, 148.05, 149.09, 173.17, 180.86, 189.63, 192.47, 227.78, 249.11, 250.47, 278.39 20.82, 26.2 26.68, 28.34, 30.49, 35.73, 42.95, 43.43, 45.95, 46.97, 48.82, 55.47, 58.56, 59.2, 61.77, 63.68, 68.04, 68.06, 69.51, 69.78, 71.21, 75.78, 77.1, 77.38, 78.41, 83.98, 88.51, 93.78, 94.14, 96.62, 102.57, 109.76, 109.87, 114.5, 126.24, 129.74, 129.93, 138.2, 138.32, 141.14, 141.77, 148.65, 148.73, 159.78, 179.43, 203.36, 209.09, 220.38, 231.78, 285.56 Table - 3 Data set 3: β=2.3, α = 145
1.57
117.57
1.38
120.93
1.8
112.58
1.94
111.39
MLE
Sample Data 52.63, 108.33, 128.48, 135.5, 150.44, 152.67, 154.17, 157.42, 162.82, 178.17 46.89, 62.29, 67.18, 69.92, 70.79, 74.40, 76.37, 93.35, 99.61, 100.28, 108.54, 109.32, 134.88, 138.42, 145.13, 146.72, 167.52, 205.46, 243.76, 268.16 19.23, 44.11, 45.82, 53.84, 58.17, 60.3, 61.56, 65.28, 72.46, 82.25, 95.76, 97.9, 103.07, 103.85, 105.14, 108.88, 109.19, 127.49, 132.64, 136.06, 138.24, 140.65, 143.06, 150.01, 190.05, 197.26, 199.7, 201.45, 210.71, 271.15 18.13, 33.64, 54.21, 55.08, 56.34, 57.51, 61.01, 64.6, 74.15, 74.66, 75.87, 77.26, 89.48, 92.04, 93.47, 95.4, 96.82, 99.44, 106.96, 114.79, 116.49, 114.79, 116.49, 119.4, 123.86, 130.26, 138.87, 139.28, 141.47, 143.68, 147.71, 156.66, 159.7, 168.97, 175.23, 199.38, 200.75, 213.18, 232.63, 247.38, 279.06 48.58, 53.6, 57.61, 60.65, 61.32, 63.29, 65.66, 72.87, 73.07, 82.61, 84.43, 91.52, 93.78, 101.51, 103.06, 103.75, 104.67, 107.09, 110.46, 113.05, 115.04, 115.13, 128.55, 130.92, 132.64, 135.76, 135.9, 137.75, 142.05, 142.32, 147.24, 150.02, 151.15, 151.37, 155.72, 158.05, 162.76, 167.86, 172.81, 177.53, 188.86, 195.82, 201.65, 219.01, 235.77, 239.03, 280.83, 328.24
LSM
β 5.56
α 149.82
β 2.86
α 158.49
2.19
137.87
2.37
137.05
2.13
132.96
2.06
133.51
2.12
135.3
2.13
135.34
2.41
150.57
2.72
149.26
IV. RESULTS As discussed earlier, we have generated random samples with known parameters (β and α) and for each parameter set we varied sample size from 10 to 50. We have taken total deviation as the criteria for comparison of two analytical methods discussed above. The total deviation (TD) has been calculated for each method as: ^
TD
^
^
(16)
^
where β and α are the known parameters and and are the estimated parameters by any method. Table 4 shows the complete results. The remarks column of the table shows the methods whose total deviation is minimum or in other words the best method. From table 4, it is obviously seen, MLE achieves the best estimate 11 times out of 15 which is approximately 73% of the time. 3 time out of 15, LSM estimate the best result and 1 out of 15 times both methods give the same total deviation.
Sl. no.
1
2
3
Sample Size 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50
Table - 4 Comparison between MLE and LSM MLE α β (known) (known) β α TD 1 15 0.86 16.8 0.26 1 15 1.21 14.86 0.22 1 15 0.92 14.59 0.11 1 15 0.93 17.26 0.22 1 15 0.97 14.65 0.05 1.5 125 1.5 149.32 0.19 1.5 125 1.36 138.19 0.20 1.5 125 1.87 153.57 0.48 1.5 125 1.57 117.57 0.11 1.5 125 1.8 112.58 0.3 2.3 145 5.56 149.82 1.45 2.3 145 2.19 137.87 0.10 2.3 145 2.13 132.96 0.15 2.3 145 2.12 135.3 0.15 2.3 145 2.41 150.57 0.09
β 0.81 1.32 0.83 0.88 0.91 1.23 1.28 2.02 1.38 1.94 2.86 2.37 2.06 2.13 2.72
LSM α 17.13 14.78 15.11 17.67 14.88 154.37 138.76 152.36 120.93 111.39 158.49 137.05 133.51 135.34 149.26
TD 0.33 0.33 0.18 0.30 0.10 0.41 0.26 0.57 0.11 0.40 0.33 0.08 0.18 0.14 0.21
Remarks (Best) MLE MLE MLE MLE MLE MLE MLE MLE BOTH MLE LSM LSM MLE LSM MLE
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A Comparative Study of Different Methods to Estimate the Parameters of Weibull Distribution (IJIRST/ Volume 2 / Issue 10 / 093)
V. CONCLUSION In this study, the good results of the MLE in terms of the number of times for the lowest value of total deviation (TD) indicated that the MLE is a superior method to estimate Weibull parameters over LSM. Here we haven’t discussed one of the method i.e. method of moment (MOM) because of its more computation time and more calculation. If we would have included MOM in our study, result might be different as indicated in some other study [4], where the author has suggested MOM as the best one. The maximum TD is 0.57 (except one value 1.45) for both the methods. This means that at the worst case the estimated parameters are approximately within +/- 50% of their actual values. REFERENCES [1] [2] [3] [4] [5]
Al-Fawzan, M. A., “Methods for Estimating the Parameters of the Weibull Distribution” Lei. Y., Ëvaluation of Three Methods for Estimating the Weibull Distribution Parameters of Chinese pine” , Journal of Forest Science, 54 2008(12): 566-571 Stone, G. C. and G. Van Heeswijk, “Parameters estimation for the Weibull distribution”, IEEE Trans. On Elect. Insul., Vol. EI-12, No.4, August 1977 http://www.weibull.com/LifeDataWeb/characteristics_of_the_weibull_distribution.htm H. Peterlick. The validity of Weibull estimators. Journal of Material Science, Vol. 22,1995, pp. 1972-76.
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