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The Interval Estimation of MTBF Based on Markov Chain Monte Carlo Method
Yi Daia , Bin-quan Lib Mechanical engineering school, Tianjin university of Technology and Education, Tianjin 300222, China ( [email protected], [email protected])
adaptability of implementation and systemic calculation, In this paper, MCMC method is proposed to structure the Abstract - The distribution of time between failures of transition kernel and Markov chain which is suitable for numerical control (NC) system follows the Weibull Weibull distribution and it solves the problems of interval distribution, thus it’s estimation of Mean Time Between estimation of MTBF[9,10] value effectively. Compared Failures （ MTBF ） in reliability engineering is of with results by different MCMC transition kernel, it significance. But there are great difficulties in interval proves that MCMC method which has systematic solution estimation of MTBF using traditional method for Weibull distribution. After the introduction of the approximate process and good adaptability is appropriate for estimation, the Markov chain Monte Carlo (MCMC) method computing various interval estimates. It greatly increases is proposed. Combined with the specific characteristics of the flexibility and precision of the calculation. two-parameter Weibull distribution, Markov chain is established to calculate the interval estimation of MTBF, which solves the problems effectively. And MCMC is more II. INTERVAL ESTIMATION OF MTBF VALUES accurate than that of engineering approximation. By BASED ON MCMC METHOD analyzing various results in different conditions of MCMC transition kernel, the paper proves that MCMC is a good As can be seen from the description above, the direct method for solving interval estimation of Weibull simulation of distribution has a huge advantage compared distribution parameters, which has systematic solution process and good adaptability. It greatly enhanced the to indirect search of pivotal quantity in the statistical robustness, effectiveness and accuracy of the calculation. inference. The following is focuses on how to directly construct the distribution of MTBF. Keywords - Markov chain, Monte Carlo, Weibull, interval estimation, reliability evaluation A. Establishing Markov Chain of Stationary Distribution L(m,)
I. INTRODUCTION In the life test of numerical control system under In the previous research of the NC system reliability fixed time censoring with replacement [11-13], it obtains [1,2], we obtained the conclusions that the failure censored samples T (t1,t2 ,tr , L1, L2 ,Ls ) .Where, probability of NC system obeyed Weibull distribution t ,t ,...t are failure data and L , L ,L are censored after two years of data collection. It was an important 1 2 r 1 2 s object in Reliability engineering to calculate the mean data. The two-parameter Weibull probability density time between failures (MTBF) of the product. However, function f (t) and Survival function S(t) are given by: the point estimate of MTBF was easy to obtain, but it had encountered great difficulties in. the solution of interval estimation. f (t) (mt m1 / m )exp[(t /)m ] (1) MCMC sampling method [3,4], fundamentally changes the ideas in computing the point estimates and interval estimates in statistics. Through dynamic m simulation, implemented by MCMC method, the expected S(t) exp[(t /) ] (2) form of random variables with specific distribution is directly constructed. The point estimates and interval Where, t 0 , m is the shape parameter and is the scale estimation of parameters, MTBF values included, can be parameter. properly calculated, which avoids indirect, cumbersome The censored samples likelihood function can now and difficult search of asymptotic distribution and pivotal be expressed as follows: quantity. MCMC method[5-8], increasing the robustness and accuracy of computing, greatly improves the u r s If it establishes the Markov chain in the space, L(m,) f (t ) S(L ) i j L(m,) i1 j1 the objective distribution is transformed another (3) mr r r t s L t m1 exp ( i )m exp ( i )m distribution in space. The Metropolis-Hastings ratio rm i i1 i1 j1 (x, y) is given by: By MCMC method, it obtains the Markov chain L(m,) -1 which takes as stationary distribution. The L(log ( my ,h y )) q ( x , y ) | J ( y ) | -1 (6) sample obtained by Markov chain can be used for L(log ( mx ,h x )) q ( y , x ) | J ( x ) | statistical inference. For example, it obtains the samples (1) (n ) X,..., X through sampling from L(m,) . If By the process above, the Markov chain which takes (1) (n ) X,..., X are the samples of Markov chain which L(m,) as stationary distribution is established. Then it obtains m and with inverse transform. The sample takes L(m,) as stationary distribution, the Monte-Carlo obtained by Markov chain can be used for statistical integral is still valid. inference. Then the problem is converted into how to construct C. Structure and implementation stationary distribution of Markov chain. p(x, y) q(x, y)(x, y) L(m,) Set . Taking as Normal distribution is selected as proposal objective distribution, after choosing the proposal distribution. The iterative step of MCMC method can be q(| x) (x, y) distribution, , is written as follows: summarized as follows: For t 1,..., N : 禳 L( m ,h ) q ( y , x ) (t- 1) ( t - 1) y y 1. Setq=(m , h ) = ( q0 , q 1 ) min睚 1, (4) 铪 L( mx ,h x ) q ( x , y ) 2. Propose new values q from Unif (- 0.5,0.5) 3. Calculate qⅱ = q + q Where, my and y are proposal values, mx and x 4. Calculate a( q , q ⅱ ) given by (6) are initial values. 5. Update q(t ) = q ⅱwith probability a( q , q ⅱ ) or keep So, p(x, y) is the transition kernel of Markov chain the same values with the remaining probability. which is determined by stationary distribution L(m,) . Normal distribution is selected as proposal B. Optimization of distribution parameters distribution. The iterative step of MCMC method can be summarized by the following steps: Because the ratio / m of Weibull distribution is t 1,..., N : large, to ensure the iterative synchronization of two For (t- 1) ( t - 1) 2 parameters, the Jacobi's transformation [14-15] is used to 1. Setq=(m , h ) = ( q0 , q 1 ) ,s = (0.1,0.1) 2 solve the two parameters first and it obtains m and 2. Generate q from Norm(q , s ) with inverse transform. 3. Calculate a( q , q ) given by (6) Set u log( ) log(m,) .Considering the q(t ) = q a( q , q ) Jacobian J as follows: 4. Update with probability or keep the same values with the remaining probability. 抖m h Ⅲ. SIMULATION ANALYSIS (m ,h ) 抖u u J = = 1 1 (u1 , u 2 ) 抖m h The censored samples of numerical control system are obtained by the life test under fixed time censoring 抖u2 u 2 -1 (5) with replacement. The data are given by Table Ⅰ. 娑u1 饿 u 1 -1 琪 TABLEⅠ 骣(u , u ) 抖m h CENSORED SAMPLES (IN HOURS) OF NUMERICAL CONTROL SYSTEM =琪 1 2 = 琪 桫 (m ,h ) 琪抖u u 琪 2 2 Fail 37 4627 4871 848 1673 抖m h ure 桫 226 864 571 74 857 6 0 1758 3877 864 752 3458 . 0 1 0 4 4
3363 916 1714 1580 3769 . η m 1
f f o o
s s 2 0 . e
820 2074 415 1789 813 e 0 1 u u l l 0 a a 3 V V 0
2797 1381 606 853 4701 . 1 8 0 229 1130 93 931 2048 . 0 0 0 2 451 2107 356 2063 2451 Dat 0 10000 30000 50000 0 10000 30000 50000 a 999 500 4781 3329 1634 Iterations Iterations 6 0 . 0 1
(a) 0 4
351 832 80 489 477 4 6 . 0 . η m 1
0
1 f f 0 o o 4
s s 2
3042 1060 1246 3629 615 0 . e e 4 0 . 1 u u l η l 0 m 1
a
a 3 f f V o V o
0
0 s
549 228 2949 65 1301 . 0 s 2 0 . e 1 0 e F 0 1 u 4 u l l 0 B a a 3 T V 8 V 0 .
2361 1741 M 0 0
0 . f 0 1 o 2 0
0 s 0 e 3 u 46 339 4429 1182 1743 8 0 10000 30000 50000 0 10000 30000 50000 0 l . 0 a 0 0 V 36 4273 1743 2101 495 2 Iterations Iterations 0 10000 30000 50000 0 0 10000 30000 50000 0 0 2 1947 297 256 1301 2354 Iterations Iterations Censored 0 10000 (b) 30000 50000 2915 400 25 1415 3551 Data Iterations 0
2011 130 744 3794 3641 0 0 F 4 B T M
f
5048 5275 2642 46 3696 0 o 0 0
0 0 s F 4 0 e B 3 u l T
1545 a M V
f o 0
0 0 s 0 0 e 0 3 u l 2 a Hollander and Proschan method[16] is selectedV for 0 10000 30000 50000 0 0 goodness of fit test of the data in Table 1. It is proved0 that 2 the data obey the Weibull distribution. According to the Iterations 0 10000 30000 50000 data in Table 1, the estimation obtained by MLE and (c) Iterations MCMC method is shown in Table Ⅱ. Fig 1. Iterations of the Weibull distribution parameters for censored samples of numerical control system: (a) shape m and (b) scale TABLEⅡ η.and (c) MTBF value ESULTS OF OINT STIMATION ND NTERVAL STIMATION
R P E A I E 2 1 0 0 . 0 3 . interval MTBF 0
m η 8
（ ） value 0 estimation 90% 0 0 . 0 2 . y y t t 0 i i
Lower confidence s s n n e 0.89 2071 1925.4 e 4 D D
limit 0 0 0 . 0 1 . MLE Point estimation 1.06 2520 2463.0 0 0 0 0 0 . 0 0 .
Upper confidence 0 1.26 3066 3246 limit 0.8 1.0 1.2 1.4 1.6 1.8 2000 3000 4000 5000 Lower confidence m η 0.89 2120.0 2073.6 limit Proposal (a)
distribution 2 1 0
Point estimation 1.06 2592.9 2548.8 0 . 0 3 (Uniform . 0 0 . distribution) 3
Upper confidence 8 0 0 0
1.26 3141.6 3143.8. 4 0 2 . y y limit 0 t t - 0 i MCMC i 0 e s s . 8 n n 2 y e e t
method 4 Lower confidence i D D 0 s 0 0 . X
0.89 2121.1 2074.0 n 0 1 e .
limit 4 0 D proposal 0 - 0 . e 1 4 distribution 0 0 0 0 Point estimation 1.06 2600.4 2559.6. 0 0 (Normal . 0 0 0 0 . 0.8 1.0 1.2 1.4 1.6 1.8 + 2000 3000 4000 5000
distribution) 0 Upper confidence e 1.25 3169.7 3190.5 0 limit m 1500 2500 η 3500 4500 0.8 1.0 1.2 1.4 1.6 1.8
(b) MTBF b 0
The iteration of m, η and MTBF value are shown in . 3
Fig. 1. The histogram and distribution of m, η and MTBF4 0 - 0 e . 8 2 value are shown in Fig. 2. y t i s X n e 4 D 0 - 0 . e 1 4 0 0 0 . + 0 e 0 1500 2500 3500 4500 0.8 1.0 1.2 1.4 1.6 1.8
MTBF b 2 1 0 0 . 0 3 . 0 8 0 0 0 . 0 2 . y y t t 0 i i s s n n e e 4 D D 0 0 0 . 0 1 . 0 0 0 0 0 . 0 0 . 0 0.8 1.0 1.2 1.4 1.6 1.8 2000 3000 4000 5000
m η
， ， 0 [7] Michael S. Hamada Alyson G. Wilson C. Shane . 3 Reese ， Harry F. Martz. Bayesian Reliability[M]. New 4
0 ： ， - York Springer 2008 0 e . 8 2 y t i [8] Pate-Cornell M E. “Uncertainties in risk analysis：six level s X n e 4 D
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0 [9] Erto P ， ” New practical Bayes estimates for the 2- 0 0 . + 0 e
0 parameter Weibull distribution.” Microelectronis 1500 2500 3500 4500 0.8 1.0 1.2 1.4reliability1.6 1.8 ，Vol 34(5)：pp.789-802,1994. MTBF [10]b Athreya, K., Doss, H. and Sethuraman, J.. “On the convergence of the Markov chainsimulation method. ”Ann. (c) Statist., Vol 24 pp.69–100.,1996. Fig 2. The histogram and r distribution of the Weibull [11] Coolen F P A. On Bayesian reliability analysis with distribution parameters: (a) shape m , (b) scale η and (c) informative priors and censoring[J]. Reliability Engineering MTBF value and System Safety， Vol 53：pp.91-98, 1996 [12] Arturo J F ， ” Bayesian inference from type II doubly censored Raleigh data[,” statistics & Probability letters ， Vol 48：pp.393-399, 2000. Ⅳ. CONCLUSION [13] Coolen F P A，Newby M J. “Bayesian reliability analysis with imprecise prior Probabilities.” Reliability Engineering 1. Markov chain was established to calculate the and System Safety， Vol 43：pp.75-85, 1994. interval estimation of MTBF value, which solved the [14] Ioannis N tzoufras. Bayesian Modeling Using WinBUGS. difficult problem effectively. It proves that it’s more Wiley，2009. accurate than the approximation by MLE and it ensures [15] Geof H. Givens, Jennifer A. Hoeting, Computational Statistics, translation by Wang Zhaojun, Liu Minqian, Zou the smooth implementation of reliability assessment. Changling, Yang Jianfeng. Posts & Telecom Press, Beijing, 2. Compared with results in the different conditions 2009. of MCMC transition kernel, it proves that MCMC method [16] ELISA T. LEE. Statistical Methods for Survival Data is appropriate for computing various interval estimates of Analysis. translation byChen Jiading, Dai Zhongwei. characteristics in reliability engineering, which has Beijing：China Statistics Press，1998. systematic solution process and good adaptability. MCMC method has unparalleled advantage in calculating interval estimates and it may basically replace the traditional methods. 3. MCMC method solves the interval estimation of MTBF in the reliability assessment of numerical control system effectively.
ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China (50875186) and Major projects of national science and technology (2009ZX04014-013).
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