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Statistical methods are usually used in reliability analysis due to the uncertainty and distribution nature of reliability data. Bayesian analysis has been part of statistical analysis from the very ...

Statistical methods are usually used in reliability analysis due to the uncertainty and distribution nature of reliability data. Bayesian analysis has been part of statistical analysis from the very beginning when the foundations of modern statistics were established. Bayesian methods, however, are rarely used in the analysis of reliability data, mainly due to the lack of user friendly and efficient computation tools. With the development of freely available and efficient software package WinBUGs and OpenBUGs, there are more and more statisticians and engineers using Bayesian’s idea to combine useful prior information and the field data. In this talk, we first briefly review several main areas in statistical reliability analysis, then introduce the basic ideas of the Bayesian method and WinBUGs / OpenBUGs software. Next we will show how to apply Bayesian methods to several typical reliability problems through WinBUGs / OpenBUGs. Finally some common mistakes and pitfalls for Bayesian application to statistical reliability data analysis are discussed.
This is a joint work with Professor William Q. Meeker at Department of Statistics of Iowa State University.

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General bayesian methods for typical reliability data analysis Presentation Transcript

  • 1. General Bayesian  General Bayesian Methods for Typical  Methods for Typical Reliability Data Analysis y y Ming Li ©2012 ASQ & Presentation Li ©2012 ASQ & Presentation Li Presented live on Jun 14th, 2012http://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_liability Calendar/Webinars ‐_English/Webinars_‐_English.html
  • 2. ASQ Reliability Division  ASQ Reliability Division English Webinar Series English Webinar Series One of the monthly webinars  One of the monthly webinars on topics of interest to  reliability engineers. To view recorded webinar (available to ASQ Reliability  Division members only) visit asq.org/reliability ) / To sign up for the free and available to anyone live  webinars visit reliabilitycalendar.org and select English  Webinars to find links to register for upcoming eventshttp://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_liability Calendar/Webinars ‐_English/Webinars_‐_English.html
  • 3. General Bayesian Methods forTypical Reliability Data Analysis Ming Li GE Global Research A joint work with William Q. Meeker at Iowa State University. Webinar ASQ Reliability Division June 14 2012
  • 4. Outline • Traditional Reliability Framework  Problems / Concepts / Methods • Bayesian Reliability Framework  Prior knowledge / Concepts / Methods • Bayesian Reliability Examples  Weibull distribution  Accelerated life test  Repeated measure degradation • Common Mistakes and Pitfalls • Conclusions 2/ 2 GE Title or job number / 5/25/2012
  • 5. Traditional Reliability Framework  Reliability Problems  Statistical Concepts  Computational Methods 3/ 3 GE Title or job number / 5/25/2012
  • 6. Reliability Problems Life of a product Degradation of performance Repairable system Warranty Prognostic Service availability or guarantee 4/ 4 GE Title or job number / 5/25/2012
  • 7. Statistical Concepts Data  Field data, Lab data, simulated data  Failure modes, system or component level data  Exact, left, right, interval and window censored data Model  Life distribution estimation (Weibull, Lognormal …)  Accelerated testing planning and analysis  Degradation modeling (physics + statistics)  Poisson process for repairable system  Non-parametric statistical models (e.g. Kaplan Meier) 5/ 5 GE Title or job number / 5/25/2012
  • 8. Computational MethodsTo calculate point estimates and confidenceintervals for statistical uncertainty:  Maximum likelihood method  Bootstrap re-sampling method  Nonparametric method About methods are pure data driven, and prior knowledge is not used.  Simulation based Bayesian method  Could integrate prior knowledge or information  Solution to certain problems that are difficult to solve by other methods (i.e. computation advantages) 6/ 6 GE Title or job number / 5/25/2012
  • 9. Bayesian Reliability Framework  Why Not the Bayesian Method?  Prior Knowledge  Concept Illustration  Implementation through BUGs 7/ 7 GE Title or job number / 5/25/2012
  • 10. Why Not the Bayesian Method?• No user friendly Bayesian computer program  Engineers do not want to write MCMC  There are many non Bayesian program  ReliaSoft’s Weibull++, ALTA, etc.  JMP, Minitab, etc.  Many companies have site licenses• Need justification of prior knowledge  Sources of prior knowledge  Management approval  Impact of biased or bad priors 8/ 8 GE Title or job number / 5/25/2012
  • 11. Prior Knowledge• Physics of failure mechanism  Activation energy is around 0.2ev• Previous empirical experience  30 year experience of Weibull shape parameter of 2.5• Sensitivity analysis and scenario test  What if the activation energy changes to a range (0.4,0.6) Bayesian method combines data and prior knowledge, big impact when data is limited. 9/ 9 GE Title or job number / 5/25/2012
  • 12. Concept Illustration Model for Data Likelihood Posterior Inference Data Distribution Bayes’ Theorem Prior Information 10 10 / GE Title or job number / 5/25/2012
  • 13. Implementation Through BUGs http://www.openbugs.info/w/ # (1) model specification model { Features }  Easy to download and install # (2) data input  Simple user interface list()  Detailed manual with a lot of examples  Many build in distributions and functions # (3) initial value list() Steps list() list()  Define the statistics problem clearly  Prepare the input data accordingly  Setup reasonable initial valuesBernoulli, Binomial, Poisson … If it converges….  Check the history plotBeta, Chi-square, Normal,  Check the density plotGamma, Weibull, Logistic …  Check BGR diagnostic plotMultinomial, Dirichlet,  Look the posterior summary statisticsMultivariate Normal, Wishart …  Extract the data for each MCMC steps 11 11 / GE Title or job number /  Mean, median and Credible intervals 5/25/2012
  • 14. Bayesian Reliability Example  Weibull Distribution  Accelerated Life Test  Repeated Measure Degradation 12 12 / GE Title or job number / 5/25/2012
  • 15. Weibull Distribution for the Bearing-Cage Field Data Data from Meeker and Escobar Example 8.16. Data Summary • 6 failures • 1697 right censored • Different censoring time • Heavy censoring • Weibull distribution • Data driven MLE Bayesian implementation • Prior on B01 (i.e., t0.01 quantile) and weibull shape parameter • Interested in estimate B10 13 13 / GE Title or job number / 5/25/2012
  • 16. Weibull Distribution  1 t   t   T ~ weibull  t ,  ,         exp          t  1 exp    t         Parameter of t p and sigma Parameter of original ME book  1   t p [ log(1  p)]  t p  [ log(1  p)]     1   1         T ~ dweib  x, v,    v x v 1 exp  x v   1  v  Parameter used in WinBUGs    1  1           t p [ log(1  p)]   t  [  log(1  p )]  p 14 14 / GE Title or job number / 5/25/2012
  • 17. OpenBUGs Implementationmodel { log.B01 ~ dunif(4.6051,8.5172) B01 <- exp(log.B01) Priors log.sigma ~ dnorm(-1.151,31.562) sigma <- exp(log.sigma) log  t0.01  ~ unif  log 100  , log  5000     v <- 1/sigma log   ~ dnorm  mean  1.151,sd  0.178   lamda <- pow(B01,-v)*0.01005034 Informative prior: 99% of the probability of  between 0.2 and 0.5. for (iii in 1:6){ 24 exact failures x.exact[iii] ~ dweib(v,lamda) } for (jjj in 1:19){ dummy[jjj] <- 0 1697 right censored dummy[jjj] ~ dloglik(logLike[jjj]) logLike[jjj] <- weight[jjj]*(-lamda*pow(lower[jjj],v)) observation in groups }} 15 15 / GE Title or job number / 5/25/2012
  • 18. Accelerated Life Test for Device-A Data from Meeker and Escobar Example 19.2. Data Summary • Three accelerated levels (10C, 40C, 60C, and 80C) • Usage level 10C • Arrhenius model for temperature. • Log-normal life distribution.  Y  log  Hours  ~ N   K  ,  2  11605   K    0  1 K 16 16 / GE Title or job number / 5/25/2012
  • 19. Re-parameterization • Replace the intercept by B01 at 40C • It will break the strong correlation between the slope and intercept  11605  B01.40  0  1  z0.01  273  40     B01.40   11605  z   0  1 273  40 0.01 • Use informative prior for 1 such that 99% of the probability will between 0.5 and 0.8. • Interested in the B10 life and the usage temperature 10C (i.e. 283K)  11605 11605    K   B01.40  z0.01  1     K 273  40  17 17 / GE Title or job number / 5/25/2012
  • 20. OpenBUGs Implementationmodel { ### For Temp=60C, i.e. 11605/(273+60) = 34.849850 B01.40 ~ dgamma(0.001,0.0001) ### 11 censored observations, and 9 exact observations b1 ~ dnorm(0.65,294.8843) ## informative prior ### 11605/(273+60) - 11605/(273+40) = -2.226827 tau ~ dgamma(0.001,0.0001) mu.60 <- B01.40 + 2.326348*sigma - b1*2.226827 sigma <- 1/sqrt(tau) for (i in 1:11){ dummy.60[i] <- 0 b0 <- B01.40 + 2.326348*sigma - b1*37.076677 dummy.60[i] ~ dloglik(logLike.60[i]) B10.10 <- mu.10 - 1.281552*sigma logLike.60[i] <- ( 1-phi((8.517193-mu.60)*sqrt(tau)) ) }### For Temp=10C, i.e. 11605/(273+10) = 41.007067 for (j in 1:9){### All 30 observations are censored. Y.log.60[j] ~ dnorm(mu.60,tau)### 11605/(273+10) - 11605/(273+40) = 3.93039 }mu.10 <- B01.40 + 2.326348*sigma + b1*3.93039for (i in 1:30){ dummy.10[i] <- 0 ### For Temp=80C, i.e. 11605/(273+80) = 32.875354 dummy.10[i] ~ dloglik(logLike.10[i]) ### 11 censored observations, and 9 exact observations logLike.10[i] <- ( 1-phi((8.517193-mu.10)*sqrt(tau)) ) ### 11605/(273+80) - 11605/(273+40) = -4.201323} mu.80 <- B01.40 + 2.326348*sigma - b1*4.201323 for (i in 1:1){ ### For Temp=40C, i.e. 11605/(273+40) = 37.076677 dummy.80[i] <- 0 ### 90 censored observations, and 10 exact observations dummy.80[i] ~ dloglik(logLike.80[i]) ### 11605/(273+40) - 11605/(273+40) = 0 logLike.80[i] <- ( 1-phi((8.517193-mu.80)*sqrt(tau)) ) mu.40 <- B01.40 + 2.326348*sigma } for (i in 1:90){ for (j in 1:14){ dummy.40[i] <- 0 Y.log.80[j] ~ dnorm(mu.80,tau) dummy.40[i] ~ dloglik(logLike.40[i]) } logLike.40[i] <- ( 1-phi((8.517193-mu.40)*sqrt(tau)) ) } } for (j in 1:10){ Y.log.40[j] ~ dnorm(mu.40,tau) } 18 18 / GE Title or job number / 5/25/2012
  • 21. Repeated Measure for Device B Degradation Data from Meeker and Escobar Example 21.1. Data Summary • 3 levels of temperature • Usage temperature 80C • ~ 10 devices per temp. • Interval of 125 hours to measure the degradation • Mixed effect model • Nonlinear path • Normal distribution for residuals 19 19 / GE Title or job number / 5/25/2012
  • 22. Model Details yij ~ Dij   ij  1 i=1,. . . ,n: index for device  ij ~ N  0,  2   j=1,…,m : index of time of observation for each device   j   Dij  tij ; temp   Di ,  1  exp  Ri 195   AF  temp   tij    11605 11605   In stable AF  temp   exp  Ea    parameterization   195  273 temp  273   Di , : The asymptote for each device. Ri 195  The reaction rate at 195C for each device  1,i   1,i  log  Ri 195     ~ MVN  mean.β,prec.β     2,i     2,i  log   D ,i  mean.β : 2x1 mean vector of a bivariate normal   3  Ea  prec.β : 2x2 precision matrix of a bivariate normal sigma  inv  prec.β  : Variance and covariance matrix 20 / 20 for the bivariate normal. or job5/25/2012 GE Title number /
  • 23. OpenBUGs Implementation *(1-exp(-R195[iii+7]*data[231+(iii-1)*17+jjj,2])) model { data[231+(iii-1)*17+jjj,1] ~ dnorm(mu.195[(iii-1)*17+jjj],tau) } for(iii in 1:34){ } bbb[iii,1:2] ~ dmnorm(mean.bbb[1:2],prec.bbb[1:2,1:2]) Dinf[iii] <- -exp(bbb[iii,2]) R195[iii] <- exp(bbb[iii,1]) #### Data and Model for Temp=237 ### } #### 11605/(195+273) - 11605/(237+273) = 2.042107 #### Index shift for data is: 33*7 + 12*17 = 435 sigma[1:2,1:2] <- inverse(prec.bbb[1:2,1:2]) #### Index shift for group is: 7+12=19 mean.bbb[1:2] ~ dmnorm(M[1:2], A[1:2,1:2]) for(iii in 1:15){ prec.bbb[1:2,1:2] ~ dwish(B[1:2,1:2 ], 2) for(jjj in 1:9){ mu.237[(iii-1)*9+jjj] <- Dinf[iii+19] b3 ~ dnorm(0.7,663.5) *(1-exp(-R195[iii+19]*exp(b3*2.042107) tau ~ dgamma(0.001,0.001) *data[435+(iii-1)*9+jjj,2]) ) sigma.error <- 1/sqrt(tau) data[435+(iii-1)*9+jjj,1] ~ dnorm(mu.237[(iii-1)*9+jjj],tau) } #### Data and Model for Temp=150C ### } #### 11605/(195+273) - 11605/(150+273) = -2.637980 for(iii in 1:7){ } Priors for(jjj in 1:33){ mu.150[(iii-1)*33+jjj] <- Dinf[iii]*(1-exp(-R195[iii]   0  106 0  mean.β ~ dmnorm    ,  6   *exp(-b3*2.637980) *data[(iii-1)*33+jjj,2]) ) 0  data[(iii-1)*33+jjj,1] ~ dnorm(mu.150[(iii-1)*33+jjj],tau)   0 10   } }  103 0   prec.β ~ dwish    0 103   ,2  #### Data and Model for Temp=195 ###     #### 11605/(195+273) - 11605/(195+273) = 0 #### Index shift for data is: 33*7=231  ~ dgamma  0.001, 0.001 #### Index shift for group is: 7 3 ~ dnorm  0.7, 663.5  for(iii in 1:12){ for(jjj in 1:17){ mu.195[(iii-1)*17+jjj] <- Dinf[iii+7] Informative prior: put 99% of the 21 21 / probability between 0.6 and 0.8 for  35/25/2012 . GE Title or job number /
  • 24. Cautious and Pitfalls• Be aware of the effect of prior selection• Do a sensitivity analysis and compare with non-informative priors• Inappropriate priors for biased results• Understand the assumptions 22 22 / GE Title or job number / 5/25/2012
  • 25. Conclusions• Reliability engineers have prior knowledge for the model parameters• Bayesian analysis provides a formal way to implement prior knowledge• OpenBUGs/WinBUGs provides user-friendly tool for Bayesian reliability analysis• Most reliability models can be implemented through OpenBUGs/WinBUGs 23 23 / GE Title or job number / 5/25/2012
  • 26. Thank you! 24 24 / GE Title or job number / 5/25/2012
  • 27. Zero-trick in OpenBUGs Reason for quick convergence: The likelihood contribution for censored observation is determined by the censoring time and use the OpenBUGs zero-trick to include the censored observation likelihood contribution.For Weibull right censored observation at censor time T, the likelihood is:  T  f  x  dx  1   f  x  dx T 0   T    exp      ME Book parameterization         exp T v  OpenBUGs parameterization 25 25 / GE Title or job number / 5/25/2012
  • 28. Traditional method in OpenBUGs• C( , ): the build-in censoring function in OpenBUGs• Very slow in convergence for heavy censoring!• Reason for slow convergence: each censor data point is treated as a random node in OpenBUGs and a stochastic MCMC chain will be established for each random node. 26 26 / GE Title or job number / 5/25/2012