The document summarizes a Bayesian webinar on general Bayesian methods for reliability data analysis. It provides an outline of the webinar covering traditional vs Bayesian reliability frameworks, examples of applying Bayesian methods to Weibull distribution, accelerated life test data and repeated measure degradation data. OpenBUGS code is presented for the examples. The webinar aims to illustrate how Bayesian methods allow incorporating prior knowledge and provide advantages over traditional methods in certain applications.

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, 2012
http://reliabilitycalendar.org/The_Re
liability_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 events
http://reliabilitycalendar.org/The_Re
liability_Calendar/Webinars_
liability Calendar/Webinars ‐
_English/Webinars_‐_English.html

3. General Bayesian Methods for
Typical 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
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5. Traditional Reliability Framework
 Reliability Problems
 Statistical Concepts
 Computational Methods
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6. Reliability Problems
 Life of a product
 Degradation of performance
 Repairable system
 Warranty
 Prognostic
 Service availability or guarantee
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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)
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8. Computational Methods
To calculate point estimates and confidence
intervals 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)
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9. Bayesian Reliability Framework
 Why Not the Bayesian Method?
 Prior Knowledge
 Concept Illustration
 Implementation through BUGs
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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
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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.
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12. Concept Illustration
Model for
Data
Likelihood
Posterior
Inference
Data Distribution
Bayes’
Theorem
Prior
Information
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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 values
Bernoulli, Binomial, Poisson … If it converges….
 Check the history plot
Beta, Chi-square, Normal,  Check the density plot
Gamma, Weibull, Logistic …  Check BGR diagnostic plot
Multinomial, Dirichlet,  Look the posterior summary statistics
Multivariate Normal, Wishart …  Extract the data for each MCMC steps 11 11
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 Mean, median and Credible intervals 5/25/2012

14. Bayesian Reliability Example
 Weibull Distribution
 Accelerated Life Test
 Repeated Measure Degradation
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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
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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
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17. OpenBUGs Implementation
model {
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
}
}
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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
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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 
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20. OpenBUGs Implementation
model {
### 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*3.93039
for (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)
}
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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
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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 /
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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
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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
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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
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26. Thank you!
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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
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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.
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