Objectives  To understand Weibull distribution  To be able to use Weibull plot for failure time analysis and diagnosis  To be able to use software to do data analysis Organization  Distribution model  Parameter estimation  Regression analysis

1. An Introduction to Weibull
Analysis (威布尔分析引论)
Rong Pan
©2014 ASQ
http://www.asqrd.org

2. RONG PA N
ASSOCIATE PROFESSOR
A RIZONA ST A T E U NIVERSIT Y
EM A IL: RONG.PA N@ A SU .EDU
An Introduction to Weibull
Analysis

3. Outlines
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 Objectives
 To understand Weibull distribution
 To be able to use Weibull plot for failure time analysis and
diagnosis
 To be able to use software to do data analysis
 Organization
 Distribution model
 Parameter estimation
 Regression analysis

4. A Little Bit of History
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 Waloddi Weibull (1887-1979)
 Invented Weibull distribution in 1937
 Publication in 1951
 A statistical distribution function of wide
applicability, Journal of Mechanics, ASME,
September 1951, pp. 293-297.
 Was professor at the Royal Institute of
Technology, Sweden
 Research funded by U.S. Air Force

5. Weibull Distribution
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 A typical Weibull distribution function has two
parameters
 Scale parameter (characteristic life)
 Shape parameter
 A different parameterization
 Intrinsic failure rate
 Common in survival analysis
 3-parameter Weibull distribution
 Mean time to failure
 Percentile of a distribution
 “B” life or “L” life




 














t
e
t
tf
1
)(
.0,,0,1)( 










tetF
t

t
etF 
1)(








 


t
etF 1)(
)/11(  MTTF

6. Functions Related to Reliability
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 Define reliability
 Is the probability of life time longer than t
 Hazard function and Cumulative hazard
function
 Bathtub curve
)(1)(1)()( tFtTPtTPtR 
)(
)(
)(
tR
tf
th  
t
dxxhtH
0
)()( )(
)( tH
etR 

Time
Hazard

7. Understanding Hazard Function
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 Instantaneous failure
 Is a function of time
 Weibull hazard could be
either increasing function of
time or decreasing function
of time
 Depending on shape
parameter
 Shape parameter <1 implies
infant mortality
 =1 implies random failures
 Between 1 and 4, early wear
out
 >4, rapid wear out

8. Connection to Other Distributions
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 When shape parameter = 1
 Exponential distribution
 When shape parameter is known
 Let , then Y has an exponential distribution
 Extreme value distribution
 Concerns with the largest or smallest of a set of random
variables
 Let , then Y has a smallest extreme value
distribution
 Good for modeling “the weakest link in a system”

TY 
TY log

9. Weibull Plot
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 Rectification of Weibull distribution
 If we plot the right hand side vs. log failure time, then we
have a straight line
 The slope is the shape parameter
 The intercept at t=1 is
 Characteristic life
 When the right hand side equals to 0, t=characteristic
life
 F(t)=1-1/e=0.63
 At the characteristic life, the failure probability does not
depend on the shape parameter
   loglog))(1log(log  ttF
 log

10. Weibull Plot Example
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 A complementary
log-log vs log plot
paper
 Estimate failure
probability (Y) by
median rank
method
 Regress X on Y
 Find
characteristic life
and “B” life on the
plot

11. Complete Data
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 Order failure times from smallest to largest
 Check median rank table for Y
 Calculation of rank table uses binomial distribution
 Y is found by setting the cumulative binomial function
equal to 0.5 for each value of sequence number
 Can be generated in Excel by BETAINV(0.5,J,N-J+1)
 J is the rank order
 N is sample size
 By Bernard’s approximation
Order
number
Failure
time
Median rank %
(Y)
1 30 12.94
2 49 31.38
3 82 50.00
4 90 68.62
5 96 87.06
)4.0/()3.0(  NJY

12. Censored Data
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 Compute reverse rank
 Compute adjusted rank
 Adjusted rank = (reverse rank * previous adjusted rank
+N+1)/(reverse rank+1)
 Find the median rank
Rank Time Reverse rank Adjusted
rank
Median rank
%
1 10S 8 Suspended
2 30F 7 1.125 9.8
3 45S 6 Suspended
4 49F 5 2.438 25.5
5 82F 4 3.750 41.1
6 90F 3 5.063 56.7
7 96F 2 6.375 72.3
8 100S 1 suspended

13. Diagnosis using Weibull Plot
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 Small sample uncertainty

14. Diagnosis using Weibull Plot
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 Low failure times

15. Diagnosis using Weibull Plot
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 Effect of suspensions

16. Diagnosis using Weibull Plot
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 Effect of outlier

17. Diagnosis using Weibull Plot
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 Initial time correction

18. Diagnosis using Weibull Plot
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 Multiple failure modes

19. Maximum Likelihood Estimation
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 Maximum likelihood estimation (MLE)
 Likelihood function
 Find the parameter estimate such that the chance of having such failure
time data is maximized
 Contribution from each observation to likelihood function
 Exact failure time
 Failure density function
 Right censored observation
 Reliability function
 Left censored observation
 Failure function
 Interval censored observation
 Difference of failure functions
)(tR
)(tF
)()( 
 tFtF
)(tf

20. Plot by Software
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 Minitab
 Stat  Reliability/Survival  Distribution analysis  Parametric
distribution analysis
 JMP
 Analyze  Reliability and Survival  Life distribution
 R
 Needs R codes such as
 data <- c(….)
 n <- length(data)
 plot(data, log(-log(1-ppoints(n,a=0.5))), log=“x”, axes=FALSE,
frame.plot=TRUE, xlab=“time”, ylab=“probability”)
 Estimation of scale and shape parameters can also be found by
 res <- survreg(Surv(data) ~1, dist=“weibull”)
 theta <- exp(res$coefficient)
 alpha <- 1/res$scale

21. Compare to Other Distributions
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 Choose a distribution model
 Fit multiple distribution models
 Criteria (smaller the better)
 Negative log-likelihood values
 AICc (corrected Akaike’s information criterion)
 BIC (Baysian information criterion)

22. Weibull Regression
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 When there is an explanatory variable
(regressor)
 Stress variable in the accelerated life testing (ALT)
model
 Shape parameter of Weibull distribution is often
assumed fixed
 Scale parameter is changed by regressor
 Typically a log-linear function is assumed
 Implementation in Software

23. Final Remarks
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 Weibull distribution
 2 parameters
 3 parameters
 Shape of hazard function
 Different stages of bathtub curve
 Weibull plot
 Find the parameter estimation
 Interpretation