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
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
4/12/2014Webinar for ASQ Reliability Division
3
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
4/12/2014Webinar for ASQ Reliability Division
4
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
4/12/2014Webinar for ASQ Reliability Division
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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
4/12/2014Webinar for ASQ Reliability Division
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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
4/12/2014Webinar for ASQ Reliability Division
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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
4/12/2014Webinar for ASQ Reliability Division
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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
4/12/2014Webinar for ASQ Reliability Division
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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
4/12/2014Webinar for ASQ Reliability Division
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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
13. Diagnosis using Weibull Plot
4/12/2014Webinar for ASQ Reliability Division
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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
4/12/2014Webinar for ASQ Reliability Division
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Initial time correction
18. Diagnosis using Weibull Plot
4/12/2014Webinar for ASQ Reliability Division
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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
4/12/2014Webinar for ASQ Reliability Division
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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
4/12/2014Webinar for ASQ Reliability Division
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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