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# An introduction to weibull analysis

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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

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### An introduction to weibull analysis

1. 1. An Introduction to Weibull Analysis (威布尔分析引论) Rong Pan ©2014 ASQ http://www.asqrd.org
2. 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. 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. 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. 5. Weibull Distribution 4/12/2014Webinar for ASQ Reliability Division 5  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. 6. Functions Related to Reliability 4/12/2014Webinar for ASQ Reliability Division 6  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. 7. Understanding Hazard Function 4/12/2014Webinar for ASQ Reliability Division 7  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. 8. Connection to Other Distributions 4/12/2014Webinar for ASQ Reliability Division 8  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. 9. Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 9  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. 10. Weibull Plot Example 4/12/2014Webinar for ASQ Reliability Division 10  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. 11. Complete Data 4/12/2014Webinar for ASQ Reliability Division 11  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. 12. Censored Data 4/12/2014Webinar for ASQ Reliability Division 12  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. 13. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 13  Small sample uncertainty
14. 14. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 14  Low failure times
15. 15. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 15  Effect of suspensions
16. 16. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 16  Effect of outlier
17. 17. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 17  Initial time correction
18. 18. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 18  Multiple failure modes
19. 19. Maximum Likelihood Estimation 4/12/2014Webinar for ASQ Reliability Division 19  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. 20. Plot by Software 4/12/2014Webinar for ASQ Reliability Division 20  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. 21. Compare to Other Distributions 4/12/2014Webinar for ASQ Reliability Division 21  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. 22. Weibull Regression 4/12/2014Webinar for ASQ Reliability Division 22  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. 23. Final Remarks 4/12/2014Webinar for ASQ Reliability Division 23  Weibull distribution  2 parameters  3 parameters  Shape of hazard function  Different stages of bathtub curve  Weibull plot  Find the parameter estimation  Interpretation