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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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- 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 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 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. 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. 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. 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. 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. 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. 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. 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. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 13 Small sample uncertainty
- 14. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 14 Low failure times
- 15. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 15 Effect of suspensions
- 16. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 16 Effect of outlier
- 17. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 17 Initial time correction
- 18. Diagnosis using Weibull Plot 4/12/2014Webinar for ASQ Reliability Division 18 Multiple failure modes
- 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. 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. 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. 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. 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

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