Bayesian Methods in                   Reliability Engineering                  R li bili E i       i                      ...
ASQ Reliability Division                   ASQ Reliability Division                  English Webinar Series               ...
Bayesian Methods in Reliability Engineering      ASQ Reliability Division Webinar Program                   Nov 15th 2012 ...
BAYESIAN METHODS IN RELIABILITY ENGINEERING With product reliability demonstration test planning and execution interacting...
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors  Conjugate Prio...
11/15/2012   ASQ RD Webinar   4
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors  Conjugate Prio...
A d                     Agenda• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors  Conjugate Prio...
When reliability follows the exponential TTFWhen reliability follows the exponential TTF model (eg the flat               ...
exponential distribution               ti l di t ib ti“non‐intuitive” non intuitiveBrains wired with                      ...
ti l di t ib ti             exponential distribution                                                 Planet = Laitnenopxe ...
Confidence vs. Credibility Intervals   C fid          C dibilit I t      l11/15/2012        ASQ RD Webinar          10
For and Against use of Bayesian Methodology            g               y               gy               PROs              ...
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors  Conjugate Prio...
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors  Conjugate Prio...
Toy Hyperbolic Example                      y yp             pFirst day on the job as reliability engineer, you overhear t...
Bayesian Core Idea                        Bayesian Core Idea       What you knew        What you knew       before WYKB.  ...
λ   KOtG           λ   1        λ   2             Failure Rate λ  (1/sec)         0.0022          0.0100       0.0020     ...
λ   KOtG               λ   1      λ   2             Failure Rate λ  (1/sec)       0.0022              0.0100     0.0020   ...
λ   KOtG           λ   1        λ   2             Failure Rate λ  (1/sec)         0.0022          0.0100       0.0020     ...
λ   KOtG            λ   1        λ   2             Failure Rate λ  (1/sec)        0.0022           0.0100       0.0020    ...
λ   KOtG           λ   1        λ   2             Failure Rate λ  (1/sec)         0.0022          0.0100       0.0020     ...
λ   KOtG           λ   1        λ   2             Failure Rate λ  (1/sec)         0.0022          0.0100       0.0020     ...
λ   KOtG           λ   1        λ   2             Failure Rate λ  (1/sec)         0.0022          0.0100       0.0020     ...
λ   KOtG            λ   1        λ   2           Failure Rate λ  (1/sec)     0.0022              0.0100       0.0020      ...
λ   KOtG           λ   1        λ   2             Failure Rate λ  (1/sec)         0.0022          0.0100       0.0020     ...
go to the spreadsheet             go to the spreadsheet11/15/2012          ASQ RD Webinar   25
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors  Conjugate Prio...
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example  Conjugate Priors• Conjugate Prio...
Conjugate Prior             C j t Pi11/15/2012        ASQ RD Webinar   28
Mean        λave = a/b                    Variance   σ2 = a/b2             In hierarchical Bayesian models, these hyperpar...
Bayesian assumptions for the gamma       y           p       f      g         exponential system model1. Failure times for...
Gamma prior parameter method 1             G       i         t     th d 11. If you have actual data from previous testing ...
Gamma prior parameter method 2             Gamma prior parameter method 22. A consensus method for determining2 A consensu...
Gamma prior parameter method 3  G       i         t     th d 33. Weak Prior Obtain consensus is on a reasonable    expecte...
Comments                   C     tMany variations are possible, based on the above three methods. For example, you might h...
New data is collected …N d t i       ll t dNew information is combined with the gamma prior model topproduce a gamma poste...
Reliability estimation with Bayesian gamma prior model11/15/2012                    ASQ RD Webinar                     36
Example                            p• A group of engineers, discussing the reliability of a new  piece of equipment decide...
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example  Conjugate Priors• Conjugate Prio...
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors  Conjugate Prio...
Bayesian test plan                                 y           pGamma prior parameters a and b have already been determine...
Special Case: a = 1 (The "Weak" Prior)   p                  (                )When the prior is a weak prior with a = 1, t...
Calculating a Bayesian Test Time                       g     yA new piece of equipment has to meet a MTBF requirement of 5...
Post‐Test Analysis Example             P t T tA l i E          l• A system has completed a reliability test aimed at   con...
Bayesian solutions for arbitrary F(t)  B    i     l ti    f     bit     F(t)What about WeibullWh t b t W ib ll or other no...
References and Further Reading      R f          d F th R di• NIST/SEMATECH e‐Handbook of Statistical Methods,   http://ww...
11/15/2012   ASQ RD Webinar   46
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors  Conjugate Prio...
Agenda                     A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors  Conjugate Prio...
Q&A11/15/2012   ASQ RD Webinar   49
Upcoming SlideShare
Loading in...5
×

Bayesian methods in reliability engineering

1,465

Published on

With product reliability demonstration test planning and execution weighing heavily on cost, availability and schedule factors, Bayesian methods offer an intelligent way of incorporating engineering knowledge based on historical information into data analysis and interpretation, resulting in an overall more precise and less resource intensive failure rate estimation. This talk consists of three parts
1. Introduction to Bayesian vs Frequentist statistical approaches
2. Bayesian formalism for reliability estimation
3. Product/component case studies and examples

Published in: Technology
0 Comments
2 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
1,465
On Slideshare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
106
Comments
0
Likes
2
Embeds 0
No embeds

No notes for slide

Transcript of "Bayesian methods in reliability engineering"

  1. 1. Bayesian Methods in  Reliability Engineering R li bili E i i Charles Recchia ©2012 ASQ & Presentation Charles Presented live on Nov 15th, 2012http://reliabilitycalendar.org/The_Reliability Calendar/Webinars_‐ y_ /_English/Webinars_‐_English.html
  2. 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  ( y Division members only) visit asq.org/reliability To sign up for the free and available to anyone live webinars  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 eventshttp://reliabilitycalendar.org/The_Reliability Calendar/Webinars_‐ y_ /_English/Webinars_‐_English.html
  3. 3. Bayesian Methods in Reliability Engineering ASQ Reliability Division Webinar Program Nov 15th 2012 Charles H. Recchia, MBA, PhD Quality Support Group, Inc http://www.qualitysupportgroup.com/ http://www qualitysupportgroup com/
  4. 4. BAYESIAN METHODS IN RELIABILITY ENGINEERING With product reliability demonstration test planning and execution interacting  heavily with cost, availability and schedule considerations, Bayesian methods  heavily with cost availability and schedule considerations Bayesian methods offer an intelligent way of incorporating engineering knowledge based on  y p g historical information into data analysis and interpretation, resulting in an  overall more precise and less resource intensive failure rate estimation.  This talk  consists of three parts Introduction to Bayesian vs Frequentist statistical approaches Bayesian formalism for reliability estimation Product/component case studies and examplesCharles Recchia has more than two dozen years of fundamental research, technology/product development, and management experience with a special focus on reliability statistics of complex systems. He earned a doctorate in Condensed Matter Physics from Ohio State University, and a Master of Business Administration degree from Babson College. Dr. Recchia accrued reliability engineering  fB i Ad i i i d f B b C ll D R hi d li bili i iexpertise at Intel, MKS Instruments and Saint‐Gobain Innovative Materials R&D, has served as adjunct professor at Wittenberg University, and is author of numerous peer‐reviewed technical papers and patents.  He is a senior member of ASQ, the American Physical Society, and serves on the Advisory  f fCommittee for the Boston Chapter of the IEEE Reliability Society.11/15/2012 ASQ RD Webinar 2
  5. 5. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors Conjugate Priors• Test Time Examples Test Time Examples• Q ti Question and Answer dA11/15/2012 ASQ RD Webinar 3
  6. 6. 11/15/2012 ASQ RD Webinar 4
  7. 7. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors Conjugate Priors• Test Time Examples Test Time Examples• Q ti Question and Answer dA11/15/2012 ASQ RD Webinar 5
  8. 8. A d Agenda• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors Conjugate Priors• Test Time Examples Test Time Examples• Q ti Question and Answer dA11/15/2012 ASQ RD Webinar 6
  9. 9. When reliability follows the exponential TTFWhen reliability follows the exponential TTF model (eg the flat the flat constant failure rate portion of Bathtub Curve):Classical Framework – The MTBF is one fixed unknown value ‐ there is no “probability” associated  with it ith it – Failure data from a test or observation period allows you to make  inferences about the value of the true unknown MTBF – No other data are used and no “judgment” ‐ the procedure is objective and  based solely on the test data and the assumed HPP modelBayesian FrameworkBayesian Framework – The MTBF is a random quantity with a probability distribution – The particular piece of equipment or system you are testing “chooses” an  MTBF from this distribution and you observe failure data that follow an  MTBF f hi di ib i d b f il d h f ll HPP model with that MTBF – Prior to running the test, you already have some idea of what the MTBF  probability distribution looks like based on prior test data or an consensus  engineering judgment11/15/2012 ASQ RD Webinar 7
  10. 10. exponential distribution ti l di t ib ti“non‐intuitive” non intuitiveBrains wired with  Planet = EarthNormal Distribution σ ~ 0.1μ 6 draw sample.   Population mean height 5’11” Sample mean = ______ 8 5’9” 6’3” 5’10” 5’9” 6’0” 5’9”
  11. 11. ti l di t ib ti exponential distribution Planet = Laitnenopxe 6 draw sample.   Population mean height 5’11” Sample mean = ______11/15/2012 ASQ RD Webinar 94’11” 13’8” 7’11” 5’9” 2’5” 4”
  12. 12. Confidence vs. Credibility Intervals C fid C dibilit I t l11/15/2012 ASQ RD Webinar 10
  13. 13. For and Against use of Bayesian Methodology g y gy PROs CONs Uses prior information ‐ thi U i i f ti this  Prior information may not be  Pi i f ti tb"makes sense“ accurate ‐ generating misleading  Less new testing may be  Less new testing may be conclusionsneeded to confirm a desired  Way of inputting prior information MTBF at a given confidenceMTBF at a given confidence (choice of prior) may not be correct (choice of prior) may not be correct Confidence intervals are really  Customers may not accept validity of intervals for the (random) MTBF intervals for the (random) MTBF prior data or engineering judgements prior data or engineering judgements‐ sometimes called "credibility  Risk of perception that results intervals“ arent objective and dont stand by  j y themselves11/15/2012 ASQ RD Webinar 11
  14. 14. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors Conjugate Priors• Test Time Examples Test Time Examples• Q ti Question and Answer dA11/15/2012 ASQ RD Webinar 12
  15. 15. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors Conjugate Priors• Test Time Examples Test Time Examples• Q ti Question and Answer dA11/15/2012 ASQ RD Webinar 13
  16. 16. Toy Hyperbolic Example y yp pFirst day on the job as reliability engineer, you overhear three colleagues debating the MTBF for a product.Evidently the engineer you are replacing had kept all his data on his now‐destroyed C:drive and all that remains is “word of mouth” among his three remaining coworkers.  Waloddi: “I remember seeing 500 seconds written down  on his whiteboard.  I can still see it in my head.” Gertrude: “No W, that was 100 seconds!  His  handwriting was atrocious, but that definitely was a 1  not a 5. not a 5 ” Taiichi: “Agree with Gertrude.  It was 100 seconds.”2 against 1.  That is all you have to go by.  Your manager needs an answer by end of day.  The ink on your badge hasn’t even dried yet. fd Th i k b d h ’t di d tWhat shall you do?  They measure product MTBF in seconds?11/15/2012 ASQ RD Webinar 14
  17. 17. Bayesian Core Idea Bayesian Core Idea What you knew  What you knew before WYKB.   New Data “Prior” Best possible update of WYKB  adjusted by the New Data. adjusted by the New Data “Posterior”11/15/2012 ASQ RD Webinar 15
  18. 18. λ KOtG λ 1 λ 2 Failure Rate λ  (1/sec) 0.0022 0.0100 0.0020 MTTF (sec) 450 100 500 Prior g (λ ) 0.667 0.333 P t i g (λ | t i ) Posterior 0.204 0 204 0.796 0 796 Prior*Likelihood 3.66E‐21 1.43E‐20 Lik lih d Π f ( i ) Likelihood  (t 5.49E‐21 5 49E 21 1.43E‐18 1 43E 18 average {t }  average {t i } = 317 i TTF data t i  (sec) f 1(t i ) f 2(t i ) 1 133 2.65E‐03 1.53E‐03 2 888 1.39E‐06 3.39E‐04 3 619 2.05E‐05 5.80E‐04 4 8 9.23E‐03 9 23E 03 1.97E‐03 1 97E 03 5 97 3.78E‐03 1.65E‐03 6 157 2.08E‐03 1.46E‐0311/15/2012 ASQ RD Webinar 16
  19. 19. λ KOtG λ 1 λ 2 Failure Rate λ  (1/sec) 0.0022 0.0100 0.0020 MTTF (sec) 450 100 500 Prior g (λ ) 0.667 0.333 P t i g (λ | t i ) Posterior 0.204 0 204 0.796 0 796 Prior*Likelihood 3.66E‐21 1.43E‐20 There is just enough time  1.43E‐18 There is justLikenough) time21 1 43E 18 Likelihood Π f ( lih d (t 5.49E‐21 5 49E i before end of day to collect 6  average {t }  average {t } = 317 i y i TTF data t  (sec) f (t ) f (t ) time‐to‐fail (TTF) data points.1.53E‐03 1 133 i 2.65E‐03 1 i 2 i 2 888 1.39E‐06 3.39E‐04 3 619 2.05E‐05 5.80E‐04 4 8 9.23E‐03 9 23E 03 1.97E‐03 1 97E 03 Let’s do that. 5 97 3.78E‐03 1.65E‐03 6 157 2.08E‐03 1.46E‐0311/15/2012 ASQ RD Webinar 17
  20. 20. λ KOtG λ 1 λ 2 Failure Rate λ  (1/sec) 0.0022 0.0100 0.0020 MTTF (sec) 450 100 500 Prior g (λ ) 0.667 0.333 P t i g (λ | t i ) Posterior 0.204 0 204 0.796 0 796 Prior*Likelihood 3.66E‐21 1.43E‐20 Lik lih d Π f ( i ) Likelihood  (t 5.49E‐21 5 49E 21 1.43E‐18 1 43E 18 average {t }  average {t i } = 317 i TTF data t i  (sec) f 1(t i ) f 2(t i ) 1 133 2.65E‐03 1.53E‐03 2 888 1.39E‐06 3.39E‐04 3 619 2.05E‐05 5.80E‐04 4 8 9.23E‐03 9 23E 03 1.97E‐03 1 97E 03 5 97 3.78E‐03 1.65E‐03 6 157 2.08E‐03 1.46E‐0311/15/2012 ASQ RD Webinar 18
  21. 21. λ KOtG λ 1 λ 2 Failure Rate λ  (1/sec) 0.0022 0.0100 0.0020 MTTF (sec) 450 100 500 Prior g (λ ) 0.667 0.333 P t i g (λ | t i ) Posterior 0.204 0 204 0.796 0 796 How likely is this data? Prior*Likelihood 3.66E‐21 1.43E‐20 Lik lih d Π f ( i ) 5 49E 21 what λ 18 Likelihood  Depends on Depends on what43E is! (t 5.49E‐21 1.43E‐18 1 average {t }  average {t i } = 317 i TTF data t i  (sec) f 1(t i ) f 2(t i ) 1 133 2.65E‐03 1.53E‐03 2 888 1.39E‐06 3.39E‐04 3 619 2.05E‐05 5.80E‐04 4 8 9.23E‐03 9 23E 03 1.97E‐03 1 97E 03 5 97 3.78E‐03 1.65E‐03 6 157 2.08E‐03 1.46E‐0311/15/2012 ASQ RD Webinar 19
  22. 22. λ KOtG λ 1 λ 2 Failure Rate λ  (1/sec) 0.0022 0.0100 0.0020 MTTF (sec) 450 100 500 Prior g (λ ) 0.667 0.333 P t i g (λ | t i ) Posterior 0.204 0 204 0.796 0 796 Prior*Likelihood 3.66E‐21 1.43E‐20 Lik lih d Π f ( i ) Likelihood  (t 5.49E‐21 5 49E 21 1.43E‐18 1 43E 18 average {t }  average {t i } = 317 i TTF data t i  (sec) f 1(t i ) f 2(t i ) 1 133 2.65E‐03 1.53E‐03 2 888 1.39E‐06 3.39E‐04 3 619 2.05E‐05 5.80E‐04 4 8 9.23E‐03 9 23E 03 1.97E‐03 1 97E 03 5 97 3.78E‐03 1.65E‐03 6 157 2.08E‐03 1.46E‐0311/15/2012 ASQ RD Webinar 20
  23. 23. λ KOtG λ 1 λ 2 Failure Rate λ  (1/sec) 0.0022 0.0100 0.0020 MTTF (sec) 450 100 500 Prior g (λ ) 0.667 0.333 P t i g (λ | t i ) Posterior 0.204 0 204 0.796 0 796 Prior*Likelihood 3.66E‐21 1.43E‐20 Lik lih d Π f ( i ) Likelihood  (t 5.49E‐21 5 49E 21 1.43E‐18 1 43E 18 average {t }  average {t i } = 317 i TTF data t i  (sec) f 1(t i ) f 2(t i ) 1 133 2.65E‐03 1.53E‐03 2 888 1.39E‐06 3.39E‐04 3 619 2.05E‐05 5.80E‐04 4 8 9.23E‐03 9 23E 03 1.97E‐03 1 97E 03 5 97 3.78E‐03 1.65E‐03 6 157 2.08E‐03 1.46E‐0311/15/2012 ASQ RD Webinar 21
  24. 24. λ KOtG λ 1 λ 2 Failure Rate λ  (1/sec) 0.0022 0.0100 0.0020 MTTF (sec) 450 100 500 Prior g (λ ) 0.667 0.333 P t i g (λ | t i ) Posterior 0.204 0 204 0.796 0 796 Prior*Likelihood 3.66E‐21 1.43E‐20 Lik lih d Π f ( i ) Likelihood  (t 5.49E‐21 5 49E 21 1.43E‐18 1 43E 18 average {t }  average {t i } = 317 i TTF data t i  (sec) f 1(t i ) f 2(t i ) 1 133 2.65E‐03 1.53E‐03 2 888 1.39E‐06 3.39E‐04 3 619 2.05E‐05 5.80E‐04 4 8 9.23E‐03 9 23E 03 1.97E‐03 1 97E 03 5 97 3.78E‐03 1.65E‐03 6 157 2.08E‐03 1.46E‐0311/15/2012 ASQ RD Webinar 22
  25. 25. λ KOtG λ 1 λ 2 Failure Rate λ  (1/sec) 0.0022 0.0100 0.0020 Shouldn t these  (sec) Shouldn’t these MTTF 450 100 500 sum to 1 if they  are exhaustive  Prior g (λ ) 0.667 0.333 possibilities? P t i g (λ | t i ) Posterior 0.204 0 204 0.796 0 796 Prior*Likelihood 3.66E‐21 1.43E‐20 Lik lih d Π f ( i ) Likelihood  (t 5.49E‐21 5 49E 21 1.43E‐18 1 43E 18 average {t }  average {t i } = 317 i TTF data t i  (sec) f 1(t i ) f 2(t i ) 1 133 2.65E‐03 1.53E‐03 2 888 1.39E‐06 3.39E‐04 3 619 2.05E‐05 5.80E‐04 4 8 9.23E‐03 9 23E 03 1.97E‐03 1 97E 03 5 97 3.78E‐03 1.65E‐03 6 157 2.08E‐03 1.46E‐0311/15/2012 ASQ RD Webinar 23
  26. 26. λ KOtG λ 1 λ 2 Failure Rate λ  (1/sec) 0.0022 0.0100 0.0020 MTTF (sec) 450 100 500 Prior g (λ ) 0.667 0.333 P t i g (λ | t i ) Posterior 0.204 0 204 0.796 0 796 Prior*Likelihood 3.66E‐21 1.43E‐20 Lik lih d Π f ( i ) Likelihood  (t 5.49E‐21 5 49E 21 1.43E‐18 1 43E 18 average {t }  average {t i } = 317 i TTF data t i  (sec) f 1(t i ) f 2(t i ) 1 133 2.65E‐03 1.53E‐03 2 888 1.39E‐06 3.39E‐04 3 619 2.05E‐05 5.80E‐04 4 8 9.23E‐03 9 23E 03 1.97E‐03 1 97E 03 5 97 3.78E‐03 1.65E‐03 6 157 2.08E‐03 1.46E‐0311/15/2012 ASQ RD Webinar 24
  27. 27. go to the spreadsheet go to the spreadsheet11/15/2012 ASQ RD Webinar 25
  28. 28. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors Conjugate Priors• Test Time Examples Test Time Examples• Q ti Question and Answer dA11/15/2012 ASQ RD Webinar 26
  29. 29. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example Conjugate Priors• Conjugate Priors• Test Time Examples Test Time Examples• Q ti Question and Answer dA11/15/2012 ASQ RD Webinar 27
  30. 30. Conjugate Prior C j t Pi11/15/2012 ASQ RD Webinar 28
  31. 31. Mean λave = a/b Variance   σ2 = a/b2 In hierarchical Bayesian models, these hyperparameters  will be represented as distributions with priors/posteriors, etc will be represented as distributions with priors/posteriors etc and have hyperparameters of their own11/15/2012 ASQ RD Webinar 29
  32. 32. Bayesian assumptions for the gamma  y p f g exponential system model1. Failure times for the system under investigation can be adequately  modeled by the exponential distribution with constant failure rate.2. The MTBF for the system can be regarded as chosen from a prior  distribution model that is an analytic representation of our  distribution model that is an analytic representation of our previous information or judgments about the systems reliability.  The form of this prior model is the gamma distribution (the  conjugate prior for the exponential model).  The prior model is actually defined for λ = 1/MTBF.3. Our prior knowledge is used to choose the gamma parameters 3 O i k l d i dt h th t a and b for the prior distribution model for λ. There are a number  of ways to convert prior knowledge to gamma parameters. y p g g p11/15/2012 ASQ RD Webinar 30
  33. 33. Gamma prior parameter method 1 G i t th d 11. If you have actual data from previous testing done on the  system (or a system believed to have the same reliability as the  system (or a system believed to have the same reliability as the one under investigation), this is the most credible prior  knowledge, and the easiest to use. Simply set the gamma  knowledge, and the easiest to use. Simply set the gamma parameter a equal to the total number of failures from all the  p e ous da a, a d se e pa a e e equa o e o a o a previous data, and set the parameter b equal to the total of all  the previous test hours.11/15/2012 ASQ RD Webinar 31
  34. 34. Gamma prior parameter method 2 Gamma prior parameter method 22. A consensus method for determining2 A consensus method for determining a and b that works well is the following: that works well is the following:  Assemble a group of engineers who know the system and its sub‐components well  from a reliability viewpoint. A. Have the group reach agreement on a reasonable MTBF they expect the system to have.  They could each pick a number they would be willing to bet even money that the system  would either meet or miss, and the average or median of these numbers would be their  50% best guess for the MTBF. Or they could just discuss even‐money MTBF candidates until  a consensus is reached. p p g , g g y p B. Repeat the process again, this time reaching agreement on a low MTBF they expect the  system to exceed. A "5%" value that they are "95% confident" the system will exceed (i.e.,  they would give 19 to 1 odds) is a good choice. Or a "10%" value might be chosen (i.e., they  g ) p would give 9 to 1 odds the actual MTBF exceeds the low MTBF). Use whichever percentile  choice the group prefers. C. Call the reasonable MTBF MTBF50 and the low MTBF you are 95% confident the system will  exceedMTBF05. These two numbers uniquely determine gamma parameters a and b that . These two numbers uniquely determine gamma parameters that  have percentile values at the right locationsCalled the 50/95 method (or the 50/90 method if one uses MTBF10 , etc.)11/15/2012 ASQ RD Webinar 32
  35. 35. Gamma prior parameter method 3 G i t th d 33. Weak Prior Obtain consensus is on a reasonable  expected MTBF, called MTBF Next however the expected MTBF called MTBF50. Next, however, the  group decides they want a weak prior that will change  rapidly, based on new test data. If the prior parameter  rapidly based on new test data If the prior parameter "a" is set to 1, the gamma has a standard deviation  equal to its mean, which makes it spread out, or  equal to its mean which makes it spread out or "weak". To set the 50th percentile we have to  choose b = ln 2 × MTBF50 = ln 2 ×Note: During planning of Bayesian tests, this weak prior is actually a very friendly prior in terms of saving test time.11/15/2012 ASQ RD Webinar 33
  36. 36. Comments C tMany variations are possible, based on the above three methods. For example, you might have prior three methods For example you might have priordata from sources that you dont completely trust. Or you might question whether the data really apply to  i ht ti h th th d t ll l tthe system under investigation. You might decide to "weight" the prior data by .5, to "weaken" it. This can be implemented by setting a = .5 x the number of  p y gfails in the prior data and b = .5 times the number of test hours. That spreads out the prior distribution test hours. That spreads out the prior distributionmore, and lets it react quicker to new test data.11/15/2012 ASQ RD Webinar 34
  37. 37. New data is collected …N d t i ll t dNew information is combined with the gamma prior model topproduce a gamma posterior distribution. g pAfter a new test is run with T additional system operating hours, and r new failures,The resultant posterior distribution for failure rate λ remainsgamma (conjugate remember?), with new parametersa = a + r b = b + Tb b + T11/15/2012 ASQ RD Webinar 35
  38. 38. Reliability estimation with Bayesian gamma prior model11/15/2012 ASQ RD Webinar 36
  39. 39. Example p• A group of engineers, discussing the reliability of a new piece of equipment decide to use the 50/95 method to equipment, convert their knowledge into a Bayesian gamma prior. Consensus is reached on a likely MTBF50 value of 600 hours and a low MTBF05 value of 250.• RT is 600/250 = 2.4. Using software to find the root of a / g univariate function, the gamma prior parameters were found to be a = 2.863 and b = 1522.46. The parameters will have ( h (approximately) a probability of 50% of b i b l i t l ) b bilit f f being below 1/600 = 0.001667 hours‐1 and a probability of 95% of being below 1/250 = 0 004 hours‐1. (The probabilities are based on 0.004 the 0.001667 and 0.004 quantiles of a gamma distribution with shape parameter a = 2.863 and scale parameter b = p p p 1522.46 hours)11/15/2012 ASQ RD Webinar 37
  40. 40. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example Conjugate Priors• Conjugate Priors• Test Time Examples Test Time Examples• Q ti Question and Answer dA11/15/2012 ASQ RD Webinar 38
  41. 41. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors Conjugate Priors• Test Time Examples Test Time Examples• Question and Answer Q ti dA11/15/2012 ASQ RD Webinar 39
  42. 42. Bayesian test plan y pGamma prior parameters a and b have already been determined. Assume we have a given MTBF objective, M, and a desired confidence level of 100×(1‐ α). We want to confirm the system will have an MTBF of at least at the 100×(1‐ α) confidence level. Pick a number of failures, r, that we can an MTBF of at least M at the 100×(1 α) confidence level Pick a number of failures r that we canallow on the test. We need a test time T such that we can observe up to r failures and still "pass" the test. If the test time is too long (or too short), we can iterate with a different choice of the test ends, the posterior gamma distribution will have (worst case ‐ assuming exactly r failures) new parameters of a  = a + r, b = b + T and passing the test means that the failure rate λ1‐ α, the upper 100×(1‐ α) percentile for the posterior gamma, has to equal the target failure rate 1/M. By definition, this is G ‐1(1‐ α; a, b), with G ‐1 denoting the inverse of the gamma CDF distribution .We can find the value of that satisfies 1(1‐ α;a b) = 1/M by trial and error. However, based on We can find the value of T that satisfies G ‐1(1 α;a, b ) = 1/M by trial and error However based onthe properties of the gamma distribution, it turns out that we can calculate T directly by using T = M×(G ‐1(1‐ α; a, 1)) ‐ b 11/15/2012 ASQ RD Webinar 40
  43. 43. Special Case: a = 1 (The "Weak" Prior) p ( )When the prior is a weak prior with a = 1, the Bayesian test is always shorter than the classical test. There is a very simple way to calculate the required Bayesian test time when the prior is a weak prior with a = 1. First calculate the classical/frequentist test time Call this Tc. The Bayesian test time T is test time. Call this The Bayesian test time is just Tc minus the prior parameter b (i.e.,T = Tc ‐ b). If the b parameter was set equal to (ln 2) × MTBF50(where MTBF50 is the consensus choice for an "even money" MTBF), then T = Tchoice for an "even money" MTBF) then T Tc ‐ (ln 2) × MTBF50This 2) × This shows that when a weak prior is used, the Bayesian test time is always less than the corresponding classical test time. That is why this prior is  p g y palso known as a friendly prior.This prior essentially sets the order of magnitude for the MTTF11/15/2012 ASQ RD Webinar 41
  44. 44. Calculating a Bayesian Test Time g yA new piece of equipment has to meet a MTBF requirement of 500 hours at 80 % confidence. A group of engineers decide to use their collective experience to determine a Bayesian gamma prior using the 50/95 method described determine a Bayesian gamma prior using the 50/95 method describedin Section 2. They determine that the gamma prior parameters are a = 2.863 and b = 1522.46 hrs.Now they want to determine an appropriate test time so that they can confirm N th tt d t i i t t t ti th t th fia MTBF of 500 with at least 80 % confidence, provided they have no more than two failures (r = 2).We obtain a test time of 1756.117 hours using 500×(G ‐1(1‐0.2; 2.863+2, 1)) ‐1522.46 To compare this result to the classical test time required, which is 2140 hours To compare this result to the classical test time required, which is 2140 hoursfor a non‐Bayesian test. The Bayesian test saves about 384 hours, or an 18 % savings. If the test is run for 1756 hours, with no more than two failures, then an MTBF of at least 500 hours has been confirmed at 80 % confidence.an MTBF of at least 500 hours has been confirmed at 80 % confidence.If, instead, the engineers had decided to use a weak prior with an MTBF50 of 600, the required test time would have been 2140 ‐ 600 × ln 2 = 1724 hours11/15/2012 ASQ RD Webinar 42
  45. 45. Post‐Test Analysis Example P t T tA l i E l• A system has completed a reliability test aimed at  confirming a 600 hour MTBF at an 80% confidence  confirming a 600 hour MTBF at an 80% confidence level. Before the test, a gamma prior with a = 2, b =  1400 was agreed upon, based on testing at the  1400 was agreed upon, based on testing at the vendors location. Bayesian test planning  calculations, allowing up to 2 new failures, called  calculations, allowing up to 2 new failures, called for a test of 1909 hours. • When that test was run there actually were exactly When that test was run, there actually were exactly  two failures. What can be said about the reliability?  The posterior gamma CDF has parameters a = 4 The posterior gamma CDF has parameters a = 4  and b = 3309. 11/15/2012 ASQ RD Webinar 43
  46. 46. Bayesian solutions for arbitrary F(t) B i l ti f bit F(t)What about WeibullWh t b t W ib ll or other non‐exponential variable failure rate TTF  th ti l i bl f il t TTFdistributions? 3.50E‐02 3.00E‐02 2.50E‐02 β = 0.6 2.00E‐02 2 00E 02 β=08 0.8 g(λ, β|data) β = 1.0 1.50E‐02 β = 1.2 1.00E‐02 β = 1.4 5.00E‐03 β = 1.6 0.00E+00 0.0000 0.0020 0.0040 0.0060 0.0080 λ (1/sec)Conjugate priors only exist for Weibull when a subset of hyperparameters are Conjugate priors only exist for Weibull when a subset of hyperparameters areknown.  MCMC and Gibbs methods exist for sampling from higher dimensional posteriors  CDFs in multiple dimensions not as straightforward.  11/15/2012 ASQ RD Webinar 44
  47. 47. References and Further Reading R f d F th R di• NIST/SEMATECH e‐Handbook of Statistical Methods,  http://www.itl.nist.gov/div898/handbook/, April (2012)• Statistical Methods for Reliability Data, WQ Meeker and LA  Escobar (1998)• Applied Reliability, 2nd edition, PA Tobias and DC Trindade (1995)• Bayesian Reliability Analysis, HF Martz and RA Waller (1982)• Methods for Statistical Analysis of Reliability and Life Data,  NR Mann, RE Schafer, and ND Singpurwalla (1974) NR M RE S h f d ND Si ll• Bayes is for the birds, RA Evans, IEEE Transactions on  Reliability R‐38, 401 (1989). R li bilit R 38 401 (1989)11/15/2012 ASQ RD Webinar 45
  48. 48. 11/15/2012 ASQ RD Webinar 46
  49. 49. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors Conjugate Priors• Test Time Examples Test Time Examples• Question and Answer Q ti dA11/15/2012 ASQ RD Webinar 47
  50. 50. Agenda A d• Bayesian vs. Frequentist Comparison• Preliminary Example• Conjugate Priors Conjugate Priors• Test Time Examples Test Time Examples• Q ti dA Question and Answer11/15/2012 ASQ RD Webinar 48
  51. 51. Q&A11/15/2012 ASQ RD Webinar 49

×