Bayesian Methods in Reliability Engineering ASQ Reliability Division Webinar Program Nov 15 th 2012 Charles H. Recchia, MBA, PhD Quality Support Group, Inc http://www.qualitysupportgroup.com/
B AYESIAN M ETHODS IN R ELIABILITY E NGINEERING With product reliability demonstration test planning and execution interacting heavily with cost, availability and schedule considerations, 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 Introduction to Bayesian vs Frequentist statistical approaches Bayesian formalism for reliability estimation Product/component case studies and examples Charles 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 expertise 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 Committee for the Boston Chapter of the IEEE Reliability Society. 11/15/2012 ASQ RD Webinar 2
Agenda • Bayesian vs. Frequentist Comparison • Preliminary Example • Conjugate Priors • Test Time Examples • Question and Answer 11/15/2012 ASQ RD Webinar 3
11/15/2012 ASQ RD Webinar 4
Agenda • Bayesian vs. Frequentist Comparison • Preliminary Example • Conjugate Priors • Test Time Examples • Question and Answer 11/15/2012 ASQ RD Webinar 5
Agenda • Bayesian vs. Frequentist Comparison • Preliminary Example • Conjugate Priors • Test Time Examples • Question and Answer 11/15/2012 ASQ RD Webinar 6
When reliability follows the exponential TTF model (eg 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 – 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 model Bayesian 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 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 judgment 11/15/2012 ASQ RD Webinar 7
exponential distribution “non - intuitive” Brains wired with Planet = Earth Normal Distribution s ~ 0.1 m 6 draw sample. Population mean height 5’11” Sample mean = ______ 8 5’9” 6’0” 5’9” 5’9” 6’3” 5’10”
exponential distribution Planet = Laitnenopxe 6 draw sample. Population mean height 5’11” Sample mean = ______ 11/15/2012 ASQ RD Webinar 9 4’11” 2’5” 5’9” 4 ” 13’8” 7’11”
Confidence vs. Credibility Intervals 11/15/2012 ASQ RD Webinar 10
For and Against use of Bayesian Methodology PROs CONs Uses prior information - Prior information may not be this "makes sense“ accurate - generating misleading Less new testing may be conclusions Way of inputting prior needed to confirm a desired MTBF at a given confidence information (choice of prior) may Confidence intervals are not be correct really intervals for the Customers may not accept validity (random) MTBF - sometimes of prior data or engineering called "credibility intervals“ judgements Risk of perception that results aren't objective and don't stand by themselves 11/15/2012 ASQ RD Webinar 11
Agenda • Bayesian vs. Frequentist Comparison • Preliminary Example • Conjugate Priors • Test Time Examples • Question and Answer 11/15/2012 ASQ RD Webinar 12
Agenda • Bayesian vs. Frequentist Comparison • Preliminary Example • Conjugate Priors • Test Time Examples • Question and Answer 11/15/2012 ASQ RD Webinar 13
Toy Hyperbolic Example First 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.” 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. What shall you do? They measure product MTBF in seconds ? 11/15/2012 ASQ RD Webinar 14
Bayesian Core Idea What you knew before WYKB. New Data “Prior” Best possible update of WYKB adjusted by the New Data. “Posterior” 11/15/2012 ASQ RD Webinar 15
l KOtG l 1 l 2 Failure Rate l (1/sec) 0.0100 0.0020 0.0022 450 100 500 MTTF (sec) Prior g ( l ) 0.667 0.333 Posterior g ( l | t i ) 0.204 0.796 Prior*Likelihood 3.66E-21 1.43E-20 Likelihood P f ( t i ) 5.49E-21 1.43E-18 average { t i } = 317 TTF data t i (sec) f 1 ( t i ) f 2 ( t i ) 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 1.97E-03 5 97 3.78E-03 1.65E-03 6 157 2.08E-03 1.46E-03 11/15/2012 ASQ RD Webinar 16
l KOtG l 1 l 2 Failure Rate l (1/sec) 0.0100 0.0020 0.0022 450 100 500 MTTF (sec) Prior g ( l ) 0.667 0.333 Posterior g ( l | t i ) 0.204 0.796 Prior*Likelihood 3.66E-21 1.43E-20 There is just enough time Likelihood P f ( t i ) 5.49E-21 1.43E-18 before end of day to collect 6 average { t i } = 317 TTF data t i (sec) f 1 ( t i ) f 2 ( t i ) i time-to-fail (TTF) data points. 1 133 2.65E-03 1.53E-03 2 888 1.39E-06 3.39E-04 3 619 2.05E-05 5.80E-04 Let’s do that. 4 8 9.23E-03 1.97E-03 5 97 3.78E-03 1.65E-03 6 157 2.08E-03 1.46E-03 11/15/2012 ASQ RD Webinar 17
l KOtG l 1 l 2 Failure Rate l (1/sec) 0.0100 0.0020 0.0022 450 100 500 MTTF (sec) Prior g ( l ) 0.667 0.333 Posterior g ( l | t i ) 0.204 0.796 Prior*Likelihood 3.66E-21 1.43E-20 Likelihood P f ( t i ) 5.49E-21 1.43E-18 average { t i } = 317 TTF data t i (sec) f 1 ( t i ) f 2 ( t i ) 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 1.97E-03 5 97 3.78E-03 1.65E-03 6 157 2.08E-03 1.46E-03 11/15/2012 ASQ RD Webinar 18
l KOtG l 1 l 2 Failure Rate l (1/sec) 0.0100 0.0020 0.0022 450 100 500 MTTF (sec) Prior g ( l ) 0.667 0.333 Posterior g ( l | t i ) 0.204 0.796 How likely is this data? Prior*Likelihood 3.66E-21 1.43E-20 Depends on what l is! Likelihood P f ( t i ) 5.49E-21 1.43E-18 average { t i } = 317 TTF data t i (sec) f 1 ( t i ) f 2 ( t i ) 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 1.97E-03 5 97 3.78E-03 1.65E-03 6 157 2.08E-03 1.46E-03 11/15/2012 ASQ RD Webinar 19
l KOtG l 1 l 2 Failure Rate l (1/sec) 0.0100 0.0020 0.0022 450 100 500 MTTF (sec) Prior g ( l ) 0.667 0.333 Posterior g ( l | t i ) 0.204 0.796 Prior*Likelihood 3.66E-21 1.43E-20 Likelihood P f ( t i ) 5.49E-21 1.43E-18 average { t i } = 317 TTF data t i (sec) f 1 ( t i ) f 2 ( t i ) 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 1.97E-03 5 97 3.78E-03 1.65E-03 6 157 2.08E-03 1.46E-03 11/15/2012 ASQ RD Webinar 20
l KOtG l 1 l 2 Failure Rate l (1/sec) 0.0100 0.0020 0.0022 450 100 500 MTTF (sec) Prior g ( l ) 0.667 0.333 Posterior g ( l | t i ) 0.204 0.796 Prior*Likelihood 3.66E-21 1.43E-20 Likelihood P f ( t i ) 5.49E-21 1.43E-18 average { t i } = 317 TTF data t i (sec) f 1 ( t i ) f 2 ( t i ) 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 1.97E-03 5 97 3.78E-03 1.65E-03 6 157 2.08E-03 1.46E-03 11/15/2012 ASQ RD Webinar 21
l KOtG l 1 l 2 Failure Rate l (1/sec) 0.0100 0.0020 0.0022 450 100 500 MTTF (sec) Prior g ( l ) 0.667 0.333 Posterior g ( l | t i ) 0.204 0.796 Prior*Likelihood 3.66E-21 1.43E-20 Likelihood P f ( t i ) 5.49E-21 1.43E-18 average { t i } = 317 TTF data t i (sec) f 1 ( t i ) f 2 ( t i ) 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 1.97E-03 5 97 3.78E-03 1.65E-03 6 157 2.08E-03 1.46E-03 11/15/2012 ASQ RD Webinar 22
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