Checking the Reliability of Reliability Models.

Transcription

1 Checking the Reliability of Reliability Models. Seminario de Estadística, CIMAT Abril 3, 007. Víctor Aguirre Torres Departmento de Estadística, ITAM Área de Probabilidad y Estadística, CIMAT

2 Credits. Partially Sponsored by Asociación Mexicana de la Cultura A. C. Joint work with Humberto Gutiérrez, University of Guadalajara Andrés Christen, CIMAT

3 Are these data normal? 3

4 No, they are i.i.d. exponential data. 4

5 Assessing specification in reliability Probability plots with confidence bands typically used. Useful in detecting patterns in the data. Large discrepancies from the model. But they are subjective and may have little power to detect misspecification. Small samples and censoring. Goodness-of-fit tests or non-nested hypotheses not useful. 5

6 Bayesian Model Selection. Pr M j X J i 1 P X M j Pr M j P X M i Pr M i j 1,,...,J P X M j Θj f X j,m j j M j d j Integrated likelihood. Likelihood, censoring, proper prior, hyper parameters. aguirre@itam.mx 6

7 Approach Consider usual models in reliability: N, LN, EV, W, Exp J=5. Several models at a time. Develop an elicitation procedure which is easy to communicate and use in applications. X=Time to failure Requires from the user only prior info on: E X L m,u m Var X L s, U s aguirre@itam.mx 7

8 f Normal Model N. 1/ τ τ x μ, τ = exp x μ π EX μ, τ = μ Var X μ, τ = 1 L m < μ < U m L s < < U s τ 1 τ aguirre@itam.mx 8

9 Normal-gama prior N. π μ, τ m, k, α, β = N μ m, kτ Γ τ α, β aguirre@itam.mx 9

10 10 Normal-gama prior N.,,,,,, β α τ τ μ β α τ μ π Γ = k m N k m m L m U m m E E E E + = = = = τ μ μ

11 11 Normal-gama prior N.,,,,,, β α τ τ μ β α τ μ π Γ = k m N k m m L m U m m E E E E + = = = = τ μ μ = = + = + = α β τ τ τ μ τ μ μ k E k m Var k E E Var Var E Var

12 Normal-gama prior N. β U = = m Lm Var μ k α 1 6 k = 36β U m L m α 1 aguirre@itam.mx 1

13 Normal-gama prior N. 1 E τ β = α 1 Var 1 τ β α 1 α = β α 1 = L + s U s β α 1 α = U s L s 6 and so on aguirre@itam.mx 13

14 Prior Distributions Peaked. Normal-gamma 1, j, j,m j,k j j j Γ j k j 1/ j 1/ exp k j 1 m j j Flat. Uniform. 1, a j,b j,d j, e j 1 a j,b j d j,e j 1 b j a j 1 e j d j aguirre@itam.mx 14

15 Hyper parameters. For each model transform occurrence rectangle into hyper parameter values. h 1 1, E X 1, h 1, V X 1,, D L m,u m L s, U s 1, and, solve h 1 1,,, h 1,,, h1, h : Θ θ 1., θ. : D D Θ aguirre@itam.mx 15

16 Hyper parameters. Let 0 L m U m and 0 L s U s E l l 0, 0 ; l 1, θ 1., θ. : D Θ V l max l, min l,, D, D 6 ; l 1, For each θ transform the mean and variance into hyperparameters of the prior. aguirre@itam.mx 16

17 Hyperparameters, Log-normal model. Normal-gama prior. m log 0 k 0 1/ log Um Us Lm 1/ L m L s Um 1/ log 9log U log s Lm 0 Lm log aguirre@itam.mx 17

18 Implementation Prior predictive densities are used to assess the method. f x M j,h j f x M j, j j M j,h j d j Simulate first θ j from prior, then x from model with parameter equal to θ j. aguirre@itam.mx 18

19 Implementation Computation of Integrated likelihood by means of Monte Carlo. j i : i 1,, K from j M j P X M j 1 K K i 1 f X j i, M j Laplace s approximation did not worked good. aguirre@itam.mx 19

20 Shock Absorber Experiment. O Connor, Practical Reliability Engineering, nd ed. Wiley. 38 shock absorbers, 7 random right censored observations in kilometers. aguirre@itam.mx 0

21 Shock Absorber Experiment. Probability Probability a Normal b Lognormal Probability ? Probability c Extreme value d Weibull aguirre@itam.mx 1

22 Shock Absorber Experiment. D=[0,000; 35,000] [5,000; 15,000] Prior predictive densities.

23 Shock Absorber Experiment. Posterior probabilities. PM X Prior Distribution Model NG U N LN EV W Exp 0 0 Exp, LN and EV are in serious doubt. aguirre@itam.mx 3

24 Airplane Air Conditioner Experiment. Proschan, Technometrics, 5, uncensored observations. 4

25 Airplane Air Conditioner Experiment.? Probability a Normal Probability b Lognormal? Probability c Extreme value Probability d Weibull w e Exponential aguirre@itam.mx 5

26 Airplane Air Conditioner Experiment. D=[50,70] [50,90]. Prior predictive densities. 6

27 Airplane Air Conditioner Experiment. Posterior Probabilities. PM X Prior Distribution Model NG U N 0 0 LN EV 0 0 W Exp N, LN and EV are in serious doubt. aguirre@itam.mx 7

28 Exponential Simulated Data. 15 uncensored observations. D = [0.5,.0] [0.5,.0]. PM X Uniform prior Model Sample 1 Sample Sample 3 Sample 4 N LN EV W Exp aguirre@itam.mx 8

29 Final Remarks For reliability experiments probability plots have strong limitations. It is useful to use a Bayesian model selection for reliability data. Elicitation depends on observable characteristics of the time to failure. Procedure available in S-plus. aguirre@itam.mx 9