Parametric and Topological Inference for Masked System Lifetime Data

Transcription

1 Parametric and for Masked System Lifetime Data Rang Louis J M Aslett and Simon P Wilson Trinity College Dublin 9 th July 2013

2 Structural Reliability Theory Interest lies in the reliability of systems composed of numerous components

3 Structural Reliability Theory Interest lies in the reliability of systems composed of numerous components Lifetime of the system, T, is determined by: the lifetime of the components,y i F Y ( ; ψ i ) the structure of the system the possible presence of a repair process via either the structure function or signature

4 Structural Reliability Theory Interest lies in the reliability of systems composed of numerous components Lifetime of the system, T, is determined by: the lifetime of the components,y i F Y ( ; ψ i ) the structure of the system the possible presence of a repair process via either the structure function or signature Probabilistic Analysis Statistical Inference

5 Structural Reliability Theory Interest lies in the reliability of systems composed of numerous components Lifetime of the system, T, is determined by: the lifetime of the components,y i F Y ( ; ψ i ) the structure of the system the possible presence of a repair process via either the structure function or signature Probabilistic Analysis Statistical Inference

6 Structure Functions & Signatures The structure function (Birnbaum et al, 1961) is a mapping φ( ) : {0, 1} n {0, 1} which determines operation of the system given the state of the n components

7 Structure Functions & Signatures The structure function (Birnbaum et al, 1961) is a mapping φ( ) : {0, 1} n {0, 1} which determines operation of the system given the state of the n components The signature (Samaniego, 1985) is less widely used, but in some ways more elegant Definition (Signature) The signature of a system is the n-dimensional probability vector s = (s 1,, s n ) with elements: s i = P(T = Y i:n ) where T is the failure time of the system and Y i:n is the ith order statistic of the n component failure times

8 Structure Functions & Signatures Definition (Signature) The signature of a system is the n-dimensional probability vector s = (s 1,, s n ) with elements: s i = P(T = Y i:n ) where T is the failure time of the system and Y i:n is the ith order statistic of the n component failure times eg = s = ( 0, 2 3, 1 3) and φ(x 1, X 2, X 3 ) = 1 (1 X 1 )(1 X 2 X 3 )

9 Masked System Lifetime Data Traditionally, one may have failure time data on components and then infer the parameters ψ of the lifetime distribution y 1 =24 y 2 > 31 y 3 =31

10 Masked System Lifetime Data Traditionally, one may have failure time data on components and then infer the parameters ψ of the lifetime distribution y 1 =24 y 2 > 31 y 3 =31 Straight-forward Bayesian inference: { } f Ψ Y (ψ y) f Y (y 1 ; ψ) f Y (y 3 ; ψ) (1 F Y (y 2 ; ψ)) f Ψ (ψ)

11 Masked System Lifetime Data Traditionally, one may have failure time data on components and then infer the parameters ψ of the lifetime distribution y 1 =24 y 2 > 31 y 3 =31 t = y 3 = y 2:3 =31

12 Masked System Lifetime Data Traditionally, one may have failure time data on components and then infer the parameters ψ of the lifetime distribution y 1 =24 y 2 > 31 y 3 =31 t = y 3? = y 2:3?:3 =31 Masked system lifetime data means only the failure time of the system as a whole is known, not the component failure times or indeed which components had failed

13 The literature on inference for masked system lifetime data is extensive, but: heavily focused on specific structures (eg series/competing risk systems, see Reiser et al (1995) or Kuo and Yang (2000)) or focused on specific lifetime distributions (eg Exponential, see Gåsemyr and Natvig (2001)) or does not focus on inferring the parameters of the model (eg infer hazard, see Ng et al (2012)) Why?

14 The literature on inference for masked system lifetime data is extensive, but: heavily focused on specific structures (eg series/competing risk systems, see Reiser et al (1995) or Kuo and Yang (2000)) or focused on specific lifetime distributions (eg Exponential, see Gåsemyr and Natvig (2001)) or does not focus on inferring the parameters of the model (eg infer hazard, see Ng et al (2012)) Why? Likelihood can be complex: m L(ψ; y) = t F T(t; ψ) = i=1 m i=1 t=ti [ ] 1 {1 F Y2 (t)} {1 F Y3 (t)} F Y1 (t) t t=ti

15 Masked System Lifetime Data (Repair) Traditionally, one may have full schedule of failure and repair time data on components and then infer the parameters ψ of the lifetime and repair time distributions {t} t System Failed

16 Masked System Lifetime Data (Repair) Traditionally, one may have full schedule of failure and repair time data on components and then infer the parameters ψ of the lifetime and repair time distributions {t}? t System Failed See Aslett (2012a)

17 Missing Data Clearly the missing data is what makes the inference hard Tanner and Wong (1987) is a classic solution to this in a Bayesian framework assuming the missing data can be simulated Iteratively simulate: f Y,T (y 1,,y m, t) f Y,T( y 1,,y m, t) Then, in the usual way the marginal samples from the Gibbs step are the required estimates: f Ψ T (ψ t) = f Ψ,Y T (ψ, y t) dy R +

18 Missing Data Clearly the missing data is what makes the inference hard Tanner and Wong (1987) is a classic solution to this in a Bayesian framework assuming the missing data can be simulated Iteratively simulate: f Y,T (y 1,,y m, t)? f Y,T( y 1,,y m, t) Then, in the usual way the marginal samples from the Gibbs step are the required estimates: f Ψ T (ψ t) = f Ψ,Y T (ψ, y t) dy R +

19 Simulating the Missing Data Consider the system from the introduction, with observed system failure times: t = {11, 42} Need realisations concordant with each observation: = 1

20 Simulating the Missing Data Consider the system from the introduction, with observed system failure times: t = {11, 42} Need realisations concordant with each observation: = 1

21 Simulating the Missing Data Consider the system from the introduction, with observed system failure times: t = {11, 42} Need realisations concordant with each observation: t 1 =11 09 =

22 Simulating the Missing Data Consider the system from the introduction, with observed system failure times: t = {11, 42} Need realisations concordant with each observation: y = {09, 27, 11} t 1 =11 09 =

23 Simulating the Missing Data Consider the system from the introduction, with observed system failure times: t = {11, 42} Need realisations concordant with each observation: y = {09, 27, 11} = 1

24 Simulating the Missing Data Consider the system from the introduction, with observed system failure times: t = {11, 42} Need realisations concordant with each observation: y = {09, 27, 11} = 1 t 2 =

25 Simulating the Missing Data Consider the system from the introduction, with observed system failure times: t = {11, 42} Need realisations concordant with each observation: y = {09, 27, 11, 32, 42, 13} = 1 t 2 =

26 Simulating the Missing Data Consider the system from the introduction, with observed system failure times: t = {11, 42} Need realisations concordant with each observation: y = {09, 27, 11, 32, 42, 13} = 1

27 Simulating the Missing Data Consider the system from the introduction, with observed system failure times: t = {11, 42} Need realisations concordant with each observation: y = {09, 27, 11, 32, 42, 13} = 2

28 Missing Data What is the challenge? f Y,T (y 1,,y m, t)? f Y,T( y 1,,y m, t) P( 1,t 1 )=? = 1 P( 1,t 1 )=? P( 1,t 1 )=?

29 Sampling Latent Failure Times f Y T (y i1,, y in ; ψ t) n [ f Y Y<t (y i(1),, y i(j 1) ; ψ) j=1 f Y Y>t (y i(j+1),, y i(n) ; ψ) I {t} (y i(j) ) ( ) ] n 1 F Y (t; ψ) j F Y (t; ψ) n j+1 s j j 1

30 Signature based data augmentation 1 With probability P(j) ( ) n 1 F Y (t i ; ψ) j F Y (t i ; ψ) n j+1 s j j 1 it was the jth failure that caused system failure 2 Having drawn a random j, sample j 1 values, y i1,, y i(j 1), from F Y Y<ti ( ; ψ), the distribution of the component lifetime conditional on failure before t i n j values, y i(j+1),, y in, from F Y Y>ti ( ; ψ), the distribution of the component lifetime conditional on failure after t i and set y ij = t i

31 Prerequisites This is a very general method The prerequisites for use are, 1 The signature of the system; 2 The ability to perform standard Bayesian inference with the full data; 3 The ability to sample from F Y Y<ti ( ; ψ) and F Y Y>ti ( ; ψ)

32 Prerequisites This is a very general method The prerequisites for use are, 1 The signature of the system; Easy for systems that are not huge ReliabilityTheory R package (Aslett, 2012b) 2 The ability to perform standard Bayesian inference with the full data; Easy for common lifetime distributions 3 The ability to sample from F Y Y<ti ( ; ψ) and F Y Y>ti ( ; ψ) Depends!

33 Canonical Exponential Component Lifetime Example 5 Parameter Value 4 3 2

34 Unknown Topologies A little blue skies thinking y 2 > 31 y 1 =24? y 3 =31 t = y 3? = y 2:3?:3 =31

35 Uniqueness of the Signature Signature repetition Type Order Unique Total Coherent systems Coherent systems /w graph

36 Signature & Topology Order 4 coherent systems with graph representation System Topology Signature System Topology Signature ( (1, 0, 0, 0) 0, 1 3, 2 3, 0) ( 1 2, 1 2, 0, 0) ( 0, 1 2, 1 4, 1 4) ( 1 4, 7 12, 1 6, 0) ( 0, 1 6, 7 12, 1 ) 4 ( 1 4, 1 4, 1 2, 0) ( 0, 0, 1 2, 1 2) ( 0, 2 3, 1 3, 0) (0, 0, 0, 1) ( 0, 1 2, 1 2, 0)

37 Jointly Inferring the Topology f Y,T (y 1,,y m, t) f Y,T( y 1,,y m, t)

38 Jointly Inferring the Topology f Y,T (y 1,,y m, t, t) s) f Y,T( y 1,,y m, t) t, s)

39 Jointly Inferring the Topology f Y,T (y 1,,y m, t, t) s) f Y,T( y 1,,y m, t) t, s) Let M be a collection of signatures, then naïvely we might presume random scan Gibbs between: f Y M,Ψ,T (y 1,, y m M j, ψ, t) f Ψ M,Y,T (ψ M j, y 1,, y m, t) f M Ψ,Y,T (M j ψ, y 1,, y m, t) explores the posterior of: f M,Ψ,Y T (M j, ψ, y t) But, positivity & Harris ergodicity concerns

40 Toy Example of Problem Assume t comprises 100 masked system lifetimes and M is all order 3 coherent systems with graph representation { M = (1,0,0), ( 1 3, 2 3,0), (0, 2 3, 1 3 ), (0,0,1) }

41 Toy Example of Problem Assume t comprises 100 masked system lifetimes and M is all order 3 coherent systems with graph representation { M = (1,0,0), ( 1 3, 2 3,0), (0, 2 3, 1 3 ), (0,0,1) Iteration 1 Let starting topology be M 1 = = s = (1, 0, 0) Let ψ have sensible starting value }

42 Toy Example of Problem Assume t comprises 100 masked system lifetimes and M is all order 3 coherent systems with graph representation { M = (1,0,0), ( 1 3, 2 3,0), (0, 2 3, 1 3 ), (0,0,1) Iteration 1 Let starting topology be M 1 = = s = (1, 0, 0) Let ψ have sensible starting value } Then f Y M,Ψ,T (y 1,, y 100 simulations st t i = y i(1:3) i, ψ, t) will produce

43 Toy Example of Problem Assume t comprises 100 masked system lifetimes and M is all order 3 coherent systems with graph representation { M = (1,0,0), ( 1 3, 2 3,0), (0, 2 3, 1 3 ), (0,0,1) Iteration 1 Let starting topology be M 1 = = s = (1, 0, 0) Let ψ have sensible starting value } Then f Y M,Ψ,T (y 1,, y 100 simulations st t i = y i(1:3) i, ψ, t) will produce Thus, a move to M 2 is harder Moreover, moves to M 3 or M 4 are impossible

44 Toy Example of Problem Assume t comprises 100 masked system lifetimes and M is all order 3 coherent systems with graph representation { M = (1,0,0), ( 1 3, 2 3,0), (0, 2 3, 1 3 ), (0,0,1) Iteration 1 Let starting topology be M 1 = = s = (1, 0, 0) Let ψ have sensible starting value } Then f Y M,Ψ,T (y 1,, y 100 simulations st t i = y i(1:3) i, ψ, t) will produce Thus, a move to M 2 is harder Moreover, moves to M 3 or M 4 are impossible Assume a move to M 2 = is made though

45 Toy Example of Problem Assume t comprises 100 masked system lifetimes and M is all order 3 coherent systems with graph representation { M = (1,0,0), ( 1 3, 2 3,0), (0, 2 3, 1 3 ), Iteration 2 Topology is M 2 = = s = ( 1 3, 2 3, 0) (0,0,1) }

46 Toy Example of Problem Assume t comprises 100 masked system lifetimes and M is all order 3 coherent systems with graph representation { M = (1,0,0), ( 1 3, 2 3,0), (0, 2 3, 1 3 ), Iteration 2 Topology is M 2 = = s = ( 1 3, 2 3, 0) (0,0,1) Now f Y M,Ψ,T (y 1,, y 100, ψ, t) will produce simulations where t i is either y i(1:3) or y i(2:3) However, very low probability that t i = y i(1:3) i or t i = y i(2:3) i }

47 Toy Example of Problem Assume t comprises 100 masked system lifetimes and M is all order 3 coherent systems with graph representation { M = (1,0,0), ( 1 3, 2 3,0), (0, 2 3, 1 3 ), Iteration 2 Topology is M 2 = = s = ( 1 3, 2 3, 0) (0,0,1) Now f Y M,Ψ,T (y 1,, y 100, ψ, t) will produce simulations where t i is either y i(1:3) or y i(2:3) However, very low probability that t i = y i(1:3) i or t i = y i(2:3) i = f M Ψ,Y,T ( ψ, y 1,, y 100, t) = 1 is likely }

48 Toy Example of Problem Assume t comprises 100 masked system lifetimes and M is all order 3 coherent systems with graph representation { M = (1,0,0), ( 1 3, 2 3,0), (0, 2 3, 1 3 ), Iteration 2 Topology is M 2 = = s = ( 1 3, 2 3, 0) (0,0,1) Now f Y M,Ψ,T (y 1,, y 100, ψ, t) will produce simulations where t i is either y i(1:3) or y i(2:3) However, very low probability that t i = y i(1:3) i or t i = y i(2:3) i = f M Ψ,Y,T ( ψ, y 1,, y 100, t) = 1 is likely Indeed: ({ } f M Ψ,Y,T, ψ, y1,, y 100, t) > 0 t i = y i(1:3) i or t i = y i(2:3) i }

49 The problem can be avoided by using the following full conditionals instead: f M,Y Ψ,T (M j, y 1,, y m ψ, t) f Ψ M,Y,T (ψ M j, y 1,, y m, t) since the block marginals are concordant with positivity and ensure Harris ergodicity Sampling the former sequentially: f M Ψ,T (M j ψ, t) f Y M,Ψ,T (y 1,, y m M j, ψ, t) The latter is unchanged from before For the former: { m } f M Ψ,T (M j ψ, t) f T Ψ,M (t i ψ, M j ) f M (M j ) i=1

50 Canonical Exponential Component Lifetime Example Parameter Value Parameter Probability Topology

51 Phase-type Component Lifetime Example Parameter Value Parameter Value Failure Rate Repair Rate Probability Topology

52 Exchangeable Systems The iid systems assumption easily relaxed to exchangeability Density λ Plot Ground truth Prior predictive Posterior predictive Probability

53 Phase-type Component Lifetimes Extreme generality of the solution allows wide variety of component lifetime distributions Solutions to the prerequisites have been derived for Phase-type distributed components 06 Failure Rate Repair Rate Density Plot Ground truth Prior predictive Posterior predictive λ May interpret as: Repairable redundant subsystems; Theoretically dense in function space of all positively supported continuous distributions

54 Work A couple of the many important avenues to be pursued: Many partial information scenarios between full information and the extreme presented here There can be many lifetime forms, but with the restrictive assumption of the same components survival signature (Coolen and Coolen-Maturi, 2012) an avenue to pursue (see 14:30 tomorrow)

55 I Aslett, L J M (2012a), MCMC for Inference on Phase-type and Masked System Lifetime Models, PhD thesis, Trinity College Dublin Aslett, L J M (2012b), ReliabilityTheory: Tools for structural reliability analysis R package version 010 Birnbaum, Z W, Esary, J D and Saunders, S C (1961), Multi-component systems and structures and their reliability, Technometrics 3(1), Coolen, F P and Coolen-Maturi, T (2012), Generalizing the signature to systems with multiple types of components, in Complex Systems and Dependability, Springer, pp Gåsemyr, J and Natvig, B (2001), Bayesian inference based on partial monitoring of components with applications to preventative system maintenance, Naval Research Logistics 48(7), Kuo, L and Yang, T Y (2000), Bayesian reliability modeling for masked system lifetime data, Statistics & Probability Letters 47(3), Ng, H K T, Navarro, J and Balakrishnan, N (2012), Parametric inference from system lifetime data under a proportional hazard rate model, Metrika 75(3), Reiser, B, Guttman, I, Lin, D K J, Guess, F M and Usher, J S (1995), Bayesian inference for masked system lifetime data, Journal of the Royal Statistical Society, Series C 44(1), Samaniego, F J (1985), On closure of the IFR class under formation of coherent systems, IEEE Transactions on Reliability R-34(1), Tanner, M A and Wong, W H (1987), The calculation of posterior distributions by data augmentation, Journal of the American Statistical Association 82(398),