Reliability Modelling Incorporating Load Share and Frailty Vincent - PowerPoint PPT Presentation

Reliability Modelling Incorporating Load Share and Frailty Vincent Raja Anthonisamy Department of Mathematics, Physics and Statistics Faculty of Natural Sciences, University of Guyana Georgetown, Guyana, South America. jointly with G. Asha Nalini Ravishanker Department of Statistics Department of Statistics CUSAT, Cochin, Kerala, India. University of Connecticut, USA. A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Organization of the talk Introduction to Load Sharing Models and its 1 Generalizations. Frailty Models 2 Generalized Bivariate Load Sharing Model with Frailty and 3 Covariates. Properties. 4 Model with Positive Stable Frailty and Weibull Baseline 5 Hazard Parameter Estimation. 6 Data Analysis. 7 Conclusion. 8 A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Load Sharing System In a load sharing system, the probability of failure of any compo- nent will depend on the working status of the other component. See Daniels (1945, Proc. Royal Society of London A) and Rosen (1964). AIAA journal. A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Load Sharing System A Load share rule dictates how the load is distributed to the surviving components. Not all load sharing rules are monotone (Kim and Kvam (2004), Drummond et. al (2000)). A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Load Sharing Systems - Applications Paired Organs in biological sciences (Gross et al. (1971), Lemke et al. (2004)). Power transmission (Gosselin et al. (1995)). Computer Networking and Software Reliability (Epema et al.(1996)). Failures and Acquisitions of financial institutions and banks (Wheelock et al. (2000)). Collapsing of bridges (Komatsu and Sakimoto (1977)). An excellent review on Load sharing systems is provided by Dewan and Nimbalkar (2010). A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Two component load sharing system Two component parallel system. Freund’s (1961) bivariate exponential distribution is an effective model for load sharing systems. A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Two component load sharing system In a two component load sharing system, Let T 1 and T 2 be non-negative random variables representing the lifetimes of A and B . the lifetime random variable of the surviving component changes to T ∗ i if T j < T i , i = 1 , 2 , i � = j . Hence we observe ( T ∗ 1 , T 2 ) if component B fails first or ( T 1 , T ∗ 2 ) if component A fails first. A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

To set notation, if we denote the lifetimes of the components A and B as non-negative random variable ( Y 1 , Y 2 ) , then one observes Y 1 = T ∗ 1 , Y 2 = T 2 , if Y 1 > Y 2 , Y 1 = T 1 , Y 2 = T ∗ 2 , if Y 1 < Y 2 . A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

If T 1 and T 2 are independent, exp ( θ i ) , i = 1 , 2 respectively. T ∗ 1 and T ∗ 2 are exp ( θ ′ i ) , i = 1 , 2 respectively. Then the joint probability density function of ( Y 1 , Y 2 ) is (Freund (1961)). � θ ′ 1 θ 2 e − θ ′ 1 y 1 e − ( θ 1 + θ 2 − θ ′ 1 ) y 2 , y 1 > y 2 f ( y 1 , y 2 ) = (1) θ 1 θ ′ 2 e − ( θ 1 + θ 2 − θ ′ 2 ) y 1 e − θ ′ 2 y 2 , y 2 > y 1 A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Extensions of Freund’s Model An extension to a model with Weibull component lifetime distri- butions is discussed in Lu, J.-C. ((1989), Reliability, IEEE Trans- actions on), Spurrier, J. D. and Weier, D. ((1981), Reliability, IEEE Transactions on), Shaked, M. ((1984), Operations research). A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

T 1 and T 2 are independently distributed with respective survival functions [ S ( . )] θ 1 , θ 1 > 0 and [ S ( . )] θ 2 , θ 2 > 0. Again, T ∗ 1 and T ∗ 2 are assumed to have survival functions [ S ( . )] θ ′ 1 > 0 and [ S ( . )] θ ′ 1 , θ ′ 2 , θ ′ 2 > 0, respectively. The joint probability density function of the failure times ( Y 1 , Y 2 ) under the generalized model is (Asha et. al (2016))   θ ′ 1 θ 2 f ( y 1 ) f ( y 2 )[ S ( y 2 )] ( θ 1 + θ 2 − θ ′ 1 − 1 )     [ S ( y 1 )] θ ′ 1 − 1 ; y 1 > y 2 f ( y 1 , y 2 ) =  θ 1 θ ′ 2 f ( y 1 ) f ( y 2 )[ S ( y 1 )] ( θ 1 + θ 2 − θ ′ 2 − 1 )     [ S ( y 2 )] θ ′ 2 − 1 ; y 2 > y 1 . (2) A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Frailty An alternate modelling tool for correlated data is the frailty approach. Frailty accounts for neglected covariates. Clayton (1978) first used unobserved random covariates in multivariate survival models on chronic disease incidence in families. The term frailty was introduced by Vaupel et.al (1979) in univariate survival models. A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

It is an unobserved random proportionality factor that modifies the hazard function of an individual/component. A frailty model is an extension of the Cox proportional hazard model. In addition to the observed regressors, a frailty model also accounts for the presence of a latent multiplicative effect on the hazard function. A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Cox PH Model: The failure rate λ ( y ) = r ( y ) e X β , where r ( y ) is the baseline failure rate and X are observed covariates. Unobserved covariates Z : then λ ( y | z ) = z r ( y ) e X β Conditional survival function � − z [Λ( y )] e X β � S ( y | z ) = exp � y where Λ( y ) = 0 r ( t ) dt . A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Unconditional survival function � e − z Λ( y ) e X β � S ( y ) = E z � Λ( y ) e X β � = L z , where L z ( s ) is the Laplace transform of Z . For a two component system � [Λ( y 1 ) + Λ( y 2 )] e X β � S ( y 1 , y 2 ) = L z where Y 1 and Y 2 are independent given Z . A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

When frailty variable is integrated out Y 1 and Y 2 will become dependent because of the common frailty and is called shared frailty model. This frailty could be due to some genetic factors or environmental factors shared by paired organs in humans or components in system. When Y 1 and Y 2 are not independent then � Λ( y 1 , y 2 ) e X β � S ( y 1 , y 2 ) = L z . A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

A bivariate analogue of the univariate failure rate function is pro- posed by Cox (1972) as P ( y ≤ Y i < y + ∆ y | y ≤ Y 1 , y ≤ Y 2 ) λ i 0 ( y ) = lim , y i = y ∆ y ∆ y → 0 + � � P y i ≤ Y i < y i + ∆ y i | y i ≤ Y i , Y j = y j λ ij ( y i | y j ) = lim , y j < y i . ∆ y i ∆ y i → 0 + (3) Observe that λ ij ( y i | y j ) , i , j = 1 , 2 is based on ageing as well as the load sharing features of the surviving components (Singpurwalla (2006)). A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Given X , and Z = z , z θ 1 r ( y ) e X β , λ 10 ( y | z , X ) = z θ 2 r ( y ) e X β , y 1 = y 2 = y > 0 λ 20 ( y | z , X ) = z θ ′ 1 r ( y 1 ) e X β , y 1 > y 2 λ 12 ( y 1 | y 2 , z , X ) = z θ ′ 2 r ( y 2 ) e X β , y 1 < y 2 λ 21 ( y 2 | y 1 , z , X ) = (4) A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Conditional Model Given Frailty and Covariates i θ j f ( y 1 ) f ( y 2 )[ S ( y j )] ze X β ( θ 1 + θ 2 − θ ′ f (( y 1 , y 2 ) | z , X ) = z 2 e 2 X β θ ′ i ) − 1 × [ S ( y i )] ze X β θ ′ i − 1 ; y i > y j (5) for i � = j = 1 , 2 . A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

The corresponding joint survival function conditional on X and Z is given by S ( y 1 , y 2 | z , X ) = [ 1 − k ij ][ S ( y i )] z ( θ 1 + θ 2 ) e X β � S ( y i ) � z θ ′ i e X β [ S ( y j )] z ( θ 1 + θ 2 ) e X β ; y i ≥ y j + k ij (6) S ( y j ) θ j i , when θ 1 + θ 2 � = θ ′ where, k ij = i , i � = j = 1 , 2 and by θ 1 + θ 2 − θ ′ S ( y 1 , y 2 | z , X ) = [ S ( y i )] z ( θ 1 + θ 2 ) e X β � �� � 1 + ze X β θ j log S ( y j ) − log S ( y i ) (7) when θ 1 + θ 2 = θ ′ i , for i � = j = 1 , 2 . A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

Unconditional Model In general the unconditional joint survival function is given by S ( y 1 , y 2 | X ) = [ 1 − k ij ] L z (Ψ 1 ( y i | X ))+ k ij L z (Ψ ij ( y 1 , y 2 | X )); y i ≥ y j , (8) where, Ψ 1 ( y i | X ) = ( θ 1 + θ 2 ) H ( y i ) e X β , i ) H ( y j )] e X β and Ψ ij ( y 1 , y 2 | X ) = [ θ ′ i H ( y i ) + ( θ 1 + θ 2 − θ ′ θ 1 + θ 2 � = θ ′ i , i � = j = 1 , 2. A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.

For θ 1 + θ 2 = θ ′ i we obtain the unconditional survival function for y i > y j as S ( y 1 , y 2 | X ) = L z (Ψ 1 ( y i )) � � � � ∂ e X β θ 2 + log S ( y j ) − log S ( y i ) ∂ (Ψ 1 ( y i )) L z (Ψ 1 ( y i )) (9) The corresponding density in both the cases is � ∂ 2 L z ( s ) � f ( y 1 , y 2 | X ) = θ ′ i θ j r ( y 1 ) r ( y 2 ) e 2 X β ∂ s 2 s =Ψ ij ( y 1 , y 2 ) ; y i > y j i , j = 1 , 2 , i � = j . (10) A.V. Raja QPRC 2017, University of Connecticut, Storrs, USA.