.
.
Bayesian inference for reliability of systems and networks using the survival signature Louis J. M. Aslett1 , Frank Coolen2 and Simon P. Wilson3 1 University
of Oxford, 2 Durham University, 3 Trinity College Dublin
Durham Asset Management Seminar 12th March 2014
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Problem setting Test data available on a components to be used in a system. Objective: inference on system/network reliability given component test data. 1.00
t1
0.75
{t11 , . . . , t1n1 }
Survival Probability
TT11 T1
Item System T1
0.50
T2 T3 T4
0.25
0.00 0
1
2
Time
3
4
5
1.00
K
t
1.00
Survival Probability
0.75
0.50
0.25
0.00 0
?
1
2
Time
3
4
0.75
K {tK 1 , . . . , tnK }
Survival Probability
TT11 TK
Item System T1
0.50
T2 T3 T4
0.25
0.00 0
1
2
Time
T2
T2
T3
T1
T1
Item System
T3
T1 T2 T3 T4
T4
T4
5
T2
T3
T1
3
4
5
Example .
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
A nonparametric model of components At a fixed time t, probability component of type k functions is Bernoulli(pkt ) for some unknown pkt . =⇒ number functioning at time t in iid batch of nk is Binomial(nk , pkt ).
Example .
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
A nonparametric model of components At a fixed time t, probability component of type k functions is Bernoulli(pkt ) for some unknown pkt . =⇒ number functioning at time t in iid batch of nk is Binomial(nk , pkt ). Let Skt ∈ {0, 1, . . . , nk } be number of working components in test batch of nk components of type k. Then, Skt ∼ Binomial(nk , pkt ) ∀ t > 0
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
A nonparametric model of components At a fixed time t, probability component of type k functions is Bernoulli(pkt ) for some unknown pkt . =⇒ number functioning at time t in iid batch of nk is Binomial(nk , pkt ). Let Skt ∈ {0, 1, . . . , nk } be number of working components in test batch of nk components of type k. Then, Skt ∼ Binomial(nk , pkt ) ∀ t > 0 Given test data tk = {tk1 , . . . , tknk }, for each t we can form corresponding observation from Binomial model skt ≜
nk ∑ i=1
I(tki > t)
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
Bayesian inference for nonparametric model Taking prior pkt ∼ Beta(αtk , βtk ), exploit conjugacy result pkt | skt ∼ Beta(αtk + skt , βtk + nk − skt ) Then, posterior predicitive for number of components surviving in a new batch of mk components is Ckt | skt ∼ Beta-binomial(mk , αtk + skt , βtk + nk − skt )
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
Bayesian inference for nonparametric model Taking prior pkt ∼ Beta(αtk , βtk ), exploit conjugacy result pkt | skt ∼ Beta(αtk + skt , βtk + nk − skt ) Then, posterior predicitive for number of components surviving in a new batch of mk components is Ckt | skt ∼ Beta-binomial(mk , αtk + skt , βtk + nk − skt )
Summary: for any fixed t, skt provides a minimal sufficient statistic for computing posterior predictive distribution of the number of components surviving to t in a new batch, without any parametric model for component lifetime being assumed.
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Propagating uncertainty to the system Now take collection of component types k ∈ {1, . . . , K}, each with test data t = {t1 , . . . , tk }, and corresponding collection of minimal sufficient statistics for a fixed t, {s1t , . . . sK t }.
Example .
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
Propagating uncertainty to the system Now take collection of component types k ∈ {1, . . . , K}, each with test data t = {t1 , . . . , tk }, and corresponding collection of minimal sufficient statistics for a fixed t, {s1t , . . . sK t }. Survival probability for a new system S∗ comprising these component types follows naturally via posterior predictive and surival signature: P(TS∗ > t | s1t , . . . sK t ) ∫ ∫ 1 1 K K 1 K = · · · P(TS∗ > t | p1t , . . . pK t )P(pt | st ) . . . P(pt | st ) dpt . . . dpt
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
Propagating uncertainty to the system Now take collection of component types k ∈ {1, . . . , K}, each with test data t = {t1 , . . . , tk }, and corresponding collection of minimal sufficient statistics for a fixed t, {s1t , . . . sK t }. Survival probability for a new system S∗ comprising these component types follows naturally via posterior predictive and surival signature: P(TS∗ > t | s1t , . . . sK t ) ∫ ∫ 1 1 K K 1 K = · · · P(TS∗ > t | p1t , . . . pK t )P(pt | st ) . . . P(pt | st ) dpt . . . dpt (K ) ∫ ∫ ∑ mK m1 ∑ ∩ = ··· ··· Φ(l1 , . . . , lK )P {Ckt = lk | pkt } l1 =0
lK =0
×
k=1 K 1 K P(p1t | s1t ) . . . P(pK t | st ) dpt . . . dpt
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
P(TS∗ > t | s1t , . . . sK t ) ∫
∫
∫ =
1 1 K K 1 K P(TS∗ > t | p1t , . . . pK t )P(pt | st ) . . . P(pt | st ) dpt . . . dpt
···
=
∫ ···
m1 ∑
l1 =0
···
mK ∑
( Φ(l1 , . . . , lK )P
=
m1 ∑ l1 =0
···
mK ∑ lK =0
) {Ckt = lk | pkt }
k=1
lK =0
×
K ∩
K 1 K P(p1t | s1t ) . . . P(pK t | st ) dpt . . . dpt
Φ(l1 , . . . , lK )
K ∫ ∏
P(Ckt = lk | pkt )P(pkt | skt ) dpkt
k=1
Final integral is simply the posterior predictive (Beta-binomial).
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
System survival probability
P(TS∗ > t | s1t , . . . sK t ) m m 1 K ∑ ∑ = ··· Φ(l1 , . . . , lK ) l1 =0
lK =0
×
) K ( ∏ mk B(lk + αk + sk , mk − lk + β k + nk − sk ) t
k=1
lk
B(αtk
t
+
t
skt , βtk
+ nk −
t
skt )
Incredibly easy to implement this algorithmically since survival signature has factorised the survival function by component type!
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Why not structure function?
ϕ(x) =
s ∏ j=1
1 −
∏
(1 − xi )
i∈Cj
where {C1 , . . . , Cs } is the collection of minimal cut sets of the system. Recall don’t need x ∈ {0, 1} — we can plug in probabilities. So why not?
Example .
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Why not structure function?
ϕ(x) =
s ∏ j=1
1 −
∏
(1 − xi )
i∈Cj
where {C1 , . . . , Cs } is the collection of minimal cut sets of the system. Recall don’t need x ∈ {0, 1} — we can plug in probabilities. So why not? P(TS∗ > t | s1t , . . . sK t ) ∫ ∫ K 1 K = · · · ϕ(pxt 1 , . . . , pxt n )P(p1t | s1t ) . . . P(pK t | st ) dpt . . . dpt where pxt i is the element of {p1t , . . . , pK t } corresponding to component i. Have fun with that integral for large K . . . !
Example .
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
Example system layout, K = 4, n = 11 Example system: 2 1
4
T1
5 7 9
T2
3
T2 T3
6
T2
T1
T3 8
T4 10
T3
T2 ∼ Wei(λ2 = 1.8, γ1 = 2.2) T3 ∼ Log-N(µ = 0.4, τ = 1.234)
T4 11
T1 ∼ Exp(λ1 = 0.55)
T1
T4 ∼ Gam(λ3 = 0.9, γ2 = 3.2)
Simulated test data with nk = 100 ∀ k
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Posterior predictive survival curves
Survival Probability
1.00
0.75
Item System T1
0.50
T2 T3 T4
0.25
0.00 0
1
2
Time
3
4
5
Example .
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Optimal redundancy? 1.00
Redundancy Comp 1
Survival Probability
Comp 10 Comp 11
0.75
Comp 2 Comp 3 Comp 4
0.50
Comp 5 Comp 6 Comp 7
0.25
Comp 8 Comp 9 None
0.00 0
1
2
Time
3
4
5
Example .
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Optimal redundancy? 12 11 10 9
Order
8
Redundancy
7
Comp 7
6
Comp 8 None
5 4 3 2 1 0
1
2
Time
3
4
5
Example .
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
Parametric models of components The survival signature achieves the same factorisation of system lifetime when using parametric models for the components. Model the lifetime of component k directly via likelihood function fk Tk ∼ fk (t; ψk ) As before, given test data tk = {tk1 , . . . , tknk } for component k, posterior density is: fΨk | Tk (ψk | tk ) ∝ fΨk (ψk )
nk ∏ i=1
fk (tki ; ψk )
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
P(TS∗ > t | t1 , . . . tK ) ∫ ∫ = · · · P(TS∗ > t | ψ1 , . . . ψK )fΨ1 | T1 (dψ1 | t1 ) . . . fΨK | TK (dψK | tK )
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
P(TS∗ > t | t1 , . . . tK ) ∫ ∫ = · · · P(TS∗ > t | ψ1 , . . . ψK )fΨ1 | T1 (dψ1 | t1 ) . . . fΨK | TK (dψK | tK ) (K ) ∫ ∫ ∑ mK m1 ∑ ∩ = ··· ··· Φ(l1 , . . . , lK )P {Ckt = lk | ψk } l1 =0
lK =0
k=1
× fΨ1 | T1 (dψ1 | t ) . . . fΨK | TK (dψK | tK ) 1
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
P(TS∗ > t | t1 , . . . tK ) ∫ ∫ = · · · P(TS∗ > t | ψ1 , . . . ψK )fΨ1 | T1 (dψ1 | t1 ) . . . fΨK | TK (dψK | tK ) (K ) ∫ ∫ ∑ mK m1 ∑ ∩ = ··· ··· Φ(l1 , . . . , lK )P {Ckt = lk | ψk } l1 =0
lK =0
k=1
× fΨ1 | T1 (dψ1 | t ) . . . fΨK | TK (dψK | tK ) 1
∫ =
∫ ···
m1 ∑
l1 =0
···
mK ∑
Φ(l1 , . . . , lK )
lK =0
) K ( ∏ mk × [Fk (t; ψk )]mk −lk [1 − Fk (t; ψk )]lk lk k=1
× fΨ1 | T1 (dψ1 | t1 ) . . . fΨK | TK (dψK | tK )
]
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
P(TS∗ > t | t1 , . . . tK ) ∫ ∫ ∑ mK m1 ∑ = ··· ··· Φ(l1 , . . . , lK ) l1 =0
lK =0
) K ( ∏ mk × [Fk (t; ψk )]mk −lk [1 − Fk (t; ψk )]lk lk
]
k=1
× fΨ1 | T1 (dψ1 | t1 ) . . . fΨK | TK (dψK | tK ) =
m1 ∑
···
l1 =0
mK ∑
Φ(l1 , . . . , lK )
lK =0
×
)∫ K ( ∏ mk k=1
lk
[Fk (t; ψk )]mk −lk [1 − Fk (t; ψk )]lk fΨk | Tk (dψk | tk )
Final term posterior predictive of lk components of type k surviving to t.
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Computing the integral for arbitrary models Three possibilities. The posterior, fΨk | Tk (dψk | tk ), is: 1. in closed form and integral tractable; 2. known distribution, but the integral is intractable; 3. not in closed form. Also, note the integral is just: ] [ EΨk | Tk [Fk (t; ψk )]mk −lk [1 − Fk (t; ψk )]lk
Example .
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
Computing the integral for arbitrary models Three possibilities. The posterior, fΨk | Tk (dψk | tk ), is: 1. in closed form and integral tractable; 2. known distribution, but the integral is intractable; 3. not in closed form. Also, note the integral is just: ] [ EΨk | Tk [Fk (t; ψk )]mk −lk [1 − Fk (t; ψk )]lk (1)
(N)
Thus, for samples ψk , . . . , ψk back to evaluating:
∼ Ψk | Tk we can always fall
1∑ (i) (i) [Fk (t; ψk )]mk −lk [1 − Fk (t; ψk )]lk N i=1 [ ] N→∞ −−−−→ EΨk | Tk [Fk (t; ψk )]mk −lk [1 − Fk (t; ψk )]lk N
Problem Setting .
Nonparametric Method ......
Example ....
Parametric Method ....
Example .
Posterior predictive survival curves for both methods
Survival Probability
1.00
0.75 Item Ground Truth 0.50
Non-parametric Parametric
0.25
0.00 0
1
2
Time
3
4
5
