Metropolis-Hastings algorithm
Chapter 5 Chapter 5 General Systems and Subset Simulation Method
5.3 Choice of intermediate failure events
(5.3)
(5.4)
The choice of the intermediate failure events { Fi : i = 1, ... , m -1} plays a key role in the subset simulation procedure. Two issues are basic to the choice of the intermediate failure events. The first is the parameterization of the target failure event F which allows the generation of intermediate failure events by varying the value of the defined parameter. The second is the choice of the specific sequence of values of the defined parameter, which affects the values of the conditional probabilities { P(Fi+dFi) : i = 1, ... , m-1} and hence the efficiency of the subset simulation procedure.
Generic representation of failure event
Many failure events encountered in engineering applications can be defined using a combination of union and intersection of some component failure events. In particular, consider a failure event F of the following form:
L Li
F =
U n
{Dj~c(O)>
Cj~c(O)} (5.5)j=l lc=l
where Dj~c(O) and CjJc (6) may be viewed as the demand and capacity variables of the (j, k) compo- nent of the system. The failure event F in (5.5) can be considered as the failure of a system with L sub-systems connected in series, where the j-th sub-system consists of Lj components connected in parallel.
In order to apply subset simulation to compute the failure probability PF, it is desirable to parameterize F with a single parameter so that the sequence of intermediate failure events { Fi : i = 1, ... , m - 1} can be generated by varying the parameter. This ca.1. be accomplished as follows. For the failure event in (5.5), define the 'critical demand-to-capacity ratio' (CDCR) Y as
D·~c(O)
Y ( 0) = . max min -=::-1--:-:::--
J=l, ... ,L Jc=l, ... ,Lj CjJc(O) (5.6)
Then it can be easily verified that
F
=
{Y(O)>
1} (5.7)and so the sequence of intermediate failure events can be generated as
Fi
=
{Y(O)>
Yi} (5.8)where 0
<
y1< · · · <
Ym = 1 is a sequence of (normalized) intermediate threshold values.Similarly, consider a failure event F of the form:
L Li
F =
n U
{DJ~c(O)>
Cj~c(O)} (5.9)j=lk=l
which can be considered as the failure of a system with L sub-systems connected in parallel with the j-th sub-system consisting of Lj components connected in series. One can easily verify that the definition
Y((}) . Dj~c(O)
= rmn max
j=l, ... ,L Jc=l, ... ,L; CjJc(O) (5.10)
satisfies (5.7) and hence the sequence of failure events can again be generated based on (5.8).
The foregoing discussion can be generalized to failure events consisting of multiple stacks of union and intersection. Essentially, Y is defined using 'max' and 'min' in the same order corresponding to each occurrence of union (U) and intersection (n) in F, respectively. As another example, consider the first excursion failure of the interstory drift in any one story of a n8-story building beyond a given threshold level b. Let the interstory drift response {X;(t; 0) : j
=
1, ... , n8 } be computed at the nt time instants t1 , ••. , tn, within the duration of interest. Thenn. nt
F
= u
U{IX;(t,.;O)I>
b} = {Y(O)>
1} (5.11)j=1 k=l
where
Y(O) = . max max IXi(t,.)l
J=l, ... ,n. A:=l, ... ,n, b (5.12)
Choice of intermediate threshold levels
The choice of the sequence of intermediate threshold values {y11 ••• ,ym} appearing in the pa- rameterization of intermediate failure events affects the values of the conditional probabilities and hence the efficiency of the subset simulation procedure. H the sequence increases slowly, then the conditional probabilities will be large, and so their estimation requires less samples N. A slow se- quence, however, requires more simulation levels m to reach the target failure event, increasing the total number of samples Nr = m N in the whole procedure. Conversely, if the sequence increases too rapidly that the conditional failure events become rare, it will require more samples N to obtain an accurate estimate of the conditional failure probabilities in each simulation level, which again increases the total number of samples. It can thus be seen that the choice of the intermediate threshold values is a trade-off between the number of samples required in each simulation level and the number of simulation levels required to reach the target failure event.
The choice of the intermediate threshold values {Yi : i = 1, ... , m - 1} deserves a detailed study which is left for future work. One strategy is to choose the Yi a priori, but then it is difficult to control values of the conditional probabilities P(FiiFi-1 ). In this work, the Yi are chosen 'adaptively' so that the estimated conditional probabilities are equal to a fixed value Po E (0, 1). This is accomplished by choosing the intermediate threshold level 1/i (i = 1, ... , m -1) as the (1- Po)N-th largest value (i.e., an order statistic) among the CDC& {Y(Oii-1)) : k = 1, ... ,N} where the oii-1) are the Markov chain samples generated at the (i - 1)-th conditional level for i = 2, ... , m - 1, and the
oio)
arethe samples from the initial Monte Carlo simulation. Here, Po is assumed to be chosen so that PoN and hence (1 - Po)N are positive integers, although this is not strictly necessary. This choice of the intermediate threshold levels implies that they are dependent on the conditional samples and will
vary in different simulation runs. For a target probability level of 10-3 to 10-6, choosing Po
=
0.1is found to yield good efficiency.
Using this adaptive choice of the proposal PDF, the subset simulation procedure is illustrated in Figure 5.1 for simulation Levels 0 (Monte Carlo) and 1 (Markov Chain Monte Carlo).
Response level b Monte Carlo
Simulation
•
Uncertain parameter space
•
• •
• •
Failure Probability estimate
(a) Level 0 (initial ph~e): Monte Carlo simulation
Response level b
Uncertain parameter space
•
• •
Po
Failure Probability estimate
(b) Level 0: Adaptive selection of first intermediate threshold level
Response level b
• •• ... ••
... !\
b, i ••
i ··,
• Ill \
• •••••••
Uncertain parameter space
Po
Failure Probability estimate
(c) Levell: Markov chain Monte Carlo Simulation
Response level b
;
..
i ••
. .
:, \ \••••••••
Uncertain parameter space
p;
PoFailure Probability estimate
(d) Levell: Adaptive selection of second intermediate threshold level Figure 5.1: illustration of subset simulation procedure