Evaluating dependability metrics of critical systems: Monte ... - iaria

Adaptive stochastic approximation (ASA)
ASA just uses a single sample path (y 0 ,... ,y n ).
Initial distribution for y 0 , matrix ˜P (0) and guess µ (0) (·).
At step j of the path, if y j ∉ ∆,
◮ matrix ˜P (j) used to generate y j+1 .
◮ From yj+1 , update the estimate of µ(y j ) by
µ (j+1) (y j ) = (1 − a j (y j ))µ (j) (y j )
+
h
i
a j (y j ) c(y j ,y j+1 ) + µ (j) P(yj ,y j+1 )
(y j+1 )
˜P (j) (y j ,y j+1 ) ,
where {a j (y), j ≥ 0}, sequence of step sizes
◮ For δ > 0 constant,
!
ˆc(yj
˜P (j+1) , y j+1 ) + µ (j+1) (y j+1 )˜
(y j , y j+1 ) = max P(y j ,y j+1 )
µ (j+1) , δ .
(y j )
◮ Otherwise µ (j+1) (y) = µ (j) (y), ˜P (j+1) (y, z) = P (j) (y, z).
◮ Normalize: P (j+1) (y j , y) =
˜P (j+1) (y j ,y)
Pz ˜P (j+1) (y j ,z) .
If y j ∈ ∆, y j+1 generated from initial distribution, but estimations of
P(·, ·) and µ(·) kept.
Batching techniques used to get a confidence interval.
G. Rubino (INRIA) Monte Carlo DEPEND 2010 46 / 72