Rare event simulation - a Point Process interpretation ... · Rare event simulation a Point Process...
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Rare event simulation
a Point Process interpretation withapplication in probability and quantileestimation and metamodel basedalgorithms
Séminaire S3 | Clément WALTER
March 13th 2015
Introduction
Problem setting:
X random vector with know distribution µX
g a "black-box" function representing a computer code: g : Rd → RY = g(X) the real-valued random variable which describes the state ofthe system; its distribution µY is unknown
Uncertainty Quanti�cation: F = {x ∈ Rd | g(x) > q}�nd p = P [X ∈ F ] = µX(F ) for a given q�nd q for a given p
Issues
p = µX(F ) = µY ([q; +∞[)� 1
needs to use g to get F or µY which is time costly
Monte Carlo estimator has a CV δ2 ≈ 1/Np⇒ N � 1/p
Séminaire S3 | March 13th 2015 | PAGE 1/26
Introduction
Two main directions to overcome this issue:
learn a metamodel on g
use variance-reduction techniques to estimate p
Importance samplingMultilevel splitting
Multilevel Splitting (Subset Simulations)
Write the sought probability p as a product of less small probabilities andestimate them with MCMC: let (qi)i=0..m be an increasing sequence withq0 = −∞ and qm = q:
p = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× P [g(X) > qm−1]
= P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× · · · × P [g(X) > q1]
⇒ how to choose (qi)i: beforehand or on-the-go? Optimality? Bias?
Séminaire S3 | March 13th 2015 | PAGE 2/26
Introduction
Two main directions to overcome this issue:
learn a metamodel on g
use variance-reduction techniques to estimate p
Importance samplingMultilevel splitting
Multilevel Splitting (Subset Simulations)
Write the sought probability p as a product of less small probabilities andestimate them with MCMC: let (qi)i=0..m be an increasing sequence withq0 = −∞ and qm = q:
p = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× P [g(X) > qm−1]
= P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× · · · × P [g(X) > q1]
⇒ how to choose (qi)i: beforehand or on-the-go? Optimality? Bias?
Séminaire S3 | March 13th 2015 | PAGE 2/26
Introduction
Two main directions to overcome this issue:
learn a metamodel on g
use variance-reduction techniques to estimate p
Importance sampling
Multilevel splitting
Multilevel Splitting (Subset Simulations)
Write the sought probability p as a product of less small probabilities andestimate them with MCMC: let (qi)i=0..m be an increasing sequence withq0 = −∞ and qm = q:
p = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× P [g(X) > qm−1]
= P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× · · · × P [g(X) > q1]
⇒ how to choose (qi)i: beforehand or on-the-go? Optimality? Bias?
Séminaire S3 | March 13th 2015 | PAGE 2/26
Introduction
Two main directions to overcome this issue:
learn a metamodel on g
use variance-reduction techniques to estimate p
Importance samplingMultilevel splitting
Multilevel Splitting (Subset Simulations)
Write the sought probability p as a product of less small probabilities andestimate them with MCMC: let (qi)i=0..m be an increasing sequence withq0 = −∞ and qm = q:
p = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× P [g(X) > qm−1]
= P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× · · · × P [g(X) > q1]
⇒ how to choose (qi)i: beforehand or on-the-go? Optimality? Bias?
Séminaire S3 | March 13th 2015 | PAGE 2/26
Introduction
Two main directions to overcome this issue:
learn a metamodel on g
use variance-reduction techniques to estimate p
Importance samplingMultilevel splitting
Multilevel Splitting (Subset Simulations)
Write the sought probability p as a product of less small probabilities andestimate them with MCMC: let (qi)i=0..m be an increasing sequence withq0 = −∞ and qm = q:
p = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× P [g(X) > qm−1]
= P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× · · · × P [g(X) > q1]
⇒ how to choose (qi)i: beforehand or on-the-go? Optimality? Bias?
Séminaire S3 | March 13th 2015 | PAGE 2/26
Introduction
Two main directions to overcome this issue:
learn a metamodel on g
use variance-reduction techniques to estimate p
Importance samplingMultilevel splitting
Multilevel Splitting (Subset Simulations)
Write the sought probability p as a product of less small probabilities andestimate them with MCMC: let (qi)i=0..m be an increasing sequence withq0 = −∞ and qm = q:
p = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× P [g(X) > qm−1]
= P [g(X) > q] = P [g(X) > qm | g(X) > qm−1]× · · · × P [g(X) > q1]
⇒ how to choose (qi)i: beforehand or on-the-go? Optimality? Bias?Séminaire S3 | March 13th 2015 | PAGE 2/26
Introduction
A typical MS algorithm works as follows:
1 Sample a Monte-Carlo population (Xi)i of size N ;y = (g(X1), · · · , g(XN )); j = 0
2 Estimate the conditional probability P [g(X) > qj+1 | g(X) > qj ]
3 Resample the (Xi)i such that g(Xi) ≤ qj+1 conditionally to begreater than qj+1 (the other ones don't change)
4 j ← j + 1 and repeat until j = m
⇒ Parallel computation at each iteration in the resampling step
Minimal variance when all conditional probabilities are equal [4]
Adaptive Multilevel Splitting: qj+1 ← y(p0N) or qj+1 ← y(k)empirical quantiles of order p0 ∈ (0, 1) ⇒ bias [4, 1]; the number ofsubsets converges toward a constant log p/ log p0empirical quantiles of order k/N ⇒ no bias and CLT [3, 2]minimal variance with k = 1 (Last Particle Algorithm [5, 6]); thenumber of subsets follows a Poisson law with parameter −N log p
⇒ disables parallel computation
Séminaire S3 | March 13th 2015 | PAGE 3/26
Introduction
A typical MS algorithm works as follows:
1 Sample a Monte-Carlo population (Xi)i of size N ;y = (g(X1), · · · , g(XN )); j = 0
2 Estimate the conditional probability P [g(X) > qj+1 | g(X) > qj ]
3 Resample the (Xi)i such that g(Xi) ≤ qj+1 conditionally to begreater than qj+1 (the other ones don't change)
4 j ← j + 1 and repeat until j = m
⇒ Parallel computation at each iteration in the resampling step
Minimal variance when all conditional probabilities are equal [4]
Adaptive Multilevel Splitting: qj+1 ← y(p0N) or qj+1 ← y(k)empirical quantiles of order p0 ∈ (0, 1) ⇒ bias [4, 1]; the number ofsubsets converges toward a constant log p/ log p0empirical quantiles of order k/N ⇒ no bias and CLT [3, 2]minimal variance with k = 1 (Last Particle Algorithm [5, 6]); thenumber of subsets follows a Poisson law with parameter −N log p
⇒ disables parallel computation
Séminaire S3 | March 13th 2015 | PAGE 3/26
Introduction
A typical MS algorithm works as follows:
1 Sample a Monte-Carlo population (Xi)i of size N ;y = (g(X1), · · · , g(XN )); j = 0
2 Estimate the conditional probability P [g(X) > qj+1 | g(X) > qj ]
3 Resample the (Xi)i such that g(Xi) ≤ qj+1 conditionally to begreater than qj+1 (the other ones don't change)
4 j ← j + 1 and repeat until j = m
⇒ Parallel computation at each iteration in the resampling step
Minimal variance when all conditional probabilities are equal [4]
Adaptive Multilevel Splitting: qj+1 ← y(p0N) or qj+1 ← y(k)empirical quantiles of order p0 ∈ (0, 1) ⇒ bias [4, 1]; the number ofsubsets converges toward a constant log p/ log p0empirical quantiles of order k/N ⇒ no bias and CLT [3, 2]minimal variance with k = 1 (Last Particle Algorithm [5, 6]); thenumber of subsets follows a Poisson law with parameter −N log p
⇒ disables parallel computation Séminaire S3 | March 13th 2015 | PAGE 3/26
Outline
1 Increasing random walk
2 Probability estimation
3 Quantile estimation
4 Design points
5 Conclusion
Séminaire S3 | March 13th 2015 | PAGE 4/26
Outline
1 Increasing random walk
2 Probability estimation
3 Quantile estimation
4 Design points
5 Conclusion
Séminaire S3 | March 13th 2015 | PAGE 5/26
Increasing random walkDe�nition
De�nition
Let Y be a real-valued random variable with distribution µY and cdf F(assumed to be continuous). One considers the Markov chain (withY0 = −∞) such that:
∀n ∈ N : P [Yn+1 ∈ A | Y0, · · · , Yn] =µY (A ∩ (Yn,+∞))
µY ((Yn,+∞))
i.e. Yn+1 is randomly greater than Yn: Yn+1 ∼ µY (· | Y > Yn)
the Tn = − log(P (Y > Yn)) are distributed as the arrival times of aPoisson process with parameter 1 [5, 7]Time in the Poisson process is linked with rarity in probabilityThe number of events My before y ∈ R is related to P [Y > y] = py
My =Mt=− log pyL∼ P(− log py)
Séminaire S3 | March 13th 2015 | PAGE 6/26
Increasing random walkDe�nition
De�nition
Let Y be a real-valued random variable with distribution µY and cdf F(assumed to be continuous). One considers the Markov chain (withY0 = −∞) such that:
∀n ∈ N : P [Yn+1 ∈ A | Y0, · · · , Yn] =µY (A ∩ (Yn,+∞))
µY ((Yn,+∞))
i.e. Yn+1 is randomly greater than Yn: Yn+1 ∼ µY (· | Y > Yn)
the Tn = − log(P (Y > Yn)) are distributed as the arrival times of aPoisson process with parameter 1 [5, 7]
Time in the Poisson process is linked with rarity in probabilityThe number of events My before y ∈ R is related to P [Y > y] = py
My =Mt=− log pyL∼ P(− log py)
Séminaire S3 | March 13th 2015 | PAGE 6/26
Increasing random walkDe�nition
De�nition
Let Y be a real-valued random variable with distribution µY and cdf F(assumed to be continuous). One considers the Markov chain (withY0 = −∞) such that:
∀n ∈ N : P [Yn+1 ∈ A | Y0, · · · , Yn] =µY (A ∩ (Yn,+∞))
µY ((Yn,+∞))
i.e. Yn+1 is randomly greater than Yn: Yn+1 ∼ µY (· | Y > Yn)
the Tn = − log(P (Y > Yn)) are distributed as the arrival times of aPoisson process with parameter 1 [5, 7]Time in the Poisson process is linked with rarity in probability
The number of events My before y ∈ R is related to P [Y > y] = py
My =Mt=− log pyL∼ P(− log py)
Séminaire S3 | March 13th 2015 | PAGE 6/26
Increasing random walkDe�nition
De�nition
Let Y be a real-valued random variable with distribution µY and cdf F(assumed to be continuous). One considers the Markov chain (withY0 = −∞) such that:
∀n ∈ N : P [Yn+1 ∈ A | Y0, · · · , Yn] =µY (A ∩ (Yn,+∞))
µY ((Yn,+∞))
i.e. Yn+1 is randomly greater than Yn: Yn+1 ∼ µY (· | Y > Yn)
the Tn = − log(P (Y > Yn)) are distributed as the arrival times of aPoisson process with parameter 1 [5, 7]Time in the Poisson process is linked with rarity in probabilityThe number of events My before y ∈ R is related to P [Y > y] = py
My =Mt=− log pyL∼ P(− log py)
Séminaire S3 | March 13th 2015 | PAGE 6/26
Increasing random walkDe�nition
First consequence: number of simulations to get the realisation of arandom variable above a given threshold (event with probability p) followsa Poisson law P(log 1/p) instead of a Geometric law G(p).
0 20 40 60 80 100
0.00
0.05
0.10
0.15
0.20
N
Den
sity
Figure: Comparison ofPoisson and Geometricdensities withp = 0.0228
Séminaire S3 | March 13th 2015 | PAGE 7/26
Increasing random walkExample
Y ∼ N (0, 1) ; p = P [Y > 2] ≈ 2, 28.10−2 ; 1/p ≈ 43, 96 ; − log p ≈ 3, 78
Séminaire S3 | March 13th 2015 | PAGE 8/26
Plan
1 Increasing random walk
2 Probability estimation
3 Quantile estimation
4 Design points
5 Conclusion
Séminaire S3 | March 13th 2015 | PAGE 9/26
Probability estimationDe�nition of the estimator
Concept
Number of events before time t = − log(P (Y > q)) follows a Poisson lawwith parameter t estimate the parameter of a Poisson law
Let (Mi)i=1..N be N iid. RV of the number of events at time
t = − log p: Mi ∼ P(− log p); Mq =N∑i=1
Mi ∼ P(−N log p)
−̂ log p ≈ 1
N
N∑i=1
Mi =Mq
N−→ p̂ =
(1− 1
N
)Mq
⇒ Last Particle Estimator, but with parallel implementation
⇒ One has indeed an estimator of P [Y > q0] , ∀q0 ≤ q:
FN (y) = 1−(1− 1
N
)Mya.s.−−−−→
N→∞F (y)
Séminaire S3 | March 13th 2015 | PAGE 10/26
Probability estimationDe�nition of the estimator
Concept
Number of events before time t = − log(P (Y > q)) follows a Poisson lawwith parameter t estimate the parameter of a Poisson law
Let (Mi)i=1..N be N iid. RV of the number of events at time
t = − log p: Mi ∼ P(− log p); Mq =N∑i=1
Mi ∼ P(−N log p)
−̂ log p ≈ 1
N
N∑i=1
Mi =Mq
N−→ p̂ =
(1− 1
N
)Mq
⇒ Last Particle Estimator, but with parallel implementation
⇒ One has indeed an estimator of P [Y > q0] , ∀q0 ≤ q:
FN (y) = 1−(1− 1
N
)Mya.s.−−−−→
N→∞F (y)
Séminaire S3 | March 13th 2015 | PAGE 10/26
Probability estimationDe�nition of the estimator
Concept
Number of events before time t = − log(P (Y > q)) follows a Poisson lawwith parameter t estimate the parameter of a Poisson law
Let (Mi)i=1..N be N iid. RV of the number of events at time
t = − log p: Mi ∼ P(− log p); Mq =N∑i=1
Mi ∼ P(−N log p)
−̂ log p ≈ 1
N
N∑i=1
Mi =Mq
N−→ p̂ =
(1− 1
N
)Mq
⇒ Last Particle Estimator, but with parallel implementation
⇒ One has indeed an estimator of P [Y > q0] , ∀q0 ≤ q:
FN (y) = 1−(1− 1
N
)Mya.s.−−−−→
N→∞F (y)
Séminaire S3 | March 13th 2015 | PAGE 10/26
Probability estimationDe�nition of the estimator
Concept
Number of events before time t = − log(P (Y > q)) follows a Poisson lawwith parameter t estimate the parameter of a Poisson law
Let (Mi)i=1..N be N iid. RV of the number of events at time
t = − log p: Mi ∼ P(− log p); Mq =N∑i=1
Mi ∼ P(−N log p)
−̂ log p ≈ 1
N
N∑i=1
Mi =Mq
N−→ p̂ =
(1− 1
N
)Mq
⇒ Last Particle Estimator, but with parallel implementation
⇒ One has indeed an estimator of P [Y > q0] , ∀q0 ≤ q:
FN (y) = 1−(1− 1
N
)Mya.s.−−−−→
N→∞F (y)
Séminaire S3 | March 13th 2015 | PAGE 10/26
Probability estimationDe�nition of the estimator
Concept
Number of events before time t = − log(P (Y > q)) follows a Poisson lawwith parameter t estimate the parameter of a Poisson law
Let (Mi)i=1..N be N iid. RV of the number of events at time
t = − log p: Mi ∼ P(− log p); Mq =N∑i=1
Mi ∼ P(−N log p)
−̂ log p ≈ 1
N
N∑i=1
Mi =Mq
N−→ p̂ =
(1− 1
N
)Mq
⇒ Last Particle Estimator, but with parallel implementation
⇒ One has indeed an estimator of P [Y > q0] , ∀q0 ≤ q:
FN (y) = 1−(1− 1
N
)Mya.s.−−−−→
N→∞F (y)
Séminaire S3 | March 13th 2015 | PAGE 10/26
Probability estimationExample
Y ∼ N (0, 1) ; p = P [Y > 2] ≈ 2, 28.10−2 ; 1/p ≈ 43, 96 ; − log p ≈ 3, 78
Séminaire S3 | March 13th 2015 | PAGE 11/26
Probability estimationPractical implementation
Ideally, one knows how to generate the Markov chainIn practice, Y = g(X) and one can use Markov chain sampling (e.g.Metropolis-Hastings or Gibbs algorithms)
requires to work with a population to get starting points
⇒ batches of k random walks are generated together
Generating k random walks
Require: k, qGenerate k copies (Xi)i=1..k according to µX ; Y ← (g(X1), · · · , g(Xk)); M = (0, · · · , 0)while minY < q do
3: ind← whichY < qfor i in ind do
Mi =Mi + 16: Generate X∗ ∼ µX(· | X > g(Xi))
Xi ← X∗; Yi = g(X∗)end for
9: end whileReturn M, (Xi)i=1..N , (Yi)i=1..N
⇒ each sample is resampled according to its own level
Séminaire S3 | March 13th 2015 | PAGE 12/26
Probability estimationPractical implementation
Ideally, one knows how to generate the Markov chainIn practice, Y = g(X) and one can use Markov chain sampling (e.g.Metropolis-Hastings or Gibbs algorithms)
requires to work with a population to get starting points
⇒ batches of k random walks are generated together
Generating k random walks
Require: k, qGenerate k copies (Xi)i=1..k according to µX ; Y ← (g(X1), · · · , g(Xk)); M = (0, · · · , 0)while minY < q do
3: ind← whichY < qfor i in ind do
Mi =Mi + 16: Generate X∗ ∼ µX(· | X > g(Xi))
Xi ← X∗; Yi = g(X∗)end for
9: end whileReturn M, (Xi)i=1..N , (Yi)i=1..N
⇒ each sample is resampled according to its own level
Séminaire S3 | March 13th 2015 | PAGE 12/26
Probability estimationPractical implementation
Ideally, one knows how to generate the Markov chainIn practice, Y = g(X) and one can use Markov chain sampling (e.g.Metropolis-Hastings or Gibbs algorithms)
requires to work with a population to get starting points
⇒ batches of k random walks are generated together
Generating k random walks
Require: k, qGenerate k copies (Xi)i=1..k according to µX ; Y ← (g(X1), · · · , g(Xk)); M = (0, · · · , 0)while minY < q do
3: ind← whichY < qfor i in ind do
Mi =Mi + 16: Generate X∗ ∼ µX(· | X > g(Xi))
Xi ← X∗; Yi = g(X∗)end for
9: end whileReturn M, (Xi)i=1..N , (Yi)i=1..N
⇒ each sample is resampled according to its own level
Séminaire S3 | March 13th 2015 | PAGE 12/26
Probability estimationPractical implementation
Ideally, one knows how to generate the Markov chainIn practice, Y = g(X) and one can use Markov chain sampling (e.g.Metropolis-Hastings or Gibbs algorithms)
requires to work with a population to get starting points
⇒ batches of k random walks are generated together
Generating k random walks
Require: k, qGenerate k copies (Xi)i=1..k according to µX ; Y ← (g(X1), · · · , g(Xk)); M = (0, · · · , 0)while minY < q do
3: ind← whichY < qfor i in ind do
Mi =Mi + 16: Generate X∗ ∼ µX(· | X > g(Xi))
Xi ← X∗; Yi = g(X∗)end for
9: end whileReturn M, (Xi)i=1..N , (Yi)i=1..N
⇒ each sample is resampled according to its own levelSéminaire S3 | March 13th 2015 | PAGE 12/26
Probability estimationPractical implementation
Convergence of the Markov chain (for conditional sampling) is increasedwhen the starting point already follows the targeted distribution; 2possibilities:
store each state (Xi)i and its corresponding level
re-draw only the smallest Xi (⇒ Last Particle Algorithm)
⇒ LPA is only one possible implementation of this estimator
Computing time with LPA implementation
Let tpar be the random time of generating N random walks by batches ofsize k = N/nc (nc standing for a number of cores) with burn-in T
tpar = max of nc RV ∼ P(−k log p)
E [tpar] =T (log p)2
ncδ2
1 +
√ncδ
2
(log p)2
√2 log nc +
1
T log 1/p
Séminaire S3 | March 13th 2015 | PAGE 13/26
Probability estimationPractical implementation
Convergence of the Markov chain (for conditional sampling) is increasedwhen the starting point already follows the targeted distribution; 2possibilities:
store each state (Xi)i and its corresponding level
re-draw only the smallest Xi (⇒ Last Particle Algorithm)
⇒ LPA is only one possible implementation of this estimator
Computing time with LPA implementation
Let tpar be the random time of generating N random walks by batches ofsize k = N/nc (nc standing for a number of cores) with burn-in T
tpar = max of nc RV ∼ P(−k log p)
E [tpar] =T (log p)2
ncδ2
1 +
√ncδ
2
(log p)2
√2 log nc +
1
T log 1/p
Séminaire S3 | March 13th 2015 | PAGE 13/26
Probability estimationComparison
Mean computer time against coe�cient of variation: the cost of analgorithm is the number of generated samples in a row by a core. Weassume nc ≥ 1 cores and burn-in = T for Metropolis-Hastings
Algorithm Time Coef. of var. δ2 Times VS δ
Monte Carlo N/nc 1/Np 1/pδ2
AMS T log plog p0
N(1−p0)nc
log plog p0
1−p0Np0
(1−p0)2p0(log p0)2
T (log p)2
ncδ2
LPA −TN log p − log pN
T (log p)2
δ2
Random walk −T Nnc
log p − log pN
T (log p)2
ncδ2
best AMS when p0 → 1
LPA brings the theoretically best AMS but is not parallel
Random walk allows for taking p0 → 1 while keeping the parallelimplementation
Séminaire S3 | March 13th 2015 | PAGE 14/26
Plan
1 Increasing random walk
2 Probability estimation
3 Quantile estimation
4 Design points
5 Conclusion
Séminaire S3 | March 13th 2015 | PAGE 15/26
Quantile estimationDe�nition
Concept
Approximate a time t = − log p with times of a Poisson process; the higherthe rate, the denser the discretisation of [0; +∞[
The center of the interval [TMt ;TMt+1] converges toward a randomvariable centred in t with symmetric pdf
q̂ =1
2(Ym + Ym+1) with m = bE[Mq]c = b−N log pc
CLT:√N (q̂ − q) L−→
m→∞N(0,−p2 log pf(q)2
)Bounds on bias on O(1/N)
Séminaire S3 | March 13th 2015 | PAGE 16/26
Quantile estimationDe�nition
Concept
Approximate a time t = − log p with times of a Poisson process; the higherthe rate, the denser the discretisation of [0; +∞[
The center of the interval [TMt ;TMt+1] converges toward a randomvariable centred in t with symmetric pdf
q̂ =1
2(Ym + Ym+1) with m = bE[Mq]c = b−N log pc
CLT:√N (q̂ − q) L−→
m→∞N(0,−p2 log pf(q)2
)Bounds on bias on O(1/N)
Séminaire S3 | March 13th 2015 | PAGE 16/26
Quantile estimationDe�nition
Concept
Approximate a time t = − log p with times of a Poisson process; the higherthe rate, the denser the discretisation of [0; +∞[
The center of the interval [TMt ;TMt+1] converges toward a randomvariable centred in t with symmetric pdf
q̂ =1
2(Ym + Ym+1) with m = bE[Mq]c = b−N log pc
CLT:√N (q̂ − q) L−→
m→∞N(0,−p2 log pf(q)2
)Bounds on bias on O(1/N)
Séminaire S3 | March 13th 2015 | PAGE 16/26
Quantile estimationDe�nition
Concept
Approximate a time t = − log p with times of a Poisson process; the higherthe rate, the denser the discretisation of [0; +∞[
The center of the interval [TMt ;TMt+1] converges toward a randomvariable centred in t with symmetric pdf
q̂ =1
2(Ym + Ym+1) with m = bE[Mq]c = b−N log pc
CLT:√N (q̂ − q) L−→
m→∞N(0,−p2 log pf(q)2
)Bounds on bias on O(1/N)
Séminaire S3 | March 13th 2015 | PAGE 16/26
Quantile estimationExample
Y ∼ N (0, 1) ; p = P [Y > 2] ≈ 2, 28.10−2 ; 1/p ≈ 43, 96 ; − log p ≈ 3, 78
Séminaire S3 | March 13th 2015 | PAGE 17/26
Plan
1 Increasing random walk
2 Probability estimation
3 Quantile estimation
4 Design points
5 Conclusion
Séminaire S3 | March 13th 2015 | PAGE 18/26
Design pointsAlgorithm
No need for exact sampling if the goal is only to get failing samples⇒ use of a metamodel for conditional sampling
Getting Nfail failing samples
Sample a minimal-sized DoELearn a �rst metamodel with trend = failure
3: for Nfail times do . Simulate the random walks one after the otherSample X1 ∼ µX ; y1 = g(X1); m = 1; train the metamodelwhile ym < q do
6: Xm+1 = Xm; ym+1 = ymfor T times do . Pseudo burn-in
X∗ ∼ K(Xm+1, ·); g̃(X∗) = y∗ . K is a kernel for Markov chain sampling9: If y∗ > ym+1, ym+1 = y∗ and Xm+1 = X∗
end forym+1 = g(Xm+1); train the metamodel
12: If ym+1 < ym, Xm+1 = Xm; ym+1 = ym; m = m+ 1end while
end for
Séminaire S3 | March 13th 2015 | PAGE 19/26
Design pointsExample
Parabolic limit-state function: g : x ∈ R2 7−→ 5− x2 − 0.5(x1 − 0.1)2
Séminaire S3 | March 13th 2015 | PAGE 20/26
Design pointsExample
A two-dimensional four branches serial system:
g : x ∈ R2 7−→ min
(3 +
(x1 − x2)2
10− | x1 + x2 |√
2,7√2− | x1 − x2 |
)
Séminaire S3 | March 13th 2015 | PAGE 21/26
Plan
1 Increasing random walk
2 Probability estimation
3 Quantile estimation
4 Design points
5 Conclusion
Séminaire S3 | March 13th 2015 | PAGE 22/26
Conclusion
Conclusion
One considers Markov chains instead of samples
⇒ N is a number of processes
Lets de�ne parallel estimators for probabilities and quantiles (andmoments [8])
Twins of Monte Carlo estimators with a "log attribute": similarstatistical properties but adding a log to the 1/p factor:
var [p̂MC] ≈p2
Np→ var [p̂] ≈ p2 log 1/p
N
var [q̂MC] ≈p2
Nf(q)2p→ var [q̂] ≈ p2 log 1/p
Nf(q)2
Séminaire S3 | March 13th 2015 | PAGE 23/26
Conclusion
Perspectives
Adaptation of quantile estimator for optimisation problem (min ormax)
Problem of conditional simulations (Metropolis-Hastings)
Best use of a metamodel
Adaptation for discontinuous RV
Séminaire S3 | March 13th 2015 | PAGE 24/26
Bibliography I
S-K Au and J L Beck.Estimation of small failure probabilities in high dimensions by subsetsimulation.Probabilistic Engineering Mechanics, 16(4):263�277, 2001.
Charles-Edouard Bréhier, Ludovic Goudenege, and Loic Tudela.Central limit theorem for adaptative multilevel splitting estimators inan idealized setting.arXiv preprint arXiv:1501.01399, 2015.
Charles-Edouard Bréhier, Tony Lelievre, and Mathias Rousset.Analysis of adaptive multilevel splitting algorithms in an idealized case.arXiv preprint arXiv:1405.1352, 2014.
F Cérou, P Del Moral, T Furon, and A Guyader.Sequential Monte Carlo for rare event estimation.Statistics and Computing, 22(3):795�808, 2012.
Séminaire S3 | March 13th 2015 | PAGE 25/26
Bibliography II
A Guyader, N Hengartner, and E Matzner-Løber.Simulation and estimation of extreme quantiles and extremeprobabilities.Applied Mathematics & Optimization, 64(2):171�196, 2011.
Eric Simonnet.Combinatorial analysis of the adaptive last particle method.Statistics and Computing, pages 1�20, 2014.
Clement Walter.Moving Particles: a parallel optimal Multilevel Splitting method withapplication in quantiles estimation and meta-model based algorithms.To appear in Structural Safety, 2014.
Clement Walter.Point process-based estimation of kth-order moment.arXiv preprint arXiv:1412.6368, 2014.
Séminaire S3 | March 13th 2015 | PAGE 26/26
Merci !
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