Clément Walter

(CEA, DAM, DIF and Université Paris Diderot, Paris, FR)

https://s3-seminar.github.io/seminars/clement-walter

Title — Rare event simulation: a Point Process interpretation with application in probability and quantile estimation

Abstract — Clément Walter, 25, graduated from Mines ParisTech in 2013. Beforehand he attended preparatory classes in Lycée Sainte-Geneviève (branch Maths and Physics). For his master degree he specialised in Geostatistics and started working with CEA as an intern on emulation of complex computer codes (especially kriging) for rare event simulation and estimation. He has then pursued his work in a PhD under the direction of Pr. Josselin Garnier, focusing on multilevel splitting methods.

Transcript

1. Rare event simulation
   a Point Process interpretation with
   application in probability and quantile
   estimation and metamodel based
   algorithms
   Séminaire S3 | Clément WALTER
   March 13th 2015
   

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2. Introduction
   Problem setting:
   X random vector with know distribution µX
   g a "black-box" function representing a computer code: g : Rd → R
   Y = g(X) the real-valued random variable which describes the state of
   the system; its distribution µY is unknown
   Uncertainty Quantication: 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
   

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3. Introduction
   Two main directions to overcome this issue:
   learn a metamodel on g
   use variance-reduction techniques to estimate p
   Séminaire S3 | March 13th 2015 | PAGE 2/26
   

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4. Introduction
   Two main directions to overcome this issue:
   learn a metamodel on g
   use variance-reduction techniques to estimate p
   Séminaire S3 | March 13th 2015 | PAGE 2/26
   

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5. Introduction
   Two main directions to overcome this issue:
   learn a metamodel on g
   use variance-reduction techniques to estimate p
   Importance sampling
   Séminaire S3 | March 13th 2015 | PAGE 2/26
   

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6. Introduction
   Two main directions to overcome this issue:
   learn a metamodel on g
   use variance-reduction techniques to estimate p
   Importance sampling
   Multilevel splitting
   Séminaire S3 | March 13th 2015 | PAGE 2/26
   

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7. 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 and
   estimate them with MCMC: let (qi)i=0..m
   be an increasing sequence with
   q0 = −∞ 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]
   Séminaire S3 | March 13th 2015 | PAGE 2/26
   

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8. 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 and
   estimate them with MCMC: let (qi)i=0..m
   be an increasing sequence with
   q0 = −∞ 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
   

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9. 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 be
   greater than qj+1
   (the other ones don't change)
   4 j ← j + 1 and repeat until j = m
   Séminaire S3 | March 13th 2015 | PAGE 3/26
   

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10. 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 be
    greater 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
    Séminaire S3 | March 13th 2015 | PAGE 3/26
    

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11. 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 be
    greater 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 of
    subsets converges toward a constant log p/ log p0
    empirical quantiles of order k/N ⇒ no bias and CLT [3, 2]
    minimal variance with k = 1 (Last Particle Algorithm [5, 6]); the
    number of subsets follows a Poisson law with parameter −N log p
    ⇒ disables parallel computation
    Séminaire S3 | March 13th 2015 | PAGE 3/26
    

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12. Outline
    1 Increasing random walk
    2 Probability estimation
    3 Quantile estimation
    4 Design points
    5 Conclusion
    Séminaire S3 | March 13th 2015 | PAGE 4/26
    

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13. Outline
    1 Increasing random walk
    2 Probability estimation
    3 Quantile estimation
    4 Design points
    5 Conclusion
    Séminaire S3 | March 13th 2015 | PAGE 5/26
    

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14. Increasing random walk
    Denition
    Denition
    Let Y be a real-valued random variable with distribution µY and cdf F
    (assumed to be continuous). One considers the Markov chain (with
    Y0 = −∞) 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)
    Séminaire S3 | March 13th 2015 | PAGE 6/26
    

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15. Increasing random walk
    Denition
    Denition
    Let Y be a real-valued random variable with distribution µY and cdf F
    (assumed to be continuous). One considers the Markov chain (with
    Y0 = −∞) 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 a
    Poisson process with parameter 1 [5, 7]
    Séminaire S3 | March 13th 2015 | PAGE 6/26
    

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16. Increasing random walk
    Denition
    Denition
    Let Y be a real-valued random variable with distribution µY and cdf F
    (assumed to be continuous). One considers the Markov chain (with
    Y0 = −∞) 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 a
    Poisson process with parameter 1 [5, 7]
    Time in the Poisson process is linked with rarity in probability
    Séminaire S3 | March 13th 2015 | PAGE 6/26
    

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17. Increasing random walk
    Denition
    Denition
    Let Y be a real-valued random variable with distribution µY and cdf F
    (assumed to be continuous). One considers the Markov chain (with
    Y0 = −∞) 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 a
    Poisson 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 py
    L
    ∼ P(− log py)
    Séminaire S3 | March 13th 2015 | PAGE 6/26
    

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18. Increasing random walk
    Denition
    First consequence: number of simulations to get the realisation of a
    random variable above a given threshold (event with probability p) follows
    a 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
    Density
    Figure: Comparison of
    Poisson and Geometric
    densities with
    p = 0.0228
    Séminaire S3 | March 13th 2015 | PAGE 7/26
    

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19. Increasing random walk
    Example
    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
    -4 -2 0 2 4
    0.0 0.1 0.2 0.3 0.4
    x
    Density
    

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20. Plan
    1 Increasing random walk
    2 Probability estimation
    3 Quantile estimation
    4 Design points
    5 Conclusion
    Séminaire S3 | March 13th 2015 | PAGE 9/26
    

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21. Probability estimation
    Denition of the estimator
    Concept
    Number of events before time t = − log(P(Y > q)) follows a Poisson law
    with parameter t estimate the parameter of a Poisson law
    Séminaire S3 | March 13th 2015 | PAGE 10/26
    

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22. Probability estimation
    Denition of the estimator
    Concept
    Number of events before time t = − log(P(Y > q)) follows a Poisson law
    with 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)
    Séminaire S3 | March 13th 2015 | PAGE 10/26
    

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23. Probability estimation
    Denition of the estimator
    Concept
    Number of events before time t = − log(P(Y > q)) follows a Poisson law
    with 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
    −→ p = 1 −
    1
    N
    Mq
    Séminaire S3 | March 13th 2015 | PAGE 10/26
    

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24. Probability estimation
    Denition of the estimator
    Concept
    Number of events before time t = − log(P(Y > q)) follows a Poisson law
    with 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
    −→ p = 1 −
    1
    N
    Mq
    ⇒ Last Particle Estimator, but with parallel implementation
    Séminaire S3 | March 13th 2015 | PAGE 10/26
    

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25. Probability estimation
    Denition of the estimator
    Concept
    Number of events before time t = − log(P(Y > q)) follows a Poisson law
    with 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
    −→ p = 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
    My
    a.s.
    −
    −
    −
    −
    →
    N→∞
    F(y)
    Séminaire S3 | March 13th 2015 | PAGE 10/26
    

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26. Probability estimation
    Example
    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
    -4 -2 0 2 4
    0.0 0.4 0.8
    y
    Empirical cdf
    N = 2 ; p = 0.0078
    

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27. Probability estimation
    Practical implementation
    Ideally, one knows how to generate the Markov chain
    In 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
    Séminaire S3 | March 13th 2015 | PAGE 12/26
    

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28. Probability estimation
    Practical implementation
    Ideally, one knows how to generate the Markov chain
    In 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
    Séminaire S3 | March 13th 2015 | PAGE 12/26
    

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29. Probability estimation
    Practical implementation
    Ideally, one knows how to generate the Markov chain
    In 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, q
    Generate k copies (Xi)i=1..k
    according to µX ; Y ← (g(X1), · · · , g(Xk)); M = (0, · · · , 0)
    while min Y < q do
    3: ind ← which Y < q
    for
    i in ind do
    Mi = Mi + 1
    6: Generate X∗ ∼ µX (· | X > g(Xi))
    Xi ← X∗; Yi = g(X∗)
    end for
    9: end while
    Return M, (Xi)i=1..N
    , (Yi)i=1..N
    Séminaire S3 | March 13th 2015 | PAGE 12/26
    

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30. Probability estimation
    Practical implementation
    Ideally, one knows how to generate the Markov chain
    In 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, q
    Generate k copies (Xi)i=1..k
    according to µX ; Y ← (g(X1), · · · , g(Xk)); M = (0, · · · , 0)
    while min Y < q do
    3: ind ← which Y < q
    for
    i in ind do
    Mi = Mi + 1
    6: Generate X∗ ∼ µX (· | X > g(Xi))
    Xi ← X∗; Yi = g(X∗)
    end for
    9: end while
    Return 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
    

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31. Probability estimation
    Practical implementation
    Convergence of the Markov chain (for conditional sampling) is increased
    when the starting point already follows the targeted distribution; 2
    possibilities:
    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
    Séminaire S3 | March 13th 2015 | PAGE 13/26
    

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32. Probability estimation
    Practical implementation
    Convergence of the Markov chain (for conditional sampling) is increased
    when the starting point already follows the targeted distribution; 2
    possibilities:
    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 of
    size 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 +
    ncδ2
    (log p)2
    2 log nc +
    1
    T log 1/p
    
    
    Séminaire S3 | March 13th 2015 | PAGE 13/26
    

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33. Probability estimation
    Comparison
    Mean computer time against coecient of variation: the cost of an
    algorithm is the number of generated samples in a row by a core. We
    assume 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 p
    log p0
    N(1−p0)
    nc
    log p
    log p0
    1−p0
    Np0
    (1−p0)2
    p0(log p0)2
    T(log p)2
    ncδ2
    LPA −TN log p −log p
    N
    T(log p)2
    δ2
    Random walk −T N
    nc
    log p −log p
    N
    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 parallel
    implementation
    Séminaire S3 | March 13th 2015 | PAGE 14/26
    

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34. Plan
    1 Increasing random walk
    2 Probability estimation
    3 Quantile estimation
    4 Design points
    5 Conclusion
    Séminaire S3 | March 13th 2015 | PAGE 15/26
    

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35. Quantile estimation
    Denition
    Concept
    Approximate a time t = − log p with times of a Poisson process; the higher
    the rate, the denser the discretisation of [0; +∞[
    Séminaire S3 | March 13th 2015 | PAGE 16/26
    

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36. Quantile estimation
    Denition
    Concept
    Approximate a time t = − log p with times of a Poisson process; the higher
    the rate, the denser the discretisation of [0; +∞[
    The center of the interval [TMt
    ; TMt+1] converges toward a random
    variable centred in t with symmetric pdf
    Séminaire S3 | March 13th 2015 | PAGE 16/26
    

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37. Quantile estimation
    Denition
    Concept
    Approximate a time t = − log p with times of a Poisson process; the higher
    the rate, the denser the discretisation of [0; +∞[
    The center of the interval [TMt
    ; TMt+1] converges toward a random
    variable centred in t with symmetric pdf
    q =
    1
    2
    (Ym + Ym+1) with m = E[Mq] = −N log p
    Séminaire S3 | March 13th 2015 | PAGE 16/26
    

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38. Quantile estimation
    Denition
    Concept
    Approximate a time t = − log p with times of a Poisson process; the higher
    the rate, the denser the discretisation of [0; +∞[
    The center of the interval [TMt
    ; TMt+1] converges toward a random
    variable centred in t with symmetric pdf
    q =
    1
    2
    (Ym + Ym+1) with m = E[Mq] = −N log p
    CLT:
    √
    N (q − q) L
    −→
    m→∞
    N 0,
    −p2 log p
    f(q)2
    Bounds on bias on O(1/N)
    Séminaire S3 | March 13th 2015 | PAGE 16/26
    

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39. Quantile estimation
    Example
    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
    -4 -2 0 2 4
    -0.10 0.00 0.10
    y
    N = 2 ; q = 1.348
    

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40. Plan
    1 Increasing random walk
    2 Probability estimation
    3 Quantile estimation
    4 Design points
    5 Conclusion
    Séminaire S3 | March 13th 2015 | PAGE 18/26
    

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41. Design points
    Algorithm
    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 DoE
    Learn a rst metamodel with trend = failure
    3: for Nfail
    times do
    Simulate the random walks one after the other
    Sample X1 ∼ µX ; y1 = g(X1); m = 1; train the metamodel
    while ym < q do
    6: Xm+1 = Xm
    ; ym+1 = ym
    for T times do
    Pseudo burn-in
    X∗ ∼ K(Xm+1, ·); g(X∗) = y∗ K is a kernel for Markov chain sampling
    9: If y∗ > ym+1
    , ym+1 = y∗ and Xm+1 = X∗
    end for
    ym+1 = g(Xm+1); train the metamodel
    12: If ym+1 < ym
    , Xm+1 = Xm
    ; ym+1 = ym
    ; m = m + 1
    end while
    end for
    Séminaire S3 | March 13th 2015 | PAGE 19/26
    

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42. Design points
    Example
    Parabolic limit-state function: g : x ∈ R2 −→ 5 − x2 − 0.5(x1 − 0.1)2
    Séminaire S3 | March 13th 2015 | PAGE 20/26
    −5 0 5
    −5 0 5
    x
    y
    LSF
    LSF
    

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43. Design points
    Example
    A two-dimensional four branches serial system:
    g : x ∈ R2 −→ min 3 +
    (x1 − x2)2
    10
    −
    | x1 + x2 |
    √
    2
    ,
    7
    √
    2
    − | x1 − x2 |
    Séminaire S3 | March 13th 2015 | PAGE 21/26
    −5 0 5
    −5 0 5
    x
    y
    LSF
    

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44. Plan
    1 Increasing random walk
    2 Probability estimation
    3 Quantile estimation
    4 Design points
    5 Conclusion
    Séminaire S3 | March 13th 2015 | PAGE 22/26
    

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45. Conclusion
    Conclusion
    One considers Markov chains instead of samples
    ⇒ N is a number of processes
    Lets dene parallel estimators for probabilities and quantiles (and
    moments [8])
    Twins of Monte Carlo estimators with a "log attribute": similar
    statistical properties but adding a log to the 1/p factor:
    var [pMC] ≈
    p2
    Np
    → var [p] ≈
    p2 log 1/p
    N
    var [qMC] ≈
    p2
    Nf(q)2p
    → var [q] ≈
    p2 log 1/p
    Nf(q)2
    Séminaire S3 | March 13th 2015 | PAGE 23/26
    

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46. Conclusion
    Perspectives
    Adaptation of quantile estimator for optimisation problem (min or
    max)
    Problem of conditional simulations (Metropolis-Hastings)
    Best use of a metamodel
    Adaptation for discontinuous RV
    Séminaire S3 | March 13th 2015 | PAGE 24/26
    

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47. Bibliography I
    S-K Au and J L Beck.
    Estimation of small failure probabilities in high dimensions by subset
    simulation.
    Probabilistic Engineering Mechanics, 16(4):263277, 2001.
    Charles-Edouard Bréhier, Ludovic Goudenege, and Loic Tudela.
    Central limit theorem for adaptative multilevel splitting estimators in
    an 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):795808, 2012.
    Séminaire S3 | March 13th 2015 | PAGE 25/26
    

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48. Bibliography II
    A Guyader, N Hengartner, and E Matzner-Løber.
    Simulation and estimation of extreme quantiles and extreme
    probabilities.
    Applied Mathematics & Optimization, 64(2):171196, 2011.
    Eric Simonnet.
    Combinatorial analysis of the adaptive last particle method.
    Statistics and Computing, pages 120, 2014.
    Clement Walter.
    Moving Particles: a parallel optimal Multilevel Splitting method with
    application 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
    

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49. Merci !
    Commissariat à l'énergie atomique et aux énergies alternatives
    CEA, DAM, DIF, F-91297 Arpajon, France
    Établissement public à caractère industriel et commercial | RCS Paris B 775 685 019
    CEA
    DAM
    DIF
    

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