Quantile Estimation in Structural Reliability with Incomplete Dependence Structure Nazih Benoumechiara 12 Gérard Biau 1 Bertrand Michel 3 Philippe Saint-Pierre 4 Roman Sueur 2 Nicolas Bousquet 1,5 Bertrand Iooss 2,4 1 Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université / UPMC, Paris. 2 Performances, Risques Industriels et Surveillance pour la Maintenance et l’Exploitation (PRISME), EDF R&D, Chatou. 3 Ecole Centrale de Nantes (ECN), Nantes. 4 Institut de Mathématique de Toulouse (IMT), Université Paul Sabatier, Toulouse. 5 Quantmetry R&D, Paris. Wednesday, March 21th 2018
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Quantile Estimation in Structural Reliability with Incomplete DependenceStructure
Nazih Benoumechiara 1 2
Gérard Biau 1 Bertrand Michel 3 Philippe Saint-Pierre 4
Roman Sueur 2 Nicolas Bousquet 1,5 Bertrand Iooss 2,4
1Laboratoire de Probabilités, Statistique et Modélisation,Sorbonne Université / UPMC, Paris.
2Performances, Risques Industriels et Surveillance pour la Maintenance et l’Exploitation (PRISME),EDF R&D, Chatou.
3Ecole Centrale de Nantes (ECN),Nantes.
4Institut de Mathématique de Toulouse (IMT),Université Paul Sabatier, Toulouse.
5Quantmetry R&D,Paris.
Wednesday, March 21th 2018
Contents
1 Industrial context
2 Methodology
3 Discussion
Contents
1 Industrial context
2 Methodology
3 Discussion
Industrial context Industrial Application
Industrial Context
Important to safety component exposed toan ageing phenomenon
[5] Roger B Nelsen. An introduction to copulas. Springer Science & Business Media, 2007.[6] Abe Sklar. Fonctions de répartition à n dimensions et leurs marges. Vol. 8. ISUP, 1959, pp. 229–231.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 3 / 26
Industrial context Industrial Application
Copulas
The dependence structure is described by a parametric copula Cθ with θ ∈ Θ ⊆ Rp suchas[5,6]
FX(x) = Cθ(F1(x1), . . . ,Fd (xd )).
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0Normal copula with θ= 0. 81
0.0 0.2 0.4 0.6 0.8 1.0
Clayton copula with θ= 3. 00
0.0 0.2 0.4 0.6 0.8 1.0
Gumbel copula with θ= 7. 93
Figure: Example of copula densities with τ = 0.6.
[5] Roger B Nelsen. An introduction to copulas. Springer Science & Business Media, 2007.[6] Abe Sklar. Fonctions de répartition à n dimensions et leurs marges. Vol. 8. ISUP, 1959, pp. 229–231.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 3 / 26
Industrial context Industrial Application
Copulas
The dependence structure is described by a parametric copula Cθ with θ ∈ Θ ⊆ Rp suchas[5,6]
FX(x) = Cθ(F1(x1), . . . ,Fd (xd )).
3 2 1 0 1 2 33
2
1
0
1
2
3
3 2 1 0 1 2 33
2
1
0
1
2
3
3 2 1 0 1 2 33
2
1
0
1
2
3
Figure: Example of joints p.d.f with Gaussian margins and τ = 0.6.
[5] Roger B Nelsen. An introduction to copulas. Springer Science & Business Media, 2007.[6] Abe Sklar. Fonctions de répartition à n dimensions et leurs marges. Vol. 8. ISUP, 1959, pp. 229–231.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 3 / 26
Industrial context The worst case scenario
Cθ
Copula
IncompleteUnknown
Known
Q(α)θ
ℙ[Y≥t] θ
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
We are interested on the output quantile.For a given probability α ∈ (0, 1) and a parametric copula Cθ such that θ ∈ Θ ⊆ Rp, westate the maximization problem
θ∗ = argmaxθ∈Θ
Qθ(α),
which gives the upper boundQθ∗ (α) ≥ Q⊥(α).
Because Y is not explicitly known, Qθ(α) is estimated:
We are interested on the output quantile.For a given probability α ∈ (0, 1) and a parametric copula Cθ such that θ ∈ Θ ⊆ Rp, westate the maximization problem1
θ∗ = argmaxθ∈Θ
Qθ(α),
which gives the upper boundQθ∗ (α) ≥ Q⊥(α).
Because Y is not explicitly known, Qθ(α) is estimated:
θn = argmaxθ∈Θ
Qn,θ(α).
1 Or minimizationNazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 4 / 26
Industrial context The worst case scenario
We are interested on the output quantile.For a given probability α ∈ (0, 1) and a parametric copula Cθ such that θ ∈ Θ ⊆ Rp, westate the maximization problem1
θ∗ = argmaxθ∈Θ
Qθ(α),
which gives the upper boundQθ∗ (α) ≥ Q⊥(α).
Because Y is not explicitly known, Qθ(α) is estimated:
θn = argmaxθ∈Θ
Qn,θ(α).
1 Or minimizationNazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 4 / 26
Contents
1 Industrial context
2 Methodology
3 Discussion
Methodology References
Related Studies
Several studies showed the influence of dependencies[3,7].
Figure: Variation of the output probability for different copula families.
[3] Mircea Grigoriu and Carl Turkstra. “Safety of structural systems with correlated resistances”. In: AppliedMathematical Modelling 3.2 (1979), pp. 130–136.[7] Xiao-Song Tang et al. “Impact of copulas for modeling bivariate distributions on system reliability”. In:Structural safety 44 (2013), pp. 80–90.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 5 / 26
Methodology References
The worst case is not always at the edge
“Fallacy 3. The worst case VaR (quantile) for a linear portfolio X + Y occurswhen ρ(X ,Y ) is maximal, i.e. X and Y are comonotonic.[2] ”
For example, we consider:X1 ∼ N (0, 1), X2 ∼ N (−2, 1) and different copula families with τ ∈ [−1, 1]η(x1, x2) = x2
1 x22 − x1x2
0.5 0.0 0.5
Kendall τ
Qθ(α
)
Independence
Normal
Clayton
Gumbel
Figure: Variation of the output quantile for different copula families and α = 5%.
[2] Paul Embrechts, Alexander McNeil, and Daniel Straumann. “Correlation and dependence in risk manage-ment: properties and pitfalls”. In: Risk management: value at risk and beyond (2002), pp. 176–223.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 6 / 26
Methodology Grid Search
In practice, θ is discretized using a thin grid ΘK of size K :
θn,K = argmaxθ∈ΘK
Qn,θ(α).
Theorem 1 (Consistency of θn,K )As K tends to infinity and under regularity assumptions of η and FX, for a givenα ∈ (0, 1) and for all ε > 0 we have
P(∣∣∣Qθn,K
(α)− Qθ∗ (α)∣∣∣ > ε
)n→∞−−−→ 0.
Moreover, if Qθ is uniquely minimized at θ∗, then for all h > 0 we have
Elliptical Copulas: the worst case correlation matrixPros: intuitive, simple to implementCons: assumption of linear correlations, no tails dependencies, ...
Archimedian CopulasPros: simple, tail dependenciesCons: not flexibile with θ
Elliptical Copulas: the worst case correlation matrixPros: intuitive, simple to implementCons: assumption of linear correlations, no tails dependencies, ...
Archimedian CopulasPros: simple, tail dependenciesCons: not flexibile with θ
Elliptical Copulas: the worst case correlation matrixPros: intuitive, simple to implementCons: assumption of linear correlations, no tails dependencies, ...
Archimedian CopulasPros: simple, tail dependenciesCons: not flexibile with θ
[4] Harry Joe. “Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependenceparameters”. In: Lecture Notes-Monograph Series (1996), pp. 120–141.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 9 / 26
Methodology Vine Copula
Regular Vines
The joint density f (x1, . . . , xd ) can be represented by a product of pair-copula densitiesand marginal densities[4].For example in d = 4. One possible decomposition of f (x1, x2, x3, x4) is:
f (x1, x2, x3, x4) = f1(x1)f2(x2)f3(x3)f4(x4)(margins)
[4] Harry Joe. “Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependenceparameters”. In: Lecture Notes-Monograph Series (1996), pp. 120–141.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 9 / 26
Methodology Vine Copula
Regular Vines
The joint density f (x1, . . . , xd ) can be represented by a product of pair-copula densitiesand marginal densities[4].For example in d = 4. One possible decomposition of f (x1, x2, x3, x4) is:
f (x1, x2, x3, x4) = f1(x1)f2(x2)f3(x3)f4(x4)(margins)
[4] Harry Joe. “Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependenceparameters”. In: Lecture Notes-Monograph Series (1996), pp. 120–141.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 9 / 26
Methodology Vine Copula
Regular Vines
The joint density f (x1, . . . , xd ) can be represented by a product of pair-copula densitiesand marginal densities[4].For example in d = 4. One possible decomposition of f (x1, x2, x3, x4) is:
f (x1, x2, x3, x4) = f1(x1)f2(x2)f3(x3)f4(x4)(margins)
[4] Harry Joe. “Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependenceparameters”. In: Lecture Notes-Monograph Series (1996), pp. 120–141.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 9 / 26
Methodology Vine Copula
Regular Vines
The joint density f (x1, . . . , xd ) can be represented by a product of pair-copula densitiesand marginal densities[4].For example in d = 4. One possible decomposition of f (x1, x2, x3, x4) is:
f (x1, x2, x3, x4) = f1(x1)f2(x2)f3(x3)f4(x4)(margins)
[4] Harry Joe. “Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependenceparameters”. In: Lecture Notes-Monograph Series (1996), pp. 120–141.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 9 / 26
Methodology Vine Copula
Regular Vines
The joint density f (x1, . . . , xd ) can be represented by a product of pair-copula densitiesand marginal densities[4].For example in d = 4. One possible decomposition of f (x1, x2, x3, x4) is:
f (x1, x2, x3, x4) = f1(x1)f2(x2)f3(x3)f4(x4)(margins)
[4] Harry Joe. “Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependenceparameters”. In: Lecture Notes-Monograph Series (1996), pp. 120–141.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 9 / 26
Methodology Vine Copula
Regular Vines
The joint density f (x1, . . . , xd ) can be represented by a product of pair-copula densitiesand marginal densities[4].For example in d = 4. One possible decomposition of f (x1, x2, x3, x4) is:
f (x1, x2, x3, x4) = f1(x1)f2(x2)f3(x3)f4(x4)(margins)
[4] Harry Joe. “Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependenceparameters”. In: Lecture Notes-Monograph Series (1996), pp. 120–141.Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 9 / 26
Methodology Vine Copula
Regular Vines
Problem: There is very large number of possible Vine decompositions :(d2
)× (n − 2)!× 2(d−2
2 ).For example, when d = 6, there are 23.040 possible R-vines.
f (x1, x2, x3, x4) = f1(x1)f2(x2)f3(x3)f4(x4)(margins)
Figure: Quantile variation with the correlation in dimension d = 2Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 11 / 26
Methodology Vine Copula
Example: Grid-search
The model:
Y = −d∑
j=1
βjXj ,
where βj = 10j
d−1 .The marginal distributions: Generalized Pareto with σ = 10 and ξ = 0.75.The copula families: d(d − 1)/2 Gaussian copulas.The vine structure: a C-vine.