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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
Potential manufacturing defects
Extreme thermo-mechanical constraints
Jeopardize the structure
Several parameters influence the structuralsafety
The parameters are uncertain
€€€IM
PORTAN
T !
AGEIN
G
Steal Component
Coating
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 1 / 26
Industrial context Industrial Application
Industrial Context
Important to safety component exposed toan ageing phenomenon
Potential manufacturing defects
Extreme thermo-mechanical constraints
Jeopardize the structure
Several parameters influence the structuralsafety
The parameters are uncertain
€€€IM
PORTAN
T !
AGEIN
G
Steal Component
Coating
Defect
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 1 / 26
Industrial context Industrial Application
Industrial Context
Important to safety component exposed toan ageing phenomenon
Potential manufacturing defects
Extreme thermo-mechanical constraints
Jeopardize the structure
Several parameters influence the structuralsafety
The parameters are uncertain
€€€IM
PORTAN
T !
AGEIN
G
Steal Component
Coating
Defect
Hot
Water
Cold
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 1 / 26
Industrial context Industrial Application
Industrial Context
Important to safety component exposed toan ageing phenomenon
Potential manufacturing defects
Extreme thermo-mechanical constraints
Jeopardize the structure
Several parameters influence the structuralsafety
The parameters are uncertain
€€€IM
PORTAN
T !
AGEIN
G
Steal Component
Coating
Defect
Hot
Water
Cold
Thermal Loading
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 1 / 26
Industrial context Industrial Application
Industrial Context
Important to safety component exposed toan ageing phenomenon
Potential manufacturing defects
Extreme thermo-mechanical constraints
Jeopardize the structure
Several parameters influence the structuralsafety
The parameters are uncertain
€€€IM
PORTAN
T !
AGEIN
G
Steal Component
Coating
Defect
Hot
Water
Cold
Thermal Loading
L H
Width
T (°C)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 1 / 26
Industrial context Industrial Application
Industrial Context
Important to safety component exposed toan ageing phenomenon
Potential manufacturing defects
Extreme thermo-mechanical constraints
Jeopardize the structure
Several parameters influence the structuralsafety
The parameters are uncertain
€€€IM
PORTAN
T !
AGEIN
G
Steal Component
Coating
Defect
Hot
Water
Cold
Thermal Loading
L H
Width
T (°C)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 1 / 26
Industrial context Industrial Application
Structural Reliability
Q(α)⟂
ℙ[Y≥t] ⟂
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Question:How important the dependence structure of X is on the quantity of interest of Y ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].
How to describe the dependence structure?
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 2 / 26
Industrial context Industrial Application
Structural Reliability
?
Dependence
IncompleteUnknown
Known
Q(α)
ℙ[Y≥t]
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Question:How important the dependence structure of X is on the quantity of interest of Y ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].
How to describe the dependence structure?
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 2 / 26
Industrial context Industrial Application
Structural Reliability
?
Dependence
IncompleteUnknown
Known
Q(α)
ℙ[Y≥t]
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Question:How important the dependence structure of X is on the quantity of interest of Y ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].
How to describe the dependence structure?
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 2 / 26
Industrial context Industrial Application
Structural Reliability
?
Dependence
IncompleteUnknown
Known
Q(α)
ℙ[Y≥t]
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Question:How important the dependence structure of X is on the quantity of interest of Y ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].
How to describe the dependence structure?
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 2 / 26
Industrial context Industrial Application
Structural Reliability
?
Dependence
IncompleteUnknown
Known
Q(α)
ℙ[Y≥t]
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Question:How important the dependence structure of X is on the quantity of interest of Y ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].
How to describe the dependence structure?
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 2 / 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 )).
Family Cθ(u, v) Θ Kendall’s τ
Independent uv / /Gaussian Φθ(Φ−1(u),Φ−1(v)) [−1, 1] 2 arcsin θ
π
Clayton (u−θ + v−θ − 1)−1/θ [0,∞] θ2+θ
Gumbel exp−[(− ln u)θ + (− ln v)θ]1/θ
[1,∞] 1− 1
θ
Table: Example of copula families.
[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:
θn = argmaxθ∈Θ
Qn,θ(α).
Nazih 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 problem
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).
Nazih 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
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
P[|θn,K − θ∗| > h] n→∞−−−→ 0.
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 7 / 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
P[|θn,K − θ∗| > h] n→∞−−−→ 0.
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 7 / 26
Methodology Multi-dimensional problem
How to model a multivariate copula?
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
Vine CopulasPros: flexibilityCons: complexity
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 8 / 26
Methodology Multi-dimensional problem
How to model a multivariate copula?
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
Vine CopulasPros: flexibilityCons: complexity
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 8 / 26
Methodology Multi-dimensional problem
How to model a multivariate copula?
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
Vine CopulasPros: flexibilityCons: complexity
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 8 / 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)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) · c34(F3(x3),F4(x4))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2)) · c24|3(F2|3(x2|x3),F4|3(x4|x3))(conditional pair)× c14|23(F1|23(x1|x2, x3),F4|23(x4|x2, x3))(conditional pair)
[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)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) · c34(F3(x3),F4(x4))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2)) · c24|3(F2|3(x2|x3),F4|3(x4|x3))(conditional pair)× c14|23(F1|23(x1|x2, x3),F4|23(x4|x2, x3))(conditional pair)
1 2 3 4
[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)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) · c34(F3(x3),F4(x4))(unconditional pairs)
× c13|2(F1|2(x1|x2),F3|2(x3|x2)) · c24|3(F2|3(x2|x3),F4|3(x4|x3))(conditional pair)× c14|23(F1|23(x1|x2, x3),F4|23(x4|x2, x3))(conditional pair)
1,2 2,3 3,41 2 3 4
[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)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) · c34(F3(x3),F4(x4))(unconditional pairs)
× c13|2(F1|2(x1|x2),F3|2(x3|x2)) · c24|3(F2|3(x2|x3),F4|3(x4|x3))(conditional pair)× c14|23(F1|23(x1|x2, x3),F4|23(x4|x2, x3))(conditional pair)
3,42,31,2
1,2 2,3 3,41 2 3 4
[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)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) · c34(F3(x3),F4(x4))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2)) · c24|3(F2|3(x2|x3),F4|3(x4|x3))(conditional pair)
× c14|23(F1|23(x1|x2, x3),F4|23(x4|x2, x3))(conditional pair)
1,3|2 2,4|3 3,42,31,2
1,2 2,3 3,41 2 3 4
[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)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) · c34(F3(x3),F4(x4))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2)) · c24|3(F2|3(x2|x3),F4|3(x4|x3))(conditional pair)
× c14|23(F1|23(x1|x2, x3),F4|23(x4|x2, x3))(conditional pair)
2,4|31,3|2
1,3|2 2,4|3 3,42,31,2
1,2 2,3 3,41 2 3 4
[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)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) · c34(F3(x3),F4(x4))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2)) · c24|3(F2|3(x2|x3),F4|3(x4|x3))(conditional pair)× c14|23(F1|23(x1|x2, x3),F4|23(x4|x2, x3))(conditional pair)
1,4|2,3 2,4|31,3|2
1,3|2 2,4|3
3,42,31,2
1,2 2,3 3,41,2 2,3 3,41 2 3 4
[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)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) · c34(F3(x3),F4(x4))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2)) · c24|3(F2|3(x2|x3),F4|3(x4|x3))(conditional pair)× c14|23(F1|23(x1|x2, x3),F4|23(x4|x2, x3))(conditional pair)
1,4|2,3 2,4|31,3|2
1,3|2 2,4|3
3,42,31,2
1,2 2,3 3,41,2 2,3 3,41 2 3 4
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 10 / 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)
× c12(F1(x1),F2(x2)) · c24(F2(x2),F4(x4)) · c34(F3(x3),F4(x4))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2)) · c23|4(F2|4(x2|x4),F3|4(x3|x4))(conditional pair)× c14|23(F1|23(x1|x2, x3),F4|23(x4|x2, x3))(conditional pair)
1,4|2,3 2,3|41,3|2
1,3|2 2,3|4 3,42,41,2
1,2 2,4 4,34 31 2
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 10 / 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.
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00Correlation
42
41
40
39
38
Qua
ntile
at
=5%
QuantileComonotonicityIndependence
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.
Figure: C-vine illustration in dimension d = 5
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 11 / 26
Methodology Vine Copula
Figure: Quantile estimations with the dependence of each pairs in dimension d = 3.
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 12 / 26
Methodology Vine Copula
Figure: Quantile estimations with the dependence of each pairs in dimension d = 4.
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 12 / 26
Methodology Vine Copula
3 4 5 6 7 8Dimension
25
20
15
10
5
0
5
10
15
20Q
uant
ile d
iffer
ence
with
inde
pend
ence Grid-search
1000
2000
3000
4000
5000
6000
7000
Gri
d-si
ze
Figure: Q⊥⊥(α)− Qθn(α)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 13 / 26
Methodology Vine Copula
Figure: Sampling of X at the obtained minimum for d = 8
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 14 / 26
Methodology Vine Copula
Grid-search with R-vines
How can we chose the vine structure?Are all the pairs influential on the minimization?Can we simplify the dependence structure?
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 15 / 26
Methodology Iterative Construction of Dependence Structure
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 16 / 26
Methodology Iterative Construction of Dependence Structure
Illustration in d = 4 with d(d − 1)/2 = 6 pairs
Set of all pairs: Ω = (i , j) : 1 ≤ i , j ≤ dSelected pairs: Ωk
Candidate pairs: Ω−k = Ω\Ωk
k = 0: Ωk = ∅, Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (1, 4), (2, 4)k = 1: Ωk = (1, 4), Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (2, 4)k = 2: Ωk = (1, 4), (3, 4), Ω−k = (1, 2), (2, 3), (1, 3), (2, 4)Results: Ωk = (1, 4), (3, 4), (2, 4)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 17 / 26
Methodology Iterative Construction of Dependence Structure
Illustration in d = 4 with d(d − 1)/2 = 6 pairs
Set of all pairs: Ω = (i , j) : 1 ≤ i , j ≤ dSelected pairs: Ωk
Candidate pairs: Ω−k = Ω\Ωk
k = 0: Ωk = ∅, Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (1, 4), (2, 4)
k = 1: Ωk = (1, 4), Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (2, 4)k = 2: Ωk = (1, 4), (3, 4), Ω−k = (1, 2), (2, 3), (1, 3), (2, 4)Results: Ωk = (1, 4), (3, 4), (2, 4)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 17 / 26
Methodology Iterative Construction of Dependence Structure
Illustration in d = 4 with d(d − 1)/2 = 6 pairs
Set of all pairs: Ω = (i , j) : 1 ≤ i , j ≤ dSelected pairs: Ωk
Candidate pairs: Ω−k = Ω\Ωk
k = 0: Ωk = ∅, Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (1, 4), (2, 4)
k = 1: Ωk = (1, 4), Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (2, 4)k = 2: Ωk = (1, 4), (3, 4), Ω−k = (1, 2), (2, 3), (1, 3), (2, 4)Results: Ωk = (1, 4), (3, 4), (2, 4)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 17 / 26
Methodology Iterative Construction of Dependence Structure
Illustration in d = 4 with d(d − 1)/2 = 6 pairs
Set of all pairs: Ω = (i , j) : 1 ≤ i , j ≤ dSelected pairs: Ωk
Candidate pairs: Ω−k = Ω\Ωk
k = 0: Ωk = ∅, Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (1, 4), (2, 4)k = 1: Ωk = (1, 4), Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (2, 4)
k = 2: Ωk = (1, 4), (3, 4), Ω−k = (1, 2), (2, 3), (1, 3), (2, 4)Results: Ωk = (1, 4), (3, 4), (2, 4)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 17 / 26
Methodology Iterative Construction of Dependence Structure
Illustration in d = 4 with d(d − 1)/2 = 6 pairs
Set of all pairs: Ω = (i , j) : 1 ≤ i , j ≤ dSelected pairs: Ωk
Candidate pairs: Ω−k = Ω\Ωk
k = 0: Ωk = ∅, Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (1, 4), (2, 4)k = 1: Ωk = (1, 4), Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (2, 4)
k = 2: Ωk = (1, 4), (3, 4), Ω−k = (1, 2), (2, 3), (1, 3), (2, 4)Results: Ωk = (1, 4), (3, 4), (2, 4)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 17 / 26
Methodology Iterative Construction of Dependence Structure
Illustration in d = 4 with d(d − 1)/2 = 6 pairs
Set of all pairs: Ω = (i , j) : 1 ≤ i , j ≤ dSelected pairs: Ωk
Candidate pairs: Ω−k = Ω\Ωk
k = 0: Ωk = ∅, Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (1, 4), (2, 4)k = 1: Ωk = (1, 4), Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (2, 4)k = 2: Ωk = (1, 4), (3, 4), Ω−k = (1, 2), (2, 3), (1, 3), (2, 4)
Results: Ωk = (1, 4), (3, 4), (2, 4)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 17 / 26
Methodology Iterative Construction of Dependence Structure
Illustration in d = 4 with d(d − 1)/2 = 6 pairs
Set of all pairs: Ω = (i , j) : 1 ≤ i , j ≤ dSelected pairs: Ωk
Candidate pairs: Ω−k = Ω\Ωk
k = 0: Ωk = ∅, Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (1, 4), (2, 4)k = 1: Ωk = (1, 4), Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (2, 4)k = 2: Ωk = (1, 4), (3, 4), Ω−k = (1, 2), (2, 3), (1, 3), (2, 4)Results: Ωk = (1, 4), (3, 4), (2, 4)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 17 / 26
Methodology Iterative Construction of Dependence Structure
Illustration in d = 4 with d(d − 1)/2 = 6 pairs
Set of all pairs: Ω = (i , j) : 1 ≤ i , j ≤ dSelected pairs: Ωk
Candidate pairs: Ω−k = Ω\Ωk
k = 0: Ωk = ∅, Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (1, 4), (2, 4)k = 1: Ωk = (1, 4), Ω−k = (1, 2), (2, 3), (3, 4), (1, 3), (2, 4)k = 2: Ωk = (1, 4), (3, 4), Ω−k = (1, 2), (2, 3), (1, 3), (2, 4)Results: Ωk = (1, 4), (3, 4), (2, 4)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 17 / 26
Methodology Iterative Construction of Dependence Structure
Back to the example in d = 8
1 2 3Iterations
670
660
650
640
630
620
610
Out
put q
uant
ile
IndependenceGrid-searchIterative
Figure: Resulting output quantiles of each method
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 18 / 26
Methodology Iterative Construction of Dependence Structure
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 19 / 26
Methodology Iterative Construction of Dependence Structure
How it changes the output distributions?
1200 1000 800 600 400 200y
0.0000
0.0005
0.0010
0.0015
0.0020
p.d.
f
Grid-searchIterativeIndependence
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 20 / 26
Methodology Iterative Construction of Dependence Structure
3 4 5 6 7 8Dimension
25
20
15
10
5
0
5
10
15
20Q
uant
ile d
iffer
ence
with
inde
pend
ence Iterative
Grid-search
1000
2000
3000
4000
5000
6000
7000
Gri
d-si
ze
Figure: Q⊥⊥(α)− Qθn(α)
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 21 / 26
Methodology Iterative Construction of Dependence Structure
Results
For d = 6, p = 13 (2 known dependences) and bounds on Θ. Moreover, Qθ(α) ismonotonic with θ.
1 2 3Number of considered pairs
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Quan
tile
at
=0.
01
independencegrid-search with K = 1000
Costs for n = 15000.Grid Search: n × K = 15× 106
Sequential:∑lmax
l=1 (p − l + 1)× k(l) = 6.5× 106
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 22 / 26
Contents
1 Industrial context
2 Methodology
3 Discussion
Discussion
Conclusion
Dependencies can have a significant impact on the model output Y and the targetedquantity of interest.
We used a grid-search strategy to determine a parametric copula that minimizes theoutput quantile of a model
We used regular vines to model multivariate copulas
We proposed a sequential strategy to minimize the quantile by selecting the mostinfluential pairs of variables
Library dep-impact available onPyPi: https://pypi.python.org/pypi/dep-impactGithub: https://github.com/nazben/dep-impact
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 23 / 26
Discussion
To Infinity and Beyond
Stay with parametric copulas: Bayesian optimization
Using non parametric dependence structure?
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 24 / 26
Discussion
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 25 / 26
Discussion
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 25 / 26
Discussion
Thank You!
Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 26 / 26