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Impact of Dependencies Between Input Variables in Structural ReliabilityProblems Under Incomplete Probability Information
Nazih Benoumechiara 1 2
Gérard Biau 1 Bertrand Michel 3 Philippe Saint-Pierre 4
Roman Sueur 2 Nicolas Bousquet 2 Bertrand Iooss 2
1Laboratoire de Statistique Théorique et Appliquée (LSTA),Université Pierre-et-Marie-Curie, Paris.
2Département Management des Risques Industriels (MRI),EDF R&D, Chatou.
3Ecole Centrale de Nantes (ECN),Nantes.
4Institut de Mathématique de Toulouse (IMT),Université Paul Sabatier, Toulouse.
Wednesday, November 23th 2016
Contents
1 Illustration: Flood Example
2 General Framework
3 Methodology
4 Discussion
Contents
1 Illustration: Flood ExampleThe ModelOverflow Probability Estimation
2 General Framework
3 Methodology
4 Discussion
Illustration: Flood Example The Model
The Flood Model[6]
S = Zv + H + Hd − Cb
with H =
(Q
BKs√
Zm−ZvL
)
Input Description Probability Distribution
Q Maximal annual flowrate Truncated Gumbel G(1013, 558) on [500, 3000]Ks Strickler coefficient Truncated normal N (30, 8) on [15, +∞]Zv River downstream level Triangular T (49, 50, 51)Zm River upstream level Triangular T (54, 55, 56)Hd Dyke height Uniform U [7, 9]Cb Bank level Triangular T (55, 55.5, 56)L Length of the river stretch Triangular T (4990, 5000, 5010)B River width Triangular T (295, 300, 305)
[6] Bertrand Iooss and Paul Lemaître. “A review on global sensitivity analysis methods”. In: UncertaintyManagement in Simulation-Optimization of Complex Systems. Springer, 2015, pp. 101–122.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 1 / 17
Illustration: Flood Example The Model
The Flood Model[6]
S = Zv + H + Hd − Cb
with H =
(Q
BKs√
Zm−ZvL
)
ℙ[S>0]
0
Overflow S
s
fS( )s
OverflowQ
Ks
B
η
ModelJoint
Distribution
FX
Margins
[6] Bertrand Iooss and Paul Lemaître. “A review on global sensitivity analysis methods”. In: UncertaintyManagement in Simulation-Optimization of Complex Systems. Springer, 2015, pp. 101–122.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 1 / 17
Illustration: Flood Example Overflow Probability Estimation
Monte-Carlo Estimation with the Independence Assumption
Estimation of Pf = P[S > 0].
102 103 104 105
Sample Size
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014O
verf
low
Pro
babili
ty P̂f = 3. 9e− 03
15 10 5 0 5 10Overflow S
0.00
0.05
0.10
0.15
0.20
0.25
Densi
ty
Threshold
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 2 / 17
Illustration: Flood Example Overflow Probability Estimation
Monte-Carlo Estimation with One Pair of Correlated Variables
We suppose that Q and Ks are correlated. How can this correlation impact P̂f ?
1.0 0.5 0.0 0.5 1.0
Q vs Ks
0.000
0.005
0.010
0.015
0.020
0.025
0.030
P̂f
P̂ f
P̂ ∗f
P̂ INDf
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 3 / 17
Illustration: Flood Example Overflow Probability Estimation
Monte-Carlo Estimation with One Pair of Correlated Variables
We suppose that Q and Ks are correlated. How can this correlation impact P̂f ?
1.0 0.5 0.0 0.5 1.0
Q vs Ks
0.000
0.005
0.010
0.015
0.020
0.025
0.030
P̂f
P̂ f
P̂ ∗f
P̂ INDf
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 3 / 17
Contents
1 Illustration: Flood Example
2 General FrameworkContextThe Worst Case ScenarioExtremum-estimation
3 Methodology
4 Discussion
General Framework Context
Q(α)⟂
ℙ[Y≥t] ⟂
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Questions:What is the relationship between the dependence structure and the output Y ?Is it conservative to suppose independence ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].Quantify the influence of the dependence for each pair of variables.
How to describe the dependence structure?
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 4 / 17
General Framework Context
?
Dependence
IncompleteUnknown
Known
Q(α)
ℙ[Y≥t]
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Questions:What is the relationship between the dependence structure and the output Y ?Is it conservative to suppose independence ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].Quantify the influence of the dependence for each pair of variables.
How to describe the dependence structure?
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 4 / 17
General Framework Context
?
Dependence
IncompleteUnknown
Known
Q(α)
ℙ[Y≥t]
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Questions:What is the relationship between the dependence structure and the output Y ?Is it conservative to suppose independence ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].Quantify the influence of the dependence for each pair of variables.
How to describe the dependence structure?
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 4 / 17
General Framework Context
?
Dependence
IncompleteUnknown
Known
Q(α)
ℙ[Y≥t]
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Questions:What is the relationship between the dependence structure and the output Y ?Is it conservative to suppose independence ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].Quantify the influence of the dependence for each pair of variables.
How to describe the dependence structure?
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 4 / 17
General Framework Context
?
Dependence
IncompleteUnknown
Known
Q(α)
ℙ[Y≥t]
t
Output Y
y
fY( )y
X1
Xi
Xd
η
ModelJoint
Distribution
FX
Margins
Questions:What is the relationship between the dependence structure and the output Y ?Is it conservative to suppose independence ?
Objectives:Inform about the importance of the dependence structure.Bound the quantity of interest: e.g. P⊥[Y ≥ t] ≤ P∗[Y ≤ t].Quantify the influence of the dependence for each pair of variables.
How to describe the dependence structure?
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 4 / 17
General Framework Context
Copulas
“Knowing the word “copula“ as a grammatical term for a word or expression that linksa subject and predicate, I felt that this would make an appropriate name for a functionthat links a multidimensional distribution to its one-dimensional margins
Sklar, 1996 ”The dependence structure is described by a parametric copula Cθ with θ ∈ Θ ⊆ Rp suchas[9,10]
FX(x) = Cθ(FX1(x1), . . . ,FXd (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
[9] Roger B Nelsen. An introduction to copulas. Springer Science & Business Media, 2007.[10] M Sklar. Fonctions de répartition à n dimensions et leurs marges. Institut de Statistique de l’Universitéde Paris, 1959.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 5 / 17
General Framework Context
Copulas
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.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 6 / 17
General Framework Context
Copulas
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.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 6 / 17
General Framework 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
For a given α ∈ (0, 1) and a copula Cθ ,θ∗ = argmax
θ∈ΘQθ(α).
This worst case gives an upper bound such thatQθ∗ (α) ≥ Q⊥(α).
Using the estimated quantile:θ̂n = argmax
θ∈ΘQ̂n,θ(α).
In practice, θ is discretized using a thin grid ΘK of cardinality K :
θ̂n,K = argmaxθ∈ΘK
Q̂n,θ(α).
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 7 / 17
General Framework The Worst Case Scenario
For a given α ∈ (0, 1) and a copula Cθ ,θ∗ = argmax
θ∈ΘQθ(α).
This worst case gives an upper bound such thatQθ∗ (α) ≥ Q⊥(α).
Using the estimated quantile:θ̂n = argmax
θ∈ΘQ̂n,θ(α).
In practice, θ is discretized using a thin grid ΘK of cardinality K :
θ̂n,K = argmaxθ∈ΘK
Q̂n,θ(α).
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 7 / 17
General Framework The Worst Case Scenario
For a given α ∈ (0, 1) and a copula Cθ ,θ∗ = argmax
θ∈ΘQθ(α).
This worst case gives an upper bound such thatQθ∗ (α) ≥ Q⊥(α).
Using the estimated quantile:θ̂n = argmax
θ∈ΘQ̂n,θ(α).
In practice, θ is discretized using a thin grid ΘK of cardinality K :
θ̂n,K = argmaxθ∈ΘK
Q̂n,θ(α).
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 7 / 17
General Framework The Worst Case Scenario
For a given α ∈ (0, 1) and a copula Cθ ,θ∗ = argmax
θ∈ΘQθ(α).
This worst case gives an upper bound such thatQθ∗ (α) ≥ Q⊥(α).
Using the estimated quantile:θ̂n = argmax
θ∈ΘQ̂n,θ(α).
In practice, θ is discretized using a thin grid ΘK of cardinality K :
θ̂n,K = argmaxθ∈ΘK
Q̂n,θ(α).
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 7 / 17
General Framework Extremum-estimation
The maximisation of a data-dependent function is an extremum-estimator [2,5]:
θ̂n = argmaxθ∈Θ
Q̂n,θ(α).
Theorem 1 (Consistency of θ̂n )If there is a function Qθ(α), for any α ∈ (0, 1) such that
Qθ(α) is uniquely minimised at θ∗,
(→ Assumed)
Θ is compact,
(→ OK using a concordance measure, i.e. Kendall’s τ)
Qθ(α) is continuous in θ,
(→ OK under regularity assumptions of η and FX)
supθ∈Θ|Q̂n,θ(α)− Qθ(α)| P−→ 0,
(→ OK, thanks to the DKW inequality)
then θ̂nP−→ θ∗.
[2] Takeshi Amemiya. Advanced econometrics. Harvard university press, 1985.[5] Fumio Hayashi. “Econometrics”. In: (2000).
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 8 / 17
General Framework Extremum-estimation
The maximisation of a data-dependent function is an extremum-estimator [2,5]:
θ̂n = argmaxθ∈Θ
Q̂n,θ(α).
Theorem 1 (Consistency of θ̂n )If there is a function Qθ(α), for any α ∈ (0, 1) such that
Qθ(α) is uniquely minimised at θ∗, (→ Assumed)Θ is compact, (→ OK using a concordance measure, i.e. Kendall’s τ)Qθ(α) is continuous in θ, (→ OK under regularity assumptions of η and FX)
supθ∈Θ|Q̂n,θ(α)− Qθ(α)| P−→ 0, (→ OK, thanks to the DKW inequality)
then θ̂nP−→ θ∗.
[2] Takeshi Amemiya. Advanced econometrics. Harvard university press, 1985.[5] Fumio Hayashi. “Econometrics”. In: (2000).
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 8 / 17
Contents
1 Illustration: Flood Example
2 General Framework
3 MethodologyGaussian AssumptionTwo-dimensional CopulasMultivariate Copulas Using Vine CopulasIterative Construction of Dependence Structure
4 Discussion
Methodology Gaussian Assumption
Multivariate Gaussian Copula
The problem can be treated assuming a Gaussian copula with correlation matrix R ∈ [−1, 1]d×d .Example for d = 3:
R =
(1 θ12 θ13θ12 1 θ23θ13 θ23 1
).
Objective: Determine the correlation matrix maximising the quantile.
Problems:The correlation matrix R should be definite semi-positive.
The Gaussian assumption is not always adapted.
“fallacies raised from the naive assumption that dependence properties of theelliptical world also hold in the non-elliptical world[3] ”The Gaussian copula does not have tail dependence: less penalizing.
[3] 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 (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 9 / 17
Methodology Gaussian Assumption
Multivariate Gaussian Copula
The problem can be treated assuming a Gaussian copula with correlation matrix R ∈ [−1, 1]d×d .Example for d = 3:
R =
(1 θ12 θ13θ12 1 θ23θ13 θ23 1
).
Objective: Determine the correlation matrix maximising the quantile.Problems:
The correlation matrix R should be definite semi-positive.
ρ1
1.00.5
0.0
0.5
1.0
ρ 2
1.0
0.5
0.0
0.5
1.0
ρ 3
1.0
0.5
0.0
0.5
1.0
The Gaussian assumption is not always adapted.
“fallacies raised from the naive assumption that dependence properties of theelliptical world also hold in the non-elliptical world[3] ”The Gaussian copula does not have tail dependence: less penalizing.
[3] 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 (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 9 / 17
Methodology Gaussian Assumption
Multivariate Gaussian Copula
The problem can be treated assuming a Gaussian copula with correlation matrix R ∈ [−1, 1]d×d .Example for d = 3:
R =
(1 θ12 θ13θ12 1 θ23θ13 θ23 1
).
Objective: Determine the correlation matrix maximising the quantile.Problems:
The correlation matrix R should be definite semi-positive.The Gaussian assumption is not always adapted.
“fallacies raised from the naive assumption that dependence properties of theelliptical world also hold in the non-elliptical world[3] ”The Gaussian copula does not have tail dependence: less penalizing.
[3] 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 (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 9 / 17
Methodology Two-dimensional Copulas
Two-Dimensional Problems
Several studies were made to study the influence of correlations[4] and using copulas[11,12].
On the Flood Example with Q and Ks at α = 99%:
0.5 0.0 0.5
Kendall τ
8
6
4
2
0
2
4
Qθ(α
)
Threshold
Independence
Normal
Clayton
Gumbel
[4] Mircea Grigoriu and Carl Turkstra. “Safety of structural systems with correlated resistances”. In: AppliedMathematical Modelling 3.2 (1979), pp. 130–136.[11] Xiao-Song Tang et al. “Impact of copulas for modeling bivariate distributions on system reliability”. In:Structural safety 44 (2013), pp. 80–90.[12] Christoph Werner et al. “Expert judgement for dependence in probabilistic modelling: a systematic literaturereview and future research directions”. In: European Journal of Operational Research (2016).
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 10 / 17
Methodology Two-dimensional Copulas
Two-Dimensional Problems
Several studies were made to study the influence of correlations[4] and using copulas[11,12].
On the Flood Example with Q and Ks at α = 99%:
0.5 0.0 0.5
Kendall τ
8
6
4
2
0
2
4
Qθ(α
)
Threshold
Independence
Normal
Clayton
Gumbel
[4] Mircea Grigoriu and Carl Turkstra. “Safety of structural systems with correlated resistances”. In: AppliedMathematical Modelling 3.2 (1979), pp. 130–136.[11] Xiao-Song Tang et al. “Impact of copulas for modeling bivariate distributions on system reliability”. In:Structural safety 44 (2013), pp. 80–90.[12] Christoph Werner et al. “Expert judgement for dependence in probabilistic modelling: a systematic literaturereview and future research directions”. In: European Journal of Operational Research (2016).
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 10 / 17
Methodology Two-dimensional Copulas
Two-Dimensional Problems
The worst case is not always at the edge.For example, we consider:
X1 ∼ N (0, 1) and X2 ∼ N (−2, 1)η(x1, x2) = x2
1 x22 − 0.3x1x2
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%.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 11 / 17
Methodology Multivariate Copulas Using Vine Copulas
Copula Densities
Using Sklar’s Theorem, a bivariate distribution of X1 and X2 can be written using acopula C12, such that[8]
F(x1, x2) = C12(F1(x1),F2(x2)).
If F1 and F2 are continuous, C is unique and admits a density
c12(u1, u2) = ∂2C12(u1, u2)∂u1u2
.
Which impliesjoint density:
f (x1, x2) = c12(F1(x1),F2(x2)) · f1(x2) · f2(x2)
conditional density:
f (x1|x2) = c12(F1(x1),F2(x2)) · f2(x2)
[8] Nicole Krämer and Ulf Schepsmeier. “Introduction to vine copulas”. In: ().Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 12 / 17
Methodology Multivariate Copulas Using Vine Copulas
Pair-Copula Construction (R-Vines)
The joint density f (x1, . . . , xd) can be represented by a product of pair-copula densitiesand marginal densities[7].
For example in d = 3. One possible decomposition of f (x1, x2, x3) is:
f (x1, x2, x3) = f1(x1)f2(x2)f3(x3)(margins)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2))(conditional pair)
[7] 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 (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 13 / 17
Methodology Multivariate Copulas Using Vine Copulas
Pair-Copula Construction (R-Vines)
The joint density f (x1, . . . , xd) can be represented by a product of pair-copula densitiesand marginal densities[7].For example in d = 3. One possible decomposition of f (x1, x2, x3) is:
f (x1, x2, x3) = f1(x1)f2(x2)f3(x3)(margins)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2))(conditional pair)
f2
f1
f3
[7] 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 (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 13 / 17
Methodology Multivariate Copulas Using Vine Copulas
Pair-Copula Construction (R-Vines)
The joint density f (x1, . . . , xd) can be represented by a product of pair-copula densitiesand marginal densities[7].For example in d = 3. One possible decomposition of f (x1, x2, x3) is:
f (x1, x2, x3) = f1(x1)f2(x2)f3(x3)(margins)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3))(unconditional pairs)
× c13|2(F1|2(x1|x2),F3|2(x3|x2))(conditional pair)
c12 c23f2
f1
f3
[7] 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 (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 13 / 17
Methodology Multivariate Copulas Using Vine Copulas
Pair-Copula Construction (R-Vines)
The joint density f (x1, . . . , xd) can be represented by a product of pair-copula densitiesand marginal densities[7].For example in d = 3. One possible decomposition of f (x1, x2, x3) is:
f (x1, x2, x3) = f1(x1)f2(x2)f3(x3)(margins)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3))(unconditional pairs)
× c13|2(F1|2(x1|x2),F3|2(x3|x2))(conditional pair)
c23c12
c12 c23f2
f1
f3
[7] 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 (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 13 / 17
Methodology Multivariate Copulas Using Vine Copulas
Pair-Copula Construction (R-Vines)
The joint density f (x1, . . . , xd) can be represented by a product of pair-copula densitiesand marginal densities[7].For example in d = 3. One possible decomposition of f (x1, x2, x3) is:
f (x1, x2, x3) = f1(x1)f2(x2)f3(x3)(margins)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3))(unconditional pairs)× c13|2(F1|2(x1|x2),F3|2(x3|x2))(conditional pair)
c13|2 c23c12
c12 c23f2
f1
f3
[7] 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 (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 13 / 17
Methodology Multivariate Copulas Using Vine Copulas
Vine Copulas
Advantages of using Vines:Multidimensional dependence structure using bivariate copulas from various families.Graphical model: set of connected trees.
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.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 14 / 17
Methodology Multivariate Copulas Using Vine Copulas
Vine Copulas
Advantages of using Vines:Multidimensional dependence structure using bivariate copulas from various families.Graphical model: set of connected trees.
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.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 14 / 17
Methodology Iterative Construction of Dependence Structure
Iterative Vine Construction
Key point: Not all pairs are equivalently influential on Qθ(α).
Idea: Iteratively add a pair in the maximisation of Qθ(α).Init: Θc = ∅1: For each pair Xi -Xj :
θ̂∗ij = argmaxθij∈Θij∪Θc
Q̂n,θi,j (α),
2: Add the pair Xi -Xj that maximises the most Qθ(α) in Θc . And adapt the R-Vinestructure.3: Loop over 2 and 3. Stop when there is no evolution of Qθ∗ (α) or when the budget isconsumed.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 15 / 17
Methodology Iterative Construction of Dependence Structure
Iterative Vine Construction
Key point: Not all pairs are equivalently influential on Qθ(α).Idea: Iteratively add a pair in the maximisation of Qθ(α).Init: Θc = ∅1: For each pair Xi -Xj :
θ̂∗ij = argmaxθij∈Θij∪Θc
Q̂n,θi,j (α),
2: Add the pair Xi -Xj that maximises the most Qθ(α) in Θc . And adapt the R-Vinestructure.3: Loop over 2 and 3. Stop when there is no evolution of Qθ∗ (α) or when the budget isconsumed.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 15 / 17
Methodology Iterative Construction of Dependence Structure
Iterative Vine Construction
Key point: Not all pairs are equivalently influential on Qθ(α).Idea: Iteratively add a pair in the maximisation of Qθ(α).
1 2 3 4
Number of pairs
2
0
2
4
6
Wors
t quanti
le a
t 0.9
9 %
Construction of Worst Case Dependence Structure
Threshold
IndependenceQ−Ks
Hd −LZv −Hd
Zm −L
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 15 / 17
Contents
1 Illustration: Flood Example
2 General Framework
3 Methodology
4 Discussion
Discussion
ConclusionDependencies can have an significant impact on the model output Y .
The worst case scenario θ∗ gives an idea of the influence of dependencies.
Using Vine Copulas we can determine a penalized dependence structure.
Structured methodology with a discretized Θ, parametric copula families and Vines.
PerspectivesAggregate expert feedback to restrict the set Θ to more realistic dependencestructures.
Apply a more greedy algorithm using Quantile Regression Forests.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 16 / 17
Discussion
ConclusionDependencies can have an significant impact on the model output Y .
The worst case scenario θ∗ gives an idea of the influence of dependencies.
Using Vine Copulas we can determine a penalized dependence structure.
Structured methodology with a discretized Θ, parametric copula families and Vines.
PerspectivesAggregate expert feedback to restrict the set Θ to more realistic dependencestructures.
Apply a more greedy algorithm using Quantile Regression Forests.
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 16 / 17
Discussion
Thank You!
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 17 / 17
Contents
5 Additional ContentImpact of Dependence
Additional Content Impact of Dependence
How to measure the impact of potential dependencies?
Definition 1 (Price of Correlation,[1])[a] Shipra Agrawal et al. “Price of correlations in stochastic optimization”. In: Operations Research60.1 (2012), pp. 150–162.
Given a decision z ∈ Z, a collection of joint densities X and a cost function h,
η(z) = supX∈X
EX[h(X, z)].
Let zI = argminz∈Z EXI [h(XI , z)], zR = argminz∈Z η(z). Then Price of Correlation(POC) is defined as:
POC = η(zI)η(zR)
The application is different, but a common question remains: how much risk it involvesto ignore the correlations?
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 18 / 17
Additional Content Impact of Dependence
To estimate the influence of a pair of variables Xi -Xj on Qθ(α), we define the quantity
Iij =|Qθ∗
ij(α)− Q⊥(α)|
|Q∗θ (α)− Q⊥(α)| ,
where θ∗ij is the solution of the max(min)imisation of Qθ(α) when only the pair Xi -Xj iscorrelated and the others are considered independent.
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1.0
Indic
ies
Impact of dependence on the quantile
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 19 / 17
Additional Content Impact of Dependence
To estimate the influence of a pair of variables Xi -Xj on Qθ(α), we define the quantity
Iij =|Qθ∗
ij(α)− Q⊥(α)|
|Q∗θ (α)− Q⊥(α)| ,
where θ∗ij is the solution of the max(min)imisation of Qθ(α) when only the pair Xi -Xj iscorrelated and the others are considered independent.
BZm
ZmQ
CbHd
ZvKs
LZv
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Hd
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Ks
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0.0
0.2
0.4
0.6
0.8
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Indic
ies
Impact of dependence on the quantile
Nazih Benoumechiara (LSTA – MRI) Rencontres Mexico-MascotNum Wednesday, November 23th 2016 19 / 17