quantile estimation in structural reliability with incomplete ... · mathematical modelling...

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

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

Steal Component

Coating

Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 1 / 26

Industrial Context

€€€IM

PORTAN

Steal Component

Coating

Defect

Industrial Context

€€€IM

PORTAN

Steal Component

Coating

Defect

Industrial Context

€€€IM

PORTAN

Steal Component

Coating

Defect

Thermal Loading

Industrial Context

€€€IM

PORTAN

Steal Component

Coating

Defect

Thermal Loading

T (°C)

Industrial Context

€€€IM

PORTAN

Steal Component

Coating

Defect

Thermal Loading

T (°C)

Structural Reliability

Q(α)⟂

ℙ[Y≥t] ⟂

Output Y

fY( )y

ModelJoint

Distribution

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?

Dependence

IncompleteUnknown

ℙ[Y≥t]

Output Y

fY( )y

ModelJoint

Distribution

Margins

Dependence

IncompleteUnknown

ℙ[Y≥t]

Output Y

fY( )y

ModelJoint

Distribution

Margins

Dependence

IncompleteUnknown

ℙ[Y≥t]

Output Y

fY( )y

ModelJoint

Distribution

Margins

Dependence

IncompleteUnknown

ℙ[Y≥t]

Output Y

fY( )y

ModelJoint

Distribution

Margins

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

Clayton (u−θ + v−θ − 1)−1/θ [0,∞] θ2+θ

Gumbel exp−[(− ln u)θ + (− ln v)θ]1/θ

[1,∞] 1− 1

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

Copulas

FX(x) = Cθ(F1(x1), . . . ,Fd (xd )).

0.0 0.2 0.4 0.6 0.8 1.00.0

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.

Copulas

FX(x) = Cθ(F1(x1), . . . ,Fd (xd )).

3 2 1 0 1 2 33

Figure: Example of joints p.d.f with Gaussian margins and τ = 0.6.

Industrial context The worst case scenario

Copula

IncompleteUnknown

Q(α)θ

ℙ[Y≥t] θ

Output Y

fY( )y

ModelJoint

Distribution

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).

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

Qθ(α),

θn = argmaxθ∈Θ

Qn,θ(α).

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

Qθ(α),

θn = argmaxθ∈Θ

Qn,θ(α).

1 Or minimizationNazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 4 / 26

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

Qθ(α),

θn = argmaxθ∈Θ

Qn,θ(α).

1 Or minimizationNazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 4 / 26

Contents

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.

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.

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

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

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:

1 2 3 4

Regular Vines

× 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

Regular Vines

× 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

Regular Vines

× 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

Regular Vines

× 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

Regular Vines

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

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.

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

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.

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

Example: Grid-search

The model:

Y = −d∑

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

QuantileComonotonicityIndependence

Figure: Quantile variation with the correlation in dimension d = 2Nazih Benoumechiara (LPSM – PRISME) Mascot-Num 2018 Wednesday, March 21th 2018 11 / 26

Example: Grid-search

The model:

Y = −d∑

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

Figure: Quantile estimations with the dependence of each pairs in dimension d = 3.

Figure: Quantile estimations with the dependence of each pairs in dimension d = 4.

3 4 5 6 7 8Dimension

ence Grid-search

Figure: Q⊥⊥(α)− Qθn(α)

Figure: Sampling of X at the obtained minimum for d = 8

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?

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)

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)

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)

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)

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)

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)

Back to the example in d = 8

1 2 3Iterations

IndependenceGrid-searchIterative

Figure: Resulting output quantiles of each method

How it changes the output distributions?

1200 1000 800 600 400 200y

0.0000

0.0005

0.0010

0.0015

0.0020

Grid-searchIterativeIndependence

3 4 5 6 7 8Dimension

ence Iterative

Grid-search

Figure: Q⊥⊥(α)− Qθn(α)

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

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

Contents

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

Discussion

To Infinity and Beyond

Stay with parametric copulas: Bayesian optimization

Using non parametric dependence structure?

Discussion

Thank You!

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