impact of dependencies between input variables in

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.


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


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


Contents


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?



?

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?



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.



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.



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.


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).


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).


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).


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).


Contents


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.





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.







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.




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).


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%.


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


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.




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:



f2

f1

f3







× 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







× 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








c13|2 c23c12

c12 c23f2

f1

f3




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.


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.




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.




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


Contents


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.


Discussion

Thank You!


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?


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

CbKs

BZv

Hd

QBL

LZm

CbQHd

Ks

LHd

ZmZv

BKs

LQ

BCb

Hd

Zv

Ks

QLCb

CbZm

BQ

LKs

ZmKs

BHd

ZvQHd

Zm

CbZv

0.0

0.2

0.4

0.6

0.8

1.0

Indic

ies

Impact of dependence on the quantile


impact of dependencies between input variables in

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