On quantile oriented sensitivity analysis
OOn quantile oriented sensitivity analysis
Kévin Elie-Dit-Cosaque ∗1, 2 and Véronique Maume-Deschamps †11
Institut Camille Jordan, Université Claude Bernard Lyon 1, Lyon, FRANCE Actuarial Modelling, SCOR, Paris, FRANCE
February 12, 2021
Abstract
We propose to study quantile oriented sensitivity indices (QOSA indices) and quantile ori-ented Shapley effects (QOSE). Some theoretical properties of QOSA indices will be given andseveral calculations of QOSA indices and QOSE will allow to better understand the behaviourand the interest of these indices.
Keywords: Quantile Oriented Sensisitivy Analysis, Shapley effects.
Sensitivity Analysis (SA) is defined by Saltelli et al. (2004) as “the study of how the uncertainty inthe output of a model can be apportioned to different sources of uncertainty in the model input”.Various tools exist today to perform a SA (see e.g. Iooss and Lemaître (2015) for a review of SAmethods).We are especially interested in
Global Sensitivity Analysis methods - GSA - which allow to studythe effects of simultaneous variation of the inputs on the model output in their entire domain. Fora detailed description of sensitivity analysis methods, the interested reader can refer to the varioussurvey papers dedicated to this topic (Saltelli et al., 2004, 2008; Faivre et al., 2016; Borgonovo andPlischke, 2016; Borgonovo et al., 2017). Variance-based methods are common tools in the analysisof complex physical phenomenons. Most of them rest on an ANalysis Of VAriance (ANOVA) ofthe model output and give information on the sensitivity around the mean (as it is variance based).In this paper, we are interested in Quantile Oriented indices, in order to obtain informations onthe sensitivity around quantiles. Much less work has been done on Quantile Oriented SensitivityAnalysis (QOSA). We shall focus on indices defined firstly in Fort et al. (2016).For completeness, Section 2 shortly recalls variance-based methods dealing both with indepedentand dependent inputs. Section 3 introduces QOSA indices which allows to quantify the sensitivityover a quantile. Note that in Kucherenko et al. (2019), other quantile oriented indices have been ∗ [email protected] † [email protected] a r X i v : . [ s t a t . M E ] F e b efined. Nevertheless, QOSA indices have more natural interpretations as will be seen in Section 3.Some properties of QOSA indices are also proposed within this section. Several calculations ofQOSA indices are done in Section 4 and a preliminary work is carried out in order to understandthe impact of the statistical dependence between the inputs over these indices. Facing some inter-pretation issues, we finally propose in Section 5 new indices based on Shapley values (Shapley, 1953)which seem to be a promising alternative. Section 6 presents some further perspective of research.Consider a model Y = η ( X ) with d random inputs denoted by X = ( X , X , . . . , X d ). Let X J indicate the vector of inputs corresponding to the index set J ⊆ D where D = { , , . . . , d } . GSAaims at quantifying the impact of the inputs X , X , . . . , X d on the output Y . We shall briefly recall the framework of Sobol indices and Shapley effects.
Sobol’ sensitivity indices stem from the works of Fisher and Mackenzie (1923) and Hoeffding (1948)on the U-statistics taken up by various authors over time such as Efron and Stein (1981). Thoseultimately lead to a functional ANOVA expansion of the model output η : η ( X ) = η + d X i =1 η i ( X i ) + X (cid:54) i Quantile Oriented Sensitivity Analysis (QOSA). Let us focus on QOSA indices measuring the impact of the inputs over the α -quantile of the outputdistribution. Given a level of quantile α ∈ ]0 , S αi = min θ ∈ R E [ ψ α ( Y, θ )] − E (cid:20) min θ ∈ R E [ ψ α ( Y, θ ) | X i ] (cid:21) min θ ∈ R E [ ψ α ( Y, θ )]= E [ ψ α ( Y, q α ( Y ))] − E [ ψ α ( Y, q α ( Y | X i ))] E [ ψ α ( Y, q α ( Y ))] , (3.1)where ψ α : ( y, θ ) ( y − θ ) (cid:16) α − { y (cid:54) θ } (cid:17) is the contrast function associated to the α -quantile andthe q ’s are the quantiles q α ( Y ) = arg min θ ∈ R E [ ψ α ( Y, θ )] and q α ( Y | X i = x i ) = arg min θ ∈ R E [ ψ α ( Y, θ ) | X i = x i ] . Remark that replacing ψ α in the above equation by ( y, θ ) ( y − θ ) leads to the definition offirst-order Sobol indices. In order to interpret QOSA indices, one has to consider ψ α ( Y, θ ) as adispersion measure of Y which is minimized for θ = q α ( Y ). So that QOSA indices compare thedispersion of Y around its quantile with its conditional counterpart.The first-order QOSA indices have been defined in Fort et al. (2016), studied and estimatedin Browne et al. (2017); Maume-Deschamps and Niang (2018); Elie-Dit-Cosaque and Maume-Deschamps (2021). They may be rewritten as follows S αi = E h Y { Y (cid:54) q α ( Y | X i ) } i − E h Y { Y (cid:54) q α ( Y ) } i α E [ Y ] − E h Y { Y (cid:54) q α ( Y ) } i = 1 − α E [ Y ] − E h Y { Y (cid:54) q α ( Y | X i ) } i α E [ Y ] − E h Y { Y (cid:54) q α ( Y ) } i . Let us mention that Kucherenko et al. (2019) have recently proposed new indices to assess theimpact of inputs over the α -quantile of the output distribution. Instead of considering the expressionof the first-order Sobol index based on a contrast function as in (3.1), they consider the expressionof Sobol’ indices with numerator Var ( E [ Y | X i ]) = E h ( E [ Y | X i ] − E [ Y ]) i and simply replace theexpectations by α -quantiles to define the following indices¯ q αi, = E [ | q α ( Y ) − q α ( Y | X i ) | ] and ¯ q αi, = E h ( q α ( Y ) − q α ( Y | X i )) i . They also provide the normalized versions as follows Q αi, = ¯ q αi, d P j =1 ¯ q αj, and Q αi, = ¯ q αi, d P j =1 ¯ q αj, . These measures thereby quantify the mean distance between quantiles q α ( Y ) and q α ( Y | X i ) ratherthan the mean distance between average contrast functions like in the first-order QOSA index givenin (3.1). The indices ¯ q αi, and Q αi, will be called below absolute value indices ; ¯ q αi, and Q αi, willbe called below squared indices . The example below shows that their interpretation as sensitivityindices is questionnable, it is why we shall focus on QOSA indices.Consider the simple model also studied in Fort et al. (2016): Y = X − X where X and X Y follows a Laplacedistribution. QOSA indices have closed form formulas that may be found in Fort et al. (2016). Theindices proposed in Kucherenko et al. (2019) may also be computed. Indeed, with γ = ( − log (2 α (1 − α )) if α ≥ log(2) if α < , and γ = ( log(2) if α ≥ − log (2 α (1 − α )) if α < , we have ¯ q α , = γ + 2 e − γ − q α , = γ + 2 e − γ − q α , = γ − γ + 2 and ¯ q α , = γ − γ + 2 . Below we show the behaviour of QOSA, ¯ q αi, and ¯ q αi, indices. As expected, QOSA indices showthat X has more influence on quantiles of level higher that and X is more influent for α lowerthan . The interpretation of ¯ q αi,j is not so clear since ¯ q α ,j are constant for α less than and ¯ q α ,j are constant for α greater than . Moreover, the ¯ q αi, ’s are not monotonic.The normalized indices below show that the interpretation for Q αi,j remains questionnable while thenormalized QOSA indices keep the interpretation of the un-normalized ones.7 .2 Some elementary properties of QOSA indices Let us turn to some elementary properties of QOSA indices.The following lemma is useful, it is closely related to the proof of sub-additivity of TVaR in risktheory (see Marceau (2013) e.g.). Lemma 3.1. Consider any event E such that P ( E ) = α . Then, for any random variable X , we have E h X { X (cid:54) q α ( X ) } i (cid:54) E [ X E ] , with q α ( X ) the α -quantile of X .Proof of Lemma 3.1. We have: E h X { X (cid:54) q α ( X ) } i − E [ X E ] = E h X (cid:16) { X (cid:54) q α ( X ) } − E (cid:17)i = E h ( X − q α ( X )) (cid:16) { X (cid:54) q α ( X ) } − E (cid:17)i (cid:54) . (cid:4) As a consequence, since P ( Y (cid:54) q α ( Y | X i ) | X i ) = α and P ( Y (cid:54) q α ( Y )) = α , we get that α E [ Y ] − E h Y { Y (cid:54) q α ( Y | X i ) } i = E h ( Y − q α ( Y | X i )) (cid:16) α − { Y (cid:54) q α ( Y | X i ) } (cid:17)i (cid:62) α E [ Y ] − E h Y { Y (cid:54) q α ( Y ) } i = E h ( Y − q α ( Y )) (cid:16) α − { Y (cid:54) q α ( Y ) } (cid:17)i (cid:62) E h Y { Y (cid:54) q α ( Y | X i ) } i − E h Y { Y (cid:54) q α ( Y ) } i (cid:62) . (3.4)This implies that 0 (cid:54) S αi (cid:54) S αi also has thethree following interesting properties. Proposition 3.2. S αi is invariant with respect to translations of the output Y .2. S αi is invariant by homothety with strictly positive ratio of the output Y .3. A homothety with strictly negative ratio of the output Y gives the index S − αi associated tothe − α level.Proof of Proposition 3.2. Let us consider any model Y = η ( X ). We will denote by S αi the QOSA indices related to a r.v. Y .1. Let Y = Y + k, k ∈ R . Then, we have q α ( Y ) = q α ( Y ) + k and q α ( Y | X i ) = q α ( Y | X i ) + k .It is easy to check that S αi = S αi . 8. Let Y = k × Y, k > 0. Then we have, q α ( Y ) = k × q α ( Y ) and q α ( Y | X i ) = k × q α ( Y | X i ).We can easily show that S αi = S αi .3. Let Y = k × Y, k < 0. Then we have, q α ( Y ) = k × q − α ( Y ) and q α ( Y | X i ) = k × q − α ( Y | X i ). It leads to S αi = S − αi . (cid:4) Now, we are going to investigate the sum S of the first-order QOSA indices: S = d X i =1 S αi = d X i =1 E h Y { Y (cid:54) q α ( Y | X i ) } i − d E h Y { Y (cid:54) q α ( Y ) } i α E [ Y ] − E h Y { Y (cid:54) q α ( Y ) } i . We see that S (cid:54) d X i =1 E h Y { Y (cid:54) q α ( Y | X i ) } i − d E h Y { Y (cid:54) q α ( Y ) } i (cid:54) (cid:16) α E [ Y ] − E h Y { Y (cid:54) q α ( Y ) } i(cid:17) . (3.5)Or equivalently: α E [ Y ] + ( d − E h Y { Y (cid:54) q α ( Y ) } i − d X i =1 E h Y { Y (cid:54) q α ( Y | X i ) } i (cid:62) . As proved in the following proposition, S is smaller than 1 in the case of an additive model withindependent inputs. Unfortunately, this result is not true in the general case as showed with acounterexample in Subsection 4.1. Proposition 3.3. Let X = ( X , . . . , X d ) with independent X i ’s. Let Y = m + d P i =1 m i ( X i ) be an additive model. Then,the sum of the first-order QOSA indices S satisfies S (cid:54) .Proof of Proposition 3.3. Given a random variable X , we denote by q α ( X ) its α -quantile. For any i = 1 , . . . , d , let Xs ( − i ) = P (cid:54) j (cid:54) dj = i m j ( X j ), thanks to the independence of the X i ’s, we have q α ( Y | X i ) = m + m i ( X i ) + q α (cid:16) Xs ( − i ) (cid:17) , and { Y (cid:54) q α ( Y | X i ) } = { Xs ( − i ) (cid:54) q α (cid:16) Xs ( − i ) (cid:17) } . We have g ( α ) := α E [ Y ] + ( d − E h Y { Y (cid:54) q α ( Y ) } i − d X i =1 E h Y { Y (cid:54) q α ( Y | X i ) } i = α E [ Y ] + ( d − E h Y { Y (cid:54) q α ( Y ) } i − d X i =1 (cid:16) α × m + E h m i ( X i ) { Xs − ( i ) (cid:54) q α ( Xs ( − i ) ) } i + E h Xs ( − i ) { Xs − ( i ) (cid:54) q α ( Xs ( − i ) ) } i(cid:17) . X i ’s implies that E h m i ( X i ) { Xs ( − i ) (cid:54) q α ( Xs ( − i ) ) } i = α E [ m i ( X i )] and thus, g ( α ) = ( d − E d X j =1 m j ( X j ) { Y (cid:54) q α ( Y ) } − d X i =1 E h Xs ( − i ) { Xs ( − i ) (cid:54) q α ( Xs ( − i ) ) } i . Now, we use Lemma 3.1 which gives:( d − E d X j =1 m j ( X j ) { Y (cid:54) q α ( Y ) } = d − X i =1 (cid:16) E h m i ( X i ) { Y (cid:54) q α ( Y ) } i + E h Xs ( − i ) { Y (cid:54) q α ( Y ) } i(cid:17) (cid:62) d − X i =1 (cid:16) E h m i ( X i ) { Y (cid:54) q α ( Y ) } i + E h Xs ( − i ) { Xs ( − i ) (cid:54) q α ( Xs ( − i ) ) } i(cid:17) . As a consequence, g ( α ) (cid:62) d − X i =1 E h m i ( X i ) { Y (cid:54) q α ( Y ) } i − E h Xs ( − d ) { Xs ( − d ) (cid:54) q α ( Xs ( − d ) ) } i = E h Xs ( − d ) { Y (cid:54) q α ( Y ) } i − E h Xs ( − d ) { Xs ( − d ) (cid:54) q α ( Xs ( − d ) ) } i (cid:62) . (cid:4) First-order QOSA indices capture only the main effect of the i -th input. Following Kala (2019),we consider below higher order QOSA indices. Kala (2019) introduced second-order QOSA indices in order to assess the impact of the interactioneffect of two inputs on the α -quantile. He proposes to measure the joint effect of the pair ( X i , X j )by: S αij = min θ ∈ R E [ ψ α ( Y, θ )] − E (cid:20) min θ ∈ R E [ ψ α ( Y, θ ) | X i , X j ] (cid:21) min θ ∈ R E [ ψ α ( Y, θ )] − S αi − S αj . Higher-order QOSA indices can be expressed analogously. Hence, one obtain a variance-like decom-position for quantiles in the case of independent inputs: d X i =1 S αi + X (cid:54) i Proposition 3.4. ST αi is invariant with respect to translations of the output Y .2. ST αi is invariant by homothety with strictly positive ratio of the output Y .3. A homothety with strictly negative ratio of the output Y gives the index ST − αi associated tothe − α level.Proof of Proposition 3.4. Just adapting the steps of the proof of the Proposition 3.2 for the total QOSA index. (cid:4) Proposition 3.5 below shows that the total QOSA index is greater than or equal to the first-order one for any α -level in the case of an additive model with X that has independent marginals.This is a major difference from variance-based methods, specifically Sobol indices. Indeed, it iswell-known that for a purely additive model with independent inputs, we have for the Sobol indices ST i = S i , ∀ i ∈ D , which is not the case for the QOSA indices. It therefore appears that the totalQOSA index captures some interaction between the inputs when using an additive model. Theorigin of this phenomenom is not yet understood at this stage and requires further analysis.Besides, it should be noted that Proposition 3.5 is not verified in the general non additive case asemphasized with a counterexample in Subsection 4.3. Proposition 3.5. Let X = ( X , . . . , X d ) with the X i ’s independent. Let Y = m + d P i =1 m i ( X i ) be an additive modelwith m i , i = 1 , . . . , d , the one-dimensional nonparametric functions operating on each element ofthe vector X . Then, ∀ α ∈ ]0 , , S αi (cid:54) ST αi . roof of Proposition 3.5. We have: ST αi − S αi = (cid:16) α E [ Y ] − E h Y { Y (cid:54) q α ( Y | X − i ) } i(cid:17) − (cid:16) E h Y { Y (cid:54) q α ( Y | X i ) } i − E h Y { Y (cid:54) q α ( Y ) } i(cid:17) α E [ Y ] − E h Y { Y (cid:54) q α ( Y ) } i . As the denominator is non negative according to Equation (3.3), we just have to show that thenumerator is also non negative.Let g ( α ) = α E [ Y ] − E h Y { Y (cid:54) q α ( Y | X − i ) } i + E h Y { Y (cid:54) q α ( Y ) } i − E h Y { Y (cid:54) q α ( Y | X i ) } i , For any i = 1 , . . . , d , let Xs ( − i ) = P (cid:54) j (cid:54) dj = i m j ( X j ), thanks to the independence, we have q α ( Y | X i ) = m + m i ( X i ) + q α (cid:16) Xs ( − i ) (cid:17) , and { Y (cid:54) q α ( Y | X i ) } = { Xs ( − i ) (cid:54) q α (cid:16) Xs ( − i ) (cid:17) } ,q α ( Y | X − i ) = m + Xs ( − i ) + q α ( m i ( X i )) , and { Y (cid:54) q α ( Y | X − i ) } = { m i ( X i ) (cid:54) q α ( m i ( X i )) } . Then, g ( α ) = α E [ Y ] − E h Y { Y (cid:54) q α ( Y | X − i ) } i + E h Y { Y (cid:54) q α ( Y ) } i − E h Y { Y (cid:54) q α ( Y | X i ) } i = α E [ Y ] − αm − E h m i ( X i ) { m i ( X i ) (cid:54) q α ( m i ( X i )) } i − E h Xs ( − i ) { m i ( X i ) (cid:54) q α ( m i ( X i )) } i + E h Y { Y (cid:54) q α ( Y ) } i − αm − E h Xs ( − i ) { Xs ( − i ) (cid:54) q α ( Xs ( − i ) ) } i − E h m i ( X i ) { Xs ( − i ) (cid:54) q α ( Xs ( − i ) ) } i . Now, the independence of the X i ’s implies that E h Xs ( − i ) { m i ( X i ) (cid:54) q α ( m i ( X i )) } i = α E h Xs ( − i ) i and E h m i ( X i ) { Xs ( − i ) (cid:54) q α ( Xs ( − i ) ) } i = α E [ m i ( X i )]. Thus, g ( α ) = E d X j =1 m j ( X j ) { Y (cid:54) q α ( Y ) } − E h m i ( X i ) { m i ( X i ) (cid:54) q α ( m i ( X i )) } i − E h Xs ( − i ) { Xs ( − i ) (cid:54) q α ( Xs ( − i ) ) } i = (cid:16) E h m i ( X i ) { Y (cid:54) q α ( Y ) } i − E h m i ( X i ) { m i ( X i ) (cid:54) q α ( m i ( X i )) } i(cid:17) + (cid:16) E h Xs ( − i ) { Y (cid:54) q α ( Y ) } i − E h Xs ( − i ) { Xs ( − i ) (cid:54) q α ( Xs ( − i ) ) } i(cid:17) . Now, the two last terms are positive according to Lemma 3.1 which concludes the proof. (cid:4) Several calculations below will help to better understand the behaviour, the usefulness and thelimitations of QOSA indices. 12 QOSA calculations on special cases In this section, we compute for some distributions the first-order and total QOSA indices for whichwe obtain the expressions in a closed or nearly closed-form. In particular, the examples withGaussian inputs allow to investigate the behavior of the indices when there is some statisticaldependence between the inputs. Let Y = X · X , with X ∼ E ( λ ), X ∼ E ( δ ), these two variables being independent. After simplecalculations, we get the first-order QOSA indices S α = S α = 1 − ( α − 1) log (1 − α ) α − λδ · E h Y { Y (cid:54) q α ( Y ) } i , (4.1)and the total QOSA indices ST α = ST α = ( α − 1) log (1 − α ) α − λδ · E h Y { Y (cid:54) q α ( Y ) } i . (4.2)The term E h Y { Y (cid:54) q α ( Y ) } i can be approximated by using a Monte-Carlo estimation or a numericalintegration.The equality of the first-order and total QOSA indices for both inputs is a particular case dueto the exponential distribution. Indeed, let X L = λδ X with X an independent copy of X . Then,the model writes Y L = λδ X · X L = kZ with Z = X · X and k = λδ > 0. As the inputs X and X have the same distribution, theirimpact over the α -quantile of the model output Z is identical. Therefore, by using item 2. ofPropositions 3.2 and 3.4 (because Y is just a homothety with strictly positive ratio of Z ), thatexplains why both first-order and total QOSA indices of the inputs are equal.Figure 1 below presents the behavior of the indices as a function of the level α for the modelcomputed with λ = 1 / 10 and δ = 1. The truncated expectation is estimated with a Monte-Carloalgorithm and a sample of size n = 10 . We observe that the first-order and total QOSA indicesvary in opposite directions. The first-order QOSA indices go to 1 when α tends to 1 while the totalones go to 0 when α tends to 1. It is interesting to notice that from α ≈ . 96 the total QOSAindices are lower than the first-order ones and the sum of the first-order ones is greater than 1. Thatcorroborates that Propositions 3.2 and 3.4 are not verified outside the additive model context. We study in this subsection a linear model with Gaussian inputs which implies that the resultingoutput is also Gaussian. This framework facilitates calculations to obtain the analytical values givenbelow. 13 .0 0.2 0.4 0.6 0.8 1.0Values of 0.00.20.40.60.81.01.21.4 QO S A i n d i c e s S , S Sum first-order QOSA indices ST , ST Figure 1: Evolution of the first-order and total QOSA indices at different levels α for the productof two exponentials with λ = 1 / 10 for the fisrt input and δ = 1 for the second one. Proposition 4.1. Let Y = η ( X ) = β + β T X with β ∈ R , β ∈ R d and X ∼ N ( µ , Σ ) where Σ ∈ R d × d is apositive-definite matrix, then the first-order and total QOSA indices for the variable i at the α -levelare S αi = 1 − r β T − i (cid:16) Σ − i, − i − Σ − i,i Σ − i,i Σ i, − i (cid:17) β − i σ Y , (4.3) ST αi = | β i | q Σ i,i − Σ i, − i Σ − − i, − i Σ − i,i σ Y , (4.4) with σ Y = Var ( Y ) = β T Σ β . We observe that as β and µ are translation parameters, they do not have any influence. Never-theless, no general conclusion can be drawn from Equations (4.3) and (4.4) except that the valuesof the first-order and total QOSA indices are the same for all levels α . This phenomenon is specificto the Gaussian linear model and will be detailed hereafter in dimension 2. Indeed, a look at thecase d = 2 may help to understand why QOSA indices does not depend on α -level in the Gaussianframework. This feature will be analyzed by studying only the impact of the variable X .We have that Y | X ∼ N (cid:16) β X + β E [ X | X ] , β Var ( X | X ) (cid:17) . As we work with Gaussian distributions, the conditional variance Var ( X | X ) does not depend onthe specific value of X and is Var ( X | X ) = σ (cid:0) − ρ (cid:1) . The conditional quantile of Y given X has the following expression q α ( Y | X ) = β X + β E [ X | X ] + | β | q Var ( X | X )Φ − ( α )14ith Φ the standard normal distribution function. One way to assess the impact of the variable X on the quantile q α ( Y ) would be to calculate the ratio E [ q α ( Y | X )] − E [ Y ] q α ( Y ) − E [ Y ] . Thus, by using that q α ( Y ) = E [ Y ] + σ Y Φ − ( α ), the previous ratio equals E [ q α ( Y | X )] − E [ Y ] q α ( Y ) − E [ Y ] = | β | σ p (1 − ρ ) σ Y . A simple calculation shows that S α = 1 − | β | σ p − ρ σ Y and gives the relation with QOSA indices.In a more general way, the following equality holds for all variables i = 1 , . . . , d when using a linearGaussian model and explains why the first-order and total QOSA indices do not depend on the α -level: α E [ Y ] − E h Y { Y (cid:54) q α ( Y | X i ) } i α E [ Y ] − E h Y { Y (cid:54) q α ( Y ) } i = E [ q α ( Y | X i )] − E [ Y ] q α ( Y ) − E [ Y ] . We now study the particular case µ = µ = 0 , β = β = 1 , σ = 1 and σ = 2. The analyticalvalues of the indices are depicted in Figure 2 on the left-hand graph for independent inputs and onthe right-hand plot as a function of the correlation coefficient between the two inputs in order toinvestigate the influence of the dependence. QO S A i n d i c e s Independent inputs 1.0 0.5 0.0 0.5 1.0Correl. coef. between X and X Dependent inputs -level S ST S ST Figure 2: First-order and total QOSA indices with independent (resp. dependent) inputs on theleft (resp. right) graph.For the independent case, it appears that the variable X has the higher impact over the α -quantile, which is consistent with the setting established. Besides, we have S αi (cid:54) ST αi , i = 1 , | ρ | → ST αi (cid:54) S αi for some correlation coefficients. The behaviour ofthese indices is similar to that of the Sobol indices in the context of dependent inputs as studiedin Kucherenko et al. (2012); Iooss and Prieur (2019). Indeed, by making an analogy with themethod proposed by Mara et al. (2015) based on four Sobol indices, we could say that in the caseof dependent inputs:• the first-order QOSA index describes the influence of a variable including its dependence withother variables,• the total QOSA index describes the influence of a variable without its dependence with othervariables. We analyze in this subsection a model with Gaussian inputs whose output is a Log-normal distri-bution so that we no longer have identical indices for any α -level. Using Gaussian inputs makescalculations possible and we obtain the following analytical values. Proposition 4.2. Let Y = η ( X ) = exp (cid:16) β + β T X (cid:17) with β ∈ R , β ∈ R d and X ∼ N ( µ , Σ ) where Σ ∈ R d × d is apositive-definite matrix, then the first-order and total QOSA indices for the variable i at the α -levelare S αi = 1 − α − Φ (cid:18) Φ − ( α ) − r β T − i (cid:16) Σ − i, − i − Σ − i,i Σ − i,i Σ i, − i (cid:17) β − i (cid:19) α − Φ (Φ − ( α ) − σ ) , (4.5) ST αi = α − Φ (cid:16) Φ − ( α ) − | β i | q Σ i,i − Σ i, − i Σ − − i, − i Σ − i,i (cid:17) α − Φ (Φ − ( α ) − σ ) , (4.6) with σ = β T Σ β and Φ the standard normal distribution function. We observe that β and µ do not play any role as these are scale parameters in this example.Let us consider the particular case d = 2 with µ = µ µ ! , β = β β ! and Σ = σ ρσ σ ρσ σ σ ! , − ≤ ρ ≤ , σ > , σ > . We have σ = β σ + 2 ρβ β σ σ + β σ and obtain from Equations (4.5) and (4.6) S α = 1 − α − Φ (cid:16) Φ − ( α ) − | β | σ p − ρ (cid:17) α − Φ (Φ − ( α ) − σ ) ,S α = 1 − α − Φ (cid:16) Φ − ( α ) − | β | σ p − ρ (cid:17) α − Φ (Φ − ( α ) − σ ) , (4.7)16nd ST α = α − Φ (cid:16) Φ − ( α ) − | β | σ p − ρ (cid:17) α − Φ (Φ − ( α ) − σ ) ,ST α = α − Φ (cid:16) Φ − ( α ) − | β | σ p − ρ (cid:17) α − Φ (Φ − ( α ) − σ ) . (4.8)In all further tests, we take µ = µ = 0 , β = β = 1 , σ = 1 and σ = 2.Figure 3 presents the analytical values of the first-order and total QOSA indices for both independentinputs and correlated inputs with ρ , = 0 . 75. In the independent setting, the influence of thevariable X is close to 0 except for large values of α . We also note that the first-order and totalQOSA indices vary in reverse direction and from some α -level, ST αi (cid:54) S αi , i = 1 , 2. This supportsthat Proposition 3.5 is not true outside the additive framework with independent inputs.The behavior of the indices is similar in the dependent case. However, the influence of the input X is reinforced in this scheme due to its large correlation with X that is an influent variable. Indeed,the index S α increases faster than in independent case. On the contrary, the index ST α decreasesto 0 quicker than in the independent case because of its high dependence with X . QO S A i n d i c e s Independent inputs 0.0 0.2 0.4 0.6 0.8 1.0Values of Dependent inputs: 1, 2 = 0.75 S ST S ST Figure 3: First-order and total QOSA indices with independent (resp. dependent) inputs on theleft (resp. right) graph.To get another perspective on the impact of the dependence over the indices, we plot in Figure4, for several levels α , the evolution of the latter as a function of the correlation coefficient. As forthe linear Gaussian model, we observe that the total QOSA indices tend to zero as | ρ | → S i (cid:54) ST αi , i = 1 , . . . , d for additive models with independent inputs. But thiscontext is far from reality for many concrete examples and this inequality is no longer valid outsidethis framework as outlined by examples presented in Subsections 4.1 and 4.3. This therefore makes17 .00.20.40.60.81.0 QO S A i n d i c e s = 0.10 = 0.301.0 0.5 0.0 0.5 1.0Correl. coef. between X and X QO S A i n d i c e s = 0.70 1.0 0.5 0.0 0.5 1.0Correl. coef. between X and X = 0.90 S ST S ST Figure 4: Evolution of the first-order and total QOSA indices at different values of ρ for severallevels α .the interpretation of the indices complicated.Furthermore, in the case of dependent inputs, the behaviour of the QOSA indices should be com-pared to that of Sobol indices. Indeed, whatever the model (additive or not), it may happen inthis scheme that the first-order QOSA indices are higher than the total ones depending on thecorrelation level. We have also observed that total indices tend to zero as the absolute value of thecorrelation goes to 1.We could establish a strategy similar to Mara et al. (2015) in order to better understand theimpact of inputs in case of statistical dependence over the α -quantile, i.e., if their contributionderives from their marginal importance or their dependence with another variable. But, we preferto turn to Shapley values which present good properties for both independent and dependent inputs.Indeed, they allocate fairly to each input the interaction and/or dependency effect in which it isinvolved. 18 Quantile oriented Shapley effects In this section, we propose to use Shapley values defined in Equation (2.5), and recalled below, inorder to quantify the impact of each input over the α -quantile of the output distribution v i = X J ⊆D\{ i } ( d − |J | − |J | ! d ! ( c ( J ∪ { i } ) − c ( J )) , (5.1)with c ( · ) a generic cost function which maps the exploratory power generated by each subset J ⊆ D .Shapley value was first adapted within the framework of variance-based sensitivity measures tomeasure how much of Var ( Y ) can be attributed to each X i . Indeed, Owen (2014) and Song et al.(2016) proposed to use the two following unnormalized cost functions to measure the variance of Y caused by the uncertainty of the inputs in the subset J ⊆ D also named as being the explanatorypower created by J : ˜ c ( J ) = Var ( E [ Y | X J ]) and c ( J ) = E [Var ( Y | X −J )] . (5.2)Measuring the variance of Y caused by the uncertainty of the inputs in J is equivalent to assessthe impact of the inputs over the expected output. Thus, when using the cost functions given in(5.2), the feature of interest of the output considered is the expectation denoted by θ ∗ ( Y ) = E [ Y ].We show in the left-hand column in Table 1 that both cost functions may be rewritten accordingto the contrast function related to the expectation as well as the conditional feature θ ∗ ( Y | X J ) = E [ Y | X J ] for the first cost function and θ ∗ ( Y | X −J ) = E [ Y | X −J ] for the second one.Thus, in order to define indices for another feature of interest, the idea is to substitute thecontrast function of the expectation by that associated with the feature of interest required.Let us now use θ ∗ ( Y ) , θ ∗ ( Y | X J ) and θ ∗ ( Y | X −J ) for J ⊆ D as generic expressions to designatea feature of interest and the conditional ones related to a contrast function ψ . The impact of theinputs over θ ∗ ( Y ) is therefore assessed by measuring their contribution to the averaged contrastfunction E [ ψ ( Y, θ ∗ ( Y ))]. This one can be seen as a relevant distance allowing to quantify thevariability around the feature of interest. The contributions of the inputs are then calculated withthe following cost functions measuring the explanatory power of the subset J ⊆ D ˜ c ( J ) = E [ ψ ( θ ∗ ( Y | X J ) , θ ∗ ( Y ))] and c ( J ) = E [ ψ ( Y, θ ∗ ( Y | X −J ))] . (5.3)These cost functions are valid choices if they satisfy that the empty set creates no value, and thatall inputs generate E [ ψ ( Y, θ ∗ ( Y ))]. This is, for example, verified for all contrast functions listed inFort et al. (2016) which allow to propose new indices named Goal oriented Shapley effects (GOSE) .In our context, this property is verified in particular for cost functions related to quantiles presentedin Table 1. Hence we may propose Shapley effects subordinated to quantiles . However, we definethe Quantile oriented Shapley effects (QOSE) denoted by Sh αi with the second cost function becauseit verifies that the incremental cost c ( J ∪ { i } ) − c ( J ) is non negative, which is also a desirableproperty for cost functions. Indeed, for J ⊆ D\{ i } , we have c ( J ∪ i ) − c ( J ) = (cid:16) α E [ Y ] − E h Y { Y (cid:54) q α ( Y | X −J ∪ i ) } i(cid:17) − (cid:16) α E [ Y ] − E h Y { Y (cid:54) q α ( Y | X −J ) } i(cid:17) = E h ( Y − q α ( Y | X −J ∪ i )) (cid:16) { Y (cid:54) q α ( Y | X −J ) } − { Y (cid:54) q α ( Y | X −J ∪ i ) } (cid:17)i (cid:62) . eature of interest θ ∗ ( Y ) = E [ Y ] θ ∗ ( Y ) = q α ( Y ) Contrast function ψ ( y, θ ) = ( y − θ ) ψ ( y, θ ) = ( y − θ ) (cid:16) α − { y (cid:54) θ } (cid:17) Average contrast function Var ( Y ) = E [ ψ ( Y, E [ Y ])]= E [ ψ ( Y, θ ∗ ( Y ))] Υ ( Y ) = E [ ψ ( Y, q α ( Y ))]= E [ ψ ( Y, θ ∗ ( Y ))] First cost function ˜ c ( J ) = Var ( E [ Y | X J ])= E [ ψ ( E [ Y | X J ] , E [ Y ])]= E [ ψ ( θ ∗ ( Y | X J ) , θ ∗ ( Y ))]˜ c ( ∅ ) = 0 ˜ c ( D ) = Var ( Y ) ˜ c ( J ) = E [ ψ ( θ ∗ ( Y | X J ) , θ ∗ ( Y ))]= E [ ψ ( q α ( Y | X J ) , q α ( Y ))]˜ c ( ∅ ) = 0 ˜ c ( D ) = Υ ( Y ) Second cost function c ( J ) = E [Var ( Y | X −J )]= E [ ψ ( Y, E [ Y | X −J ])]= E [ ψ ( Y, θ ∗ ( Y | X −J ))] c ( ∅ ) = 0 c ( D ) = Var ( Y ) c ( J ) = E [ ψ ( Y, θ ∗ ( Y | X −J ))]= E [ ψ ( Y, q α ( Y | X −J ))] c ( ∅ ) = 0 c ( D ) = Υ ( Y )Table 1: Analogy of the cost functions used for quantifying the impact of the inputs over theexpectation for the case where the quantile is the feature of interest.At this stage, this property has not yet been demonstrated for the first cost function ˜ c .We study in the next subsections examples whose analytical values of the index Sh αi are computedby using the cost function normalized by the quantity Υ ( Y ) introduced in Table 1, so that d X i =1 Sh αi =1. Our aim is to show that these new indices give sensible answers compared to the classical QOSAindices defined in Section 3. We obtain the following analytical values for the linear model with Gaussian inputs. Proposition 5.1. If Y = η ( X ) = β + β T X with β ∈ R , β ∈ R d and X ∼ N ( µ , Σ ) where Σ ∈ R d × d is a positive-definite matrix, then the Quantile oriented Shapley effect for the variable i at the α -level is Sh αi = 1 d · σ Y X J ⊆D\{ i } d − |J | ! − "r β T J + i (cid:16) Σ J + i, J + i − Σ J + i, −J − i Σ − −J − i, −J − i Σ −J − i, J + i (cid:17) β J + i − r β T J (cid:16) Σ J , J − Σ J , −J Σ − −J , −J Σ −J , J (cid:17) β J (5.4)20 ith σ Y = Var ( Y ) = β T Σ β , and J + i (resp. −J − i ), a notational compression for J ∪ { i } (resp. −J ∪ { i } ). As for the QOSA index, we may notice that β and µ do not play any role as translationparameters and that the index does not depend on the α -level which is a specificity of the linearGaussian model as explained previously. Let us consider the case d = 2 with µ = µ µ ! , β = β β ! and Σ = σ ρσ σ ρσ σ σ ! , − ≤ ρ ≤ , σ > , σ > . We have σ Y = V ar ( Y ) = β σ + 2 ρβ β σ σ + β σ and obtain from(5.4) Sh α = 12 − | β | σ p − ρ · σ Y + | β | σ p − ρ · σ Y ,Sh α = 12 − | β | σ p − ρ · σ Y + | β | σ p − ρ · σ Y . (5.5)We observe that the correlation effects on the first-order QOSA indices (e.g. σ Y − | β | σ p − ρ for X ) and on the total QOSA indices (e.g. | β | σ p − ρ for X ) are allocated half to the Quantileoriented Shapley effects - QOSE. We also see that the Shapley effects are equal when the correlationis maximum (i.e. | ρ | = 1).Figure 5 presents the first-order and total QOSA indices as well as the QOSE for the particularcase µ = µ = 0 , β = β = 1 , σ = 1 and σ = 2.On the left-hand graph of the figure, we see that the Shapley effects are also constant and they arebrackected by the first-order and total QOSA indices : S αi (cid:54) Sh αi (cid:54) ST αi , i = 1 , QO S A i n d i c e s Independent inputs 1.0 0.5 0.0 0.5 1.0Correl. coef. between X and X Dependent inputs -level S ST Sh S ST Sh Figure 5: First-order and total QOSA indices as well as the QOSE with independent (resp. depen-dent) inputs on the left (resp. right) graph.We illustrate on the right-hand graph the evolution of the indices as a function of the correlationbetween the two inputs. As X is the more uncertain variable, its sensitivity indices are larger21han those of X . Then, although the values are not identical, we can note that the shape of thecurves is exactly the same as that observed for the variance-based Shapley effects calculated for thetwo-dimensional Gaussian linear model (with the same setting) in Iooss and Prieur (2019). Indeed,we observe that in the presence of correlation, the QOSE lie between the first-order QOSA indicesand the total ones with either S αi (cid:54) Sh αi (cid:54) ST αi or ST αi (cid:54) Sh αi (cid:54) S αi , i = 1 , 2. This phenomenonis called the “sandwich effect” within the variance framework in Iooss and Prieur (2019). Finally,as for the variance-based Shapley effects, it also seems that the dependence between the two inputslead to a rebalancing of their respective QOSE. We analyze in this subsection the analytical values below for the model with Gaussian inputs andthe resulting output Log-normal distributed. Proposition 5.2. If Y = η ( X ) = exp (cid:16) β + β T X (cid:17) with β ∈ R , β ∈ R d and X ∼ N ( µ , Σ ) where Σ ∈ R d × d is apositive-definite matrix, then the Quantile oriented Shapley effect for the variable i at the α -level is Sh αi = 1 d · A X J ⊆D\{ i } d − |J | ! − h Φ (cid:16) Φ − ( α ) − B ( J ) (cid:17) − Φ (cid:16) Φ − ( α ) − C ( J , i ) (cid:17)i (5.6) with A = α − Φ (cid:16) Φ − ( α ) − σ (cid:17) and σ = β T Σ β ,B ( J ) = r β T J (cid:16) Σ J , J − Σ J , −J Σ − −J , −J Σ −J , J (cid:17) β J ,C ( J , i ) = r β T J + i (cid:16) Σ J + i, J + i − Σ J + i, −J − i Σ − −J − i, −J − i Σ −J − i, J + i (cid:17) β J + i , where J + i (resp. −J − i ) is a notational compression for J ∪ { i } (resp. −J ∪ { i } ). As for the QOSA indices, we observe that β and µ do not play any role and that the indicesdepend on α compared to the linear Gaussian model. However, it is difficult to reach a conclusionfrom Equation (5.6). Accordingly, we consider the particular case d = 2 with µ = µ µ ! , β = β β ! and Σ = σ ρσ σ ρσ σ σ ! , − ≤ ρ ≤ , σ > , σ > . We have σ = β σ + 2 ρβ β σ σ + β σ and obtain from (5.6) Sh α = 12 + 12 · Φ (cid:16) Φ − ( α ) − | β | σ p − ρ (cid:17) α − Φ (Φ − ( α ) − σ ) − · Φ (cid:16) Φ − ( α ) − | β | σ p − ρ (cid:17) α − Φ (Φ − ( α ) − σ ) ,Sh α = 12 + 12 · Φ (cid:16) Φ − ( α ) − | β | σ p − ρ (cid:17) α − Φ (Φ − ( α ) − σ ) − · Φ (cid:16) Φ − ( α ) − | β | σ p − ρ (cid:17) α − Φ (Φ − ( α ) − σ ) . (5.7)We adopt the next settings in all further tests: µ = µ = 0 , β = β = 1 , σ = 1 and σ = 2.22he analytical values of the first-order, total QOSA indices and the QOSE are illustrated inFigure 6 for both independent inputs and correlated inputs with ρ , = 0 . 75. The “sandwich effect”which was noticed in the linear Gaussian model in the presence of correlation is also observed here.Indeed, both in the dependent and independent cases and for all the levels α , the QOSE lie betweenthe first-order and total QOSA indices.Besides, with the three indices, we obtain the same ranking of the inputs for all α -levels butthe QOSE is easier to interpret because it properly condenses all the information (dependence andinteraction effects). For instance, let us focus over the input X on the right-hand graph at the level α = 0 . 2. If we use the first-order QOSA index S α , we conclude that the impact of the input X is low, but not so small because, conversely its total QOSA index is high enough. But, ultimately,it is difficult to quantify precisely on the basis of these two indices the contribution of the input X at level α = 0 . 2. The Shapley index, in contrast, contains the marginal contribution of thevariable but also those due to dependence and interaction effects that are correctly allocated to it.It therefore makes easier to express an opinion on the impact of the variable by taking into accountall possible contributions. This observation is valid for all the levels α . QO S A i n d i c e s Independent inputs 0.0 0.2 0.4 0.6 0.8 1.0Values of Dependent inputs: 1, 2 = 0.75 S ST Sh S ST Sh Figure 6: First-order and total QOSA indices as well as the Quantile oriented Shapley effects withindependent (resp. dependent) inputs on the left (resp. right) graph.Again, to get another insight on the impact of the dependence over the indices, we plot inFigure 7, for several levels α , the evolution of the latter as a function of the correlation coefficient.As explained before, the QOSE give a condensed information of all contributions. That explainswhy we observe that the Shapley effects of both variables are almost equal for small values of α .Conversely, for large values, the variable X is the most influential overall except when | ρ | → .00.20.40.60.81.0 QO S A i n d i c e s = 0.10 = 0.301.0 0.5 0.0 0.5 1.0Correl. coef. between X and X QO S A i n d i c e s = 0.70 1.0 0.5 0.0 0.5 1.0Correl. coef. between X and X = 0.90 S ST Sh S ST Sh Figure 7: Evolution of the first-order and total QOSA indices as well as the Quantile orientedShapley effects at different values of ρ for several levels α . As a conclusion of this preliminary work, QOSE appear to be a good alternative to the classicalQOSA indices. Indeed, they make possible to overcome the various problems encountered with theQOSA indices such as ST αi (cid:54) S αi outside the additive framework or when using dependent inputs.Besides, the Sobol indices stem, for example, from the functional ANOVA decomposition but thereis no such a decomposition for the quantiles. 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