text: (cid:104) (cid:105) FIG. 7: The ﬁnal value of δφ¯2 as a function of ξ. | | Palatini formalism, with the increase of ξ the growth of the ﬂuctuation is delayed and the (cid:104) (cid:105) duration of parameter resonance era increases. The ﬁnal values of the ﬂuctuation ( δφ¯2 ) f are almost the same as that in the ξ = 0 case, which can also be seen from Tab. (II). For the metric formalism, since ω will play a dominating role during the whole preheating with 2 | | the increase of ξ , the duration of the parameter resonance era does not increase forever | | with the increase of ξ and the numerical calculation shows that it has a maximum at about (cid:39) − − ξ 70, if 250 < ξ < 0. In addition, one can see that several short plateau appear (cid:104) (cid:105) − in the evolution of δφ2 when ξ = 50, which arises from two eﬀective resonance bands. (cid:104) (cid:105) Tab. (II) and Fig. 6 show that the variation of δφ¯2 is dramatical in the metric formalism, f which is diﬀerent from that in the Palatini one. In order to reveal the characteristics of (cid:104) (cid:105) | | (cid:104) (cid:105) | | δφ¯2 with the increase of ξ , we plot δφ¯2 as a function of ξ in Fig. 7. We ﬁnd that in f f (cid:104) (cid:105) − (cid:46) the Palatini formalism the ﬁnal value of δφ¯2 is almost a constant within 1000 ξ < 0. (cid:104) (cid:105) | | While in the metric formalism δφ¯2 decreases ﬁrstly with the increase of ξ , and then f − (cid:104) (cid:105) reaches a minimum when ξ is about 70. After that, δφ¯2 increases fast until it reaches f − its maximum at about ξ = 120, and then decreases slowly. ˜ Tab. (II) indicates that in the Palatini formalism the oscillation amplitude φ of the 0 inﬂaton ﬁeld at the termination time x of the parameter resonance decreases slowly, while f ˜ in the metric formalism φ (x ) does not vary monotonously. Even so, in both formalisms 0 f (cid:104) (cid:105) | | δφ¯2 /φ˜2(x ) increases with the increase of ξ and it increases faster in the metric formalism. f 0 f This indicates that the energy transfer from the inﬂaton ﬁeld to δφ becomes more eﬃcient k | | with the increase of ξ and the eﬃciency in the Palatini formalism does not increase as fast as that in the metric one. 17 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: The

text: ﬁnal

text: value

text: of

text: δφ¯2

text: as

text: a

text: function

text: of

text: ξ.

text: Palatini

text: formalism,

text: with

text: the

text: increase

text: of

text: ξ

text: the

text: growth

text: of

text: the

text: ﬂuctuation

text: is

text: delayed

text: and

text: the

text: duration

text: of

text: parameter

text: resonance

text: era

text: increases.

text: The

text: ﬁnal

text: values

text: of

text: the

text: ﬂuctuation

text: δφ¯2

text: f

text: are

text: almost

text: the

text: same

text: as

text: that

text: in

text: the

text: ξ

text: case,

text: which

text: can

text: also

text: be

text: seen

text: from

text: Tab.

text: (II).

text: For

text: the

text: metric

text: formalism,

text: since

text: ω

text: will

text: play

text: a

text: dominating

text: role

text: during

text: the

text: whole

text: preheating

text: with

text: the

text: increase

text: of

text: ξ

text: the

text: duration

text: of

text: the

text: parameter

text: resonance

text: era

text: does

text: not

text: increase

text: forever

text: with

text: the

text: increase

text: of

text: ξ

text: and

text: the

text: numerical

text: calculation

text: shows

text: that

text: it

text: has

text: a

text: maximum

text: at

text: about

text: ξ

text: if

text: ξ

text: In

text: addition,

text: one

text: can

text: see

text: that

text: several

text: short

text: plateau

text: appear

text: in

text: the

text: evolution

text: of

text: δφ2

text: when

text: ξ

text: which

text: arises

text: from

text: two

text: eﬀective

text: resonance

text: bands.

text: Tab.

text: (II)

text: and

text: Fig.

text: show

text: that

text: the

text: variation

text: of

text: δφ¯2

text: is

text: dramatical

text: in

text: the

text: metric

text: formalism,

text: f

text: which

text: is

text: diﬀerent

text: from

text: that

text: in

text: the

text: Palatini

text: one.

text: In

text: order

text: to

text: reveal

text: the

text: characteristics

text: of

text: δφ¯2

text: with

text: the

text: increase

text: of

text: ξ

text: we

text: plot

text: δφ¯2

text: as

text: a

text: function

text: of

text: ξ

text: in

text: Fig.

text: We

text: ﬁnd

text: that

text: in

text: f

text: f

text: the

text: Palatini

text: formalism

text: the

text: ﬁnal

text: value

text: of

text: δφ¯2

text: is

text: almost

text: a

text: constant

text: within

text: ξ

text: While

text: in

text: the

text: metric

text: formalism

text: δφ¯2

text: decreases

text: ﬁrstly

text: with

text: the

text: increase

text: of

text: ξ

text: and

text: then

text: f

text: reaches

text: a

text: minimum

text: when

text: ξ

text: is

text: about

text: After

text: that,

text: δφ¯2

text: increases

text: fast

text: until

text: it

text: reaches

text: f

text: its

text: maximum

text: at

text: about

text: ξ

text: and

text: then

text: decreases

text: slowly.

text: Tab.

text: (II)

text: indicates

text: that

text: in

text: the

text: Palatini

text: formalism

text: the

text: oscillation

text: amplitude

text: φ

text: of

text: the

text: inﬂaton

text: ﬁeld

text: at

text: the

text: termination

text: time

text: x

text: of

text: the

text: parameter

text: resonance

text: decreases

text: slowly,

text: while

text: f

text: in

text: the

text: metric

text: formalism

text: φ

text: (x

text: does

text: not

text: vary

text: monotonously.

text: Even

text: so,

text: in

text: both

text: formalisms

text: f

text: δφ¯2

text: increases

text: with

text: the

text: increase

text: of

text: ξ

text: and

text: it

text: increases

text: faster

text: in

text: the

text: metric

text: formalism.

text: f

text: f

text: This

text: indicates

text: that

text: the

text: energy

text: transfer

text: from

text: the

text: inﬂaton

text: ﬁeld

text: to

text: δφ

text: becomes

text: more

text: eﬃcient

text: k

text: with

text: the

text: increase

text: of

text: ξ

text: and

text: the

text: eﬃciency

text: in

text: the

text: Palatini

text: formalism

text: does

text: not

text: increase

text: as

text: fast

text: as

text: that

text: in

text: the

text: metric

text: one.

text: ##LTFigure##

text: ##LTLine##

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text: arXiv:1612.06515. [21] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010). [22] A. De Felice and S. Tsujikawa, Living Rev. Relativ. 13, 3 (2010). [23] F. Bauer and D. A. Demir, Phys. Lett. B 665, 222 (2008). [24] B. Li, D. F. Mota and D. J. Shaw, Class. Quant. Grav. 26, 055018 (2009). [25] N. Tamanini and C. R. Contaldi, Phys. Rev. D 83, 044018 (2011). [26] C. P. Burgess, H. M. Lee, and M. Trott, JHEP 0909, 103 (2009). [27] F. Bezrukov, A. Magnin, M. Shaposhnikov and S. Sibiryakov, JHEP 1101, 016 (2011). [28] M. Atkins and X. Calmet, Phys. Lett. B 697, 37 (2011). [29] R. N. Lerner and J. McDonald, JCAP 1004, 015 (2010). [30] J. L. F. Barbon and J. R. Espinosa, Phys. Rev. D 79, 081302 (2009). [31] F. Bauer and D. A. Demir, Phys. Lett. B 698, 425 (2011). [32] S. Rasanen, and P. Wahlman, arXiv: 1709.07853. [33] T. Futamase and K.i. Maeda, Phys. Rev. D 39, 399 (1989). [34] J. P. Zibin, R. Brandenberger and D. Scott, Phys. Rev. D 63, 043511 (2001). [35] S. Y. Khlebnikov and I. I. Tkachev, Phys. Rev. D 56, 653 (1997); R. Easther and E. A. Lim, JCAP 0604, 010 (2006). 23

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text: V.

text: Faraoni,

text: Rev.

text: Mod.

text: Phys.

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text: De

text: Felice

text: and

text: S.

text: Tsujikawa,

text: Living

text: Rev.

text: Relativ.

text: F.

text: Bauer

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text: D.

text: A.

text: Demir,

text: Phys.

text: Lett.

text: B

text: B.

text: Li,

text: D.

text: F.

text: Mota

text: and

text: D.

text: J.

text: Shaw,

text: Class.

text: Quant.

text: Grav.

text: N.

text: Tamanini

text: and

text: C.

text: R.

text: Contaldi,

text: Phys.

text: Rev.

text: D

text: C.

text: P.

text: Burgess,

text: H.

text: M.

text: Lee,

text: and

text: M.

text: Trott,

text: JHEP

text: F.

text: Bezrukov,

text: A.

text: Magnin,

text: M.

text: Shaposhnikov

text: and

text: S.

text: Sibiryakov,

text: JHEP

text: M.

text: Atkins

text: and

text: X.

text: Calmet,

text: Phys.

text: Lett.

text: B

text: R.

text: N.

text: Lerner

text: and

text: J.

text: McDonald,

text: JCAP

text: J.

text: L.

text: F.

text: Barbon

text: and

text: J.

text: R.

text: Espinosa,

text: Phys.

text: Rev.

text: D

text: F.

text: Bauer

text: and

text: D.

text: A.

text: Demir,

text: Phys.

text: Lett.

text: B

text: S.

text: Rasanen,

text: and

text: P.

text: Wahlman,

text: arXiv:

text: T.

text: Futamase

text: and

text: K.i.

text: Maeda,

text: Phys.

text: Rev.

text: D

text: J.

text: P.

text: Zibin,

text: R.

text: Brandenberger

text: and

text: D.

text: Scott,

text: Phys.

text: Rev.

text: D

text: S.

text: Y.

text: Khlebnikov

text: and

text: I.

text: I.

text: Tkachev,

text: Phys.

text: Rev.

text: D

text: R.

text: Easther

text: and

text: E.

text: A.

text: Lim,

text: JCAP


text: † | | where a and a are the annihilation and creation operators respectively and k = k , k k from Eq. (15), we ﬁnd that δφ satis(cid:20)ﬁes the following equation(cid:21)of motion: k k2 (cid:104) (cid:105) δφ¨ + 3Hδφ˙ + + 3λ(φ2 + δφ2 ) + ξRˆ δφ = 0 (18) k k a2 0 with the expectation values of δφ2 taking t(cid:90)he form (cid:104) (cid:105) 1 | | δφ2 = k2 δφ 2dk . (19) (cid:82) 2π2 k ≡ − ≡ Introducing the conformal time η a 1dt and deﬁning a new scalar ﬁeld ϕ aφ and 0 0 ≡ its perturbation δϕ aδφ, Eqs. (14) and (18) can be rewritten as d2ϕ (cid:104) (cid:105) − 1 d2a 0 + λϕ (ϕ2 + 3 δϕ2 ) + (ξRˆa2 )ϕ = 0 , (20) dη2 0 0 a dη2 0 d2 δϕ + ω2δϕ = 0 , (21) dη2 k k k with ≡ (cid:104) (cid:105) − 1 d2a ω2 k2 + 3λ(ϕ2 + δϕ2 ) + ξRˆa2 . (22) k 0 a dη2 In the non-minimal coupling model, the ξRˆa2 term aﬀects not only the oscillation mode of the inﬂaton ﬁeld but also the parametric resonance. Furthermore, the contributions of this term in the metric formalism are diﬀerent from those in the Palatini formalism. We choose the conformal vacuum as the initial state and the following initial conditions of the ﬂuctuation (cid:112) 1 δϕ (0) = , (23) k 2ω (0) k − δϕ˙ (0) = iω (0)δϕ (0) . (24) k k k 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: where

text: a

text: and

text: a

text: are

text: the

text: annihilation

text: and

text: creation

text: operators

text: respectively

text: and

text: k

text: k

text: k

text: k

text: from

text: Eq.

text: we

text: ﬁnd

text: that

text: δφ

label: equation

text: the

text: following

text: motion:

text: k

text: δφ¨

text: 3Hδφ˙

text: δφ2

text: ξRˆ

text: δφ

text: k

text: k

text: with

text: the

text: expectation

text: values

text: of

text: δφ2

text: taking

text: form

text: δφ2

text: δφ

text: k

text: Introducing

text: the

text: conformal

text: time

text: η

text: a

text: and

text: deﬁning

text: a

text: new

text: scalar

text: ﬁeld

text: ϕ

text: aφ

text: and

text: its

text: perturbation

text: δϕ

text: aδφ,

text: Eqs.

text: and

text: can

text: be

text: rewritten

text: as

text: λϕ

text: δϕ2

text: )ϕ

text: a

text: δϕ

text: ω2δϕ

text: k

text: k

text: k

text: with

text: ω2

text: δϕ2

text: k

text: a

text: In

text: the

text: non-minimal

text: coupling

text: model,

text: the

text: term

text: aﬀects

text: not

text: only

text: the

text: oscillation

text: mode

text: of

text: the

text: inﬂaton

text: ﬁeld

text: but

text: also

text: the

text: parametric

text: resonance.

text: Furthermore,

text: the

text: contributions

text: of

text: this

text: term

text: in

text: the

text: metric

text: formalism

text: are

text: diﬀerent

text: from

text: those

text: in

text: the

text: Palatini

text: formalism.

text: We

text: choose

text: the

text: conformal

text: vacuum

text: as

text: the

text: initial

text: state

text: and

text: the

text: following

text: initial

text: conditions

text: of

text: the

text: ﬂuctuation

text: δϕ

text: k

text: 2ω

text: k

text: δϕ˙

text: iω

text: k

text: k

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Shapley value originates from game theory in [Sha53]. This quantity can be use- ful in very diﬀerent ﬁelds (see for example [MvLG+08] or [HI03]). Hence, there is an active strand of litterature focusing on the computation of Shapley values ([CBSV16], [FWJ08]...) but they are adapted to a more general framework than sensitivity analysis. Only a few articles focus on this computation in the ﬁeld of global sensitivity analysis (see [Owe14], [SNS16], [OP17], [IP17]). In this paper, we focus on two popular variance-based sensitivity indices (Sobol indices and Shapley eﬀects) when the input variables form independent groups and the model is additive. We ﬁrst show an analytical formula on the variances of the conditional expectations. Then, we use this formula to prove that the sensitivity indices have much simpler expressions in this framework. In particular, we show that many Sobol indices are equal to zero and that the computation of the Shapley eﬀects is much easier in this case than in the general framework. Finally, we provide an algorithm for evaluating these sensitivity indices in the linear Gaussian case, which is particularly eﬃcient when the covariance matrix of the input Gaussian distribution is block diagonal. The paper is organized as follows: in Section 2, we recall the deﬁnition of Sobol indices and Shapley eﬀects and their properties. In Section 3, we give our main theoretical results about these sensitivity indices when there are indepen- dent groups of input variables. In Section 4, we focus on the linear Gaussian framework and give an algorithm to compute the variance-based sensitivity indices in this case. Then, we apply our theoretical results about the variance- based sensitivity indices to give an eﬃcient computation when the covariance matrix is block diagonal. In Section 5, we highlight our results comparing these algorithms with the existing algorithm of [SNS16] designed to compute the Shapley eﬀects. The proofs are postponed to the appendix. 2 Background and notations Let (Ω, A, P) be a probability space, E = ×p E be endowed with the E ⊗ E i=1 i product σ-algebra = p and X be a random variable from Ω to E in L2. i=1 i The (X ) ∈ will be the input variables of the model. i i [E1:p]→ R B R Let f : (E, ) ( , ( )) be in L2 and Y = f(X). We will call Y the output (cid:54) variable. We assume that Var(Y ) = 0. ⊂ ∈ R Let [1 : p] be the set of all integers between 1 and p. If u [1 : p] and x p, we will write x := (x ) ∈ . ⊂ u i i u | For all u [1 : p], we write V := Var(E(Y X )). We can now deﬁne the Sobol u (cid:88) u indices (see [SCSo00]) for a group of variable X as: u 1 − | |−| | S := ( 1) u v V , (1) u Var(Y ) ⊂ v v u | | where u is the cardinal of u. This quantity assesses the impact of the subset of inputs X on the model output variability. These sensitivity indices u were the ﬁrst ones induced in [Sob93] for non-linear models. The Sobol indices ⊂ are well deﬁned and the sum of the Sobol indices over u [1 : p] is still equal to one, even if the inputs are dependent. Moreover, these sensitivity indices 2 ##LTLine##

text: Shapley

text: value

text: originates

text: from

text: game

text: theory

text: in

text: This

text: quantity

text: can

text: be

text: use-

text: ful

text: in

text: very

text: diﬀerent

text: ﬁelds

text: (see

text: for

text: example

text: or

text: Hence,

text: there

text: is

text: an

text: active

text: strand

text: of

text: litterature

text: focusing

text: on

text: the

text: computation

text: of

text: Shapley

text: values

text: but

text: they

text: are

text: adapted

text: to

text: a

text: more

text: general

text: framework

text: than

text: sensitivity

text: analysis.

text: Only

text: a

text: few

text: articles

text: focus

text: on

text: this

text: computation

text: in

text: the

text: ﬁeld

text: of

text: global

text: sensitivity

text: analysis

text: (see

text: In

text: this

text: paper,

text: we

text: focus

text: on

text: two

text: popular

text: variance-based

text: sensitivity

text: indices

text: (Sobol

text: indices

text: and

text: Shapley

text: eﬀects)

text: when

text: the

text: input

text: variables

text: form

text: independent

text: groups

text: and

text: the

text: model

text: is

text: additive.

text: We

text: ﬁrst

text: show

text: an

text: analytical

text: formula

text: on

text: the

text: variances

text: of

text: the

text: conditional

text: expectations.

text: Then,

text: we

text: use

text: this

text: formula

text: to

text: prove

text: that

text: the

text: sensitivity

text: indices

text: have

text: much

text: simpler

text: expressions

text: in

text: this

text: framework.

text: In

text: particular,

text: we

text: show

text: that

text: many

text: Sobol

text: indices

text: are

text: equal

text: to

text: zero

text: and

text: that

text: the

text: computation

text: of

text: the

text: Shapley

text: eﬀects

text: is

text: much

text: easier

text: in

text: this

text: case

text: than

text: in

text: the

text: general

text: framework.

text: Finally,

text: we

text: provide

text: an

text: algorithm

text: for

text: evaluating

text: these

text: sensitivity

text: indices

text: in

text: the

text: linear

text: Gaussian

text: case,

text: which

text: is

text: particularly

text: eﬃcient

text: when

text: the

text: covariance

text: matrix

text: of

text: the

text: input

text: Gaussian

text: distribution

text: is

text: block

text: diagonal.

text: The

text: paper

text: is

text: organized

text: as

text: follows:

text: in

text: Section

text: we

text: recall

text: the

text: deﬁnition

text: of

text: Sobol

text: indices

text: and

text: Shapley

text: eﬀects

text: and

text: their

text: properties.

text: In

text: Section

text: we

text: give

text: our

text: main

text: theoretical

text: results

text: about

text: these

text: sensitivity

text: indices

text: when

text: there

text: are

text: indepen-

text: dent

text: groups

text: of

text: input

text: variables.

text: In

text: Section

text: we

text: focus

text: on

text: the

text: linear

text: Gaussian

text: framework

text: and

text: give

text: an

text: algorithm

text: to

text: compute

text: the

text: variance-based

text: sensitivity

text: indices

text: in

text: this

text: case.

text: Then,

text: we

text: apply

text: our

text: theoretical

text: results

text: about

text: the

text: variance-

text: based

text: sensitivity

text: indices

text: to

text: give

text: an

text: eﬃcient

text: computation

text: when

text: the

text: covariance

text: matrix

text: is

text: block

text: diagonal.

text: In

text: Section

text: we

text: highlight

text: our

text: results

text: comparing

text: these

text: algorithms

text: with

text: the

text: existing

text: algorithm

text: of

text: designed

text: to

text: compute

text: the

text: Shapley

text: eﬀects.

text: The

text: proofs

text: are

text: postponed

text: to

text: the

text: appendix.

text: Background

text: and

text: notations

text: Let

text: (Ω,

text: A,

text: P)

text: be

text: a

text: probability

text: space,

text: E

text: ×p

text: E

text: be

text: endowed

text: with

text: the

text: E

text: E

text: i

text: product

text: σ-algebra

text: p

text: and

text: X

text: be

text: a

text: random

text: variable

text: from

text: Ω

text: to

text: E

text: in

text: i

text: The

text: (X

text: will

text: be

text: the

text: input

text: variables

text: of

text: the

text: model.

text: i

text: i

text: R

text: B

text: R

text: Let

text: f

text: (E,

text: be

text: in

text: and

text: Y

text: f(X).

text: We

text: will

text: call

text: Y

text: the

text: output

text: variable.

text: We

text: assume

text: that

text: Var(Y

text: R

text: Let

text: p]

text: be

text: the

text: set

text: of

text: all

text: integers

text: between

text: and

text: p.

text: If

text: u

text: p]

text: and

text: x

text: p,

text: we

text: will

text: write

text: x

text: (x

text: u

text: i

text: i

text: u

text: For

text: all

text: u

text: p],

text: we

text: write

text: V

text: Var(E(Y

text: X

text: We

text: can

text: now

text: deﬁne

text: the

text: Sobol

text: u

text: u

text: indices

text: (see

text: for

text: a

text: group

text: of

text: variable

text: X

text: as:

text: u

text: S

text: u

text: v

text: V

text: u

text: Var(Y

text: v

text: v

text: u

text: where

text: u

text: is

text: the

text: cardinal

text: of

text: u.

text: This

text: quantity

text: assesses

text: the

text: impact

text: of

text: the

text: subset

text: of

text: inputs

text: X

text: on

text: the

text: model

text: output

text: variability.

text: These

text: sensitivity

text: indices

text: u

text: were

text: the

text: ﬁrst

text: ones

text: induced

text: in

text: for

text: non-linear

text: models.

text: The

text: Sobol

text: indices

text: are

text: well

text: deﬁned

text: and

text: the

text: sum

text: of

text: the

text: Sobol

text: indices

text: over

text: u

text: p]

text: is

text: still

text: equal

text: to

text: one,

text: even

text: if

text: the

text: inputs

text: are

text: dependent.

text: Moreover,

text: these

text: sensitivity

text: indices

text: ##LTLine##


text: (cid:18) (cid:19) (cid:88) − − 1 p 1 1 − η = (cid:18) | | (cid:19) (V ∪{ } V ) i pVar(Y ) (cid:88)⊂− u (cid:88)u i u u i − − 1 p 1 1 k −   = pVar(Y ) (cid:88)⊂− (cid:18) |u| (cid:19) (V(cid:88)(u∪i)∩Cj Vu∩Cj) u i j=1   − − 1 p 1 1 − − = (cid:18) | | (cid:19) V ∩ V ∩ + (V ∪{ } ∩ V ∩ ) pVar(Y ) (cid:88)⊂− u (cid:54) u Cj u Cj (u i ) Cj(i) u Cj(i) u i j=j(i) − − = pVa1r(Y ) ⊂−(cid:88) p|u|1 (cid:88)1 (V((cid:18)u∪{i})∩C(cid:19)j(i) − Vu∩Cj(i)) u i − − 1 p 1 1 − = (cid:18)| ∪ | (cid:19) (V ∪ ∪{ } ∩ V ∪ ∩ ) pVar(Y ) ⊂(cid:88) − ⊂(cid:88)− u v (u v i ) Cj(i) u v Cj(i) u Cj(i) i v Cj(i) − − 1 p 1 1 − = pVar(Y ) ⊂(cid:88) − (cid:88)⊂− (cid:18) |u| + |v|(cid:19) (cid:18) (Vu∪{(cid:19)i} Vu) u Cj(i) i v Cj(i) 1 p−|Cj(i)| p − |C | p − 1 −1 − = j(i) | | (V ∪{ } V ). pVar(Y ) ⊂ − j u + j u i u u Cj(i) i j=0 It remains to prove the following equation: (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:88) 1 p−|Cj(i)| p − |C | p − 1 −1 1 C − 1 −1 j(i) | | = | | j(|i) | . (17) p j u + j C u j(i) j=0 In the interest of simplifying notation, until the end of the proof, we will | | write u (resp. c) instead of u (resp. C ). We can verify that the equation j(i) (17) is equivalent to the following equations: (cid:88) − − − − − − − p c 1 (p c)! (u + j)!(p 1 u j)! 1 u!(c 1 u)! (cid:18) −(cid:19) (cid:18)− (cid:19) − (cid:18) (cid:19) = − (cid:88) p j!(p c j)! (p 1)! c (c 1)! j=0 p−c − − − ⇐⇒ u + j p 1 u j p − − = . (18) u c u 1 c j=0 We will show (18). Now, we can remark that we have: − − xc xu xc u 1 − = x − − − . (19) (1 x)c+1 (1 x)u+1 (1 x)c u 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: p

text: η

text: (V

text: V

text: i

text: pVar(Y

text: u

text: i

text: u

text: u

text: i

text: p

text: k

text: pVar(Y

text: |u|

text: Vu∩Cj)

text: u

text: i

text: p

text: V

text: V

text: (V

text: V

text: pVar(Y

text: u

font: default

text: u

text: Cj

text: u

text: Cj

text: (u

text: i

text: Cj(i)

text: u

text: Cj(i)

text: u

text: i

text: j=j(i)

text: Vu∩Cj(i))

text: u

text: i

text: p

text: (V

text: V

text: pVar(Y

text: u

text: v

text: (u

text: v

text: i

text: Cj(i)

text: u

text: v

text: Cj(i)

text: u

text: Cj(i)

text: i

text: v

text: Cj(i)

text: p

text: pVar(Y

text: |u|

text: Vu)

text: u

text: Cj(i)

text: i

text: v

text: Cj(i)

text: p−|Cj(i)|

text: p

text: |C

text: p

text: −1

text: j(i)

text: (V

text: V

text: pVar(Y

text: j

text: u

text: j

text: u

text: i

text: u

text: u

text: Cj(i)

text: i

text: It

text: remains

text: to

text: prove

text: the

text: following

text: equation:

text: p−|Cj(i)|

text: p

text: |C

text: p

text: −1

text: C

text: −1

text: j(i)

text: j(|i)

text: p

text: j

text: u

text: j

text: C

text: u

text: j(i)

text: In

text: the

text: interest

text: of

text: simplifying

text: notation,

text: until

text: the

text: end

text: of

text: the

text: proof,

text: we

text: will

text: write

text: u

text: (resp.

text: c)

text: instead

text: of

text: u

text: (resp.

text: C

text: We

text: can

text: verify

text: that

text: the

text: equation

text: j(i)

text: is

text: equivalent

text: to

text: the

text: following

text: equations:

text: p

text: c

text: (p

text: c)!

text: (u

text: j)!(p

text: u

text: j)!

text: u!(c

text: u)!

text: p

text: j!(p

text: c

text: j)!

text: (p

text: c

text: (c

text: p−c

text: u

text: j

text: p

text: u

text: j

text: p

text: u

text: c

text: u

text: c

text: We

text: will

text: show

text: Now,

text: we

text: can

text: remark

text: that

text: we

text: have:

text: xc

text: xu

text: xc

text: u

text: x

text: x)c

text: u

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ⊂ Proposition 2. For all u [1 : p] such that u is not a subset of a C , we have: j S = 0. (5) u Remark 1. We have the same result for the ”generalized sensitivity indices” suggested in [Cha13] but the proof is not a consequence of Proposition 1 and is straig(cid:80)htforward. This proposition gives the majority of Sobol indices immediately. It remains − − only k 2Cj 1 unknown non-zero Sobol indices instead of 2p 1. j=1 We also provide a consequence of Proposition 1 on Shapley eﬀects: ∈ ∈ N ∈ Proposition 3. Let i [1 : p]. Let j(i) be so that i C . Then, we (cid:18) (cid:19) j(i) have: (cid:88) (cid:0) (cid:1) | | − − 1 C 1 1 − η = | | j(|i) | V ∪{ } V . (6) i Cj(i) Var(Y ) ⊂ \{ } u u i u u Cj(i) i In an equivalent manner, let us write η˜ for the Shapley eﬀect for the follow- i ing output: Y := g (X ) Cj(i) j Cj (we just focus on the variables which are in the same group as X ). Then, the i Shapley eﬀect η for the whole output (Y = f(X)) is equal to : i Var(Y ) η = Cj(i) η˜ . (7) i Var(Y ) i For example, if we apply this proposition in the case where X is the only i variable in its group, then we have η = S . i i To compute the Shapley eﬀect η , the last proposition reduces the sum from all \{ } i \{ } the subsets of [1 : p] i to all the subset of C i . Then, the computational j(i) gain is the same as in Proposition 1. 4 Explicit computation of sensitivity indices for Gaussian linear models 4.1 Linear Gaussian framework ∼ N In this section, we assume that X (µ, Γ), that Γ is invertible and that (cid:55)−→ ∈ R f : x β + βT x, for a ﬁxed β and a ﬁxed vector β. This framework is 0 0 widely used to model physical phenomena (see for example [KHF+06], [HT11], [Ros04]). Indeed, uncertainties are often modelled as Gaussian variables and an unknown function is commonly estimated by its linear approximation. We 4 ##LTLine## ##LTLine##

text: Proposition

text: For

text: all

text: u

text: p]

text: such

text: that

text: u

text: is

text: not

text: a

text: subset

text: of

text: a

text: C

text: we

text: have:

text: j

text: S

text: u

text: Remark

text: We

text: have

text: the

text: same

text: result

text: for

text: the

text: ”generalized

text: sensitivity

text: indices”

text: suggested

text: in

text: but

text: the

text: proof

text: is

text: not

text: a

text: consequence

text: of

text: Proposition

text: and

text: is

text: This

text: proposition

text: gives

text: the

text: majority

text: of

text: Sobol

text: indices

text: immediately.

text: It

text: remains

text: only

text: k

text: unknown

text: non-zero

text: Sobol

text: indices

text: instead

text: of

text: We

text: also

text: provide

text: a

text: consequence

text: of

text: Proposition

text: on

text: Shapley

text: eﬀects:

text: N

text: Proposition

text: Let

text: i

text: p].

text: Let

text: j(i)

text: be

text: so

text: that

text: i

text: C

text: Then,

text: we

text: j(i)

text: have:

text: C

text: η

text: j(|i)

text: V

text: V

text: i

text: Cj(i)

text: Var(Y

text: u

text: u

text: i

text: u

text: u

text: Cj(i)

text: i

text: In

text: an

text: equivalent

text: manner,

text: let

text: us

text: write

text: η˜

text: for

text: the

text: Shapley

text: eﬀect

text: for

text: the

text: follow-

text: i

text: ing

text: output:

text: Y

text: g

text: (X

text: Cj(i)

text: j

text: Cj

text: (we

text: just

text: focus

text: on

text: the

text: variables

text: which

text: are

text: in

text: the

text: same

text: group

text: as

text: X

text: Then,

text: the

text: i

text: Shapley

text: eﬀect

text: η

text: for

text: the

text: whole

text: output

text: (Y

text: f(X))

text: is

text: equal

text: to

text: i

text: Var(Y

text: η

text: Cj(i)

text: η˜

text: i

text: Var(Y

text: i

text: For

text: example,

text: if

text: we

text: apply

text: this

text: proposition

text: in

text: the

text: case

text: where

text: X

text: is

text: the

text: only

text: i

text: variable

text: in

text: its

text: group,

text: then

text: we

text: have

text: η

text: S

text: i

text: i

text: To

text: compute

text: the

text: Shapley

text: eﬀect

text: η

text: the

text: last

text: proposition

text: reduces

text: the

text: sum

text: from

text: all

text: i

text: the

text: subsets

text: of

text: p]

text: i

text: to

text: all

text: the

text: subset

text: of

text: C

text: i

text: Then,

text: the

text: computational

text: j(i)

text: gain

text: is

text: the

text: same

text: as

text: in

text: Proposition

text: Explicit

text: computation

text: of

text: sensitivity

text: indices

text: for

text: Gaussian

text: linear

text: models

text: Linear

text: Gaussian

text: framework

text: N

text: In

text: this

text: section,

text: we

text: assume

text: that

text: X

text: (µ,

text: Γ),

text: that

text: Γ

text: is

text: invertible

text: and

text: that

text: R

text: f

text: x

text: β

text: βT

text: x,

text: for

text: a

text: ﬁxed

text: β

text: and

text: a

text: ﬁxed

text: vector

text: β.

text: This

text: framework

text: is

text: widely

text: used

text: to

text: model

text: physical

text: phenomena

text: (see

text: for

text: example

text: Indeed,

text: uncertainties

text: are

text: often

text: modelled

text: as

text: Gaussian

text: variables

text: and

text: an

text: unknown

text: function

text: is

text: commonly

text: estimated

text: by

text: its

text: linear

text: approximation.

text: We

text: ##LTLine##

text: ##LTLine##


text: 4.2 Linear Gaussian framework with independent groups of variables Despite the analytical formula (10), the computational cost remains an issue when the number of input variables p is too large. Based on an implementation ≤ in the R software, LG-Indices provides almost instantaneous results for p 15, ≥ but becomes impracticable for p 30. Indeed, we have to store 2p values, | namely the (Var(Y X )) ⊂ , and this can be a signiﬁcant issue. Fortunately, u u [1:p] when p is large, it can frequently be the case that there are independent groups of random variables. That is, after a permutation of the variables, the covariance matrix Γ is a block diagonal matrix. In this case, Propositions 2 and 3 show that this high dimensional computational problem boils down to a collection of (cid:80) lower dimensional problems. In this framework, we have seen in Proposition 1 that we only have to cal- | | { | ⊂ ∈ } culate the kj{=1 2 Cj |values ⊂Var(Y X}u), u Cj, j [1 : k] instead of all the 2p values Var(Y X ), u [1 : p] . We detail this idea in the algorithm u ”LG-GroupsIndices”. LG-GroupsIndices: Inputs : β, Γ. 1. Let C , ..., C be the independent groups of variables, for example using 1 k the function ”graph from adjacency matrix” of the R package ”igraph” (see [CN06]). 2. Let η be a vector of size p. 3. For j = 1, ..., k, do the following: (a) Let (S˜, η˜) be the output of the algorithm LG-Indices with the inputs β and Γ . Cj Cj,Cj (b) Let βT Γ β Sj = Cj C,j,Cj Cj S˜ β Γβ T T (c) Let βT Γ β η = Cj C,j,Cj Cj η˜. Cj β Γβ T T Ouputs : (S1, ..., Sk, η). We have used LG-GroupsIndices for computing Shapley eﬀects on an in- dustrial study in the ﬁeld of nuclear safety. In this model, the twelve inputs were modelled by a Gaussian vector with two independent groups of six vari- ables. Proposition 3 enables us to compute the Shapley eﬀects computing only 26 + 26 = 128 conditional variances instead of 212 = 4096. These results have been presented by Pietro Mosca in the sixteen International Symposium on Re- actor Dosimetry. The computational time does not really depend of the coeﬃcients of β and 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: Linear

text: Gaussian

text: framework

text: with

text: independent

text: groups

text: of

text: variables

text: Despite

text: the

text: analytical

text: formula

text: the

text: computational

text: cost

text: remains

text: an

text: issue

text: when

text: the

text: number

text: of

text: input

text: variables

text: p

text: is

text: too

text: large.

text: Based

text: on

text: an

text: implementation

text: in

text: the

text: R

text: software,

text: LG-Indices

text: provides

text: almost

text: instantaneous

text: results

text: for

text: p

text: but

text: becomes

text: impracticable

text: for

text: p

text: Indeed,

text: we

text: have

text: to

text: store

text: values,

text: namely

text: the

text: (Var(Y

text: X

text: and

text: this

text: can

text: be

text: a

text: signiﬁcant

text: issue.

text: Fortunately,

text: u

text: u

text: when

text: p

text: is

text: large,

text: it

text: can

text: frequently

text: be

text: the

text: case

text: that

text: there

text: are

text: independent

text: groups

text: of

text: random

text: variables.

text: That

text: is,

text: after

text: a

text: permutation

text: of

text: the

text: variables,

text: the

text: covariance

text: matrix

text: Γ

text: is

text: a

text: block

text: diagonal

text: matrix.

text: In

text: this

text: case,

text: Propositions

text: and

text: show

text: that

text: this

text: high

text: dimensional

text: computational

text: problem

text: boils

text: down

text: to

text: a

text: collection

text: of

text: lower

text: dimensional

text: problems.

text: In

text: this

text: framework,

text: we

text: have

text: seen

text: in

text: Proposition

text: that

text: we

text: only

text: have

text: to

text: cal-

text: culate

text: the

text: Cj

text: |values

text: ⊂Var(Y

text: X}u),

text: u

text: Cj,

text: j

text: k]

text: instead

text: of

text: all

text: the

text: values

text: Var(Y

text: X

text: u

text: p]

text: We

text: detail

text: this

text: idea

text: in

text: the

text: algorithm

text: u

text: ”LG-GroupsIndices”.

text: LG-GroupsIndices:

text: Inputs

text: β,

text: Γ.

text: Let

text: C

text: C

text: be

text: the

text: independent

text: groups

text: of

text: variables,

text: for

text: example

text: using

text: k

text: the

text: function

text: ”graph

text: from

text: adjacency

text: matrix”

text: of

text: the

text: R

text: package

text: ”igraph”

text: (see

text: Let

text: η

text: be

text: a

text: vector

text: of

text: size

text: p.

text: For

text: j

text: k,

text: do

text: the

text: following:

text: (a)

text: Let

text: (S˜,

text: η˜)

text: be

text: the

text: output

text: of

text: the

text: algorithm

text: LG-Indices

text: with

text: the

text: inputs

text: β

text: and

text: Γ

text: Cj

text: Cj,Cj

text: (b)

text: Let

text: βT

text: Γ

text: β

text: Sj

text: Cj

text: C,j,Cj

text: Cj

text: S˜

text: β

text: Γβ

text: T

text: T

text: (c)

text: Let

text: βT

text: Γ

text: β

text: η

text: Cj

text: C,j,Cj

text: Cj

text: η˜.

text: Cj

text: β

text: Γβ

text: T

text: T

text: Ouputs

text: Sk,

text: η).

text: We

text: have

text: used

text: LG-GroupsIndices

text: for

text: computing

text: Shapley

text: eﬀects

text: on

text: an

text: in-

text: dustrial

text: study

text: in

text: the

text: ﬁeld

text: of

text: nuclear

text: safety.

text: In

text: this

text: model,

text: the

text: twelve

text: inputs

text: were

text: modelled

text: by

text: a

text: Gaussian

text: vector

text: with

text: two

text: independent

text: groups

text: of

text: six

text: vari-

text: ables.

text: Proposition

text: enables

text: us

text: to

text: compute

text: the

text: Shapley

text: eﬀects

text: computing

text: only

text: conditional

text: variances

text: instead

text: of

text: These

text: results

text: have

text: been

text: presented

text: by

text: Pietro

text: Mosca

text: in

text: the

text: sixteen

text: International

text: Symposium

text: on

text: Re-

text: actor

text: Dosimetry.

text: The

text: computational

text: time

text: does

text: not

text: really

text: depend

text: of

text: the

text: coeﬃcients

text: of

text: β

text: and

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text: [SNS16]. ”shapleyPermEx” diﬀers from ”shapleyPermRand” by computing the sum over all the permutations when the former algorithm was only estimating the sum thanks to a Monte-Carlo method. We used the linear Gaussian frame- work to replace the expectation of conditional variances by their theoretical values, such that the algorithm gives the exact values of the Shapley eﬀects. We will write ”exact permutations Algorithm” for this algorithm. This method − still remains very long due to the computation of (p 1)! conditional variances. For example, it spends more than ten minutes computing the Shapley eﬀects for only p = 10. 5.2 Shapley eﬀects in the linear Gaussian framework From now on, as we focus on the Shapley eﬀects, we do not carry out the steps related to the computation of Sobol indices in the algorithms ”LG-indices” and ”LG-GroupsIndices”. N Let us consider a simulated toy example. We generate β by a (0, I ) p random variable and Γ by writing Γ = AAT , where the coeﬃcients of A are generated independently with a standard normal distribution. First, we compare LG-Indices with exact permutations Algorithm. Both provide the exact Shapley values but with diﬀerent computational times. Table 1 provides the computation times in seconds for diﬀerent values of p, the number of input variables. We remark that LG-indices is faster than exact permutations Table 1: Computational time (in seconds) for exact permutations Algorithm and LG-Indices for diﬀerent values p. p = 6 p = 7 p = 8 p = 9 p = 10 exact permutations 0.11 0.78 7.31 77.6 925 LG-Indices 0.004 0.008 0.018 0.039 0.086 Algorithm. We can also compare LG-Indices with random permutations Algorithm. For the latter, we choose m, the number of permutations generated in Song, so that the computational time is the same as LG-Indices. Yet, while our algorithm gives the exact Shapley eﬀect, the random permutations Algorithm provides an estimation of it. Hence, the performance of Song is evaluated by computing the coeﬃcients of variation in %, for diﬀerent values of p. We give in Table 2 the average of the p coeﬃcients of variations. We recall that the coeﬃcient of variation corresponds to the ratio of the standard deviation over the mean value. We see that the algorithm suggested in [SNS16] has quite large coeﬃcients of variation when we choose m so that the computational time is the same as our algorithm. However, this variation decreases with the number of inputs p. We can explain that saying that the computational time of LG-Indices is exponential with p. So, we can see that the precision of random permutations 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ”shapleyPermEx”

text: diﬀers

text: from

text: ”shapleyPermRand”

text: by

text: computing

text: the

text: sum

text: over

text: all

text: the

text: permutations

text: when

text: the

text: former

text: algorithm

text: was

text: only

text: estimating

text: the

text: sum

text: thanks

text: to

text: a

text: Monte-Carlo

text: method.

text: We

text: used

text: the

text: linear

text: Gaussian

text: frame-

text: work

text: to

text: replace

text: the

text: expectation

text: of

text: conditional

text: variances

text: by

text: their

text: theoretical

text: values,

text: such

text: that

text: the

text: algorithm

text: gives

text: the

text: exact

text: values

text: of

text: the

text: Shapley

text: eﬀects.

text: We

text: will

text: write

text: ”exact

text: permutations

text: Algorithm”

text: for

text: this

text: algorithm.

text: This

text: method

text: still

text: remains

text: very

text: long

text: due

text: to

text: the

text: computation

text: of

text: (p

text: conditional

text: variances.

text: For

text: example,

text: it

text: spends

text: more

text: than

text: ten

text: minutes

text: computing

text: the

text: Shapley

text: eﬀects

text: for

text: only

text: p

text: Shapley

text: eﬀects

text: in

text: the

text: linear

text: Gaussian

text: framework

text: From

text: now

text: on,

text: as

text: we

text: focus

text: on

text: the

text: Shapley

text: eﬀects,

text: we

text: do

text: not

text: carry

text: out

text: the

text: steps

text: related

text: to

text: the

text: computation

text: of

text: Sobol

text: indices

text: in

text: the

text: algorithms

text: ”LG-indices”

text: and

text: ”LG-GroupsIndices”.

text: N

text: Let

text: us

text: consider

text: a

text: simulated

text: toy

text: example.

text: We

text: generate

text: β

text: by

text: a

text: I

text: p

text: random

text: variable

text: and

text: Γ

text: by

text: writing

text: Γ

text: AAT

text: where

text: the

text: coeﬃcients

text: of

text: A

text: are

text: generated

text: independently

text: with

text: a

text: standard

text: normal

text: distribution.

text: First,

text: we

text: compare

text: LG-Indices

text: with

text: exact

text: permutations

text: Algorithm.

text: Both

text: provide

text: the

text: exact

text: Shapley

text: values

text: but

text: with

text: diﬀerent

text: computational

text: times.

text: Table

text: provides

text: the

text: computation

text: times

text: in

text: seconds

text: for

text: diﬀerent

text: values

text: of

text: p,

text: the

text: number

text: of

text: input

text: variables.

text: We

text: remark

text: that

text: LG-indices

text: is

text: faster

text: than

text: exact

text: permutations

text: Table

text: Computational

text: time

text: (in

text: seconds)

text: for

text: exact

text: permutations

text: Algorithm

text: and

text: LG-Indices

text: for

text: diﬀerent

text: values

text: p.

text: p

text: p

text: p

text: p

text: p

text: exact

text: permutations

text: LG-Indices

text: Algorithm.

text: We

text: can

text: also

text: compare

text: LG-Indices

text: with

text: random

text: permutations

text: Algorithm.

text: For

text: the

text: latter,

text: we

text: choose

text: m,

text: the

text: number

text: of

text: permutations

text: generated

text: in

text: Song,

text: so

text: that

text: the

text: computational

text: time

text: is

text: the

text: same

text: as

text: LG-Indices.

text: Yet,

text: while

text: our

text: algorithm

text: gives

text: the

text: exact

text: Shapley

text: eﬀect,

text: the

text: random

text: permutations

text: Algorithm

text: provides

text: an

text: estimation

text: of

text: it.

text: Hence,

text: the

text: performance

text: of

text: Song

text: is

text: evaluated

text: by

text: computing

text: the

text: coeﬃcients

text: of

text: variation

text: in

text: for

text: diﬀerent

text: values

text: of

text: p.

text: We

text: give

text: in

text: Table

text: the

text: average

text: of

text: the

text: p

text: coeﬃcients

text: of

text: variations.

text: We

text: recall

text: that

text: the

text: coeﬃcient

text: of

text: variation

text: corresponds

text: to

text: the

text: ratio

text: of

text: the

text: standard

text: deviation

text: over

text: the

text: mean

text: value.

text: We

text: see

text: that

text: the

text: algorithm

text: suggested

text: in

text: has

text: quite

text: large

text: coeﬃcients

text: of

text: variation

text: when

text: we

text: choose

text: m

text: so

text: that

text: the

text: computational

text: time

text: is

text: the

text: same

text: as

text: our

text: algorithm.

text: However,

text: this

text: variation

text: decreases

text: with

text: the

text: number

text: of

text: inputs

text: p.

text: We

text: can

text: explain

text: that

text: saying

text: that

text: the

text: computational

text: time

text: of

text: LG-Indices

text: is

text: exponential

text: with

text: p.

text: So,

text: we

text: can

text: see

text: that

text: the

text: precision

text: of

text: random

text: permutations

text: ##LTLine##

text: ##LTLine##

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text: 6 5 7 7 4 8 5 4 6 8 4 3 9 3 9 5 7 9 10 6 8 2 10 3 2 10 2 1 1 11 0 11 1 11 Figure 2: On the left, an example of a noncrossing tree in a 12-gon and on the middle the in- duced labelling of the noncrossing tree. On the right, the Hasse diagram of the corresponding poset where the maximal elements are 1 and 11. ◦ grafting diagonal. If the diagonal is present in both f and g, then it is kept in f g. If it is ◦ i present in exactly one of the two trees, then it is not kept in f g. Otherwise, the result is i not a noncrossing tree. One ‘denominator’ diagonal is added and the result is a noncrossing plant. See Section 5.2 of [Cha07] for more details. It was shown in Paragraph 5.1 of [CHNT08] that one can construct a poset from a noncrossing tree. Let us recall this construction. Let T be a noncrossing tree in a based regular n + 1-gon. Here by based we mean that we choose one side of the gon and call it the base. We can label the edges of the n + 1-gon by assigning the number 0 to the base, and then assigning the numbers 1 to n to the edges in a clockwise order. If an edge of T is a boundary edge we assign to it the number of the boundary edge. Otherwise, the label of the edge of the noncrossing tree is the number of (cid:67) the unique open boundary edge that it separates from the base. Then, we set i j if the T edge i is separated from the base by the edge j. An example is given in Figure 2. Lemma 3.1. Let T be a non-crossing tree in a based regular n + 1-gon. Then, the poset (cid:67) ([1, n], ) is an interval-poset. T Proof. We label the boundary edges of a based regular n + 1-gon as above. We use the (cid:54) notation [i , i ] where i i for the edge that goes from the left side of the boundary 1 2 1 2 edge i to the right side of the boundary edge i . For example, in Figure 2, the edge with 1 2 label 6 corresponds to [4, 6] and the edge labelled by 4 corresponds to [4, 4]. Note that by construction of our labelling, the edge [i , i ] labelled by the number i separates the boundary 1 2 (cid:54) (cid:54) edge i of the regular n + 1-gon from the base. In particular this implies that 1 i i and (cid:54) (cid:54) 1 i i n. 2 (cid:67) (cid:54) We want to check that the poset (P , ) is an interval poset. Let 0 < i < j < k n (cid:67) T such that i k. This means that the edge [i , i ] labelled by i is separated from the base 1 2 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: On

label: caption

text: the

text: left,

text: an

text: example

text: of

text: a

text: noncrossing

text: tree

text: in

text: a

text: and

text: on

text: the

text: middle

text: the

text: in-

text: duced

text: labelling

text: of

text: the

text: noncrossing

text: tree.

text: On

text: the

text: right,

text: the

text: Hasse

text: diagram

text: of

text: the

text: corresponding

text: poset

text: where

text: the

text: maximal

text: elements

text: are

text: and

text: grafting

text: diagonal.

text: If

text: the

text: diagonal

text: is

text: present

text: in

text: both

text: f

text: and

text: g,

text: then

text: it

text: is

text: kept

text: in

text: f

text: g.

text: If

text: it

text: is

text: i

text: present

text: in

text: exactly

text: one

text: of

text: the

text: two

text: trees,

text: then

text: it

text: is

text: not

text: kept

text: in

text: f

text: g.

text: Otherwise,

text: the

text: result

text: is

text: i

text: not

text: a

text: noncrossing

text: tree.

text: One

text: ‘denominator’

text: diagonal

text: is

text: added

text: and

text: the

text: result

text: is

text: a

text: noncrossing

text: plant.

text: See

text: Section

text: of

text: for

text: more

text: details.

text: It

text: was

text: shown

text: in

text: Paragraph

text: of

text: that

text: one

text: can

text: construct

text: a

text: poset

text: from

text: a

text: noncrossing

text: tree.

text: Let

text: us

text: recall

text: this

text: construction.

text: Let

text: T

text: be

text: a

text: noncrossing

text: tree

text: in

text: a

text: based

text: regular

text: n

text: Here

text: by

text: based

text: we

text: mean

text: that

text: we

text: choose

text: one

text: side

text: of

text: the

text: gon

text: and

text: call

text: it

text: the

text: base.

text: We

text: can

text: label

text: the

text: edges

text: of

text: the

text: n

text: by

text: assigning

text: the

text: number

text: to

text: the

text: base,

text: and

text: then

text: assigning

text: the

text: numbers

text: to

text: n

text: to

text: the

text: edges

text: in

text: a

text: clockwise

text: order.

text: If

text: an

text: edge

text: of

text: T

text: is

text: a

text: boundary

text: edge

text: we

text: assign

text: to

text: it

text: the

text: number

text: of

text: the

text: boundary

text: edge.

text: Otherwise,

text: the

text: label

text: of

text: the

text: edge

text: of

text: the

text: noncrossing

text: tree

text: is

text: the

text: number

text: of

text: the

text: unique

text: open

text: boundary

text: edge

text: that

text: it

text: separates

text: from

text: the

text: base.

text: Then,

text: we

text: set

text: i

text: j

text: if

text: the

text: T

text: edge

text: i

text: is

text: separated

text: from

text: the

text: base

text: by

text: the

text: edge

text: j.

text: An

text: example

text: is

text: given

text: in

text: Figure

text: Lemma

text: Let

text: T

text: be

text: a

text: non-crossing

text: tree

text: in

text: a

text: based

text: regular

text: n

text: Then,

text: the

text: poset

text: n],

text: is

text: an

text: interval-poset.

text: T

text: Proof.

text: We

text: label

text: the

text: boundary

text: edges

text: of

text: a

text: based

text: regular

text: n

text: as

text: above.

text: We

text: use

text: the

text: notation

text: [i

text: i

text: where

text: i

text: i

text: for

text: the

text: edge

text: that

text: goes

text: from

text: the

text: left

text: side

text: of

text: the

text: boundary

text: edge

text: i

text: to

text: the

text: right

text: side

text: of

text: the

text: boundary

text: edge

text: i

text: For

text: example,

text: in

text: Figure

text: the

text: edge

text: with

text: label

text: corresponds

text: to

text: and

text: the

text: edge

text: labelled

text: by

text: corresponds

text: to

text: Note

text: that

text: by

text: construction

text: of

text: our

text: labelling,

text: the

text: edge

text: [i

text: i

text: labelled

text: by

text: the

text: number

text: i

text: separates

text: the

text: boundary

text: edge

text: i

text: of

text: the

text: regular

text: n

text: from

text: the

text: base.

text: In

text: particular

text: this

text: implies

text: that

text: i

text: i

text: and

text: i

text: i

text: n.

text: We

text: want

text: to

text: check

text: that

text: the

text: poset

text: (P

text: is

text: an

text: interval

text: poset.

text: Let

text: i

text: j

text: k

text: n

text: T

text: such

text: that

text: i

text: k.

text: This

text: means

text: that

text: the

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text: [i

text: i

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text: by

text: i

text: is

text: separated

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text: the

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text: IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, XXX 2017 5 Fig. 2: Data processing pipeline of ST-Link. coarse-grained grid cells using a modiﬁed version of quad towards the sums for the neighboring cell(s)2 as well. We trees [14]. Each entity is assigned to one (an in rare cases use a strip around the border of the cell to determine to a few) of the grid cells, which becomes that entity’s the notion of ‘close to the border’. The width of the strip dominating grid. The dominating grid of an entity is the cell is taken as the 1/8th of the minimum cell’s edge width. that contains the most events from the entity. Entities that This means that around 43% of a grid overlaps with one do not share their dominating grid cells are not considered or more neighboring grids. This adjustment resembles the for linkage. The intuition behind this ﬁltering step is that, loose quad trees [33]. E I • if entity x from dataset and entity y from dataset have An entity can potentially have multiple dominating differing dominating grids, then they cannot be the same grid cells. We have found this to be rare for users in practice. entity. 3.2.1 Coarse Partitioning Figure 3b shows the resulting grids over selected areas in For quad-tree generation in the ST-Link algorithm, we con- Turkey, and the most popular venues from our dataset. Red tinue splitting the space until the grid cells size hits a given pins are showing the venues and the blue ones are showing minimum. For our experiments, we make sure that the area the ones that count towards neighboring grids. of the grid cells is at least 100 km squares. For users, the grid 3.2.3 Forming Partitioned Datasets cells should be big enough to cover a typical user’s mobility Once the dominating grid cells of users are determined, range around his home and work location. If the minimum we create grid cell speciﬁc datasets. For a given grid cell grid cell size is too small, then the spatial ﬁltering can c, we take only the events of the entities who has c as incorrectly eliminate potential matches, as the dominating a dominating grid cell. These events may or may not be grids from different datasets may end up being different. A in the grid cell c. Determination of the dominating grid concrete example is a user that checks in to coffee shops and cells of entities requires a single scan over the time sorted restaurants around his work location, but uses a location- events from entities. The forming of the partitioned datasets based match-making application only when he is at home. requires a second scan. 3.3 Temporal Filtering Temporal ﬁltering aims at creating a small set of candi- date user pairs on which the full linkage algorithm can be executed. To create this set, temporal ﬁltering looks for user pairs that have co-occurring events, as expressed by Eq. 3. Importantly, temporal ﬁltering also detects alibi events, based on Eq. 5, and prevents user pairs that have (a) Grid cells. (b) Most popular venues. such alibi events from taking part in the candidate pair set. Fig. 3: Grids cells and top 1K venues Temporal ﬁltering is based on two main ideas. First, a temporal window is slided over the events from two We also do not split grid cells that do not contain any different datasets to detect user pairs with co-occurring events. As a result, not all grid cells are the same size. events. Since co-occurring events must appear within a Figure 3 shows the grid cells for two selected areas in Turkey given time duration, the window approach captures all co- and the top 1K venues in those areas in terms of check-in occurring events. Second, as the window slides, alibi events counts, based on Foursquare check-in data. are tracked to prune the candidate user pair set. However, since the number of alibis is potentially very large, alibis 3.2.2 Determining Dominating Grids are only tracked for the user pairs that are currently in the candidate set. This means that some relevant alibis can be The determination of the dominating grid for an entity is missed if the user pair was added into the candidate set simply done by counting the entity’s events for different after an alibi event occurred. To process such alibis properly, cells and picking the cell with the highest count. A subtle a reverse window scan is performed, during which no new issue here is about entities whose events end up being close candidate pairs are added, but only alibis are processed. to the border areas of the grid cells. As a speciﬁc example, Algorithm 1 gives the pseudo-code of temporal ﬁltering. consider a user who lives in one cell and works in another. 3.3.1 Data Structures In this case, it is quite possible that a majority of the user’s check-ins happen in one cell and the majority of the calls in A window of size α (see Eq. 2) is slided jointly over both time sorted datasets. Figure 4a depicts this visually. Each another cell. This will result in missing some of the potential matches. To avoid this situation, we make two adjustments: • 2. An event can count towards at most 3 neighbors, in case it is at the If an event is close to the border, then it is counted corner of the grid. ##LTFigure## ##LTFigure## ##LTFigure##

text: IEEE

font: RNZHUR+NimbusSanL-Regu

text: TRANSACTIONS

text: ON

text: MOBILE

text: COMPUTING,

text: VOL.

text: XX,

text: NO.

text: XX,

text: XXX

text: Fig.

font: MIDUUM+URWPalladioL-Roma

text: Data

text: processing

text: pipeline

text: of

text: ST-Link.

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text: ##LTFigure##

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text: IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, XXX 2017 10 Fig. 6: Reduction in the number of Fig. 7: Precision as a function of Fig. 8: Number of true positives as a possible pairs. check-in probability. function of check-in probability. Precision Recall 2-2 0.19 0.31 3-2 0.36 0.32 3-3 0.89 0.61 4-3 0.93 0.47 4-4 0.99 0.58 5-4 0.99 0.47 5-5 0.99 0.50 Fig. 11: Precision and recall using Fig. 9: Precision as a function of us- Fig. 10: Number of true positives as unweighted linkage age ratio. a function of usage ratio. Fig. 12: Precision as a function of Fig. 13: Number of true positives as Fig. 14: k-l values distribution window size. a function of window size. didate user pairs, which gets close to 5 orders of magnitude k-l settings but for 1-1. We see that using 1-1 diversity reduction in the number of pairs. results in very poor precision for low values of the check- 4.3 Quality of Linkage in probability. As the check-in probability increases, then the precision of 1-1 diversity increases as well, but never We observe the precision and the number of true positives reaches 1. The increase is understandable, as more events on as a function of the usage ratio, check-in probability, and the check-in side will help rule out incorrect candidate pairs window size. We also observe the precision and recall values via alibis. Surprisingly, the precision for higher k-l values for a variation of the ST-Link algorithm that does not use are all close to 1. This is due the impact of alibi detection, weights, thus trades off precision for better recall. and strong weight constraint. As we will see shortly, not 4.3.1 Impact of Check-in Probability using weights trades off precision for better recall. Even if Figures 7 and 8 plot the precision and number of true pos- two users have events that are only 2-2 diverse, they can be itives, respectively, for the results produced by the ST-Link correctly linked if they have no alibis. algorithm as a function of the mean check-in probability. Figure 8 shows that the number of true positives in the Different series in the ﬁgure represent different k-l settings. linkage increases with the increasing check-in probability. ≥ We set k l, as the co-occurrence counts has to be greater This is expected, as more events help in increasing the co- than the diversity counts. For these experiments the usage occurrence and diversity counts. We also observe that higher ratio is set to 50%, which means only half of the call users k-l values result in reduced number of linkages. Given that are performing check-ins. 2-2 diversity has very good precision, and has the second Figure 7 shows that precision is very close to 1 for all highest true positive count (after 1-1 diversity, which has ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: IEEE

text: TRANSACTIONS

text: ON

text: MOBILE

text: COMPUTING,

text: VOL.

text: XX,

text: NO.

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text: Figure 4. Examples of cafe images taken from the outside. Part of the cafe interior is not visible due to the reﬂection. Figure 5. Examples of the Places dataset. It has a large number of categories and images in the same category are fairly diverse. where λ = λ = 100 are used for all experiments. 1 2 3.2. Reﬂection Separation using Real Images In this section, we describe how to train a network using L (G , G , G , G , D , D ) real observations without ground truth images of the trans- t r y m t r L L mitted scene and reﬂected scene. Note that the discussed = adv(Gt, Dt) + adv(Gr, Dr) E (cid:107) − (cid:107) (cid:107) ⊗ − (cid:107) networks so far rely on synthetic images since it is difﬁcult + λ ∼ [ y G (y) ] + G (y) y G + 1 (cid:88)y pdat(cid:0)a(y) (cid:107) − y 2 ⊗ −m (cid:107) mt 2 to obtain ground truths of corresponding t and r. Even in the (1 G (y)) y G (y) ] m mr 2 recent attempt for data collection [27], the original scene be- 1 (cid:107) ⊗ − (cid:107) (cid:1) fore reﬂection cannot be obtained. It limits the performance + λ2 V fi(Gm(y) y) fi(Gmt(y)) 2+ i of reﬂection separation due to the difference between the i (cid:107) − ⊗ − (cid:107) f ((1 G (y)) y) f (G (y)) . real observation and synthesized images. i m i mr 2 (7) One of the largest gaps between real and synthesized im- ages is due to the combination weight w in the synthesis It only uses t and r to compute the adversarial loss which model. For real images, reﬂected scenes are often dominant does not need the exact ground truth data of y. Note that it in certain regions of an observed image where transmitted allows us to learn reﬂection separation problem without any signals are unrecognizable and can only be guessed. For approximations for synthetic images. 4. Experimental Results example, as shown in Figure 4, cafe interiors are not visible due to reﬂected buildings. However, state-of-the-art synthe- We ﬁrst describe the experimental settings. We use sis methods use a scalar w to combine two scenes. To alle- the Places dataset which contains 8 million images of 365 viate this issue, we predict a mask map using a new branch scenes to synthesize images. As shown in Figure 5, it has di- G instead of a scalar w as shown in Figure 2(d). For each verse images that cover large variations in the lighting con- m y location, its value is between [0, 1] which represents a conﬁ- dition, viewpoint, distance to the scene, and a number of dence score that the pixel belongs to a transmitted scene. As objects in the scene. For each experiment, two scenes are ⊗ (cid:39) ⊗ such, we have G (y) y G (y) G (y) = G (y) and selected to synthesize training images. For real images, we − ⊗ m(cid:39) − m⊗ t mt (1 G (y)) y (1 G (y)) G (y) = G (y), where collect 178 photos of a cafe taken from the outside, from ⊗ m m r mr denotes pixel-wise multiplication and G and G are the Internet. As shown in Figure 4, the images contain chal- mt mr deﬁned for notational simplicity. The conﬁdence map pre- lenging reﬂections. diction branch and other generation branches are trained it- For training, networks shown in Figure 2(b), Figure 2(c), eratively. and Figure 2(d) share weights of the ﬁrst three layers in the The network can be trained in a weakly supervised man- generator. For each mini-batch, we ﬁrst resize images into × ner, i.e., tuples of (y, t, r) are given where t and r are im- 256 256 pixels. A rectangular patch is cropped with a ages belonging to the same categories of transmitted and height and a width sampled from [192, 256] uniformly at × reﬂected scenes included in y, respectively. In this case, t random. Then, it is resized to 128 128 pixels for training. ∈ and r are not conditioned on y. Thus, (1) is changed to: For the blurring model, we pick w [0.5, 0.7] and σ of the Gaussian kernel between [2, 5]. More quantitative and qual- itative results are presented in the supplementary material. L (G , D ) =E ∼ [log D (t)]+ The source code will be made available to the public. adv t t Et pdata(t) t− (6) 4.1. Synthetic Images ∼ [log(1 D (G (y))]. y pdata(y) t t In this section, we show reﬂection separation results using a network trained on synthetic data. We evaluate In addition, a loss function in (5) should be changed for the proposed algorithm against the state-of-the-art methods weakly supervised learning. By combining all the losses, [4, 2, 24]. As they use different approximation methods to the overall loss function becomes model the reﬂection, we provide results for all cases. ##LTFigure## ##LTFigure## ##LTLine##

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text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##