Derivative-based Global Sensitivity Measures
and Their Link with Sobol’ Sensitivity Indices

arXiv:1605.07830v1 [] 25 May 2016

Sergei Kucherenko and Shugfang Song

Abstract The variance-based method of Sobol’ sensitivity indices is very popular
among practitioners due to its efficiency and easiness of interpretation. However,
for high-dimensional models the direct application of this method can be very timeconsuming and prohibitively expensive to use. One of the alternative global sensitivity analysis methods known as the method of derivative based global sensitivity
measures (DGSM) has recently become popular among practitioners. It has a link
with the Morris screening method and Sobol’ sensitivity indices. DGSM are very
easy to implement and evaluate numerically. The computational time required for
numerical evaluation of DGSM is generally much lower than that for estimation of
Sobol’ sensitivity indices. We present a survey of recent advances in DGSM and
new results concerning new lower and upper bounds on the values of Sobol’ total
sensitivity indices Stot
i . Using these bounds it is possible in most cases to get a good
practical estimation of the values of Stot
i . Several examples are used to illustrate an
application of DGSM.
Keywords: Global sensitivity analysis; Monte Carlo methods; Quasi Monte Carlo
methods; Derivative based global measures; Morris method; Sobol sensitivity indices

1 Introduction
Global sensitivity analysis (GSA) is the study of how the uncertainty in the model
output is apportioned to the uncertainty in model inputs [9],[14]. GSA can provide valuable information regarding the dependence of the model output to its input
parameters. The variance-based method of global sensitivity indices developed by
Sobol’ [11] became very popular among practitioners due to its efficiency and easSergei Kucherenko · Shugfang Song
Imperial College London, London, SW7 2AZ, UK e-mail: s.kucherenko@imperial.ac.uk, e-mail:
shufangsong@nwpu.edu.cn

1

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Sergei Kucherenko and Shugfang Song

iness of interpretation. There are two types of Sobol’ sensitivity indices: the main
effect indices, which estimate the individual contribution of each input parameter to
the output variance, and the total sensitivity indices, which measure the total contribution of a single input factor or a group of inputs [3]. The total sensitivity indices
are used to identify non-important variables which can then be fixed at their nominal values to reduce model complexity [9]. For high-dimensional models the direct
application of variance-based GSA measures can be extremely time-consuming and
impractical.
A number of alternative SA techniques have been proposed. In this paper we
present derivative based global sensitivity measures (DGSM) and their link with
Sobol’ sensitivity indices. DGSM are based on averaging local derivatives using
Monte Carlo or Quasi Monte Carlo sampling methods. These measures were briefly
introduced by Sobol’ and Gershman in [12]. Kucherenko et al [6] introduced some
other derivative-based global sensitivity measures (DGSM) and coined the acronym
DGSM. They showed that the computational cost of numerical evaluation of DGSM
can be much lower than that for estimation of Sobol’ sensitivity indices which later
was confirmed in other works [5]. DGSM can be seen as a generalization and formalization of the Morris importance measure also known as elementary effects [8].
Sobol’ and Kucherenko[15] proved theoretically that there is a link between DGSM
and the Sobol’ total sensitivity index Stot
i for the same input. They showed that
DGSM can be used as an upper bound on total sensitivity index Stot
i . They also introduced modified DGSM which can be used for both a single input and groups of
inputs [16]. Such measures can be applied for problems with a high number of input
variables to reduce the computational time. Lamboni et al [7] extended results of
Sobol’ and Kucherenko for models with input variables belonging to the class of
Boltzmann probability measures.
The numerical efficiency of the DGSM method can be improved by using the automatic differentiation algorithm for calculation DGSM as was shown in [5]. However, the number of required function evaluations still remains to be proportional to
the number of inputs. This dependence can be greatly reduced using an approach
based on algorithmic differentiation in the adjoint or reverse mode [1]. It allows estimating all derivatives at a cost at most 4-6 times of that for evaluating the original
function [4].
This paper is organised as follows: Section 2 presents Sobol’ global sensitivity
indices. DGSM and lower and upper bounds on total Sobol’ sensitivity indices for
uniformly distributed variables and random variables are presented in Sections 3 and
4, respectively. In Section 5 we consider test cases which illustrate an application of
DGSM and their links with total Sobol’ sensitivity indices. Finally, conclusions are
presented in Section 6.

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3

2 Sobol’ global sensitivity indices
The method of global sensitivity indices developed by Sobol’ is based on ANOVA
decomposition [11]. Consider the square integrable function f (x) defined in the unit
hypercube H d = [0, 1]d . The decomposition of f (x)
d

d

d

f (x) = f0 + ∑ fi (xi ) + ∑ ∑ fi j (xi , x j ) + · · · + f12···d (x1 , · · · , xd ),
where f0 =

R

Hd

(1)

i=1 j>i

i=1

f (x)dx , is called ANOVA if conditions
Z

Hd

fi1 ...is dxik = 0

(2)

are satisfied for all different groups of indices x1 , · · · , xs such that 1 ≤ i1 < i2 < ... <
is ≤ d. These conditions guarantee that all terms in (1) are mutually orthogonal with
respect to integration.
The variances of the terms in the ANOVA decomposition add up to the total
variance:
D=

Z

Hd

f 2 (x)dx − f02 =

d

d

∑ ∑

s=1 i1 <···<is

Di1 ...is ,

where Di1 ...is = H d fi21 ...is (xi1 , ..., xis )dxi1 , ..., xis are called partial variances.
Total partial variances account for the total influence of the factor xi :
R

Dtot
i =

∑ Di1 ...is ,

<i>

where the sum ∑ is extended over all different groups of indices x1 , · · · , xs satisfy<i>

ing condition 1 ≤ i1 < i2 < ... < is ≤ n, 1 ≤ s ≤ n, where one of the indices is equal
to i. The corresponding total sensitivity index is defined as

tot
D.
Stot
i = Di

Denote ui (x) the sum of all terms in ANOVA decomposition (1) that depend on
xi :
d

ui (x) = fi (xi ) +

∑

fi j (xi , x j ) + · · · + f12···d (x1 , · · · , xd ).

j=1, j6=i

From the definition of ANOVA decomposition it follows that
Z

Hd

ui (x)dx = 0.

(3)

The total partial variance Dtot
i can be computed as

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Sergei Kucherenko and Shugfang Song

Dtot
i =

Z

Hd

u2i (x)dx =

Z

Hd

u2i (xi , z)dxi dz.

Denote z = (x1 , ..., xi−1 , xi+1 , ..., xd ) the vector of all variables but xi , then x ≡
(xi , z) and f (x) ≡ f (xi , z). The ANOVA decomposition of f (x) in (1) can be presented in the following form
f (x) = ui (xi , z) + v(z),
where v(z) is the sum
of terms independent of xi . Because of (2) and (3) it is easy to
R
show that v(z) = H d f (x)dxi . Hence
ui (xi , z) = f (x) −

Z

Hd

f (x)dxi .

(4)

Then the total sensitivity index Stot
i is equal to
u2i (x)dx
.
(5)
D
We note that in the case of independent random variables all definitions of the
ANOVA decomposition remain to be correct but all derivations should be considered
in probabilistic sense as shown in [14] and presented in Section 4.
Stot
i

=

R

Hd

3 DGSM for uniformly distributed variables
Consider continuously differentiable
function f (x) defined in the unit hypercube

H d = [0, 1]d such that ∂ f ∂ xi ∈ L2 .
Theorem 1. Assume that c ≤

∂f
∂ xi

≤ C. Then

c2
C2
≤ Stot
≤
.
i
12D
12D

(6)

The proof is presented in [15].
The Morris importance measure also known as elementary effects originally defined as finite differences averaged over a finite set of random points [8] was generalized in [6]:

µi =

Z

Hd

∂ f (x)
dx.
∂ xi

(7)

Kucherenko et al [6] also introduced a new DGSM measure:

νi =

Z

Hd



∂ f (x)
∂ xi

2

dx.

(8)

In this paper we define two new DGSM measures:

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(m)

wi

5

=

Z

Hd

xm
i

∂ f (x)
dx,
∂ xi

(9)

where m is a constant, m > 0,

∂ f (x) 2
dx.
xi (1 − xi )
∂ xi
Hd
 2
We note that νi is in fact the mean value of ∂ f ∂ xi . We also note that
1
ςi =
2



Z

∂f
∂ ui
=
.
∂ xi
∂ xi

(10)

(11)

3.1 Lower bounds on Stot
i
Theorem 2. There exists the following lower bound between DGSM (8) and the
Sobol’ total sensitivity index
R

(

Hd

[ f (1, z) − f (0, z)] [ f (1, z) + f (0, z) − 2 f (x)] dx)2
< Stot
i .
4νi D

(12)

Proof. Consider an integral
Z

Hd

ui (x)

∂ ui (x)
dx.
∂ xi

(13)

Applying the Cauchy–Schwarz inequality we obtain the following result:
Z


2 Z
Z 
∂ ui (x)
∂ ui (x) 2
2
dx ≤
dx.
ui (x)dx ·
ui (x)
∂ xi
∂ xi
Hd
Hd
Hd

(14)

It is easy to prove that the left and right parts of this inequality cannot be equal.
should be linearly dependent.
Indeed, for them to be equal functions ui (x) and ∂ u∂ix(x)
i
For simplicity consider a one-dimensional case: x ∈ [0, 1]. Let’s assume

∂ u(x)
= Au(x),
∂x
where A is a constant. The general solution to this equation u(x) = B exp(Ax) , where
B is a constant. It is easy to see that this solution is not consistent with condition (3)
which should be imposed on function u(x).
R
Integral H d ui (x) ∂ u∂ix(x)
dx can be transformed as
i

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Sergei Kucherenko and Shugfang Song

R ∂ u2i (x)
H d ∂ xi

∂ ui (x)
1
dx
H d uRi (x) ∂ xi dx = 2

1
2
= 2 RH d−1 ui (1, z) − u2i (0, z) dz
= 12 H d−1 (ui (1, z) − ui (0, z)) (ui (1, z) + ui(0, z)) dz
R
= 12 H d ( f (1, z) − f (0, z)) ( f (1, z) + f (0, z) − 2v(z)) dz.

R

(15)

All terms in the last integrand are independent of xi , hence we can replace integration with respect to dz to integration with respect to dx and substitute v(z) for
f (x) in the integrand due to condition (3). Then (15) can be presented as
Z

Hd

ui (x)

1
∂ ui (x)
dx =
∂ xi
2

Z

Hd

[ f (1, z) − f (0, z)] [ f (1, z) + f (0, z) − 2 f (x)] dx (16)

From (11) ∂ u∂ix(x)
= ∂ ∂f x(x)
, hence the right hand side of (14) can be written as
i
i
tot
νi Di . Finally dividing (14) by νi D and using (16), we obtain the lower bound (12).
⊓
⊔
We call
R

(

Hd

[ f (1, z) − f (0, z)] [ f (1, z) + f (0, z) − 2 f (x)] dx)2
4νi D

the lower bound number one (LB1).
Theorem 3. There exists the following lower bound between DGSM (9) and the
Sobol’ total sensitivity index
(2m + 1)

hR

Hd

i
(m+1) 2

( f (1, z) − f (x)) dx − wi
(m + 1)2D

< Stot
i

(17)

Proof. Consider an integral
Z

Hd

xm
i ui (x)dx.

(18)

Applying the Cauchy–Schwarz inequality we obtain the following result:
Z

2

xm
i ui (x)dx
Hd

≤

Z

Hd

x2m
i dx ·

Z

Hd

u2i (x)dx.

(19)

It is easy to see that equality in (19) cannot be attained. For this to happen
functions ui (x) and xm
i should be linearly dependent. For simplicity consider a onedimensional case: x ∈ [0, 1]. Let’s assume
u(x) = Axm ,
where A 6= 0 is a constant. This solution does not satisfy condition (3) which should
be imposed on function u(x).
Further we use the following transformation:
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Title Suppressed Due to Excessive Length

Z

Hd

7


Z
Z
∂ xm+1
ui (x)
∂ ui (x)
i
dx = (m + 1)
xm
ui (x)dx +
xm+1
dx
i
i
d
d
∂ xi
∂ xi
H
H

to present integral (18) in a form:


R
R ∂ (xm+1
R
ui (x))
m+1 ∂ ui (x)
i
m u (x)dx = 1
x
x
dx
−
dx
d
d
d
H i i
H
H i
m+1
∂ xi
∂ xi
hR
i
R
m+1 ∂ ui (x)
1
= m+1 H d−1 ui (1, z)dz − H d xi
∂ xi dx
i
hR
R
m+1 ∂ ui (x)
1
= m+1
dx
.
x
(
f
(1,
z)
−
f
(x))
dx
−
d
d
H i
H
∂ xi

(20)

We notice that

Z

Hd

x2m
i dx =

1
.
(2m + 1)

(21)
⊓
⊔

Using (20) and (21) and dividing (19) by D we obtain (17).
This second lower bound on Stot
i we denote γ (m):

γ (m) =

(2m + 1)

hR

Hd

i
(m+1) 2

( f (1, z) − f (x)) dx − wi
(m + 1)2D

< Stot
i .

(22)

In fact, this is a set of lower bounds depending on parameter m. We are interested
in the value of m at which γ (m) attains its maximum. Further we use star to denote
such a value m: m∗ = arg max(γ (m)) and call

γ ∗ (m∗ ) =

(2m∗ + 1)

hR

Hd

i
(m∗ +1) 2

( f (1, z) − f (x)) dx − wi
(m∗ + 1)2 D

(23)

the lower bound number two (LB2).
We define the maximum lower bound LB∗ as
LB∗ = max(LB1, LB2).

(24)

We note that both lower and upper bounds can be estimated by a set of derivative
based measures:
(m)

ϒi = {νi , wi }, m > 0.

(25)

3.2 Upper bounds on Stot
i
Theorem 4.
Stot
i ≤

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νi
.
π 2D

macro: svmult.cls

(26)

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Sergei Kucherenko and Shugfang Song

The proof of this Theorem in given in [15].
Consider the set of values ν1 , ..., νn , 1 ≤ i ≤ n. One can expect that smaller νi
correspond to less influential variables xi .
We further call (26) the upper bound number one (UB1).
Theorem 5.
Stot
i ≤

ςi
,
D

(27)

where ςi is given by (10).
Proof. We use the following inequality [2]:
0≤

Z 1

2

u dx −

0

Z

1
0

2
Z
1 1
x(1 − x)u′2dx.
udx ≤
2 0

(28)

The inequality
R is reduced to an equality only if u is constant. Assume that u is given
by (3), then 01 udx = 0, and from (28) we obtain (27).
⊓
⊔

Further we call ςDi the upper bound number two (UB2). We note that 12 xi (1 − xi )
for 0 ≤ xi ≤ 1 is bounded: 0 ≤ 21 xi (1 − xi) ≤ 18 . Therefore, 0 ≤ ςi ≤ 81 νi .

3.3 Computational costs
All DGSM can be computed using the same set of partial derivatives
∂ f (x)
∂ xi

∂ f (x)
∂ xi , i

=

1, ..., d. Evaluation of
can be done analytically for explicitly given easilydifferentiable functions or numerically.
In the case of straightforward numerical estimations of all partial derivatives and
computation of integrals using MC or QMC methods, the number of required function evaluations for a set of all input variables is equal to N(d + 1), where N is a number of sampled points. Computing LB1 also requires values of f (0, z) , f (1, z), while
computing LB2 requires only values of f (1, z). In total, numerical computation of
∗
LB∗ for all input variables would require NFLB = N(d + 1) + 2Nd = N(3d + 1) function evaluations. Computation of all upper bounds require NFUB = N(d + 1) function
evaluations. We recall that the number of function evaluations required for compuS
tation of Stot
i is NF = N(d + 1) [10]. The number of sampled points N needed to
achieve numerical convergence can be different for DGSM and Stot
i . It is generally
lower for the case of DGSM. The numerical efficiency of the DGSM method can be
significantly increased by using algorithmic differentiation in the adjoint (reverse)
mode [1]. This approach allows estimating all derivatives at a cost at most 6 times
of that for evaluating the original function f (x) [4]. However, as mentioned above
∗
lower bounds also require computation of f (0, z) , f (1, z) so NFLB would only be
∗
LB
UB
reduced to NF = 6N + 2Nd = N(2d + 6), while NF would be equal to 6N.

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9

4 DGSM for random variables
Consider a function f (x1 , ..., xd ), where x1 , ..., xd are independent random variables
with distribution functions F1 (x1 ) , ..., Fd (xd ). Thus the point x = (x1 , ..., xd ) is defined in the Euclidean space Rd and its measure is dF1 (x1 ) · · · dFd (xd ).
The following DGSM was introduced in [15]:

νi =

Z

Rd



∂ f (x)
∂ xi

2

dF(x).

(29)

We introduce a new measure
wi =

Z

Rd

∂ f (x)
dF(x).
∂ xi

(30)

4.1 The lower bounds on Stot
i for normal variables
Assume that xi is normally distributed with the finite variance σi2 and the mean value
µi .
Theorem 6.

σi2 w2i
≤ Stot
i .
D

Proof. Consider
tain

(31)

R

Rd xi ui (x)dF(x). Applying the Cauchy–Schwarz inequality we ob-

Z

Rd

2 Z
Z
xi ui (x)dF(x) ≤
x2i dF(x) ·
u2i (x)dF(x).
Rd

Rd

(32)

Equality in (32) can be attained if functions ui (x) and xi are linearly dependent. For
simplicity consider a one-dimensional case. Let’s assume
u(x) = A(x − µ ),
where A 6= 0 is a constant. This solution Rsatisfies condition (3) for normally distributed variable x with the mean value µ : Rd u(x)dF(x) = 0.
For normally distributed variables the following equality is true [2]:
Z

Rd

2 Z
Z
xi ui (x)dF(x) =
x2i dF(x) ·
Rd

Rd

∂ ui (x)
dF(x).
∂ xi

(33)

By definition Rd x2i dF(x) = σi2 . Using (32) and (33) and dividing the resulting
inequality by D we obtain the lower bound (31).
⊓
⊔
R

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Sergei Kucherenko and Shugfang Song

4.2 The upper bounds on Stot
i for normal variables
The following Theorem 7 is a generalization of Theorem 1.
Theorem 7. Assume that c ≤

∂f
∂ xi

≤ C, then

σi2 c2
σi2C2
≤ Stot
≤
.
i
D
D

(34)

The constant factor σi2 cannot be improved.
Theorem 8.
Stot
i ≤

σi2
νi .
D

(35)

The constant factor σi2 cannot be reduced.
Proofs are presented in [15].

5 Test cases
In this section we present the results of analytical and numerical estimation of Si ,
tot
Stot
i , LB1, LB2 and UB1, UB2. The analytical values for DGSM and Si were calculated and compared with numerical results. For text case 2 we present convergence
plots in the form of root mean square error (RMSE) versus the number of sampled
points N. To reduce the scatter in the error estimation the values of RMSE were
averaged over K = 25 independent runs:
 ∗
2 ! 12
1 K Ii,k − I0
.
∑
K k=1
I0

εi =

Here Ii∗ is numerically computed values of Stot
i , LB1, LB2 or UB1, UB2, I0 is
the corresponding analytical value of Stot
,
LB1,
LB2
or UB1, UB2. The RMSE can
i
be approximated by a trend line cN −α . Values of (−α ) are given in brackets on the
plots. QMC integration based on Sobol’ sequences was used in all numerical tests.
Example 1. Consider a linear with respect to xi function:
f (x) = a(z)xi + b(z).
For this function Si =
(

R

Hd

2

2 (z)x dzdx
(a2 (z)−2a
i)
i)
R

4D H d−1

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a2 (z)dz

Stot
i ,

Dtot
i =

1 R
2
12 H d−1 a (z)dz,

= 0 and γ (m) =

job: Kucherenko_mcqmc

νi =

R

H d−1 a

macro: svmult.cls

(z)dz, LB1 =

2

(2m+1)m2 ( H d−1 a(z)dz)
4(m+2)2 (m+1)2 D
R

2

. A maximum value

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11

R
2
of γ (m) is attained at m∗ =3.745, when γ ∗ (m∗ ) = 0.0401
D ( a(z)dz) . The lower and
R
1
1
tot
2
tot
upper bounds are LB∗ ≈ 0.48Stot
i . UB1 ≈ 1.22Si . UB2 = 12D 0 a(z) dz = Si . For

this test function UB2 < UB1.

Example 2. Consider the so-called g-function which is often used in GSA for
illustration purposes:
d

f (x) = ∏ gi ,
i=1

|4xi −2|+ai
1+ai ,

where gi =

ai (i = 1, ..., d) are constants. It is easy to see that for this
d

function fi (xi ) = (gi − 1), ui (x) = (gi − 1) ∏ g j and as a result LB1=0. The total
j=1, j6=i

d



variance is D = −1 + ∏ 1 +
j=1

are given in Table 1.

1/3
(1+a j )2



. The analytical values of Si , Stot
i and LB2

Table 1 The analytical expressions for Si , Stot
i and LB2 for g-function

γ (m)

Stot
i

Si

1/3
(1+ai )2

1/3
(1 + ai )2 D

d

∏

j=1, j6=i



1/3
1 + (1+a
2
)
j

D

2

4(1−(1/2)m+1 )
(2m + 1) 1 −
m+2
(1 + ai )2 (m + 1)2 D

d γ (m)
0.0772
∗
∗
dm = 0 , we find that m =9.64, γ (m ) = (1+ai )2 D . It is
interesting to note that m∗ does not depend on ai , i = 1, 2, ..., d and d. In the extreme
∗
∗
cases: if ai → ∞ for all i, γ S(mtot ) → 0.257, SStoti → 1, while if ai → 0 for all i, γ S(mtot ) →
i
i
i

By solving equation

0.257
, Si
(4/3)d−1 Stot
i

→

1
.
(4/3)d−1

The analytical expression for Stot
i , UB1 and UB2 are given

in Table 2.

Table 2 The analytical expressions for Stot
i UB1 and UB2 for g-function
Stot
i
1/3
(1+ai )2

UB1
d



1/3
1 + (1+a
∏
2
j)

UB2
d

16

j=1, j6=i



1/3
1 + (1+a
∏
2
j)

j=1, j6=i

(1 + ai

D

Stot

)2 π 2 D

Stot

d

4

∏

j=1, j6=i



1/3
1 + (1+a
)2
j

3(1 + ai

)2 D

UB2
i
i
For this test function UB1
= π48 , UB2
= 14 , hence UB1
= π12 < 1. Values of Si ,
Stot
i , UB and LB2 for the case of a=[0,1,4.5,9,99,99,99,99], d=8 are given in Table
3 and shown in Figure 1. We can conclude that for this test function the knowledge
of LB2 and UB1, UB2 allows to rank correctly all the variables in the order of their
importance.

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12

Sergei Kucherenko and Shugfang Song

1

Si
Stot
i

0

UB
LB

log2(RMSE)

−1
−2
−3
−4
−5
0

2

4
log2(N)

Fig. 1
Values of Si ,Stot
i , LB2
a=[0,1,4.5,9,99,99,99,99], d=8.

and

6

UB1

for

8

all input

variables.

Example

2,

Fig. 2 presents RMSE of numerical estimations of Stot
i , UB1 and LB2. For an
individual input LB2 has the highest convergence rate, following by Stot
i , and UB1
in terms of the number of sampled points. However, we recall that computation of
all indices requires NFLB∗ = N(3d + 1) function evaluations for LB, while for Stot
i
this number is NFS = N(d + 1) and for UB it is also NFUB = N(d + 1).
4

"

n

Example 3. Hartmann function f (x) = − ∑ ci exp − ∑ αi j (x j − pi j
i=1

j=1

)2

#

, xi ∈

[0, 1]. For this test case a relationship between the values LB1, LB2 and Si varies
with the change of input (Table 4, Figure 3): for variables x2 and x6 LB1> Si > LB2,
while for all other variables LB1< LB2 <Si . LB* is much smaller than Stot
i for all
inputs. Values of m* also vary with the change of input. For all variables but variable
2 UB1 > UB2.
Table 3 Values of LB*, Si , Stot
i , UB1 and UB1. Example 2, a=[0,1,4.5,9,99,99,99,99], d=8.
LB∗
Si
Stot
i
UB1
UB2

Page: 12

x1
0.166
0.716
0.788
3.828
3.149

x2
0.0416
0.179
0.242
1.178
0.969

job: Kucherenko_mcqmc

x3
0.00549
0.0237
0.0343
0.167
0.137

macro: svmult.cls

x4
0.00166
0.00720
0.0105
0.0509
0.0418

x5 ...x8
0.000017
0.0000716
0.000105
0.000501
0.00042

date/time: 26-May-2016/0:33

Title Suppressed Due to Excessive Length
0

13

−4

−14

tot

tot

Si (−0.977)

log2(RMSE)

log2(RMSE)

2

−17

−10

−8

−12

−10

−14

−12
4

−16
4

UB1(−0.894)
LB2(−0.836)

−16

−8

−6

Si (−0.993)

−15

UB1(−0.844)
LB2(−1.048)

−6

−4
log (RMSE)

tot

Si (−0.953)

UB1(−0.962)
LB2(−1.134)

−2

−18
−19
−20
−21
−22

5

6

7

8
log2(N)

9

10

11

12

5

6

7

(a)

8
log2(N)

9

10

11

12

−23
4

5

6

(b)

7

8
log2(N)

9

10

11

12

(c)

Fig. 2 RMSE of Stot
i , UB and LB2 versus the number of sampled points. Example 2,
a=[0,1,4.5,9,99,99,99,99], d=8. Variable 1 (a), variable 3 (b) and variable 5 (c).

0.5

Si
Stot
i

0

UB
LB1
LB2

log2(RMSE)

−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1

2

3

4

5

6

log2(N)
Fig. 3 Values of Si ,Stot
i , UB1, LB1 and LB2 for all input variables. Example 3.
Table 4 Values of m∗ , LB1, LB2, UB1, UB2,Si and Stot
i for all input variables.
LB1
LB2
m∗
LB∗
Si
Stot
i
UB1
UB2

Page: 13

x1
0.0044
0.0515
4.6
0.0515
0.115
0.344
1.089
1.051

x2
0.0080
0.0013
10.2
0.0080
0.00699
0.398
0.540
0.550

job: Kucherenko_mcqmc

x3
0.0009
0.0011
17.0
0.0011
0.00715
0.0515
0.196
0.150

x4
0.0029
0.0418
5.5
0.0418
0.0888
0.381
1.088
0.959

macro: svmult.cls

x5
0.0014
0.0390
3.6
0.0390
0.109
0.297
1.073
0.932

x6
0.0357
0.0009
19.9
0.0357
0.0139
0.482
1.046
0.899

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14

Sergei Kucherenko and Shugfang Song

6 Conclusions
We can conclude that using lower and upper bounds based on DGSM it is possible in
most cases to get a good practical estimation of the values of Stot
i at a fraction of the
CPU cost for estimating Stot
.
Small
values
of
upper
bounds
imply
small values of
i
Stot
.
DGSM
can
be
used
for
fixing
unimportant
variables
and
subsequent
model rei
duction. For linear function and product function, DGSM can give the same variable
ranking as Stot
i . In a general case variable ranking can be different for DGSM and
variance based methods. Upper and lower bounds can be estimated using MC/QMC
integration methods using the same set of partial derivative values. Partial derivatives can be efficiently estimated using algorithmic differentiation in the reverse
(adjoint) mode.
We note that all bounds should be computed with sufficient accuracy. Standard
techniques for monitoring convergence and accuracy of MC/QMC estimates should
be applied to avoid erroneous results.

Acknowledgements The authors would like to thank Prof. I. Sobol’ his invaluable contributions
to this work. Authors also gratefully acknowledge the financial support by the EPSRC grant
EP/H03126X/1.

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