The concept of system signature was introduced by Samaniego for systems whose
components have i.i.d. lifetimes. We consider its extension to the continuous
dependent case and give an explicit expression for this extension as a
difference of weighted means of the structure function values. We then derive a
formula for the computation of the coefficients of these weighted means in the
special case of independent continuous lifetimes. Finally, we interpret this
extended concept of signature through a natural least squares approximation
problem

Boland

Esary

Grabisch

Jean-Luc Marichal

Marichal

Navarro

Pierre Mathonet

Samaniego

English

arXiv

AbstractThe concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression for this extension as a difference of weighted means of the structure function values. We then derive a formula for the computation of the coefficients of these weighted means in the special case of independent continuous lifetimes. Finally, we interpret this extended concept of signature through a natural least squares approximation problem

Marichal, Jean-Luc

Mathonet, Pierre

Elsevier - Publisher Connector 

Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations 

CiteSeerX

Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations

Journal of Multivariate Analysis 102 (2011) 931–936
Contents lists available at ScienceDirect
Journal of Multivariate Analysis
journal homepage: www.elsevier.com/locate/jmva
Extensions of system signatures to dependent lifetimes: Explicit
expressions and interpretations
Jean-Luc Marichal ∗, Pierre Mathonet
Mathematics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove-Kalergi, L-1359 Luxembourg, Grand Duchy of Luxembourg
a r t i c l e i n f o
Article history:
Received 28 April 2010
Available online 25 February 2011
AMS 2010 subject classifications:
Primary 62N05
90B25
secondary 62G30
94C10
Keywords:
System signature
System reliability
Semicoherent system
Order statistic
a b s t r a c t
The concept of system signature was introduced by Samaniego for systems whose
components have i.i.d. lifetimes. We consider its extension to the continuous dependent
case and give an explicit expression for this extension as a difference of weighted means
of the structure function values. We then derive a formula for the computation of the
coefficients of these weighted means in the special case of independent continuous
lifetimes. Finally, we interpret this extended concept of signature through a natural least
squares approximation problem.
© 2011 Elsevier Inc. All rights reserved.
1. Introduction
Consider an n-component semicoherent system. The design of such a system can be described through its structure function
φ: {0, 1}n → {0, 1}, which expresses at any time the state of the system in terms of the states of its components. Here
‘‘semicoherent’’ means that the structure function φ is nondecreasing in each variable and satisfies the boundary conditions
φ(0) = 0 and φ(1) = 1.
Let X1, . . . , Xn denote the component lifetimes and let X1:n, . . . , Xn:n be the order statistics obtained by rearranging the
variables X1, . . . , Xn in ascending order of magnitude; that is, X1:n ⩽ · · · ⩽ Xn:n.
The concept of signaturewas introduced in 1985 by Samaniego [8] for systems whose components have i.i.d. lifetimes as
the n-tuple s = (s1, . . . , sn) defined by
sk = Pr(T = Xk:n) (1)
where T denotes the system lifetime. That is, sk is the probability that the kth component failure causes the system to fail.
For general background, see Samaniego [9].
Under the i.i.d. assumption, the probability sk can be computed as the ratio of nk, the number of orderings for which
the kth component failure causes the system failure, to n!, the total number of possible orderings of the failure times. An
alternativeway to calculate sk, which does not require the exhaustive inspection of all the orderings, was found by Boland [1]
∗ Corresponding author.
E-mail addresses: jean-luc.marichal@uni.lu (J.-L. Marichal), pierre.mathonet@uni.lu (P. Mathonet).
0047-259X/$ – see front matter© 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmva.2011.01.010
932 J.-L. Marichal, P. Mathonet / Journal of Multivariate Analysis 102 (2011) 931–936
through the formula
sk = 1 n
n−k+1
 −
x∈{0,1}n
|x|=n−k+1
φ(x)− 1 n
n−k
 −
x∈{0,1}n
|x|=n−k
φ(x) (2)
where |x| =∑ni=1 xi.
Even though the i.i.d. assumption is quite reasonable inmany applications, especiallywhenwewant to compare different
system designs, it is legitimate to investigate the probability (1) in the general case of dependent lifetimes and to search for
formulaswhich extend (2) to this general framework.Weobserve that only a few results have been obtained in this direction,
assuming for instance that the component lifetimes are exchangeable or independent and exponentially distributed (see,
e.g., Samaniego [9, Section 8.3] and Navarro et al. [6]).
Equation (2) shows that, in the i.i.d. case, the n-tuple s depends only on the system design. This is no longer true in the
general dependent case, where the probability Pr(T = Xk:n)may also depend on the joint c.d.f. of the component lifetimes,
that is the function
F(t1, . . . , tn) = Pr(X1 ⩽ t1, . . . , Xn ⩽ tn).
In this general setting, we shall denote the probability Pr(T = Xk:n) by pk to emphasize that the n-tuple p = (p1, . . . , pn) is
not always a signature in the strict sense (i.e., a feature of the system design only).
In this paper, assuming only that the joint c.d.f. F is absolutely continuous, we provide a closed-form expression for
pk = Pr(T = Xk:n) as a difference of two weighted arithmetic means of the structure function values whose weights depend
only on F (Theorem 2). We show that this expression reduces to (2) as soon as the component lifetimes are exchangeable
(Remark 2). We also provide a useful expression for the weights (as a one-dimensional integral) in the special case of
independent lifetimes (Proposition 3) and examine the particular case of independentWeibull lifetimes, which includes the
exponential model (Corollary 4). Finally, we show that the n-tuple p can be obtained from a symmetric approximation of
the structure function in the sense of weighted least squares andwe point out a formal analogy between this approximation
and the concept of projected system recently introduced in [7].
Through the usual identification of the elements of {0, 1}n with the subsets of [n] = {1, . . . , n}, a pseudo-Boolean
function f : {0, 1}n → R can be equivalently described by a set function vf : 2[n] → R. We simply write vf (S) = f (1S), where
1S denotes the n-tuplewhose ith coordinate is 1, if i ∈ S, and 0, otherwise. To avoid cumbersome notation,we henceforth use
the same symbol to denote both a givenpseudo-Boolean function and its underlying set function, thuswriting f : {0, 1}n → R
or f : 2[n] → R interchangeably.
The kth order statistic function osk:n: {0, 1}n → {0, 1} is defined by osk:n(x) = xk:n. We then have osk:n(x) = 1, if
|x| ⩾ n− k+ 1, and 0, otherwise. As a matter of convenience, we also formally define os0:n ≡ 0 and osn+1:n ≡ 1.
2. Explicit expressions
Let F be the (absolutely continuous) joint c.d.f. of the component lifetimes X1, . . . , Xn. We define the associated relative
quality function q: 2[n] → [0, 1] as
q(S) = Pr

max
i∈[n]\S
Xi < min
j∈S Xj

with the convention that q(∅) = q([n]) = 1. That is, q(S) is the probability that the lifetime of every component in S
is greater than the lifetime of every component in [n] \ S. Thus defined, q(S) is a measure of the overall quality of the
components in S when compared with the components in [n] \ S.
Since the r.v.’s X1, . . . , Xn are continuous, we see that the function q can also be written as
q(S) =
−
σ∈Sn : {σ(n−|S|+1),...,σ (n)}=S
Pr(Xσ(1) < · · · < Xσ(n)) (3)
whereSn denotes the group of permutations on [n].
We then observe that, for every k ∈ [n], the values q(S) for |S| = k sum up to one. In fact, by (3), we have−
S⊆[n]
|S|=k
q(S) =
−
σ∈Sn
Pr(Xσ(1) < · · · < Xσ(n)) = 1. (4)
Remark 1. (a) The validity of (4) is especially transparent if one focuses on a particular simple case, say, when k = 1. It
then simply says that there exists (with probability 1) a unique component jwhose lifetime is maximum, that is,
n−
j=1
Pr

max
i∈[n]\{j}
Xi < Xj

= 1.
J.-L. Marichal, P. Mathonet / Journal of Multivariate Analysis 102 (2011) 931–936 933
(b) If the variables X1, . . . , Xn are exchangeable, then the relative quality function q is symmetric. In this case, by (4), we
have q(S) = 1/

n
|S|

.
(c) Equation (4) shows that comparing q(S) with q(S ′) is relevant whenever |S| = |S ′|. In general, according to (b) above,
the relative quality could be better measured by the function q˜(S) =

n
|S|

q(S).
We now give an expression for pk = Pr(T = Xk:n) as a difference of two arithmeticmeans of the structure function values
weighted by the relative quality function. We first present a lemma.
Lemma 1. For every k ∈ [n], we have
Pr(T ⩾ Xk:n) =
−
|x|=n−k+1
q(x) φ(x).
Proof. For every k ∈ [n] and every σ ∈ Sn, we have
Pr(T ⩾ Xσ(k) | Xσ(1) < · · · < Xσ(n)) = φ
{σ(k), . . . , σ (n)}. (5)
Indeed, assume Xσ(1) < · · · < Xσ(n). Using the path representation of the life function [2] of the system, we have T ⩾ Xσ(k)
if and only if
max
1⩽j⩽m
min
i∈Pj
Xi ⩾ Xσ(k) (6)
where P1, . . . , Pm denote the minimal path sets of the system. In turn, event (6) is equivalent to
‘‘There is a minimal path set P such that Xi ⩾ Xσ(k) for all i ∈ P ’’.
Equivalently, ‘‘there is a minimal path set P ⊆ {σ(k), . . . , σ (n)}’’. By monotonicity of φ, this event reduces to
‘‘φ
{σ(k), . . . , σ (n)} = 1’’, which finally proves (5).
By combining the law of total probability with (5), we obtain
Pr(T ⩾ Xk:n) =
−
σ∈Sn
φ({σ(k), . . . , σ (n)}) Pr(Xσ(1) < · · · < Xσ(n)).
Grouping the terms for which {σ(k), . . . , σ (n)} is a fixed set S and then summing over S, we obtain
Pr(T ⩾ Xk:n) =
−
|S|=n−k+1
−
σ∈Sn : {σ(k),...,σ (n)}=S
φ(S) Pr(Xσ(1) < · · · < Xσ(n)).
The result then follows from (3). 
Theorem 2. For every k ∈ [n], we have
pk =
−
|x|=n−k+1
q(x) φ(x)−
−
|x|=n−k
q(x) φ(x). (7)
Proof. We have Pr(T = Xk:n) = Pr(T ⩾ Xk:n)− Pr(T ⩾ Xk+1:n). We then conclude by Lemma 1. 
Remark 2. (a) It is noteworthy that pk can be rewritten in the form
pk =
−
x∈{0,1}n
rk(x) φ(x),
with rk(x) = q(x) (−xk+1:n + 2 xk:n − xk−1:n). This fact follows immediately from the identity−
x∈{0,1}n
q(x) φ(x) xk:n =
−
|x|⩾n−k+1
q(x) φ(x).
(b) As expected, we observe from (7) that pk combines linearly two pieces of information:
(i) the system design, which is encoded in the structure function φ, and
(ii) the component lifetimes, which are encoded in the relative quality function q.
(c) When the variables X1, . . . , Xn are exchangeable, by Remark 1(b) we see that (7) reduces to (2). Thus in this case, p is
the signature of the system. This fact was previously observed in [5, Lemma 1].
The following proposition gives a formula for q(S) as a one-dimensional integral in the special case of independent
lifetimes.
934 J.-L. Marichal, P. Mathonet / Journal of Multivariate Analysis 102 (2011) 931–936
Proposition 3. For independent continuous lifetimes X1, . . . , Xn, each Xi having p.d.f. fi and c.d.f. Fi, with Fi(0) = 0, we have
q(S) =
−
j∈S
∫ ∞
0
fj(t)
∏
i∈[n]\S
Fi(t)
∏
i∈S\{j}
(1− Fi(t)) dt.
Proof. Denote the p.d.f. and c.d.f. of a r.v. X by fX and FX , respectively. Recall that for two independent continuous r.v.’s X
and Y , we have
Pr(X < Y ) =
∫∫
x<y
fX (x)fY (y) dx dy =
∫ ∞
−∞
FX (y)fY (y) dy.
By applying this formula to q(S), we obtain
q(S) =
∫ ∞
0
Pr

max
i∈[n]\S
Xi ⩽ t
 d
dt
Pr

min
j∈S Xj ⩽ t

dt
=
∫ ∞
0
∏
i∈[n]\S
Fi(t)
d
dt

1−
∏
i∈S
(1− Fi(t))

dt
which immediately leads to the result. 
Corollary 4. For independent Weibull lifetimes, with Fi(t) = 1− e−(λit)α , we have
q(S) =
−
K⊆[n]\S
(−1)|K | λα(S)
λα(K ∪ S) (8)
for every S ≠ ∅, where λα(S) =∑i∈S λαi .
Proof. By Proposition 3, we have
q(S) =
−
j∈S
λαj
∫ ∞
0
α tα−1 e−λα(S) t
α
∏
i∈[n]\S

1− e−λα({i}) tα  dt
where the product can be expanded (by the generalized binomial theorem) as∏
i∈[n]\S

1− e−λα({i}) tα  = −
K⊆[n]\S
(−1)|K | e−λα(K) tα .
We then have
q(S) =
−
j∈S
λαj
−
K⊆[n]\S
(−1)|K |
∫ ∞
0
α tα−1 e−λα(K∪S) t
α
dt
which immediately leads to the result. 
Remark 3. (a) Given a set function q: 2[n] → [0, 1], there is an additive set function λα: 2[n] →]0,∞[ satisfying (8) if and
only if
q([n] \ {i}) > 0 ∀i ∈ [n] (9)
and
q(S) =
−
K⊆[n]\S
(−1)|K |
∑
i∈S
q([n] \ {i})∑
i∈K∪S
q([n] \ {i}) ∀S ≠ ∅. (10)
Indeed, if such an additive function exists, then by (8) we obtain
q([n] \ {i}) = 1− λα([n] \ {i})
λα([n]) =
λα({i})
λα([n]) (11)
which leads to (9) and (10). Conversely, if (9) holds, then we may choose λα({i}) = q([n] \ {i}) and we then see that
(10) leads to (8). Thus (9) and (10) provide necessary and sufficient conditions on a set function q: 2[n] → [0, 1] to be a
relative quality function obtained fromWeibull lifetimes (with a common shape parameter α).
(b) Under the assumptions of Corollary 4, by (11) the ratio λα({i})/λα([n]) is exactly the probability that Xi is the shortest
lifetime. More generally, the ratio λα(S)/λα([n]) = ∑i∈S q([n] \ {i}) is the probability that the component having the
shortest lifetime is in S.
J.-L. Marichal, P. Mathonet / Journal of Multivariate Analysis 102 (2011) 931–936 935
(c) For S ⊆ [n], the S-difference of a function f : {0, 1}n → R is defined inductively by ∆∅f = f and ∆S f = ∆{i}∆S\{i}f for
i ∈ S, with∆{i}f (x) = f (x | xi = 1)− f (x | xi = 0). It is then easy to see [3, Section 2] that (8) can be rewritten as
q(S) = (−1)n−|S| λα(S)

∆[n]\S
1
λα

(S). (12)
Moreover, the ([n] \ S)-difference in (12) can be interpreted as the marginal interaction [3, Section 2] (associated with
the function 1/λα) among the components in [n] \ S conditioned to the presence of the components in S.
3. Links with approximations of structure functions
In [4] the authors solved the problem of approximating a given pseudo-Boolean function f : {0, 1}n → R by a symmetric
one in the sense of weighted least squares.
Specifically, given a weight function w: {0, 1}n →]0,∞[, the best symmetric approximation of a function f : {0, 1}n → R
is defined as the unique symmetric function f ∗: {0, 1}n → R that minimizes the weighted squared distance
‖f − g‖2 =
−
x∈{0,1}n
w(x)

f (x)− g(x)2 (13)
among all symmetric functions g: {0, 1}n → R.
The best symmetric approximation f ∗ is actually the orthogonal projection of f , with respect to the inner product
⟨f , g⟩ =
−
x∈{0,1}n
w(x)f (x)g(x)
onto the linear subspace of symmetric functions g: {0, 1}n → R. In terms of the order statistic functions, this projection is
given by
f ∗ = f (0)+
n−
k=1
ck osk:n (14)
where
ck =
−
|x|=n−k+1
w(x) f (x)−
−
|x|=n−k
w(x) f (x) (15)
and
w(x) = w(x)∑
|z|=|x|
w(z)
which shows that ck is actually a difference of two expected values (see [4]). Since ck is the coefficient of osk:n in f ∗, it can be
interpreted as a measure of the influence of the kth smallest variable on f .
Now, consider an n-component semicoherent system defined by a structure function φ and an absolutely continuous
joint c.d.f. F of the component lifetimes. Assume that the associated relative quality function q is strictly positive. Consider
also the weighted distance (13) with w = q and apply the approximation problem above to the structure function f = φ.
By (4) we see that w = q = q. Theorem 2 then shows that the coefficient ck, as defined in (15), is precisely the probability
pk = Pr(T = Xk:n).
Moreover, from (14) it follows that the best symmetric approximation φ∗ of φ (with respect to the weighted distance
(13) withw = q) is given by
φ∗ =
n−
k=1
pk osk:n. (16)
Under the i.i.d. assumption, (16) reduces to φ∗ =∑nk=1 sk osk:n, where sk is given by (2).
Interestingly enough, we also observe a formal analogy between the orthogonal projection (16) of φ and the concept of
projected system, recently introduced in [7]. Indeed, φ∗ is a combination of the order statistic functions weighted by the n-
tuplepwhile the projected system is the systemwhichmixes the k-out-of-n systems (φk = osk:n)withmixing distributionp.
Acknowledgments
The authorswish to thank the reviewers for helpful comments and suggestions. This research is supported by the internal
research project F1R-MTH-PUL-09MRDO of the University of Luxembourg.
936 J.-L. Marichal, P. Mathonet / Journal of Multivariate Analysis 102 (2011) 931–936
References
[1] P.J. Boland, Signatures of indirect majority systems, J. Appl. Probab. 38 (2001) 597–603.
[2] J.D. Esary, A.W. Marshall, Coherent life functions, SIAM J. Appl. Math. 18 (1970) 810–814.
[3] M. Grabisch, J.-L. Marichal, M. Roubens, Equivalent representations of set functions, Math. Oper. Res. 25 (2) (2000) 157–178.
[4] J.-L. Marichal, P. Mathonet, Symmetric approximations of pseudo-Boolean functions, http://arXiv.org/abs/1004.2593.
[5] J. Navarro, T. Rychlik, Reliability and expectation bounds for coherent systems with exchangeable components, J. Multivariate Anal. 98 (1) (2007)
102–113.
[6] J. Navarro, F.J. Samaniego, N. Balakrishnan, D. Bhattacharya, On the application and extension of system signatures in engineering reliability, Naval Res.
Logist. 55 (2008) 313–327.
[7] J. Navarro, F. Spizzichino, N. Balakrishnan, Applications of average and projected systems to the study of coherent systems, J. Multivariate Anal. 101 (6)
(2010) 1471–1482.
[8] F.J. Samaniego, On closure of the IFR class under formation of coherent systems, IEEE Trans. Reliab. Theory 34 (1985) 69–72.
[9] F.J. Samaniego, System Signatures and their Applications in Engineering Reliability, in: Int. Series in Operations Research & Management Science,
vol. 110, Springer, New York, 2007.


peer reviewedThe concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression for this extension as a difference of weighted means of the structure function values. We then derive a formula for the computation of the coefficients of these weighted means in the special case of independent continuous lifetimes. Finally, we interpret this extended concept of signature through a natural least squares approximation problem.F1R-MTH-PUL-09MRDO > MRDO > 01/01/2009 - 31/12/2011 > MARICHAL Jean-Lu

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The concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression for this extension as a difference of weighted means of the structure function values. We then derive a formula for the computation of the coefficients of these weighted means in the special case of independent continuous lifetimes. Finally, we interpret this extended concept of signature through a natural least squares approximation problem

Jean-luc Marichal

peer reviewedThe concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression for this extension as a difference of weighted means of the structure function values. We then derive a formula for the computation of the coefficients of these weighted means in the special case of independent continuous
lifetimes. Finally, we interpret this extended concept of signature through a natural least squares approximation problem

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Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations

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