7 InternationalWorkshoponAppliedProbabilityJune16th,2014 J.-L.Marichal,P.Mathonet,F.Spizzichino Onstructuresignaturesandprobabilitysignaturesofgeneraldecomposablesystems

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On structure signatures and probability signatures of general decomposable systems

J.-L. Marichal, P. Mathonet, F. Spizzichino

Mathematics Research Unit, FSTC, University of Luxembourg, Luxembourg University of Liege (ULg), Faculty of sciences, Department of Mathematics

Department of Mathematics, University La Sapienza, Rome, Italy

7^th International Workshop on Applied Probability June 16th, 2014

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P. Mathonet, University of Li`ege, Faculty of Sciences, Department of Mathematics

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Coherent systems : notation

• C = {c₁, . . . , c_n} : n binary components (two possible states);

• they are connected to form a system;

• Basic examples : series, parallel, bridge, k-out-of-n systems.

c1 c2 cn

cn

c1

c2

c5

c4

c3

c2

c1

c2

c3

c1 c3

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Structure functions

• With each component c_k, (k ∈ [n] = {1, . . . , n}), we associate a Boolean variable

x_k =

0 if c_k is in a failed state 1 if c_k is in function.

TheBoolean vectorx = (x₁, . . . , x_n) encodes the states of all components.

• We can also consider the set A of components in function : x = (1, 0, 1, 0, 1) corresponds to A = {1, 3, 5}.

So the states are represented by x ∈ {0, 1}ⁿ or A ⊂ [n] = {1, . . . , n}.

• Thestructure functiondefines the state of the system :

φ : {0, 1}ⁿ→ {0, 1} : x = (x1, . . . , xn) 7→ xS = φ(x1, . . . , xn).

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Examples

The structure function can be defined with ∧ and ∨ or as apolynomial function.

c1 c2 cn

c_n c1

c₂

φ(x₁, . . . , x_n) = x₁· · · x_n φ(x₁, . . . , x_n) = 1 − (1 − x₁) · · · (1 − x_n)

c5

c4

c3

c2

c1

φ(x1, . . . , x5) = 1 − (1 − x2x4)(1 − x1x5)(1 − x2x3x5)(1 − x1x3x4).

This function can be extended as a polynomial bφ : Rⁿ→ R, ofdegree at most 1in each variable. This is themultilinear extension of φ.

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Properties of φ

• φ : {0, 1}ⁿ→ {0, 1} or φ : P([n]) → {0, 1};

• φ(0, . . . , 0) = φ(∅) = 0;

• φ(1, . . . , 1) = φ([n]) = 1;

• φ is increasing (nondecreasing) :

A ⊂ B ⇒ φ(A) 6 φ(B).

Every function with these properties is the structure function of a semi-coherent system.

This system iscoherent if in addition, all the variables areessentialin φ.

Example : ak-out-of-n systemis a system that fails with the k-th failure :

φ(A) = 1 iff |A| > n − k.

or φ(x) = xk:n(series are 1-out-of-n, parallel are n-out-of-n).

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Some notation concerning probability

1 T_k : randomlifetimeof component c_k.

2 For t > 0, X_k(t) : rand. stateof comp. c_k at time t (Bernoulli var.).

3 TS : system random lifetime.

4 XS(t) : random state of the system at time t (Bernoulli var.).

5 Joint cumulative distribution of component lifetimes : F (t₁, . . . , t_n) = Pr(T₁6 t1, . . . , T_n6 tn).

So in general a system is a triple :

S = (n, φ, F ).

Classical hypotheses :

• F isabsolutely continuous; the lifetimes arei.i.d.

• or the lifetimes areexchangeable;

• or ties have null probability (no ties) :

Pr(Tk = T`) = 0, when k 6= `.

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Structure Signature

The random variables T1, . . . , Tn induce theorder statistics T1:n, . . . , Tn:n, such that (when there are no ties)

T_1:n< . . . < T_n:n.

Consider a system S = (n, φ, F ), where F is absolutely continuous i.i.d.

Definition (Samaniego (1985))

Thestructure signatureof the system (or Samaniego signature) is the n-tuple

s = (s1, . . . , sn) where

sk = Pr(TS = Tk:n).

Remark : it is not the Barlow index (b₁, . . . , b_n) where bk = Pr(TS = Tk).

Theorem : the signature s does not depend on F (in the i.i.d. situation), butonly on the structure. It is a combinatorial object.

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Examples

For a series system : s = (1, 0, . . . , 0).

For a parallel system : s = (0, . . . , 0, 1).

The signature of a more complex system:

c2

c3

c1

c4 c5 s = (₁₂₀⁴⁸,₁₂₀³⁶,₁₂₀³⁶, 0, 0)

Why ? All the events

Eσ= (Tσ(1)< Tσ(2)< Tσ(3)< Tσ(4)< Tσ(5))

where σ is a permutation of {1, 2, 3, 4, 5} areequally likely. Since there are no ties :

Pr(T_σ(1)< T_σ(2)< T_σ(3)< T_σ(4)< T_σ(5)) = 1 120.

We just have to count the number of these events that correspond to the event (T_S = T_k:5).

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How to compute : Boland’s formula

Proposition (Boland (2001))

If the components have i.i.d. lifetimes, we have for k 6 n sk = 1

n n−k+1

X

|A|=n−k+1

φ(A) − 1

n n−k

X

|A|=n−k

φ(A)

where for A ⊂ [n], |A| is the cardinality of A.

Both terms that appear in the formula have a meaning : Sk = 1

n n−k

X

|A|=n−k

φ(A) =

n

X

i =k+1

si= Pr(TS > Tk:n).

It is the kth component of thetail structure signature.

It can be extended (for convenience) with S0= 1 and Sn= 0. We then have

s_k = S_k−1− Sk, ∀ k : 1 6 k 6 n.

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Some more examples

c2

c3

c1

c4 c5

We have S0= 1, S1= ³₅, S2= ₂₀⁶, S3= S4= S5= 0, so s = (1 −3

5,3 5− 6

20, 6

20− 0, 0 − 0, 0 − 0) = (48 120, 36

120, 36 120, 0, 0)

c2

c1

c6

c5

c3 c4

We have S₀= 1, S₁= ²₃, S₂= ₃₀⁸, S₃= S₄= S₅= S₆= 0.

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Use : Samaniego’s decomposition of reliability

The reliability is R(t) = Pr(TS > t). Set Rk:n(t) = Pr(Tk:n> t).

Proposition (Samaniego (1985))

If F is absolutely continuous and i.i.d. , then

R_S(t) =

n

X

k=1

s_kR_k:n(t), (1)

for all t > 0, and every coherent system S = (n, φ, F ).

• Use : Comparison of systems built with the same components.

• Proofs : Samaniego used probabilities, but it’s also simple algebra.

• First extensions : Navarro-Rychlik (2007) for exchangeable components.

• Marichal,M. , Waldhauser (2011) : This decomposition holds for every coherent structure φ if and only if thestate variables X₁(t), . . . , X_n(t) are exchangeable.

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The general (non i.i.d.) setting

For a system S = (n, φ, F ) such that F hasno ties, we can define

• The structure signature s (through Boland’s formula);

• Theprobability signaturep, defined as above (Navarro et al. 2010):

p = (p₁, . . . , p_n), where

p_k = Pr(T_S = T_k:n).

• The probability signature may depend both on F and φ.

Can we find an explicit expression of pk in terms of F et φ, for instance by generalizing Boland’s formula ?

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Therelative quality functionis defined by q : P([n]) → R : A 7→ q(A) = Pr

max

k6∈ATk < min

j∈ATj

,

and q(∅) = q([n]) = 1. So q(A) measures the quality of elements of A.

Proposition (Marichal,M. (2011))

If F has no ties, then the probability signature is given by

p_k = X

|A|=n−k+1

q(A) φ(A) − X

|A|=n−k

q(A) φ(A).

Here again both terms have a direct meaning

P_k = X

|A|=n−k

q(A) φ(A) =

n

X

i =k+1

p_i = Pr(T_S > T_k:n)

is the k-th coordinate of thetail probability signature.

When F is i.i.d. we have q(A) = ¹ (|A|ⁿ).

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Modular decomposition of the structure

A modular decomposition of ([n], φ, F ) intor disjoint modulesis

• a partitionC = {C1, . . . , Cr} of [n];

• for every j, a semi-coherent system Mj = (Cj, χj, Gj), where Gj is the marginal distribution of the components in Cj;

• the modules are connected according to a structure function ψ : {0, 1}^r → {0, 1}such that

φ(A) = ψ(χ1(A ∩ C1), . . . , χr(A ∩ Cr)), A ⊆ C , (2)

C₁

C2

C3

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Modular decomposition of the structure signature s

Question : does the structure signature of a modular system decompose in terms of the signatures of the modules and the organizing structure ψ ?

• A known result, expressed in terms of the tail signature :

Theorem (Gertsbakh, Shpungin, Spizzichino (2011))

Fortwo modulesC1and C2of size n1and n2connected inseries, we have

Sn−k = X

06a16n1, 06a26n2

a₁+a₂=k n1

a₁

n2

a₂

n k

S¹_n₁_−a₁S²_n₂_−a₂.

• The same kind of formula holds for a parallel of two modules;

• Independent results in the same direction by Da, Zheng and Hu (2012);

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Modular decomposition of s : general result

For r modules of C1, . . . , Cr of size n1, . . . , nr we set

Tk = {a = (a₁, . . . , a_r) ∈ N^r : 0 6 aj6 n^j for j = 1, . . . , r and Pr

j=1a_j = k}.

and for every a ∈ Tk :

c0(a1, . . . , ar) =

n₁ a1 · · · ⁿ_a^r

r

n k

.

Theorem (Marichal,M.,Spizzichino (submitted))

For every semi-coherent system (C , φ) with a modular decomposition into r disjoint modules (C_j, χ_j), j = 1, . . . , r , connected according to a semi-coherent structure ψ, we have (without assumption)

Sn−k = X

a∈Tk

c0(a) bψ S¹_n₁_−a₁, . . . , S^r_n_r_−a_r, 0 6 k 6 n. (3)

Here bψ is the multilinear extension of ψ.

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Decomposability of q

The hypotheses for the decomposition :

Definition

Denote by q^C^j the relative quality function associated with module Cj : q^C^j(A) = Pr

max

i ∈C_j\ATi< min

i ∈ATi

, A ⊆ Cj.

Definition

Given a partition C = {C1, . . . , Cr} of C , the relative quality function q is C-decomposableif there exists a function c :Qr

i =1{0, . . . , ni} → R such that

q(A) = c(|A ∩ C1|, . . . , |A ∩ Cr|)

r

Y

j=1

q^C^j(A ∩ Cj) , A ⊆ C . (4)

Remark : This is a condition on the component lifetimes.

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Modular decomposition of p

Recall the tail signature P_k = Pn

i =k+1p_i = Pr(T_S > T_k:n) for 0 6 k 6 n − 1, P⁰= 1 and Pn= 0.

Theorem (Marichal,M.,Spizzichino (submitted))

Ifthe relative quality function q is C-decomposable for some partition C = {C₁, . . . , C_r} of C ,thenfor every semi-coherent system (C , φ, F ) with a modular decomposition into (C_j, χ_j, G_j), j = 1, . . . , r , connected according to ψ : {0, 1}^r → {0, 1}, we have

Pn−k = X

a∈T_k

c(a) bψ P¹_n₁_−a₁, . . . , P^r_n_r_−a_r, 0 6 k 6 n. (5)

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Ad hoc hypothesis ?

Theorem (Marichal,M.,Spizzichino (submitted))

Consider a partition C = {C1, . . . , Cr} of C and a distribution F of the component lifetimes. Assume that there exists a function

γ : Qr

j=1{0, . . . , nj} → R such that,for every semi-coherent system (C , φ, F ) with a modular decompositioninto r disjoint modules (C_j, χ_j, G_j), j = 1, . . . , r , connected according to a semi-coherent structure ψ : {0, 1}^r→ {0, 1}, we have

Pn−k = X

a∈T_k

γ(a) bψ P¹_n

1−a₁, . . . , P^r_n

r−a_r . (6)

Then the relative quality function q associated with F is C-decomposable.

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Some cases where q is C-decomposable

• If q and q^C^j are symmetric, then it is q C-decomposable for every C, but then p = s.

• This happens when the values p_σ= Pr(T_σ(1)< · · · < T_σ(n)) are equal to _n!¹.

Definition

The function q : 2^C → R is C-symmetric for a partition C = {C1, . . . , Cr} if q(A) = q(B) for every A, B ⊆ C such that |A ∩ C_j| = |B ∩ Cj| for every j ∈ [r ].

• If the function q is C-symmetric for some partition C = {C1, . . . , Cr} and the functions q^C^j are symmetric for j = 1, . . . , r , then q is C-decomposable.

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Decomposable systems

Modular decomposition of φ and corresponding decomposability of q both play a role, so we propose a new definition :

Definition

We say that a semicoherent system (C , φ, F ) isdecomposableif, for some partition C = {C₁, . . . , C_r} of C ,

(i) thestructure φ : {0, 1}ⁿ→ {0, 1} has amodular decompositioninto r disjoint semicoherent modules (Cj, χj, Gj), j = 1, . . . , r , and (ii) the functionq is C-decomposable.

For decomposable systems, the signatures both admit a modular decomposition.

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Final remarks

• We can make the same computations for the Barlow-Proschan index (2012);

• We have a simple analytic method to compute s in terms of φ (i.i.d.

case only) (2012);

• The structure signature can be defined in a purely geometric way (2011), (least squares).

Thank you.

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A word about the proof

• Use the multilinear extension of ψ to obtain

Sn−k = X

|A|=k

q0(A)φ(A) = X

|A|=k

q0(A) ψ χ1(A ∩ C1), . . . , χr(A ∩ Cr)

= X

B⊆[r ]

ψ(B) X

|A|=k

q₀(A)Y

j∈B

χ_j(A ∩ C_j) Y

j∈[r ]\B

(1 − χ_j(A ∩ C_j)) .

where q0(A) = 1/ _|A|ⁿ.

• Decompose q₀in terms of c₀ and q₀^C^j, j 6 r .

• Arrange the sums in a suitable way (easy since we have products).

• Use the fundamental property of q0 and q₀^C^j : X

A⊆C ,|A|=k

q0(A) = 1 X

Aj⊆Cj,|Aj|=aj

q^C₀^j(Aj) = 1.

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Decomposition of reliability

How to extend Samaniego’s decomposition ?

Proposition (Samaniego (1985))

If F is absolutely continuous and i.i.d. , then

RS(t) =

n

X

k=1

skRk:n(t),

for all t > 0, and every coherent system S = (n, φ, F ).

In the i.i.d. case we have p = s, so the decomposition also writes

RS(t) =

n

X

k=1

pkRk:n(t). (7)

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Some questions

Assume that F has no ties.

1 Find necesary and sufficient conditions (on F ) so that RS(t) =

n

X

k=1

skRk:n(t), holds for every system and all t > 0.

2 Find necesary and sufficient conditions so that RS(t) =

n

X

k=1

pkRk:n(t), holds for every system and all t > 0.

3 Find necesary and sufficient conditions so that s = p,

holds for every system.

Sufficient conditions (e.g. i.i.d. or exchangeability) are known.

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Necessary and sufficient conditions I

For every t > 0, the state variables Xk(t) are defined by Xk(t) = Ind(Tk > t).

Theorem (Marichal, M., Waldhauser (2011))

For every t > 0, (Samaniego) decomposition

RS(t) =

n

X

k=1

skRk:n(t)

holds for every coherent structure φ if and only if the state variables X1(t), . . . , Xn(t) are exchangeable.

The condition is weaker than exchangeability of component lifetimes.

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Necessary and sufficient conditions II

Theorem (Marichal,M.,Waldhauser (2011))

For every t > 0, decomposition

RS(t) =

n

X

k=1

pkRk:n(t)

holds for every coherent structure φ iff Pr(X(t) = x) = q(x) ( X

|z|=|x|

Pr(X(t) = z)),

where X(t) = (X1(t), . . . , Xn(t)).

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Equality of s and p

Proposition (Marichal,M.,Waldhauser (2011))

We have p = s for every coherent structure φ if and only if q is symmetric.

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All the new results in this talk are in one of the following papers.

• J.-L. Marichal, P. M., Extension of system signatures for dependent lifetimes: Explicit expressions and interpretations, Journal of Multivariate analysis, 102, 931–936 (2011);

• J.-L. Marichal, P. M., Symmetric approximations of pseudo-Boolean functions with applications to influence indexes; Applied

Mathematics Letters (2012);

• J.-L. Marichal, P. M.,T. Waldhauser, On signature-based

expressions of system reliability, Journal of Multivariate analysis, 102 (10), 1410–1416 (2011);

• J.-L. Marichal, P. M.,F. Spizzichino, On modular decompositions of system signatures, submitted;

• J.-L. Marichal, P. M., Computing system signatures through reliability functions, Statistics & Probability Letters 83 (3), (2013);

• J.-L. Marichal, P. M., On the extensions of Barlow–Proschan importance index and system signature to dependent lifetimes, Journal of Multivariate analysis 115, (2013).

• Everything is on the ArXiv.

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