Jean-LucMarichal CumulativeDistributionFunctionsandMomentsofWeightedLatticePolynomials

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Cumulative Distribution Functions and Moments

of Weighted Lattice Polynomials

Jean-Luc Marichal

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Part I : Weighted lattice polynomials Definitions

Representation and characterization

Part II : Cumulative distribution functions of aggregation operators

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Let L be a lattice with lattice operations ∧ and ∨

We assume that L is

bounded (with bottom 0 and top 1) distributive

Definition (Birkhoff 1967)

An n-ary lattice polynomial is a well-formed expression involving n variables x1, . . . , xn∈ L linked by the lattice operations ∧ and ∨ in an arbitrary combination of parentheses

Example.

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We assume that L is

An n-ary lattice polynomial is a well-formed expression involving n variables x1, . . . , xn∈ L linked by the lattice operations ∧ and ∨ in an arbitrary combination of parentheses

Example.

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We assume that L is

An n-ary lattice polynomial is a well-formed expression involving n

variables x1, . . . , xn∈ L linked by the lattice operations ∧ and ∨ in

an arbitrary combination of parentheses

Example.

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We assume that L is

An n-ary lattice polynomial is a well-formed expression involving n variables x1, . . . , xn∈ L linked by the lattice operations ∧ and ∨ in

Example.

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We assume that L is

An n-ary lattice polynomial is a well-formed expression involving n variables x1, . . . , xn∈ L linked by the lattice operations ∧ and ∨ in

Example.

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Any lattice polynomial naturally defines alattice polynomial function(l.p.f.) p : Ln_{→ L.}

Example.

p(x1, x2, x3) = (x1∧ x2) ∨ x3

If p and q represent the same function, we say that p and q are equivalent and we write p = q

Example.

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Example.

p(x1, x2, x3) = (x1∧ x2) ∨ x3

Example.

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Example.

p(x1, x2, x3) = (x1∧ x2) ∨ x3

Example.

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Example.

p(x1, x2, x3) = (x1∧ x2) ∨ x3

Example.

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Notation. [n] := {1, . . . , n}.

Proposition (Birkhoff 1967)

Let p : Ln_{→ L be any l.p.f.}

Then there are nonconstant set functions v , w : 2[n]→ {0, 1}, with v (∅) = 0 and w (∅) = 1, such that

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Notation. [n] := {1, . . . , n}.

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Notation. [n] := {1, . . . , n}.

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Notation. [n] := {1, . . . , n}.

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x1∨ (x1∧ x2) = x1 = x1∧ (x1∨ x2)

Notation. 1S := characteristic vector of S ⊆ [n] in {0, 1}n.

Proposition (Marichal 2002)

From among all the set functions v that disjunctively generate the l.p.f. p, only one is isotone :

v (S ) = p(1S)

From among all the set functions w that conjunctively generate the l.p.f. p, only one is antitone :

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x1∨ (x1∧ x2) = x1 = x1∧ (x1∨ x2)

Notation. 1S := characteristic vector of S ⊆ [n] in {0, 1}n. Proposition (Marichal 2002)

v (S ) = p(1S)

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x1∨ (x1∧ x2) = x1 = x1∧ (x1∨ x2)

v (S ) = p(1S)

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x1∨ (x1∧ x2) = x1 = x1∧ (x1∨ x2)

v (S ) = p(1S)

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Denote by x₍₁₎, . . . , x_(n) theorder statisticsresulting from

reordering x1, . . . , xn in the nondecreasing order : x(1) 6 · · · 6 x(n). Proposition (Ovchinnikov 1996, Marichal 2002)

p is a symmetric l.p.f. ⇐⇒ p is an order statistic Notation. Denote by osk : Ln→ L the kth order statistic function.

osk(x ) := x(k)

Then we have

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p is a symmetric l.p.f. ⇐⇒ p is an order statistic

Notation. Denote by osk : Ln→ L the kth order statistic function. osk(x ) := x(k)

Then we have

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Notation. Denote by osk : Ln→ L the kth order statistic function.

osk(x ) := x(k) Then we have

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osk(x ) := x(k)

Then we have

osk(1S) = 1 ⇐⇒ |S| > n − k + 1

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osk(x ) := x(k)

Then we have

osk(1S) = 1 ⇐⇒ |S| > n − k + 1

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We can generalize the concept of l.p.f. by regarding some variables as parameters.

Example. For c ∈ L, we consider

p(x1, x2) = (c ∨ x1) ∧ x2 Definition

p : Ln→ L is an n-aryweighted lattice polynomialfunction (w.l.p.f.) if there exist parameters c1, . . . , cm ∈ L and a l.p.f. q : Ln+m → L such that

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p(x1, x2) = (c ∨ x1) ∧ x2

Definition

p : Ln→ L is an n-aryweighted lattice polynomialfunction (w.l.p.f.) if there exist parameters c1, . . . , cm ∈ L and a l.p.f. q : Ln+m → L such that

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p : Ln→ L is an n-aryweighted lattice polynomialfunction (w.l.p.f.)

if there exist parameters c1, . . . , cm ∈ L and a l.p.f. q : Ln+m → L

such that

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p : Ln→ L is an n-aryweighted lattice polynomialfunction (w.l.p.f.) if there exist parameters c1, . . . , cm ∈ L and a l.p.f. q : Ln+m → L

such that

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Proposition (Lausch & N¨obauer 1973)

Let p : Ln→ L be any w.l.p.f.

Then there are set functions v , w : 2[n]→ L such that

p(x ) = _ S ⊆[n] h v (S ) ∧^ i ∈S xi i = ^ S ⊆[n] h w (S ) ∨_ i ∈S xi i . Remarks.

p is a l.p.f. if v and w range in {0, 1}, with v (∅) = 0 and w (∅) = 1.

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Then there are set functions v , w : 2[n]→ L such that p(x ) = _ S ⊆[n] h v (S ) ∧^ i ∈S xi i = ^ S ⊆[n] h w (S ) ∨_ i ∈S xi i . Remarks.

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Any w.l.p.f. is entirely determined by 2n parameters, even if

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From among all the set functions v that disjunctively generate the w.l.p.f. p, only one is isotone :

v (S ) = p(1S)

From among all the set functions w that conjunctively generate the w.l.p.f. p, only one is antitone :

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From among all the set functions v that disjunctively generate the w.l.p.f. p, only one is isotone :

v (S ) = p(1S)

From among all the set functions w that conjunctively generate the w.l.p.f. p, only one is antitone :

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Let us generalize the concept of discrete Sugeno integral in the framework of bounded distributive lattices.

Definition (Sugeno 1974)

An L-valued fuzzy measure on [n] is an isotone set function µ : 2[n] → L such that µ(∅) = 0 and µ([n]) = 1.

TheSugeno integral of a function x : [n] → L with respect to µ is defined by S_µ(x ) := _ S ⊆[n] h µ(S ) ∧^ i ∈S xi i

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Remark. A function f : Ln_{→ L is an n-ary Sugeno integral if and}

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Remark. A function f : Ln_{→ L is an n-ary Sugeno integral if and}

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Notation. The median function is the function os2 : L3→ L. Proposition (Marichal 2006)

For any w.l.p.f. p : Ln → L, there is a fuzzy measure µ : 2[n] _{→ L}

such that

p(x ) = medianp(1_∅), Sµ(x ), p(1[n])

Corollary (Marichal 2006) Consider a function f : Ln→ L.

The following assertions are equivalent : f is a Sugeno integral

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such that

p(x ) = medianp(1_∅), Sµ(x ), p(1[n])

Corollary (Marichal 2006)

Consider a function f : Ln→ L.

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such that

p(x ) = medianp(1_∅), Sµ(x ), p(1[n])

Corollary (Marichal 2006)

Consider a function f : Ln→ L.

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Sugeno integrals

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Let f : Ln→ L and k ∈ [n] and define f0

k, fk1 : Ln→ L as

f_k0(x ) := f (x1, . . . , xk−1, 0, xk+1, . . . , xn)

f_k1(x ) := f (x1, . . . , xk−1, 1, xk+1, . . . , xn) Remark. If f is a w.l.p.f., so are f_k0 and f_k1

Consider the following system of n functional equations, called the

median based decomposition formula

f (x ) = medianf0

k(x ), xk, fk1(x )

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k, fk1 : Ln→ L as

f_k0(x ) := f (x1, . . . , xk−1, 0, xk+1, . . . , xn)

f_k1(x ) := f (x1, . . . , xk−1, 1, xk+1, . . . , xn)

Remark. If f is a w.l.p.f., so are f_k0 and f_k1

f (x ) = medianf0

k(x ), xk, fk1(x )

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k, fk1 : Ln→ L as

f_k0(x ) := f (x1, . . . , xk−1, 0, xk+1, . . . , xn)

f_k1(x ) := f (x1, . . . , xk−1, 1, xk+1, . . . , xn)

f (x ) = medianf0

k(x ), xk, fk1(x )

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k, fk1 : Ln→ L as

f_k0(x ) := f (x1, . . . , xk−1, 0, xk+1, . . . , xn)

f_k1(x ) := f (x1, . . . , xk−1, 1, xk+1, . . . , xn)

f (x ) = medianf0

k(x ), xk, fk1(x )

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Any solution of the median based decomposition formula f (x ) = medianf0 k(x ), xk, fk1(x ) (k = 1, . . . , n) is an n-ary w.l.p.f.

Example. For n = 2 we have

f (x1, x2) = medianf (x1, 0), x2, f (x1, 1)

with

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The median based decomposition formula characterizes the w.l.p.f.’s

Theorem (Marichal 2006)

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The median based decomposition formula characterizes the w.l.p.f.’s

Theorem (Marichal 2006)

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Consider

an aggregation operator A : Rn→ R

n independent random variables X1, . . . , Xn, with cumulative distribution functions F1(x ), . . . , Fn(x ) X1 .. . Xn      −→ YA = A(X1, . . . , Xn)

Problem. We are searching for the cumulative distribution function (c.d.f.) of YA :

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Consider

n independent random variables X1, . . . , Xn, with cumulative

distribution functions F1(x ), . . . , Fn(x ) X1 .. . Xn      −→ YA = A(X1, . . . , Xn)

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Consider

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Consider

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Consider

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From the c.d.f. of YA, we can calculate the expectation

Eg (YA) =

Z ∞

−∞

g (y ) dFA(y )

for any measurable function g .

Some useful examples :

g (x ) Eg (YA)

x expected value of YA

xr raw moments of YA

x − E(Y_A)r central moments of YA

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From the c.d.f. of YA, we can calculate the expectation

Eg (YA) =

Z ∞

−∞

g (y ) dFA(y )

for any measurable function g .

Some useful examples :

g (x ) Eg (YA)

x expected value of YA

xr raw moments of YA

x − E(Y_A)r central moments of YA

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AM(x1, . . . , xn) = 1 n n X i =1 xi

FAM(y ) is given by the convolution product of F1, . . . , Fn FAM(y ) = (F1∗ · · · ∗ Fn)(ny )

For uniform random variables X1, . . . , Xn on [0, 1], we have

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AM(x1, . . . , xn) = 1 n n X i =1 xi

FAM(y ) is given by the convolution product of F1, . . . , Fn

FAM(y ) = (F1∗ · · · ∗ Fn)(ny )

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AM(x1, . . . , xn) = 1 n n X i =1 xi

FAM(y ) = (F1∗ · · · ∗ Fn)(ny )

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AM(x1, . . . , xn) = 1 n n X i =1 xi

FAM(y ) = (F1∗ · · · ∗ Fn)(ny )

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Case of n = 3 uniform random variables X1, X2, X3 on [0, 1]

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TL(x1, . . . , xn) = max h 0, n X i =1 xi− (n − 1) i FT_L(y ) = Pr h max0, P_iXi− (n − 1)_{6 y}i = Prh0 6 y and P iXi − (n − 1) 6 y i = Pr[0 6 y ] PrhP iXi 6 y + n − 1 i = H0(y ) FAM y +n−1_n

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TL(x1, . . . , xn) = max h 0, n X i =1 xi− (n − 1) i FTL(y ) = Pr h max0, P_iXi− (n − 1)_{6 y}i = Prh0 6 y and P iXi − (n − 1) 6 y i = Pr[0 6 y ] PrhP iXi 6 y + n − 1 i = H0(y ) FAM y +n−1_n

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TL(x1, . . . , xn) = max h 0, n X i =1 xi− (n − 1) i FTL(y ) = Pr h max0, P_iXi− (n − 1) 6 y i = Prh0 6 y and P iXi − (n − 1) 6 y i = Pr[0 6 y ] PrhP iXi 6 y + n − 1 i = H0(y ) FAM y +n−1_n

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TL(x1, . . . , xn) = max h 0, n X i =1 xi− (n − 1) i FTL(y ) = Pr h max0, P_iXi− (n − 1) 6 y i = Prh0 6 y and P iXi − (n − 1) 6 y i = Pr[0 6 y ] PrhP iXi 6 y + n − 1 i = H0(y ) FAM y +n−1_n where Hc(y ) is theHeavisidefunction

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where Hc(y ) is theHeavisidefunction

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where Hc(y ) is theHeavisidefunction

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Graph of FTL(y )

Remark.

FTL(y ) is discontinuous

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Graph of FTL(y )

Remark.

FTL(y ) is discontinuous

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osk(x1, . . . , xn) = x(k) Fosk(y ) = X S ⊆[n] |S|>k Y i ∈S Fi(y ) Y i ∈[n]\S 1 − Fi(y )

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(see e.g. David & Nagaraja 2003)

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Let p : Ln→ L be a l.p.f. on L = [0, 1]

It can be extended to an aggregation function from Rn to R. p(x1, . . . , xn) = _ S ⊆[n] p(1S)=1 ^ i ∈S xi = ^ S ⊆[n] p(1[n]\S)=0 _ i ∈S xi

Note. ∧ = min, ∨ = max

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It can be extended to an aggregation function from Rn to R.

p(x1, . . . , xn) = _ S ⊆[n] p(1S)=1 ^ i ∈S xi = ^ S ⊆[n] p(1[n]\S)=0 _ i ∈S xi

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It can be extended to an aggregation function from Rn to R.

p(x1, . . . , xn) = _ S ⊆[n] p(1S)=1 ^ i ∈S xi = ^ S ⊆[n] p(1[n]\S)=0 _ i ∈S xi

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Example. p(x ) = (x1∧ x2) ∨ x3

Uniform random variables X1, X2, X3 on [0, 1]

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Consider

vp: 2[n] → R, defined by vp(S ) := p(1S) v_p∗: 2[n] → R, defined by v∗

p(S ) = 1 − vp([n] \ S ) mv : 2[n]→ R, the M¨obius transform of v, defined by

mv(S ) := X T ⊆S

(−1)|S|−|T |v (T )

Alternate expressions of Fp(y )

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Consider

vp: 2[n] → R, defined by vp(S ) := p(1S)

v_p∗: 2[n] → R, defined by v∗

p(S ) = 1 − vp([n] \ S ) mv : 2[n]→ R, the M¨obius transform of v, defined by

mv(S ) := X T ⊆S

(−1)|S|−|T |v (T )

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Consider

p(S ) = 1 − vp([n] \ S )

mv : 2[n]→ R, the M¨obius transform of v, defined by

mv(S ) :=

X

T ⊆S

(−1)|S|−|T |v (T )

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Consider

p(S ) = 1 − vp([n] \ S )

mv(S ) :=

X

T ⊆S

(−1)|S|−|T |v (T )

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Consider

p(S ) = 1 − vp([n] \ S )

mv(S ) :=

X

T ⊆S

(−1)|S|−|T |v (T )

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Let p : Rn→ R be a w.l.p.f. on R = [−∞, +∞]

Notation. eS := characteristic vector of S in {−∞, +∞}n p(x ) = _ S⊆[n] h p(eS) ∧^ i ∈S xi i = ^ S ⊆[n] h p(e[n]\S) ∨ _ i ∈S xi i Fp(y ) = 1 − X S⊆[n] 1 − H_p(e_S₎(y ) Y i ∈[n]\S Fi(y ) Y i ∈S 1 − F_i(y ) Fp(y ) = X S⊆[n] H_p(e_[n]\S₎(y )Y i ∈S Fi(y ) Y i ∈[n]\S 1 − Fi(y )

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Notation. eS := characteristic vector of S in {−∞, +∞}n

p(x ) = _ S⊆[n] h p(eS) ∧^ i ∈S xi i = ^ S ⊆[n] h p(e[n]\S) ∨ _ i ∈S xi i Fp(y ) = 1 − X S⊆[n] 1 − H_p(eS₎(y ) Y i ∈[n]\S Fi(y ) Y i ∈S 1 − F_i(y ) Fp(y ) = X S⊆[n] H_p(e[n]\S)(y )Y i ∈S Fi(y ) Y i ∈[n]\S 1 − Fi(y )

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p(x ) = _ S⊆[n] h p(eS) ∧ ^ i ∈S xi i = ^ S ⊆[n] h p(e[n]\S) ∨ _ i ∈S xi i Fp(y ) = 1 − X S⊆[n] 1 − H_p(e_S₎(y ) Y i ∈[n]\S Fi(y ) Y i ∈S 1 − F_i(y ) Fp(y ) = X S⊆[n] H_p(e_[n]\S₎(y )Y i ∈S Fi(y ) Y i ∈[n]\S 1 − Fi(y )

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p(x ) = _ S⊆[n] h p(eS) ∧ ^ i ∈S xi i = ^ S ⊆[n] h p(e[n]\S) ∨ _ i ∈S xi i Fp(y ) = 1 − X S⊆[n] 1 − H_p(eS₎(y ) Y i ∈[n]\S Fi(y ) Y i ∈S 1 − F_i(y ) Fp(y ) = X S⊆[n] H_p(e[n]\S)(y )Y i ∈S Fi(y ) Y i ∈[n]\S 1 − Fi(y )

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Example. p(x ) = (c ∧ x1) ∨ x2

Uniform random variables X1, X2 on [0, 1]

F (y ) = median[0, y , 1] S p(eS) ∅ −∞ {1} c {2} +∞ {1, 2} +∞ Fp(y ) = F (y ) F (y )+Hc(y )[1−F (y )]

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Example. Given a w.l.p.f. p : [0, 1]n→ [0, 1] and a measurable function g : [0, 1] → R, compute

Z

[0,1]n

gp(x) dx

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Z

[0,1]n

gp(x) dx

Solution. The integral is given by Eg (Yp), where the variables

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Z

[0,1]n

gp(x) dx

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Z

[0,1]n

gp(x) dx

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Sugeno integral Z [0,1]n S_µ(x ) dx = X S⊆[n] Z µ(S ) 0 yn−|S|(1 − y )|S|dy Example. Z [0,1]2 (c ∧ x1) ∨ x2 dx = 1 2 + 1 2c 2₋1 3c 3

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Consider a system made up of n indep. components C1, . . . , Cn Each component Ci has

a lifetimeXi a reliabilityri(t) at time t > 0 ri(t) := Pr[Xi > t] = 1 − Fi(t) X1 X2 X3 X1 X2 Assumptions :

The lifetime of a series subsystem is the minimum of the component lifetimes

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Consider a system made up of n indep. components C1, . . . , Cn

Each component Ci has

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Question. What is the lifetime of the following system ?

X1 X2

X3

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Question. What is the lifetime of the following system ?

X1 X2

X3

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For a system mixing series and parallel connections :

System lifetime:

Yp = p(X1, . . . , Xn)

where p is

an n-ary l.p.f.

an n-ary w.l.p.f. if some Xi’s are constant

We then have explicit formulas for

the c.d.f. of Yp

the expected value E[Yp]

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System lifetime:

Yp = p(X1, . . . , Xn)

where p is

an n-ary l.p.f.

We then have explicit formulas for the c.d.f. of Yp

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System lifetime:

Yp = p(X1, . . . , Xn)

where p is

an n-ary l.p.f.

We then have explicit formulas for

the c.d.f. of Yp

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System lifetime:

Yp = p(X1, . . . , Xn)

where p is

an n-ary l.p.f.

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System lifetime:

Yp = p(X1, . . . , Xn)

where p is

an n-ary l.p.f.

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System lifetime:

Yp = p(X1, . . . , Xn)

where p is

an n-ary l.p.f.

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System reliability at time t > 0

Rp(t) := Pr[Yp > t] = 1 − Fp(t)

For any measurable function g : [0, ∞[ → R such that

g (∞)ri(∞) = 0 (i = 1, . . . , n) we have Eg (Yp) = g (0) + Z ∞ 0 Rp(t) dg (t)

Mean time to failure:

E[Yp] = Z ∞

0

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Rp(t) := Pr[Yp > t] = 1 − Fp(t)

For any measurable function g : [0, ∞[ → R such that g (∞)ri(∞) = 0 (i = 1, . . . , n) we have Eg (Yp) = g (0) + Z ∞ 0 Rp(t) dg (t)

E[Yp] =

Z ∞

0

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Rp(t) := Pr[Yp > t] = 1 − Fp(t)

For any measurable function g : [0, ∞[ → R such that g (∞)ri(∞) = 0 (i = 1, . . . , n) we have Eg (Yp) = g (0) + Z ∞ 0 Rp(t) dg (t)

E[Yp] =

Z ∞

0

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Example. Assume ri(t) = e−λit (i = 1, . . . , n) EYp = X S ⊆[n] S6=∅ mvp(S ) 1 P i ∈Sλi Series system EYp = 1 P i ∈[n]λi Parallel system EYp = X S ⊆[n] S6=∅ (−1)|S|−1 P 1 i ∈Sλi

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