Hierarchical Decomposition Theorems for Choquet Integral Models Michio SUGENO and Katsushige FUJIMOTO Department of Systems Science, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 226, Japan Toshiaki MUROFUSHI Department of Communications and Systems, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu 182, Japan

Abstract In this paper, we give two necessary and sufficient conditions for a Choquet integral model to be decomposable into two equivalent hierarchical Choquet integral models constructed by hierarchical combinations of some ordinary Choquet integral models. These conditions are obtained by Inclusion-Exclusion Covering (IEC). Moreover, we show some properties on the set of IECs.

x1 x2 x3 x4 xn

1

Introduction

A subjective evaluation model related to multi-attribute objectes using Choquet integral with respect to fuzzy measures have been applied in various fields, with good results [1, 2, 3] Suppose that X is the set of all attributes, f (x) is the partially evaluated value on each attribute x ∈ X of an object f , and z is the overall evaluated value. The Choquet integral model as shown in Figure 1, is the model whose input is f (x) on RX and whose output is represented as z = (C) X f dµ. It can be regarded as the generalization of a conventional linear model whose input is f (x) on X, and whose P output is represented as a weighted sum z = x∈X ωx f (x), because the output of a linear model canRbe represented by a Lebesgue integral z = (L) X f dm if a weight ωx is regarded as a measure m({x}) of a singleton {x}. That is, the Choquet integral model replaces the measure m and Lebesgue integral of a linear model by a fuzzy measure µ and a Choquet integral, respectively. Clearly, the expressive power of Choquet integral model is higher than that of a linear

f

fdµ X

Z overall evaluation

Choquet integral

attributes

Figure 1: Choquet integral model

model. However, The Choquet Integral model is difficult to handle because 2n − 1 parameters (i.e. values of a fuzzy measure) are generally required for an n-attributes object. Accordingly, as the number of attributes increases, not only does the identification of the fuzzy measure becomes difficult, but the evaluation model obtained become quite complex and the structure becomes difficult to grasp. This is a reason we propose a hierarchical Choquet integral model constructed by hierarchical combinations of ordinary Choquet integral models, as shown in Figure 2(a)(b). Figure 2(a) shows the separated hierarchical Choquet integral model in which one attribute must belong to only one macro-attribute. Figure 2(b) shows the overlapped hierarchical Choquet integral model in which one attribute can belong to more than one macro-attributes.

In the remainder of this introduction, we shall indicate the contents of the four sections that follow. Section 2 gives definitions and basic properties of a fuzzy measure and the Choquet integral. Section 3 gives theorems for an ordinary Choquet integral model to be decomposable into equivalent hierarchical Choquet integral models. Section 4 discusses various properties and structures on the set of IECs.

2

Fuzzy measure and Choquet Integral

In this section, we give the definition of a (non-)monotonic fuzzy measure and the Choquet integral, and show their basic properties. Throughout the paper we assume that (X, X ) is a measurable space. Definition 2.1 A monotonic fuzzy measure on (X, X ) is a real-valued set function λ : X → R+ satisfying the following two conditions: 1. λ(∅) = 0, 2. λ(A) ≤ λ(B) whenever A ⊂ B and A, B ∈ X , where R+ = [0, ∞), the set of non-negative real numbers. Definition 2.2 A non-monotonic fuzzy measure on (X, X ) is a real valued set function µ : X → R satisfying µ(∅) = 0. Definition 2.3 The Choquet integral of a measurable function f : X → R with respect to a non-monotonic fuzzy measure µ is defined by Z +∞ Z µf (r)dr f dµ ≡ (C) A

−∞

whenever the integral in the right-hand side exists, where µf (r) =



µ({x|f (x) > r} ∩ A) µ({x|f (x) > r} ∩ A) − µ(A)

if if

r ≥ 0, r < 0.

A measurable function f is called integrable if the Choquet integral of f exists and its value is finite. Proposition 2.1 Suppose that µ is a nonmonotonic fuzzy measure on (X, X ), then there exist two monotonic fuzzy measures µ+ and µ− on (X, X ) such that µ(A) = µ+ (A) − µ− (A)

for every A ∈ X . Furthermore Z Z Z (C) f dµ = (C) f dµ+ − (C) f dµ− , A

A

if at least either (C) finite.

3 3.1

R

A

+

A

f dµ or (C)

R

A

f dµ− is

Hierarchical Decomposition Null set and Atom

Definition 3.1 A measurable set N ∈ X is called a null set (with respect to µ) if µ(A ∪ N ) = µ(A) for every A ∈ X . An ordinary null set, in measure theory, is a set of measure zero. The null set defined above coincides with an ordinary measure-theoretic null set when µ is additive. A set of fuzzy measure zero need not be a fuzzy measure-theoretic null set in this sense. Clearly, the empty set is a null set in both senses. Proposition 3.1 If N is a null set, then Z Z (C) f dµ = (C) f dµ X

Nc

for every measurable function f on X The above proposition means that a null set is a set which does not influence the result of integral. Hence, by removing such a set, we can get a model which is simpler than, and equivalent to, the original one. Definition 3.2 A measurable non-null set A ∈ X is called an atom (with respect to µ) if either B or A \ B is a null set for every measurable set B ⊆ A. That A ∈ X is an atom means that each measurable set B ⊆ A always plays the role either of A or of the empty set for every measurable set. Definition 3.3 A measurable non-null set S ∈ X is called a semi-atom (with respect to µ) if it satisfies the following satement: “Either of the following conditions hold for each measurable set A ⊆ S: (a) µ(A ∪ B) = µ(B) whenever S ∩ B = ∅ and B ∈ X, (b) µ(A ∪ B) = µ(S ∪ B) whenever S ∩ B = ∅ and B ∈ X .” When a measurable set A ⊆ S satisfies condition (b), we write A ≈ S.

xxx111 f

xx11 (C) (C) C f dµ f dµCC1 1 1 C1

xx222 xx333 x

((C) C) f d C2 fµ dµ C2 C 2 C2

x xx444

f (C) (C) C f dµ f dµCC1 1 1 C1

x x22

fM

xx33 (C) (C) fM fMddνν C C

zz

dµ ((C) C) f dµ C2 f dµ C C2 C2

x x44

(C(C) ) f dµ Cm f dµ Cm Cm

xxnnn x

Cm

macro-attributes attributes (a) Separated hierarchical Choquet integral model

(C) (C) fM fMddνν C C

xx n-1 n-1

x xxn-1 n-1 n-1

fM zz

(C(C) ) f dµ Cm f dµ Cm Cm

xxnn

Cm

macro-attributes

attributes (b) Overlaping hierarchical Choquet integral model

Figure 2: Hierarchical Choquet integral models That S ∈ X is a semi-atom means that each measurable set A ⊆ S always plays the role either of S or of the empty set for every measurable set B which is outside of S. Definition 3.4 A measurable non-null set Q ∈ X is called a quasi-atom (with respect to µ) if it satisfies the following condition for every mesurable set A ⊆ Q: µ(A ∪ B) = µ(B) or µ(Q ∪ B) whenever Q ∩ B = ∅ and B ∈ X . That Q ∈ X is a quasi-atom means that for each measurable set B such that Q ∩ B = ∅, each measurable set A ⊆ Q always plays the role either of Q or of the empty set. It is clear that every singleton is an atom. Clearly an atom is a semi-atom, and a semiatom is a quasi-atom. These are equivalent when µ is additive. Proposition 3.2 Let S be a semi-atom. If fS is defined by  supA≈S inf y∈A f (x) if x ∈ S, fS = f (x) if x ∈ /S for every measurable function f , then fS is a measurable function, and the following holds: Z Z (C) fS dµ = (C) f dµ. X

X

By applying the above proposition, we can regard a semi-atom S as a singleton {S}. Hence,

x1

f

x2

(C) f dµ µS S d

fS

semi-atom

x3

(C) f d µ'µ' X’

z

x4 x5 Figure 3: Sub-model on a semi-atom we can show that a sub-model of the original Choquet integral model can be constructed on a semi-atom S, as shown in Figure 3, where S = {x1 , x2 } is a semi-atom and X ′ = {S} ∪ {x3 , x4 , x5 }. Here µS and µ′ are fuzzy measures on S and X ′ defined as follows  1 if A ≈ S, µS (A) = 0 otherwise,  µ((B − {S}) ∪ S) if {S} ∈ B, µ′ (B) = µ(B) if {S} ∈ / B.

3.2

3.3

Inclusion-Exclusion Covering(IEC)

Definition 3.5 A measurable finite partition {Pi }i∈{1,···,n} of X is called an Inter-Additive Partition (IAP) of X (with respect to µ) if X µ(A) = µ(A ∩ Pi ) i∈{1,···,n}

for every A ∈ X . Definition 3.6 A finite measurable covering {Ci }i∈{1,···,n} of X is called an InclusionExclusion Covering (IEC) of X (with respect to µ) if X \ µ(A) = (−1)|I|+1 µ( Ci ∩ A) i∈I

I⊆{1,···,n} I6=∅

for every A ∈ X . Clearly an IEC C is an IAP when C is a partition of X. Definition 3.7 A finite measurable covering {Ci }i∈{1,···,n} of X is called an pre-InclusionExclusion Covering (pre-IEC) of X (with respect to µ) if it satisfies the following statement: “ There exists a partition {Ij }j∈J of {1, . . . , n} satisfying the following conditions: [ (a) { Ck }j∈J is an IEC of X (with respect k∈Ij

Hierarchical Theorems

decomposition

Definition 3.8 Suppose that C = {Ci }i∈{1,···,n} is a measurable covering, then subalgebras Si of X are defined by Si ≡ {Ci ∩ A | A ∈ X }, respectively. Let M = {µCi (·)}i∈{1,···,n} be an n-ary class of non-monotonic fuzzy measures on S = {Si }i∈{1,···,n} . The function fM on C is defined by Z fM (Ci ) ≡ (C) f dµCi Ci

for every measurable function f on X. The hierarchical Choquet integral model is defined as the model whose input is f (x) on X and R whose output z is represented as z = (C) C fM dν, where ν is a non-monotonic fuzzy measure on 2C . In the following, we will give a necessary and sufficient condition for an ordinary Choquet integral model, as shown in Figure 1, to be decomposable into an equivalent hierarchical Choquet integral model, as shown in Figure 2. In general, the condition becomes quite complex, but in cases the non-monotonic fuzzy measure ν is additive, it simplifies. Hence, we first deal with the case where the non-monotonic fuzzy measure ν additive. Theorem 3.1 (Decomposition Theorem 1)

to µ), (b) µ(RCk ) > 0 for every k ∈ [Ij whenever |Ij | = 6 1, where RCk ≡ Ck \ Cj . j∈{1,...,n}\{k}

(c)

[

[

Ck \

k∈Ij

R Ck \

k∈Ij

\

Ck is a null set for

k∈Ij

every j ∈ J,

Let C = {Ci }i∈{1,...,n} be a measurable covering of X. Then there exist an n-ary class of non-monotonic fuzzy measures M ≡ {µCi (·)}i∈{1,...,n} on S ≡ {Si }i∈{1,...,n} and an additive non-monotonic fuzzy measure ν on 2C such that Z Z f dµ = (C) fM dν (C) X

[

µ(A) − µ(A \

R Ci )

i∈Ij (A)

(d)

µ(

[

=

R Ci )

i∈Ij (A)

µ(B) − µ(B \

[

R Ci )

i∈Ij (B)

µ(

[

for every j ∈ J

R Ci )

i∈Ij (B)

whenever |Ij | = 6 1, A \

[

i∈Ij

R Ci = B \

C

for every measurable function f on X, if and only if C is an IEC.

[

R Ci ,

i∈Ij

Ij (A) 6= ∅ and Ij (B) 6= ∅, where Ij (A) ≡ {i ∈ {1, . . . , n}|µ(A) > µ(A \ RCi )} ∩ Ij . ”

By applying this theorem, an overlapping hierarchical Choquet integral model, as shown in Figure 2(b), can be hierarchically constructed by an IEC from an ordinary Choquet integral model, as shown in Figure 1. As this model is constructed by linear combinations of ordinary Choquet integral models, it can be regarded as a model situated between linear models and ordinary Choquet integral models. This is also a reason for proposing the hierarchical Choquet integral model.

Corollary 3.1 Let P = {Pi }i∈{1,...,n} be a measurable partition of X. Then there exists an n-ary class of non-monotonic fuzzy measures M ≡ {µPi (·)}i∈{1,...,n} on S ≡ {Si }i∈{1,...,n} and an additive non-monotonic fuzzy measure ν on 2P such that Z Z (C) f dµ = (C) fM dν X

P

for every measurable function f on X, if and only if P is an IAP. By applying this corollary, a separated hierarchical Choquet integral model, as shown in Figure 2(a), can be hierarchically constructed by an IAP from an ordinary Choquet integral model, as shown in Figure 1. Definition 3.9 A finite measurable covering {Ci }i∈{1,···,n} of X is called a Covering with Essential Elements(CEE) of X (with respect to µ) if it satisfies the following conditions: (a) RCi 6= ∅, ∀i ∈ {1, . . . , n} (b) µ(A) < µ(A ∪ RCi ), ∀i ∈ {1, . . . , n} whenever A ⊆ Ci \ RCi , A ∈[X , where RCi ≡ Ci \ Cj . j∈{1,...,n}\{i}

Theorem 3.2 (Decomposition Theorem 2) Let (X, X , µ) be a monotonic-fuzzy measure space, C = {Ci }i∈{1,...,n} be a CEE of X. Then there exists an n-ary class of non-monotonic fuzzy measures M ≡ {µCi (·)}i∈{1,...,n} on S ≡ {Si }i∈{1,...,n} and a monotonic fuzzy measure ν on 2C such that Z Z (C) f dµ = (C) fM dν X

(C)

Z

f dµ = (C) X

Z

fM dν P

for every measurable function f on X, if and only if Pi is a semi-atom or {Pi , Pic } is an IAP of X for every i ∈ {1, . . . , n}. Suppose that C = {Ci }i∈{1,...,n} satisfies the condition in Theorem 3.2, then subalgebras Fj of 2C are defined by [ Fi ≡ { {Ck } ∩ A|A ∈ 2C }. k∈Ij

Let L ≡ {λj (·)}j∈J be a |J|-ary class of nonmonotonic fuzzy measure [ on F ≡ {Fj }j∈J . The function fL on F ≡ { {Ck }}j∈J is defined by k∈Ij

fL (

[

k∈Ij

Z {Ck }) ≡ (C) S

fM dλj . k∈Ij

{Ck }

Corollary 3.4 Let C = {Ci }i∈{1,...,n} be a measurable covering of X satisfying the condition in Theorem 3.2. Then there exists an n-ary class of non-monotonic fuzzy measures M ≡ {µCi (·)}i∈{1,...,n} on S ≡ {Si }i∈{1,...,n} and a |J|-ary class of non-monotonic fuzzy measures L ≡ {λj (·)}j∈J on F and an additive nonmonotonic fuzzy measure ν on 2F such that Z Z fL dν f dµ = (C) (C) X

F

for every measurable function f on X.

3.4

Examples on Section 3

C

for every measurable function f on X, if and only if C is a pre-IEC. Corollary 3.2 Let C = {Ci }i∈{1,...,n} be a CEE of X. If C is a pre-IEC and a partition {Ij }j∈J of {1, . . . , n} satisfies conditions in the definition of pre-IEC.Then RCi is a semi-atom with respect to µR and a quasi-atom with respect to µ for every i ∈ Ij whenever |Ij | 1, where µR is a S restriction of µ to i∈{1,...,n} RCi . Corollary 3.3 Let P = {Pi }i∈{1,...,n} be a measurable partition of X. Then there exist an n-ary class of monotonic fuzzy measures M ≡ {µPi (·)}i∈{1,...,n} on S ≡ {Si }i∈{1,...,n} and a monotonic fuzzy measure ν on 2P such that

By applying Proposition 3.1 and Theorem 3.1, an ordinary Choquet integral model is decomposed into an equivalent hierarchical one, as shown in Figure 4, where X = {x1 , . . . , x7 } is the set of all attributes, {{x1 , x2 }, {x2 , x3 , x4 }, {x4 , x5 }, {x6 , x7 }} is an IEC, and {x7 } is a null set. By applying corollary 3.4, an ordinary Choquet integral model is decomposed into an equivalent hierarchical one, as shown in Figure 4, if it satisfies the condition in Theorem 3.2.

4

The Set of IECs

In this section, we show some properties of the set of all IECs.

xx11

xx11 f

f (C) (C) f dµ f dµ C1 C C C 1

1

1

xx22

xx22

xx33

xx33 xx44

zz

(C) f dµ X

C C

22

xx44

xx55

xx55

xx66

xx66

xx77

xx77

null set

null set

fM

(C) (C) f dµ f dµ CC

22

(C) (C) f dµ f dµ CC 3

3

(C) (C) f dµ f dµ C C 4

4

C C

33

C C 4

4

(C) (C) fMfMdνdν C

C

zz

Figure 4: Hierarchical decomposition by an IEC and a null set

4.1

xx11 xx22 xx33 xx44

Proposition 4.1 Let C = {Ci }i∈{1,···,n} be a measurable covering of X. If A is a subset of some Ci ∈ C, then the following holds: X \ µ(A) = (−1)|J|+1 µ( Cj ∩ A).

f ) fdµC1 (C (C) (C) C1 fdµ C1 C1

J⊆{1,···,n} J6=∅

fM

C1

(C) fdµ

C2 C2

(C) fdλ1 (C) (C) fdfdλ1 ∪ CK ∪ CK

C3

xx77 xx88

((C) C) fdµC4 (C)C4 C4fdµC4 C4

(C) fdµC5 (C)C5 fdµ C5 C5

fL

ΣΣ

) fdµC3 (C (C) (C)C3 C3fdµC3

xx55 xx66

Structures on the Set of IECs

(C) fdfdλ2 (C) (C)∪ CK fd λ2 ∪ CK

C5

Figure 5: Hierarchical decomposition by cororally 3.3

zz

j∈J

This proposition means that we can determine whether a measurable covering C is an IEC by only examining whether every set which is not subset of any Cj ∈ C satisfies the condition for it to be an IEC. Moreover, if X ∈ C, then C − D is an IEC for every D ⊂ C such as X ∈ / D. This shows that there is generally more than one IEC. Hence providing some sort of structure on the set of all IECs is useful and needed. Definition 4.1 Let C and D be two finite measurable coverings of X. We write D ⊑ C if for every D ∈ D, there exists C ∈ C such that D ⊆ C. As two important operations on the set of finite measurable coverings, we introduce the inter-refinement of C and D and the union of C and D denoted by C ⊓ D and C ⊔ D, respectively, and defined by C ⊓ D ≡ {C ∩ D | C ∈ C, D ∈ D} C⊔D ≡C∪D The relation ⊑ is reflexive and transitive.

C ⊓ D and C ⊔ D are clearly measurable coverings, and C ⊓ D ⊑ C, D ⊑ C ⊔ D for two measurable coverings C and D. Proposition 4.2 Let C and D be two finite measurable coverings of X. If C is an IEC and C ⊑ D, then D is also an IEC. Corollary 4.1 Let C be an IEC. If there exist C, D ∈ C such that C ⊂ 6= D then C − {C} is also an IEC. Hence all we have to do is focus on the irreducible IEC which is defined as follows. Definition 4.2 An irreducible covering C of X is a finite covering satisfying the following: C = D, whenever C ⊆ D, C, D ∈ C. Theorem 4.1 Let C and D be two IECs. Then the following two statements hold:

4.2

Structures of the Set of Irreducible IECs

Definition 4.4 Let C and D be two irreducible coverings. The irreducible inter-refinement (irreducible union) of C and D is denoted by C⊓ir D (C ⊔ir D), and defined by that C ⊓ir D ( C ⊔ir D) is the irreducible covering of C ⊓ D ( C ⊔ D). Theorem 4.3 Let IRC be the set of all measurable irreducible coverings. Then an ordered set ( IRC, ⊑ ) is a distributive lattice. Futhermore, sup{C, D} = C ⊔ir D and inf{C, D} = C ⊓ir D for every C, D ∈ IRC. Corollary 4.4 Let IRIEC be the set of all measurable irreducible IEC. Then an ordered set ( IRIEC, ⊑ ) is a distributive lattice.

4.3

An example on Section 4

(a) C ⊓ D is an IEC. (b) C ⊔ D is an IEC. In other words, the set of all IECs is closed under the formation of inter-refinement and union. Lemma 4.1 For each finite measurable covering C of X, there exists uniquely a finite measurable irreducible covering D such that C ⊑ D and D ⊑ C. This D is called the irreducible covering of C Corollary 4.2 For each IEC C of X, there exists uniquely an irreducible IEC D such that C ⊑ D and D ⊑ C. Theorem 4.2 Let X be a finite set. Then there exists uniquely an irreducible IEC C such that C ⊑ D for every IEC D. Now we give a relation ∼, on the set of all finite measurable coverings of X, defined as follows: Definition 4.3 Let C and D be two finite measurable coverings of X. We write C ∼ D if C ⊑ D and D ⊑ C. The relation ∼ is clearly an equivalent relation. Proposition 4.3 Let MC be the set of all finite measurable coverings of X and IRC the set of all irreducible coverings of X. Then MC/∼ is isomorphic to IRC. Corollary 4.3 Let IEC be the set of all IECs of X and IRIEC the set of all irreducible IECs of X. Then IEC/∼ is isomorphic to IRIEC.

By the argument above, if X = {x1 , . . . , x4 }, C = {{x1 , x4 }, {x2 , x3 }}, and D = {{x1 , x2 }, {x2 , x3 }, {x3 , x4 }}, then C ⊓ D = {{x1 }, {x2 }, {x3 }, {x4 }, {x2 , x3 }} and C ⊓ir D = {{x1 }, {x2 , x3 }, {x4 }}. Hence, if there are two hierarchical Choquet integral models, as shown in Figure 4(a)(b), then a hierarchical Choquet integral model, as shown in Figure 4(c), is obtained immediately.

5

Conclusions

We gave two necessary and sufficient conditions for an ordinary Choquet integral model to be decomposable into two eqivalent hierarchical Choquet integral models. The hierarchical decomposition based on these conditions is expected to be useful as a structural analysis method not only for subjective evaluation models, but also for various phenomena represented by Choquet integral models. Furthermore, it is also expected to be useful as a model situated between linear models and ordinary Choquet integral models. However since identified fuzzy measures generally yield errors when dealing with real problem, these hierarchical decomposition theorems in this paper can not be expected to be satisfied exactly. Accordingly, the challenge is to execute the actual hierarchical decomposition based on the results. We are studying this problem and will report on it as soon as the results are available.

xx

xx

1 1

xx

1 1

(C) (C) ffdµ dµC2 C1 C1

2 2

(C) dν (C) ffMMdν C C

xx

3 3

xx

4 4

(C) (C) D ffdµ dµDD1 D11

C2

xx

2 2

zz

1

(C) (C) ffdµ dµDD2 D2 D2

xx

2

(C) dν (C) fMfMdν D D

zz

3 3

(C) (C) ffdµ dµCC22 C1 C1

(C) (C) D ffdµ dµDD3

xx

4 4

3

D33

(b)

(a)

xx

1 1

xx

2 2

(C) (C) ffdµ dµEE1 E1 E1

1

(C) (C) E ffdµ dµE2E E22

xx

2

3 3

xx

4 4

(C) dν (C) ffMMdν EE

zz

(C) (C) E ffdµ dµEE3 E33

3

(c) Figure 6: Hierarchical decomposition and construction by irreducible IEC

Conjecture

quet integral and null sets,” J. Math. Anal. Appl., vol. 159, pp. 532-549, 1991.

We expect that the set of all measurable coverings satisfying the condition in Theorem 3.2 has some structures that are similar to those of CIEs.

[6] K.Fujimoto, T.Murofushi, and M.Sugeno, ”A hierarchical evaluation model using Choquet integral (2),” Proc. 9th Fuzzy Systems Symposium, pp. 673-676, 1993. (in Japanese)

References

[7] M.Sugeno,K.Fujimoto and T.Murofushi, ”A hierarchical decomposition of Choquet integral model,” Int. J. Uncertainty, Fuziness and Knowledge-Based Systems (to appear)

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