The Choquet Integral with Respect to Fuzzy-Valued Set Functions

Transcription

1 The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i iformatio fusio ad data miig as a aggregatio tool successfully. However, i some real-world problems, the values of the ivolved set fuctio may be ot crisp but fuzzy. So the Choquet itegral eeds to be geeralized such that it ca be take with respect to iterval-valued siged efficiecy measure ad, more geerally, fuzzy-valued siged efficiecy measure. A calculatio formula for such kid of Choquet itegral is developed whe the uiversal set, such as the set of attributes i a database, is fiite ad the fuzzy-valued set fuctio takes trapezoidal fuzzy umbers, ivolvig triagular fuzzy umbers as their special cases. Some properties of the Choquet itegrals with respect to fuzzy-valued efficiecy measures are also discussed. Keywords. Noadditive set fuctios, the Choquet itegral, fuzzificatio. 1. Itroductio As a aggregatio tool, oliear itegrals, such as the Choquet itegral with respect to oadditive set fuctios have bee widely applied i iformatio fusio ad data miig. Up to ow, i the existig oliear multiregressio models ad oliear classifiers, the oliear itegrals are take with respect to real-valued set fuctios. However, i may real problems, allowig set fuctios to assume fuzzy umbers seems more reasoable. Hece, this paper tries to establish the Choquet itegral with respect to fuzzy-valued 1

2 siged efficiecy measures. Usig such Choquet itegrals i multiregressio is oe of the approaches of fuzzy iformatio retrieval. To develop the calculatio formula of the Choquet itegral with respect to fuzzy-valued siged efficiecy measures, there is a problem whe itegrad takes both positive ad egative values sice the subtractio is ot the iverse operatio of the additio for fuzzy umbers. Decompositio theorem of fuzzy sets ad extesio priciple are used to cope with the problem. The paper is orgaized as follows. I Sectio 2, the relevat mathematical cocepts are itroduced. I Sectio 3, we show the iterval-valued siged efficiecy measure o P (X). Sectio 4 presets the fuzzy-valued siged efficiecy measure o P (X). I Sectio 5 we discuss the properties of such kid of Choquet itegral with respect to fuzzy-valued siged efficiecy measures. Fially, brief coclusios are give i Sectio Relevat mathematical cocepts Let X be a oempty set, called the uiversal set. We also let F be a σ -algebra, cosistig of subsets of X. If X is fiite, usually we take P (X) as σ -algebra F. (X,F ) is called a measurable space. Defiitio 1 [5]. Set fuctio µ :F (X) (, ) is called a mootoe measure ( is also called fuzzy measure ) if (1) µ ( ) = 0; (2) µ ( E) 0 E F ; (3) µ ( E) µ ( F) if EF F,, ad E F. If µ ( X ) = 1, mootoe measure µ is called ormalized. 2

3 Defiitio 2 [5]. Set fuctio µ :F (X) (, ) is called a efficiecy measure ( is also called geeralized measure ) if it satisfies the coditio (1) ad (2) give i Defiitio 1. Ay mootoe measure is a special case of efficiecy measures. Defiitio 3 [5]. Set fuctio µ :F (X) (, ) is called a siged efficiecy measure (is also called siged geeralized measure) if it satisfies oly coditio (1) give i Defiitio 1. Let f : X (, ) be a measurable fuctio with respect to F. Whe µ is a mootoe measure, the Choquet itegral of f with respect to µ ca be defied as follows. Defiitio 4 [5]. Let µ be a mootoe measure o (X,F ). The Choquet itegral of F with respect to µ, deoted by (C) f dµ, is defied as 0 (C) f dµ = [ µ ( F) µ ( X)] d + µ ( F) d (1) if ot both Riema itegrals i (1) are ifiite, where F = { x f( x) } for (, ) ad is called the -level set of f. 0 The -level set F is oicreasig with respect to. Sice µ is odecreasig, µ ( F ) is a oicreasig fuctio of, such that both Riema s itegral i Defiitio 4 are well defied. As a special case, whe f is oegative, formula (1) is reduced to be 3

4 (C) f dµ = µ ( F ) d. (2) 0 I most cases, formula (1) is also feasible for the Choquet itegral with respect to efficiet measures. Defiitio 5 [6]. Let µ be a siged efficiecy measure o F. A pair of two efficiecy measures, + ν ad ν -, satisfyig + - µ ( A) = ν ( A) ν ( A) A F ( simply, we + write µ = ν ν - ) is called a oegative decompositio of µ. We may omit word oegative i the above defiitio if there is o cofusio, ad simply call it a decompositio. For a give siged efficiecy measure, there are ifiitely may decompositios. Amog them, there is a smallest oe. Defiitio 6 [6]. The smallest decompositio of siged efficiecy measure µ is the decompositio, + µ ad µ - + +, such that µ ν ad µ ν for ay decompositio, + ν ad ν -, of µ. The smallest decompositio of µ is uique. It ca be expressed as ( ) ( ) 0 + µ A if µ A µ ( A) = 0 otherwise ad µ ( A) if µ ( A) 0 µ ( A) = 0 otherwise for ay A F. + µ ad µ - are called the positive part ad the egative part of µ respectively. 4

5 Defiitio 7 [5]. Let µ be a siged efficiecy measure ad f be a real-valued measurable fuctio o (X,F ). The Choquet itegral of f with respect to µ is defied as (C) fdµ = (C) fdµ 1 (C) f dµ 2 (3) if ot both Choquet itegral i formula (4) are ifiite, where µ 1 ad µ 2 are efficiecy measures ad form the smallest decompositio of µ. I ay database, the umber of attributes is always fiite, that is, X is a fiite set. I this case, there is a simple formula for calculatig the value of (C) f dµ oce f ad µ are give. First, the values of fuctio f, { f ( x1), f( x2),..., f( x )}, are rearraged ito a odecreasig order as, ( ) ( )... ( ) f x1 f x2 f x where ( x1, x2,..., x ) is a permutatio of ( x1, x2,..., x ). The the Choquet itegral of f with respect to µ ca be calculated by with a covetio f( x 0 ) = 0. (C) f dµ = [ f( x ) f( x 1)] µ ({ x, x 1,..., x + }) i= 1 i i i i Defiitio 8 [2]. Let A be a fuzzy subset of uiversal set X. For ay [0,1], crisp set { x m ( x), x X} is called the cut of A, deoted by A ; while crisp set A { x m ( x) >, x X} is called the strog cut of A, deoted by A +. It is clear that A A0 X, A 0 + = =supp A, ad A 1 + =. 5

6 Defiitio 9 [2]. For ay crisp set E ad ay real umber [0,1], E is the fuzzy set havig membership fuctio m E ( x) = 0 if x E if x E x X. Let A be a fuzzy subset of uiversal set X. Theorem 1 (Decompositio theorem I ) [2]. A = A = aa. [0,1] (0,1] Theorem 2 (Decompositio theorem II ) [2]. A = A = aa. + + [0,1] (0,1] Defiitio 10 [2]. Set { m ( x) = for some x X} is called the level-value set of A fuzzy set A, deoted by L A. Theorem 2 (Decompositio theorem III ) [2]. A =. L A A Theorem 3 (Extesio priciple) [2]. Let X1, X2,..., X, ad Y be oempty crisp sets, U = X1 X2... X be the product set of X1, X 2,..., ad X, ad f : U Y be a mappig from U toy. The mappig f ca be exteded to be f : F ( X 1) F ( X 2)... F ( X ) F ( Y ) as follows: for ay give fuzzy sets A i F ( X i ), i= 1, 2,,, fuzzy set B= f( A1, A2,..., A ) F ( Y ) has membership fuctio with covetio m ( y) = sup mi[ m ( x ), m ( x ),..., m ( x )] B A1 1 A2 2 A x1, x2,..., x y= f ( x1, x2,..., x) 6

7 sup{ xx [0, 1]} = 0. As a special case, if * is a biary operator o uiversal set X, that is, *: X X X, the, by the extesio priciple, we ca obtai a biary operator o F ( X ): for ay A, B F ( X ), m ( z) = sup[ m ( x) m ( x)] z X. AB * A B x y= z By usig classical extesio priciple, the commo operatios of real umbers ca be exteded to be operatios for iterval umbers. Ay real umber a ca be regarded as a iterval umber [ a, a ]. The set of all iterval umbers is deoted by I while the set of all oegative iterval umbers is deoted by I +. To quatify fuzzy cocepts, some types of fuzzy subsets of R = (, + ) are used. Fuzzy umbers are a most commo type of fuzzy subsets of R for this purpose. Defiitio 11 [2]. Ay closed iterval [ a, b ], where < a b< is called a iterval umber. Ay iterval umber[ a, b ] satisfyig a 0 is called a oegative iterval umber. Defiitio 12 [2]. A fuzzy umber, deoted by a capital letter with a wave such as A, is a fuzzy subset of R with membership fuctio m: R [0, 1] satisfyig the followig coditios: (FN1) A, the cut of A, is a closed iterval for ay (0, 1] ; (FN2) A 0 + is bouded. 7

8 Coditio (FN1) implies the covexity of A, i.e., ay fuzzy umber is a covex fuzzy subset of R. For ay (0, 1], the cut of a fuzzy umber is a iterval umber. The set of all fuzzy umbers is deoted by N F. Defiitio 13 [2]. Fuzzy umber A is said to be oegative if its A [ a, b] with a 0 for all (0, 1]. = Theorem 4 [2]. Coditio (FN1) is equivalet to the followig coditios: (FN1.1) there exists at least oe real umber a 0 such that ma ( 0) = 1; (FN1.2) mt () is odecreasig o (, a0 ] ad oicreasig o [ a 0, ); (FN1.3) mt () is upper semi-cotiuous, or say, mt () is right-cotiuous o (, a0) ad is left-cotiuous o ( a 0, + ). For ay fuzzy umber with membership fuctio mt ( ), there exists a closed iterval[ a, a ] such that b c 1 t [ ab, ac] mt () = lt () t (, ab ) rt () t ( ac, + ), where 0 lt ( ) < 1, called the left brach of mt ( ), is odecreasig ad 0 rt ( ) < 1, called the right brach of mt ( ), is oicreasig. Defiitio 14 [2]. A rectagular fuzzy umber is a fuzzy umber with membership fuctio havig form as 8

9 where a, a l r 1 if t [ ab, ac] mt () =, 0 otherwise R with a l a r. A fuzzy umber is rectagular iff the left brach ad right brach of its membership fuctio are zero. It is idetified with the correspodig vector [ a a ] ad is a iterval umber essetially. Ay crisp real umber a ca be regarded as a special rectagular fuzzy umber with a l = a r = a. l r Defiitio 15 [2]. A triagular fuzzy umber is a fuzzy umber with membership fuctio where a, a0, a l r mt () 1 if t = a0 t a l if t [ al, a0) a a 0 = t a 0 R with a a0 a. l 0 l ar a r r if t ( a, a ] 0 otherwise r, A triagular fuzzy umber is idetified with correspodig vector [ al a0 a r]. Ay crisp real umber a ca be regarded as a special triagular fuzzy umber with al = a0 = ar = a. Defiitio 16 [5]. A trapezoidal fuzzy umber is a fuzzy umber with membership fuctio 9

10 where al, ab, ac, ar 1 if t [ ab, ac] t a l if t [ al, ab) ab al mt () =, t ar if t ( ac, ar] ac ar 0 otherwise R with a l a b a c a r. A trapezoidal fuzzy umber is idetified with the correspodig vector [ al ab ac a r]. Fig. 1 shows the membership fuctio of a trapezoidal fuzzy umber, deoted by E, ad its -cut. Ay rectagular fuzzy umber [ a a ] ca be regarded as a special trapezoidal l r fuzzy umber with a l a a = b ad c r = a. Similarly, ay triagular fuzzy umber [ ] a = a = a. Of al a0 a r ca be regarded as a special trapezoidal fuzzy umber with b c 0 course, ay crisp real umber a ca be regarded as a special trapezoidal fuzzy umber with a l = a b = a c = a r = a. Thus, our discussio ad models ca be applicable to database ivolvig eve both crisp ad fuzzy data. Both the left brach ad the right brach of the membership fuctio of a trapezoidal fuzzy umber are piecewise liear. Notatio [ a, b] should be uderstood as a fuzzy set possessig membership fuctio m [ a, b] if t [ a, b] () t =. 0 otherwise Now we restrict i usig trapezoidal fuzzy umbers for the fuzzy-valued siged efficiecy measure i the Choquet itegral. 10

11 m () E t 1 a c d b t E Fig. 1 The membership fuctio of a trapezoidal fuzzy umber ad its -cut Let A ad B be two trapezoidal fuzzy members with membership fuctios ad t a1 if t [ a1, c1) c1 a1 1 if t [ c1, d1] ma() t = b1 t if t ( d1, b1] b1 d1 0 otherwise t a2 if t [ a2, c2) c2 a2 1 if t [ c2, d2] mb () t =, b2 t if t ( d2, b2] b2 d2 0 otherwise respectively. The, 11

12 A = [ ( c a ) + a, b ( b d )], B = [ ( c a ) + a, b ( b d )] Sice ( A+ B) = A + B = [ ( c + c a a ) + a + a, b + b ( b + b d d )], we obtai m A+ B t ( a1+ a2) if t [ a1+ a2, c1+ c2) ( c1+ c2) ( a1+ a2) 1 if t [ c1+ c2, d1+ d2] () t = ( b + b ) t (, ] 1 2 if t d1+ d2 b1+ b2 ( b1+ b2) ( d1+ d2) 0 otherwise. This meas that the sum of ay two trapezoidal fuzzy umbers is still a trapezoidal fuzzy umber. It is evidet that a trapezoidal fuzzy umber multiplied by a costat is still a trapezoidal fuzzy umber. Thus, ay liear combiatio of trapezoidal fuzzy umbers is still a trapezoidal fuzzy umber that ca be determied by the ed poits ad the top poits of the origial trapezoidal fuzzy umbers. Defiitio 17 [5]. Set fuctio µ :F (X) N I, deoted by µ, is called a iterval-valued siged efficiecy measure, if µ ( ) = 0. Defiitio 18 [5]. Set fuctio µ :F (X) N F, deoted by µ, is called a fuzzy-valued siged efficiecy measure, if µ ( ) = 0. 12

13 Defiitio 19 [5]. For ay fuzzy-valued siged efficiecy measure µ o (X, F ) ad ay (0,1], the iterval-valued siged efficiecy measure µ ( A) = [ µ ( A)] for very A F is called the cut of µ. 3. The Choquet Itegral with Iterval-valued siged efficiecy measure o P (X) Whe X is a fiite oempty set, accordig to Theorem 1 (Decomposability), the calculatio of the Choquet itegral with respect to a iterval-valued siged efficiecy measure is realizable. Let X = { x1, x2,..., x }, µ be a iterval-valued siged efficiecy measure o P (X), ad f be a give real-valued fuctio o X. Similar to Defiitio 7, rearrage { x1, x2,..., x } ito * * * X = { x1, x2,..., x } such that * * * f ( x1 ) f( x2 )... f( x ) ad let f ( x ) =0. The the Choquet itegral of f with * 0 respect to µ ca be calculated by Deote ad we have * * * * * (C) fd µ = [ f ( xi ) f ( xi 1)] µ ({ xi, xi+ 1,..., x }). i= 1 µ ({ x, x +,..., x }) = [ a, b], * * * i i 1 i i * * f( xi ) f( xi 1) = i, for i = 1, 2,...,, (C) fdµ = i [ ai, bi]. i= 1 * Notig i 0 ad = f ( x ) = mi f( x i ), we obtai i 13

14 (C) [ iai + 1a1, ibi + 1b1] if 1 0 i= 2 i= 2 fdµ =. (4) [ iai + 1b1, ibi + 1a1] if 1 0 i= 2 i= 2 Example 1 Let X = { x1, x2, x3}, µ be a iterval-valued siged efficiecy measure o P (X) defied as [0, 0] if A = [1, 2] if A = { x1} [ 1, 1] if A = { x2} [3, 5] if A { x1, x = 2} µ ( A) = [ 2, 3] if A = { x3} [1, 4] if A = { x1, x3} [ 5, 2] if A = { x2, x3} [5, 6] if A = X ad f be a real-valued fuctio o X expressed by 0.3 if x = x1 f = 0.2 if x = x2, 0.5 if x = x3 the, x = x, x = x ad x * = x. * * Furthermore, 1 = 0.5 < 0, 2 = 0.3 ( 0.5) = 0.2, 3 = 0.2 ( 0.3) = 0.5, [ a1, b 1] = [5, 6], [ a2, b 2] = [3, 5] ad [ a 3, b 3 ] = [ 1, 1]. Thus, by usig (4), 14

15 µ = [ + +, + + ] (C) fd 2a2 3a3 1b1 2b2 3b3 1a1 = [ ( 1) + ( 0.5) 6, ( 0.5) 5] = [ 2.9, 1]. 4. The Choquet itegral with fuzzy-valued siged efficiecy measure o P (X) Let µ : F N F, be a fuzzy-valued siged efficiecy measure ad f be a measurable fuctio o measurable space(x, F ). Sice the cut of ay give fuzzy umber is a closed iterval for every (0, 1] ad the Choquet itegral with respect to a iterval-valued siged efficiecy measure ca be expressed i a liear form of the values of the iterval-valued siged efficiecy measure, the Choquet itegral with respect to a siged fuzzy-valued efficiecy measure ca be expressed by usig the compositio theorem of fuzzy sets. Thus, the Choquet itegral of f with respect to µ is (C) fd ~ µ = (C) fdµ, [0, 1] where µ is the cut of µ, that is, the cut of fuzzy umber value ad it is a iterval value. Whe Let X = { x1, x2,..., x }, that is, X is fiite, we rearrage { x1, x2,..., x } * * * ito{ x, x,..., x } such that 1 2 f ( x ) f( x )... f( x ) ad let * * * 1 2 f( x ) = 0. The * 0 the Choquet itegral of f with respect to µ ca be expressed as (C) [ iai( ) + 1a1( ), ibi( ) + 1b1( )] if 1 0 [0,1] i= 2 i= 2 fd µ =, [ iai( ) + 1b1( ), ibi( ) + 1a1( )] if 1 0 [0,1] i= 2 i= 2 15

16 where ai ( ) ad bi ( ) are the ed poits of iterval * * * [ ai( ), bi( )] = µ ({ xi, xi+ 1,..., x }), ad * * i = f ( xi ) f( xi 1) for i = 1, 2,...,, with f( x ) = 0. This makes the calculatio of the Choquet itegral with respect to * 0 trapezoidal fuzzy-valued siged efficiecy measure rather easy. The formula is (C) [ iai+ 1a1, ici, idi, ibi+ 1b1] if 1 0 i= 2 i= 1 i= 1 i= 2 fd µ =, [ iai+ 1b1, ici, idi, ibi+ 1a1] if 1 0 i= 2 i= 1 i= 1 i= 2 where a, b, ad c, d are the ed poits ad top poits of trapezoidal fuzzy i i i i umber [ a, c, d, b] = ({ x, x,..., x }) ad * * * i i i i µ i i+ 1 * * i = f ( xi ) f( xi 1) for i = 1, 2,...,, respectively. Example 2 Let X = { x1, x2, x3}, µ be a fuzzy-valued siged efficiecy measure o P (X) defied as [0,0,0,0] if A = [1, 1.5, 3.5, 2] if A = { x1} [ 1, 0, 1, 1] if A = { x2} [3, 4.5, 1, 5] if A = { x1, x2} µ ( A) =, [ 2, 5, 1, 3] if A = { x3} [1, 2, 3, 4] if A = { x1, x3} [ 5, 1, 3, 2] if A = { x2, x3} [5,6,3,6] if A = X 16

17 the, * * if x = x1 f = 0.2 if x = x2, 0.5 if x = x3 x = x, x = x ad x * = x. Thus, as show i (6), fdµ is still a trapezoidal 3 2 fuzzy umber, whose ed poits are just the ed poits of the resultig iterval i Example 1, that is, 2.9 ad 1. As for the top poit of (C) fdµ is the Choquet itegral of f with respect to siged efficiecy measure, whose value at ay set i P (X) is just the top poits of the value ( a trapezoidal fuzzy umber) of µ at the same set, that is, (C) fdµ c = ( 0.5) = 2.7, (C) fdµ d = ( 0.5) ( 1) = 1.8, where 0 if A = 1.5 if A = { x1} 0 if A = { x2} 4.5 if A = { x1, x2} µ c( A) =, 5 if A = { x3} 2 if A = { x1, x3} 1 if A = { x2, x3} 6 if A = X 17

18 0 if A = 3.5 if A = { x1} 1 if A = { x2} 1 if A = { x1, x2} µ d ( A) =. 1 if A = { x3} 3 if A = { x1, x3} 3 if A = { x2, x3} 3 if A = X Cosequetly, (C) fdµ = [ 2.9, 2.7, 1.8, 1], A trapezoidal fuzzy umber with ed poits 2.9 ad 1, top poit 2.7 ad 0.8, whose membership fuctio is t if t [ 2.9, 2.7) if t [ 2.7, 1.8] mt () =. 1 t if t ( 1.8, 1] otherwise 5. Properties of the Choquet itegral with respect to fuzzy-valued siged efficiecy measures The choquet itegral of oegative measurable fuctio f with respect to a fuzzy-valued siged efficiecy measure µ has followig basic properties, where we 18

19 assume that all ivolved fuctios ad sets are measurable: (CIP1) (C) fdµ 0; (CIP2) (C) c fdµ = c (C) fdµ for ay oegative costat c. These properties ca be obtaied from the defiitio directly. However, (C) ( f + g) dµ = (C) fdµ + (C) gdµ may ot be true. This ca be see from the followig example. Example 3. Let X = { a, b}, F =P (X), ad [0,0,0,0] if E = µ ( E) =. [1,1,1,1] if otherwise I this case, ay fuctio o X is measurable. Cosiderig two fuctios, ad we have 0 if x = a f( x) = 1 if x = b 0 if x = b gx ( ) =, 1 if x = a (C) fdµ = [0, 0, 0, 0] ad (C) gdµ = [1,1,1,1]. Sice f + g = 1, a costat fuctio o X, we obtai 19

20 (C) ( f + g) dµ = (C) 1d µ = 1 µ ( X) = [1,1,1,1]. Thus, (C) ( f + g) dµ (C) fdµ + (C) gdµ. (CIP3) Whe µ is a oegative fuzzy-valued siged efficiecy measure, (C) fdµ (C) gdµ if f g. (CIP4) (C) ( f + c) dµ = (C) fdµ + c µ ( X ) for ay costat c. 6.Coclusios The paper presets that the Choquet itegral of a real-valued measurable fuctio ca be geeralized for iterval-valued siged efficiecy measures, ad further more for fuzzy-valued siged efficiecy measures. Whe the fuzzy values are restricted to trapezoidal fuzzy umbers, the calculatio formula of such kid of Choquet itegral is clearly preseted. For oegative measurable fuctios, the choquet itegral with respect to a fuzzy-valued siged efficiecy measure preserves some properties of the Choquet itegral with respect to a siged efficiecy measure. 20

21 Refereces [1] L. S. Shapley, A value for -perso games, Aals of Mathematics Studies 28. NJ: Priceto Uiv. Press, 1953, vol. II, [2] Z. Wag, Lecture for otes, Chapter 3 ad Chapter 4. [3] Z. Wag ad H. Guo, A ew geetic algorithm for oliear multiregressios based o geeralized Choquet itegrals, Proc. FUZZ-IEEE2003, [4] Z. Wag ad G. J. Klir, Fuzzy Measure Theory, Pleum, New York, [5] Z. Wag ad G. J. Klir, Geeralized Measure Theory, Spriger, New York, [6] Z. Wag, R. Yag, Ki-Hog Lee, ad Kwog-Sak Leug, The Choquet itegral with respect to fuzzy-valued siged efficiecy measures, WCCI 2008,