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Some quantities represented by the Choquet integral
Fuzzy Sets and Systems 56 (1993) 229-235 North-Holland
229
Some quantities represented by the Choquet integral Toshiaki Murofushi Department of Communications and Systems, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan
Michio Sugeno Department of Systems Science, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 227, Japan Received April 1992 Revised August 1992
Abstract." This paper shows that the Choquet integral can represent some useful quantities such as supremum, infimum, essential supremum, essential infimum, mean, median, a-quantile (100a-percentile), and L-estimators including a-trimmed mean.
Keywords." Measure theory; probability theory and statistics; fuzzy measure; Choquet integral; Sugeno integral.
1. Introduction The purpose of this paper is to point out that the Choquet integral [2, 3, 6, 7, 12] can express several important quantities in probability theory and statistics, that is, the Choquet integral is very powerful to express statistical quantities. For instance, the Choquet integral can express the mean of a random variable; it is well-known that, for a given random variable X on a probability space (t2, ~, P), the Choquet integral of X with respect to P equals the Lebesgue integral of X with respect to P, i.e., the mean of X. The Choquet integral can express much more quantities. In the sequel we shall discuss this issue. Throughout the paper we assume that (t2, ~ ) is a measurable space. A fuzzy measure on (~, ~ ) is a real-valued set function ~ : ~ [0, 1] satisfying (1) It(0) = 0 and/z(I2) = 1, (2) /z(A) <~/x(B) whenever A c B and A, B E ~. The Choquet integral of a measurable function f with respect to a fuzzy measure p~ is defined by (C) f d / z =
foo /z({o~[ f ( t o ) > x } ) d x
+
[/x({oJ [ f ( ( o ) > x } ) - 1]dx. oo
where the integrals of the right side are ordinary ones. The following properties are well-known and can be easily shown.
Proposition 1.1 [2]. Let tz and v be fuzzy measures on ~. (1) For every measurable function f,
(c) f f d . =
x
where the integral of the right side is the Stieltjes integral and F(x) = -/x({oJ I f ( t o ) > x } )
Vxe(-~,
oo).
Correspondence to: Dr. T. Murofushi, Department of Communications and Systems, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan. 0165-0114/93/$06.00 (~) 1993--Elsevier Science Publishers B.V. All rights reserved
T. Murofushi, M. Sugeno / Some quantities represented by the Choquet integral
230
(2) If v ~ IX, i.e., v(A ) <~ix(A) VA e ~, then, for every measurable function f,
(c)f f dv< (c)ff dix. 2. 0-1 fuzzy measures Definition 2.1. A 0-1 fuzzy measure is a fuzzy measure whose range is {0, 1}. Proposition 2.2. Let IX be a 0-1 fuzzy measure. For every measurable function f, ( C ) J( f dix =
sup A:/x(A)=I
inf f(o~). weA
Proof. Let a = SUpA:#(A)=I info~Af(~0). Taking into account Proposition 1.1(1), it is sufficient to prove that Ix({o~[f(o~)>x})=
1 ifxa.
Assume x < a . Then there is a measurable set A such that t z ( A ) = l and x x } , and hence that Ix({w I f ( w ) > x } ) = 1. Now assume x > a . If Ix({w I f ( w ) > x}) = 1, then it follows that x > a/> info, E{o4i(o~)>x}f(w)/> x, and, since this is a contradiction, tz({w ]f(w)>x})=O. [] The Sugeno integral [10] of a measurable function f : 1 2 ~ [0, 1] with respect to t~ is defined by ff°ix=
sup x~[0,1]
[x^M{o~lf(oo)>x})l
where ^ stands for minimum.
Proposition 2.3. If IX is a 0-1 fuzzy measure, then, for every measurable function f :g2--->[0, 1], ( c ) f f d~ = f foix. Proof. As shown in [10], it holds that
f fo/x
= sup {[ inf f(w)] ^ IX(A)}. m E o~
o~em
Since IX values only in {0, 1}, sup{[inff(w)]^ix(A)}= AE,~:
o) e A
sup
inff(o)).
[]
A : / z ( A ) = 1 o) e A
Definition 2.4 [1]. Let F be a non-empty subset of g2. The 0-1 possibility measure focused on F, denoted by [IF, is a set function on ,~ defined by fl HF(A)=~O
ifAAF¢O, ifANF=O.
T. Murofushi, M. Sugeno / S o m e quantities represented by the Choquet integral
231
The 0-1 necessity measure focused on F, denoted by NF, is a set function on ~ defined by {~ NF(A)=
ifFcA, if F C-A.
The following proposition is obvious.
Proposition 2.5. Let F be a non-empty subset of ~2 and I an any non-empty set. (1) IIF and NF are both 0-1 fuzzy measures. (2) II F and NF are dual, i.e., He(A) + NF(A c) = 1 VA • ~. (3) If{A, i • I } c o%and U i d A i • ~, then IIF(UiEIAi) = s u p i ~ l IIF(Ai). (4) If {Ai i E I } c o%and (~i~lAi • ~, then NF((-~iEiAi) infidNF(A,). (5) For every fuzzy measure tz on ~, N , <~I~ <~H~. =
Let F be a non-empty subset of ~2 and f : £ 2 ~ [0, 1] a measurable function. Tsukamoto [11] proved that ffoIIF=SUpf(w) ~oEF
and
ffONF=inff(w). ~oEF
Taking into account Proposition 2.3, the above result is a corollary of the following proposition.
Proposition 2.6. Let F be a non-empty subset of £2 and f a measurable function on 12. (1) ( C ) f f dHF = s u p ~ F f ( W ) . (2) ( C ) f f dNF = info~Ff(w). (3) For every fuzzy measure tz on ~, inf f ( w ) ~< ( C ) ( f d/z ~ sup f(w). ~o c .(2
.]
t o ~ ..(2
Proof. (1) If we write a = sup,o~Ff(t0), then it follows that {0_ F(x)=-HF({o~lf(w)>x})=
ifx>a, 1 ifx
and hence that ( C ) f f dHF = f~_=x dF(x) = a. (2) Similar to (1). (3) Immediately from (1), (2), Proposition 2.5(5), and Proposition 1.1(2).
[]
Note. (1) and (2) in the above theorem for a finite universe £2 are easily derived from Smets' result [9]: E*(f)=
~ AcX
re(A), m a x f ( x ) x~A
and
E,(f)=
~
m(A).minf(x),
A~X
xeA
where E * ( f ) and E , ( f ) are Dempster's upper and lower expected values [5] off, respectively, and m is a basic probability assignment [8]. It is well-known that E* and E , are particular cases of the Choquet integral, and that HF and NF are a plausibility function and a belief function, respectively, with the basic probability assignment m such that re(A) = 1 if A = F and re(A) = 0 if A ¢ F.
3. 0-1 fuzzy measures induced by probability measures In this section we assume (£2, o%, P) is a probability space. The concept of essential supremum is one of the most important concepts in probability and measure theory. The Choquet integral can express this quantity.
T. Murofushi, M. Sugeno / Some quantities represented by the Choquet integral
232
Definition 3.1. Let F be a measurable set for which P ( F ) > 0. The essential 0-1 possibility measure focused on F induced by P, denoted by ess IIF, is defined by /~ ess IIF(A) = _~
ifP(Z AF)>0, if P(A fq F) = O.
The essential 0-1 necessity measure focused on F induced by P, denoted by ess Nr, is defined by fl ess NF(A) = ~0
ifP(F-A)=O, if P(F - A) > O.
If the universe £2 is finite, then ess IIF = IIC and ess NF = No, where G = ~ {A e ~ [ P(A) = P(F) and A c F}. In an infinite case, however, it does not always hold. We give an example. Let £2 be the unit interval [0, 1] in the real line, ~ the family of all the Borel sets in £2, P the Lebesgue measure on ~ , F a measurable set such that P(F) > 0, and G a non-empty subset of £2. Then, for every w e G, HG({o}) = 1 but ess HF({O})= 0 since P ( { o } ) = 0, and hence ess IIF # I I c . (Therefore, Tsukamoto's G0-measure and G=-measure [11] are ess Hn and ess Nn, respectively.) The following proposition is obvious.
Proposition 3.2. Let F be a measurable set for which P(F) > O. (1) ess HF and ess NF are both 0-1 fuzzy measures. (2) ess IIF and ess NF are dual, i.e., ess IIF(A ) + ess NF(A c) = 1 VA ~ ~ . (3) ess [IF (~-Jn=a A , ) = S U p n ess HF(An). (4) ess NF(A~=I An) = infn ess NF(An). Let F be a measurable set for which P ( F ) > 0. The essential supremum of f on F, denoted by ess sup~,~Ff(OJ), is defined by ess s u p f ( w ) = inf{xlP({o~lf(w) > x} fq F ) = 0}. eo~V
The essential infimum of f on F ess info,~Ff(W) = --ess supo, ~F (--f(w)).
is
defined
as
the
dual
of
the
essential
supremum;
Proposition 3.3. Let F be a measurable set for which P(F) > O. For every random variable X on £2, ( c ) f X d(eSSHF)=esssupX(w)o~F
and
( c ) f X d(essNF)=essinfX(w).o~EF
Proof. Essentially same as the proof of Proposition 2.6.
[]
The Choquet integral can express the a-quantile, which is a basic quantity in statistics. For a random variable X and a real number ore(0, 1), a real number x such that P({oJ]X(oJ)
Definition 3.4. Let o~ be a real number such that 0 < a < 1. The upper-a-quantile measure induced by P, denoted by u~, is defined by {1
u~(A) = . v
i f P ( Z ~)<-a, if P(A c) > a.
The lower-a-quantile measure induced by P, denoted by l~, is defined by 1 if P(AC) < a, l~(A)= 0 i f P ( A c)>la. We can call ul/2 and lu2 the upper-median measure and the lower-median measure, respectively.
T. Murofushi, M. Sugeno / Some quantities represented by the Choquet integral
233
The following proposition is obvious.
Proposition 3.5. (1) u~ and I~, are both 0-1 f u z z y measures. (2) l ~ < ~ u ~ < ~ l ~ < ~ u ~ i f O < a < [ 3 < l . (3) u~, and 11-~ are dual, i.e., u,~(A) + ll_~(A ¢) = 1. (4) u, is continuous from the below, i.e., An ~ A ~ u ~ , ( A n ) ~ u~(A), and l~, is continuous from the above, i.e., A~ ~ A ~ I~(A~) ~ I~(A ). It seems inappropriate that u~ and l~ are called the upper-a-quantile measure and the Iower-aquantile measure, respectively, since u,~ is not an upper measure and l~ is not a lower measure (see the note at the end of this section). However, the adjectives 'upper' and 'lower' qualify not 'measure' but 'a-quantile'. The following proposition shows the reason why we call u~ and l~ the upper-a-quantile measure and the lower-a-quantile measure, respectively.
Proposition 3.6. Let X be a random variable on I2 and a ~ (0, 1). I f u=(C)f
Xdu,,
and
l = ( C ) f Xdl ,
then u and l are both a-quantiles o f X. Moreover, fl is an a-quantile iff l <~fl <~u; therefore if I* is a f u z z y measure on o~ for which 1~ <~t* <~u~, then ( C ) f X dr* is an a-quantile.
Proof. If we write a* = sup {x I P({to [ X(to) x } ) = _
if P({to [ X(to)~ a, 1 ifP({to IX(to)<~x})<-a,
it follows that F(x) = 0 for x > a* and F(x) = - 1 for x < a*, and hence that
u = (C) f X duo =
x dF(x) = *.
Similarly it follows that l = inf{x I P({to IX(to)
[]
Note. I f / z and v are dual fuzzy measures and if/z(A) ~< v ( A ) VA ~ ,~, then/~(A) and v(A) are called the lower uncertain value and the upper uncertain value [4], or the lower measure and the upper measure, respectively. It follows from Definition 3.4 and Proposition 3.5(3) that u, is a lower measure if a < ½ and l~ is an upper measure if a > ½. The continuity from the above and the continuity from the below are sometimes called the lower-(semi) continuity or the upper-(semi)continuity, respectively. Delgado and Moral [4] called a lower-continuous fuzzy measure and an upper-continuous fuzzy measure a lower fuzzy measure and an upper fuzzy measure, respectively. Proposition 3.5(4) implies that an upper-a-quantile measure is not an upper fuzzy measure but a lower fuzzy measure and that a lower-a-quantile measure is not a lower fuzzy measure but an upper fuzzy measure.
4. L-estimators The Choquet integral can represent the L-estimator, which is also a basic concept in statistics. In this section we assume that 12 = {to1, to2. . . . , ton} and ~ is the power set of ~2. An L-estimator is a statistic
T. Murofushi, M. Sugeno / Some quantities represented by the Choquet integral
234
T that is a linear c o m b i n a t i o n of o r d e r statistics:
T = ~ ciX(i), i=l
where X(1) ~< X(2) ~< • • • ~< X(n) and ~ni=l c i = 1. T is the arithmetic m e a n when c i = 1/n for all i. T is the m e d i a n when ci =
1 0
if i = ½(n + 1), otherwise,
Ci ~ - - -
½0 i f i = ½ n o r l n + l , otherwise,
if n is odd,
and if n is even.
T is the (k/n)-trimmed m e a n when
{~/(n-2k) ci =
if k < i < ~ n - k , otherwise.
M o r e o v e r , T = X(m), the m-th value f r o m the minimum, w h e n
ci =
{10 i f i = m , otherwise.
Now, for any function f on £2, we define a functional t by
t(f) = 2 cifii, i=1
where ~ni=lCi = 1 and fl ~
(fi,f2, . . . , fn) the functional t p r o o f is easy, of the C h o q u e t
is a p e r m u t a t i o n of (f(Wx),f(w2), . . . , f ( w n ) ) such that and L-estimator functional. W e can easily prove the following but we can consider this proposition a corollary of the integral [3, T h e o r e m 2.1].
Proposition 4.1. Let t be an L-estimator functional with coefficients ci. I f ci >! 0 for all i, then there is a f u z z y measure ix on ~ such that, for every function f on £2, t(f) = (c)ff
dix,
and tz is given as n-lAI
IX(A)= 1- ~ ci, i--1
where IAI denotes the cardinality of A. The Choquet integral with respect to a f u z z y measure IX is an L-estimator functional iff Ix(A) depends only on Iml, i.e., ix(A) = IX(B) whenever IA[ = IBI. T h e above proposition implies that the C h o q u e t integral can represent (k/n)-trimmed mean, and m - t h value fm of {f(to 0, f(t02) . . . . . f(ton)}.
the mean,
median,
Acknowledgments T h e authors would like to thank a reviewer for the suggestions. Especially, Proposition 3.5(3) was suggested by the reviewer.
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235
References [1] G. Bannon, Distinction between several subsets of fuzzy measures, Fuzzy Sets and Systems 5 (1981) 291-305. [2] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131-295. [3] L.M. de Campos, M.T. Lamata and S. Moral, A unified approach to define fuzzy integrals, Fuzzy Sets and Systems 39 (1991) 75-90. [4] M. Delgado and S. Moral, Upper and lower fuzzy measures, Fuzzy Sets and Systems 33 (1989) 191-200. [5] A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist. 38 (1967) 325-339. [6] T. Murofushi and M. Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems 29 (1989) 201-227. [7] T. Murofushi and M. Sugeno, A theory of fuzzy measures: Representations, the Choquet integral, and null sets, J. Math. Anal Appl. 159 (1991) 532-549. [8] G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, NJ 1976). [9] Ph. Smets, The degree of belief in a fuzzy event, Inform. Sci. 25 (1981) 1-19. [10] M. Sugeno, Theory of fuzzy integrals and its applications, Doctoral Thesis, Tokyo Institute of Technology (1974). [11] Y. Tsukamoto, A measure theoretic approach to evaluation of fuzzy set defined on probability space, Fuzzy Math. 2 (3) (1982) 89-98. [12] P. Wakker, A behavioral foundation for fuzzy measures, Fuzzy Sets and Systems 37 (1990) 327-350.
229
Some quantities represented by the Choquet integral Toshiaki Murofushi Department of Communications and Systems, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan
Michio Sugeno Department of Systems Science, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 227, Japan Received April 1992 Revised August 1992
Abstract." This paper shows that the Choquet integral can represent some useful quantities such as supremum, infimum, essential supremum, essential infimum, mean, median, a-quantile (100a-percentile), and L-estimators including a-trimmed mean.
Keywords." Measure theory; probability theory and statistics; fuzzy measure; Choquet integral; Sugeno integral.
1. Introduction The purpose of this paper is to point out that the Choquet integral [2, 3, 6, 7, 12] can express several important quantities in probability theory and statistics, that is, the Choquet integral is very powerful to express statistical quantities. For instance, the Choquet integral can express the mean of a random variable; it is well-known that, for a given random variable X on a probability space (t2, ~, P), the Choquet integral of X with respect to P equals the Lebesgue integral of X with respect to P, i.e., the mean of X. The Choquet integral can express much more quantities. In the sequel we shall discuss this issue. Throughout the paper we assume that (t2, ~ ) is a measurable space. A fuzzy measure on (~, ~ ) is a real-valued set function ~ : ~ [0, 1] satisfying (1) It(0) = 0 and/z(I2) = 1, (2) /z(A) <~/x(B) whenever A c B and A, B E ~. The Choquet integral of a measurable function f with respect to a fuzzy measure p~ is defined by (C) f d / z =
foo /z({o~[ f ( t o ) > x } ) d x
+
[/x({oJ [ f ( ( o ) > x } ) - 1]dx. oo
where the integrals of the right side are ordinary ones. The following properties are well-known and can be easily shown.
Proposition 1.1 [2]. Let tz and v be fuzzy measures on ~. (1) For every measurable function f,
(c) f f d . =
x
where the integral of the right side is the Stieltjes integral and F(x) = -/x({oJ I f ( t o ) > x } )
Vxe(-~,
oo).
Correspondence to: Dr. T. Murofushi, Department of Communications and Systems, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan. 0165-0114/93/$06.00 (~) 1993--Elsevier Science Publishers B.V. All rights reserved
T. Murofushi, M. Sugeno / Some quantities represented by the Choquet integral
230
(2) If v ~ IX, i.e., v(A ) <~ix(A) VA e ~, then, for every measurable function f,
(c)f f dv< (c)ff dix. 2. 0-1 fuzzy measures Definition 2.1. A 0-1 fuzzy measure is a fuzzy measure whose range is {0, 1}. Proposition 2.2. Let IX be a 0-1 fuzzy measure. For every measurable function f, ( C ) J( f dix =
sup A:/x(A)=I
inf f(o~). weA
Proof. Let a = SUpA:#(A)=I info~Af(~0). Taking into account Proposition 1.1(1), it is sufficient to prove that Ix({o~[f(o~)>x})=
1 ifx
Assume x < a . Then there is a measurable set A such that t z ( A ) = l and x
sup x~[0,1]
[x^M{o~lf(oo)>x})l
where ^ stands for minimum.
Proposition 2.3. If IX is a 0-1 fuzzy measure, then, for every measurable function f :g2--->[0, 1], ( c ) f f d~ = f foix. Proof. As shown in [10], it holds that
f fo/x
= sup {[ inf f(w)] ^ IX(A)}. m E o~
o~em
Since IX values only in {0, 1}, sup{[inff(w)]^ix(A)}= AE,~:
o) e A
sup
inff(o)).
[]
A : / z ( A ) = 1 o) e A
Definition 2.4 [1]. Let F be a non-empty subset of g2. The 0-1 possibility measure focused on F, denoted by [IF, is a set function on ,~ defined by fl HF(A)=~O
ifAAF¢O, ifANF=O.
T. Murofushi, M. Sugeno / S o m e quantities represented by the Choquet integral
231
The 0-1 necessity measure focused on F, denoted by NF, is a set function on ~ defined by {~ NF(A)=
ifFcA, if F C-A.
The following proposition is obvious.
Proposition 2.5. Let F be a non-empty subset of ~2 and I an any non-empty set. (1) IIF and NF are both 0-1 fuzzy measures. (2) II F and NF are dual, i.e., He(A) + NF(A c) = 1 VA • ~. (3) If{A, i • I } c o%and U i d A i • ~, then IIF(UiEIAi) = s u p i ~ l IIF(Ai). (4) If {Ai i E I } c o%and (~i~lAi • ~, then NF((-~iEiAi) infidNF(A,). (5) For every fuzzy measure tz on ~, N , <~I~ <~H~. =
Let F be a non-empty subset of ~2 and f : £ 2 ~ [0, 1] a measurable function. Tsukamoto [11] proved that ffoIIF=SUpf(w) ~oEF
and
ffONF=inff(w). ~oEF
Taking into account Proposition 2.3, the above result is a corollary of the following proposition.
Proposition 2.6. Let F be a non-empty subset of £2 and f a measurable function on 12. (1) ( C ) f f dHF = s u p ~ F f ( W ) . (2) ( C ) f f dNF = info~Ff(w). (3) For every fuzzy measure tz on ~, inf f ( w ) ~< ( C ) ( f d/z ~ sup f(w). ~o c .(2
.]
t o ~ ..(2
Proof. (1) If we write a = sup,o~Ff(t0), then it follows that {0_ F(x)=-HF({o~lf(w)>x})=
ifx>a, 1 ifx
and hence that ( C ) f f dHF = f~_=x dF(x) = a. (2) Similar to (1). (3) Immediately from (1), (2), Proposition 2.5(5), and Proposition 1.1(2).
[]
Note. (1) and (2) in the above theorem for a finite universe £2 are easily derived from Smets' result [9]: E*(f)=
~ AcX
re(A), m a x f ( x ) x~A
and
E,(f)=
~
m(A).minf(x),
A~X
xeA
where E * ( f ) and E , ( f ) are Dempster's upper and lower expected values [5] off, respectively, and m is a basic probability assignment [8]. It is well-known that E* and E , are particular cases of the Choquet integral, and that HF and NF are a plausibility function and a belief function, respectively, with the basic probability assignment m such that re(A) = 1 if A = F and re(A) = 0 if A ¢ F.
3. 0-1 fuzzy measures induced by probability measures In this section we assume (£2, o%, P) is a probability space. The concept of essential supremum is one of the most important concepts in probability and measure theory. The Choquet integral can express this quantity.
T. Murofushi, M. Sugeno / Some quantities represented by the Choquet integral
232
Definition 3.1. Let F be a measurable set for which P ( F ) > 0. The essential 0-1 possibility measure focused on F induced by P, denoted by ess IIF, is defined by /~ ess IIF(A) = _~
ifP(Z AF)>0, if P(A fq F) = O.
The essential 0-1 necessity measure focused on F induced by P, denoted by ess Nr, is defined by fl ess NF(A) = ~0
ifP(F-A)=O, if P(F - A) > O.
If the universe £2 is finite, then ess IIF = IIC and ess NF = No, where G = ~ {A e ~ [ P(A) = P(F) and A c F}. In an infinite case, however, it does not always hold. We give an example. Let £2 be the unit interval [0, 1] in the real line, ~ the family of all the Borel sets in £2, P the Lebesgue measure on ~ , F a measurable set such that P(F) > 0, and G a non-empty subset of £2. Then, for every w e G, HG({o}) = 1 but ess HF({O})= 0 since P ( { o } ) = 0, and hence ess IIF # I I c . (Therefore, Tsukamoto's G0-measure and G=-measure [11] are ess Hn and ess Nn, respectively.) The following proposition is obvious.
Proposition 3.2. Let F be a measurable set for which P(F) > O. (1) ess HF and ess NF are both 0-1 fuzzy measures. (2) ess IIF and ess NF are dual, i.e., ess IIF(A ) + ess NF(A c) = 1 VA ~ ~ . (3) ess [IF (~-Jn=a A , ) = S U p n ess HF(An). (4) ess NF(A~=I An) = infn ess NF(An). Let F be a measurable set for which P ( F ) > 0. The essential supremum of f on F, denoted by ess sup~,~Ff(OJ), is defined by ess s u p f ( w ) = inf{xlP({o~lf(w) > x} fq F ) = 0}. eo~V
The essential infimum of f on F ess info,~Ff(W) = --ess supo, ~F (--f(w)).
is
defined
as
the
dual
of
the
essential
supremum;
Proposition 3.3. Let F be a measurable set for which P(F) > O. For every random variable X on £2, ( c ) f X d(eSSHF)=esssupX(w)o~F
and
( c ) f X d(essNF)=essinfX(w).o~EF
Proof. Essentially same as the proof of Proposition 2.6.
[]
The Choquet integral can express the a-quantile, which is a basic quantity in statistics. For a random variable X and a real number ore(0, 1), a real number x such that P({oJ]X(oJ)
Definition 3.4. Let o~ be a real number such that 0 < a < 1. The upper-a-quantile measure induced by P, denoted by u~, is defined by {1
u~(A) = . v
i f P ( Z ~)<-a, if P(A c) > a.
The lower-a-quantile measure induced by P, denoted by l~, is defined by 1 if P(AC) < a, l~(A)= 0 i f P ( A c)>la. We can call ul/2 and lu2 the upper-median measure and the lower-median measure, respectively.
T. Murofushi, M. Sugeno / Some quantities represented by the Choquet integral
233
The following proposition is obvious.
Proposition 3.5. (1) u~ and I~, are both 0-1 f u z z y measures. (2) l ~ < ~ u ~ < ~ l ~ < ~ u ~ i f O < a < [ 3 < l . (3) u~, and 11-~ are dual, i.e., u,~(A) + ll_~(A ¢) = 1. (4) u, is continuous from the below, i.e., An ~ A ~ u ~ , ( A n ) ~ u~(A), and l~, is continuous from the above, i.e., A~ ~ A ~ I~(A~) ~ I~(A ). It seems inappropriate that u~ and l~ are called the upper-a-quantile measure and the Iower-aquantile measure, respectively, since u,~ is not an upper measure and l~ is not a lower measure (see the note at the end of this section). However, the adjectives 'upper' and 'lower' qualify not 'measure' but 'a-quantile'. The following proposition shows the reason why we call u~ and l~ the upper-a-quantile measure and the lower-a-quantile measure, respectively.
Proposition 3.6. Let X be a random variable on I2 and a ~ (0, 1). I f u=(C)f
Xdu,,
and
l = ( C ) f Xdl ,
then u and l are both a-quantiles o f X. Moreover, fl is an a-quantile iff l <~fl <~u; therefore if I* is a f u z z y measure on o~ for which 1~ <~t* <~u~, then ( C ) f X dr* is an a-quantile.
Proof. If we write a* = sup {x I P({to [ X(to)
if P({to [ X(to)~
it follows that F(x) = 0 for x > a* and F(x) = - 1 for x < a*, and hence that
u = (C) f X duo =
x dF(x) = *.
Similarly it follows that l = inf{x I P({to IX(to)
[]
Note. I f / z and v are dual fuzzy measures and if/z(A) ~< v ( A ) VA ~ ,~, then/~(A) and v(A) are called the lower uncertain value and the upper uncertain value [4], or the lower measure and the upper measure, respectively. It follows from Definition 3.4 and Proposition 3.5(3) that u, is a lower measure if a < ½ and l~ is an upper measure if a > ½. The continuity from the above and the continuity from the below are sometimes called the lower-(semi) continuity or the upper-(semi)continuity, respectively. Delgado and Moral [4] called a lower-continuous fuzzy measure and an upper-continuous fuzzy measure a lower fuzzy measure and an upper fuzzy measure, respectively. Proposition 3.5(4) implies that an upper-a-quantile measure is not an upper fuzzy measure but a lower fuzzy measure and that a lower-a-quantile measure is not a lower fuzzy measure but an upper fuzzy measure.
4. L-estimators The Choquet integral can represent the L-estimator, which is also a basic concept in statistics. In this section we assume that 12 = {to1, to2. . . . , ton} and ~ is the power set of ~2. An L-estimator is a statistic
T. Murofushi, M. Sugeno / Some quantities represented by the Choquet integral
234
T that is a linear c o m b i n a t i o n of o r d e r statistics:
T = ~ ciX(i), i=l
where X(1) ~< X(2) ~< • • • ~< X(n) and ~ni=l c i = 1. T is the arithmetic m e a n when c i = 1/n for all i. T is the m e d i a n when ci =
1 0
if i = ½(n + 1), otherwise,
Ci ~ - - -
½0 i f i = ½ n o r l n + l , otherwise,
if n is odd,
and if n is even.
T is the (k/n)-trimmed m e a n when
{~/(n-2k) ci =
if k < i < ~ n - k , otherwise.
M o r e o v e r , T = X(m), the m-th value f r o m the minimum, w h e n
ci =
{10 i f i = m , otherwise.
Now, for any function f on £2, we define a functional t by
t(f) = 2 cifii, i=1
where ~ni=lCi = 1 and fl ~
(fi,f2, . . . , fn) the functional t p r o o f is easy, of the C h o q u e t
is a p e r m u t a t i o n of (f(Wx),f(w2), . . . , f ( w n ) ) such that and L-estimator functional. W e can easily prove the following but we can consider this proposition a corollary of the integral [3, T h e o r e m 2.1].
Proposition 4.1. Let t be an L-estimator functional with coefficients ci. I f ci >! 0 for all i, then there is a f u z z y measure ix on ~ such that, for every function f on £2, t(f) = (c)ff
dix,
and tz is given as n-lAI
IX(A)= 1- ~ ci, i--1
where IAI denotes the cardinality of A. The Choquet integral with respect to a f u z z y measure IX is an L-estimator functional iff Ix(A) depends only on Iml, i.e., ix(A) = IX(B) whenever IA[ = IBI. T h e above proposition implies that the C h o q u e t integral can represent (k/n)-trimmed mean, and m - t h value fm of {f(to 0, f(t02) . . . . . f(ton)}.
the mean,
median,
Acknowledgments T h e authors would like to thank a reviewer for the suggestions. Especially, Proposition 3.5(3) was suggested by the reviewer.
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235
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