Uncertainty Carried by Fuzzy Measures in Aggregation
Zhenyuan Wang
Kwong-Sak Leung
Department of Mathematics,
Department of Computer Science and Engineering,
University of Nebraska at Omaha, Omaha,
Chinese University of Hong Kong, Shatin,
NE 68182, USA
NT, Hong Kong
[email protected]
[email protected]
Abstract
information from these information sources to be one real number that can be easily used for
Two extreme nonlinear integrals, the
decision making. However, the Choquet integral
upper integral and the lower integral,
is
are discussed. They have most common
coordination manner that binds all attributes as
properties as the Choquet integral has.
much as possible. In the literature, people have
Based on these two types of integrals, a
hardly considered an objective reason to support
degree of uncertainty carried by the
the usage of the Choquet integral. In fact, the
fuzzy measures used in the integral is
pair of the classical linear Lebesgue-like integral
introduced. It describes the maximal
[1] and the Choquet integral [2, 5] are two
possible
variation
extremes in regard to the coordination among
received
information
coordinations
in
aggregating
a
model
describing
a
special
various
attributes. Introducing more types of integrals to
information
fit the various situations in real problems is
with
among
only
sources based on given fuzzy measure.
necessary. Recently, the upper integral and the lower integral are also proposed for information
Keywords: Fuzzy measures, nonlinear integrals,
fusion [7, 8]. They are another pair of two
information fusion, uncertainty.
extremes of nonlinear integrals in regard to the aggregation amount. All of these nonlinear integrals are generalizations of the linear Lebesgue-like integral, that is, when the fuzzy
1. Introduction
measure is σ-additive, they coincide with the In information fusion, each information source
Lebesgue-like integral. Thoroughly discussing
is regarded as an attribute and a fuzzy measure
the properties and the specialties of various
[3, 4, 6] can be used to describe the interaction
types of integrals will benefit developing their
among the attributes towards the fusion [2, 6].
applications.
Based on the given fuzzy measures, people usually adopt the Choquet integral as the
Now there is a new problem that people face: in
aggregation
given real information fusion problem, once the
tool
to
fuse
the
received
fuzzy measure is determined, what type of integrals should be used as the aggregation tool.
2. The upper integral and the lower integral
When the interaction among the attributes can be ignored, a classical additive measure and the
Unless another special indication is given, let (X,
Lebesgue-like
weighted
F, μ) be a generalized fuzzy measure space.
average) may be adopted. In this case, there is
That is, X is a nonempty set, F is a σ-algebra of
no uncertainty in the resulting fusion amount.
subsets of X, and μ : F → [0, ∞) is a set
However, when the interaction cannot be
function with μ(∅) = 0 called a generalized
ignored, a fuzzy measure should be adopted to
fuzzy measure. A generalized fuzzy measure is
describe the interaction. In this case, besides
called a fuzzy measure if it is monotonic. We
determining the fuzzy measure in some way, a
assume that the generalized fuzzy measures
proper nonlinear integral should be chosen to
considered in this paper are always nontrivial,
replace the Lebesgue-like integral. In most
that is, there exists some set A ∈ F such that
information fusion problems, it is difficult to
μ ( A) > 0 .
integral
(i.e.,
the
know which type of integrals is suitable. Different types of integrals are used to describe
Definition 1 Given a measurable function f: X
different coordination manners and will result
→ [0, ∞) and a set A ∈ F, the upper integral of
different fusion amounts. This means that there
f with respect to
is some uncertainty carried by the fuzzy
( U) ∫A f dμ , is defined as
measure when the coordination manner is unknow. Fortunately, we can use the pair of two
uncertainty carried by the fuzzy measure.
on A, in symbol
( U) ∫A f dμ = lim U ε , ε →0+
extreme nonlinear integrals, the upper integral and the lower integral, to calculate such an
μ
where U ε = sup{
∞
∑ λ j ⋅ μ (E j ) j =1
∞
f ≥ ∑ λ j ⋅ χE j ≥ f − ε , j =1
E j ∈F ∩ A, λ j ≥ 0, j = 1,2,L
The paper is organized as follows. After the
}
introduction, the definitions of the upper integral
for ε > 0 , in which χ is the symbol of the
and the lower integral are given in section 2.
characteristic
The relation among some nonlinear integrals is
F ∩ A = {B ∩ A | B ∈F } ; similarly, the lower
discussed in section3. As one of the main results
integral of f with respect to μ on A, in symbol
of this paper, section 4 presents some properties
(L) ∫A f dμ , is defined as
of the upper integral and the lower integral. In section 5, a numerical measurement of the
conclusions and comments are listed in section 6.
and
(L) ∫A f dμ = lim Lε , ε →0+
uncertainty for a fuzzy measure used in information fusion is introduced. Finally, some
function
where Lε = inf {
∞
∑ λ j ⋅ μ (E j ) j =1
∞
f ≤ ∑ λ j ⋅ χEj ≤ f + ε , j =1
E j ∈ F ∩ A, λ j ≥ 0, j = 1,2, L
}
for ε > 0 .
find the maximum and the minimum in above definitions by hand.
In the above definition, functions having a ∞
form as
∑ λ j ⋅ χE
j
j =i
, where
E j ∈F
and
Example 1 Three works x1 , x2 , and x3
manufacture a certain type of toys. Their
λ j ≥ 0 for j = 1, 2, L , are called elementary
individual as well as joint efficiencies can be
functions.
expressed by a generalized fuzzy measure μ :
There
the
requirement
of
the
measurability of function f is necessary. It value of μ
guarantees the existence of some elementary
Set
functions (but not simple functions since f may
{x1}
5
not be upper bounded!) between f and f + ε . In
{x2}
6
case we allow the given function to be not
{x1, x2}
14
measurable, we may use simple functions to
{x3}
7
give relatively looser concepts of widen-upper
{x1, x3}
13
integral and widen-lower integral as follows.
{x2, x3}
9
{x1, x2, x3}
17
Definition 2 Let f be a nonnegative function on
X and set A ∈ F. The widen-upper integral,
The numbers of their working days in a
denoted by ( W ) ∫A f dμ , is defined as
specified week is a function f defined on X:
k
k
j =1
j =1
⎧6 ⎪ f ( x) = ⎨3 ⎪4 ⎩
( W ) ∫A f dμ = sup{∑ λ j ⋅ μ ( E j ) f ≥ ∑ λ j ⋅ χ E j , k ≥ 0, E j ∈ F ∩ A, λ j ≥ 0, j = 1,2, L , k};
if x = x3
( U ) ∫ f dμ = ( W ) ∫ f dμ = 88
to μ , denoted by , is defined as N →∞
if x = x2 .
From Definitions 1 and 2 directly, we have
while the widen-lower integral of f with respect
(W) ∫A f dμ = lim (W) ∫A f N dμ
if x = x1
and (L) ∫ f dμ = ( W ) ∫ f dμ = 64 .
where function f N = min( N , f ) and k
k
This means that these three works, in any
j =1
j =1
cooperative manner, can manufacture at most 88
( W ) ∫A f N dμ = inf{∑ λ j ⋅ μ ( E j ) f N ≤ ∑ λ j ⋅ χ E j , k ≥ 0, E j ∈ F ∩ A, λ j ≥ 0, j = 1,2,..., k}.
but not less than 64 toys in this week.
When f is bounded, we may use f to replace f N
A general method for calculating the value of
in above definition directly. Similar to the
the upper integral and the lower integral (as well
Lebesgue-like integral and the Choquet integral,
as the widen-upper integral and the widen-lower
we will omit the subscript A in the symbol of the
integral) on finite set has been given in [7, 8].
integral when A = X. In case X contains only a few attributes, such as 2 or 3, it is not difficult to
3. Relation among integrals
(W) ∫A f dμ ≤ (L) ∫A f dμ .
When function f is measurable, we may compare
In comparison with the Choquet integral, we have
these integrals defined in the last section. From
the following relation.
this section, we will omit all proofs for the theorems.
Theorem 3 For any set A∈F and any given measurable function f: X → [0, ∞) ,
Theorem 1 If f: X → [0, ∞) is a measurable
(L) ∫A f dμ ≤ (C) ∫A f dμ ≤ ( U) ∫A f dμ .
function on (X, F ) and A ∈ F , then
( W ) ∫A f dμ = ( U) ∫A f dμ .
Example 3 Recall Example 1, we have obtained
There is no similar result for the lower integral and the widen-lower integral. We can see this from the following counterexample.
( U) ∫ f dμ =88 , and
(L) ∫ f dμ =64 . The
relative Choquet integral is (C) ∫ f dμ =74 . This verifies the conclusion showing in Theorem 3.
Example 2 Let X = {a, b} , F = P (X), function
f = χ{a} , and μ ( A) = A (mod 2) for ∀A ∈ F , where A is the cardinality of A. For ε ≥ 1 , we
Theorem 4 If f: X → [0, ∞) is a measurable function and μ is a fuzzy measure on measurable space (X, F ), then
have Lε = 0 , with k = 1 , λ1 = 1 and E1 = X
( W ) ∫A f dμ = (L) ∫A f dμ .
reaching the infimum; while when 0 < ε < 1 , we have Lε = 1 with k = 1 , λ1 = 1 , and E1 = {a} reaching the infimum. So, we have (L) ∫ f dμ = 1 .
4. Properties of the upper integral and the lower integral
However, in this example, (W) ∫ f dμ = 0 with
k = 1 , λ1 = 1 (or larger), and E1 = X reaching
Generally, neither the upper integral nor the lower
the
integral is linear. In fact, we may have
infimum.
This
shows
that
(W) ∫ f dμ = (L) ∫ f dμ may not be true, though function f is measurable.
However, we still have the following inequality in general.
( U) ∫ ( f + g ) dμ ≠ ( U) ∫ f dμ + ( U) ∫ g dμ and
(L) ∫ ( f + g ) dμ ≠ ( L) ∫ f dμ + (L) ∫ g dμ for some fuzzy measure μ and nonnegative measurable functions f and g. Similar to the
Theorem 2 If f: X → [0, ∞) is a measurable
Choquet integral, the nonlinearity of the upper
function on (X, F ), then
integral and the lower integral comes from the
and
nonadditivity of the fuzzy measure. Example 4 Let X = {x1 , x2 , x3} and F = P(X). Fuzzy measure
μ is defined as
(L) ∫ ( f + g ) dμ = 1 ⋅ μ ({x1 , x2 , x3}) = 1 × 5 = 5 . So, we have
(L) ∫ ( f + g ) dμ < (L) ∫ f dμ + (L) ∫ g dμ .
value of μ
Set {x1}
3
{x2}
3
The above example suggests us to find two
{x1, x2}
5
inequalities as a property of the upper integral and
{x3}
1
the lower integral.
{x1, x3}
5
{x2, x3}
5
Theorem 5 Let f and g be nonnegative measurable
{x1, x2, x3}
5
functions on (X, F). Then,
( U) ∫ ( f + g ) dμ ≥ ( U) ∫ f dμ +( U) ∫ g dμ
Taking functions
⎧1 ⎪ f ( x) = ⎨1 ⎪0 ⎩
if x = x1
and
if x = x2
(L) ∫ ( f + g ) dμ ≤ (L) ∫ f dμ + (L) ∫ g dμ .
if x = x3
and
⎧0 ⎪ g ( x) = ⎨0 ⎪1 ⎩
if x = x1
The Choquet integral has a property
if x = x2
(C) ∫ 1 dμ = μ ( X ) .
if x = x3 .
Unlike the Choquet integral, the upper integral
Then,
( U ) ∫ f dμ = 1 ⋅ μ ( x1 ) + 1 ⋅ μ ( x2 ) = 1 × 3 + 1× 3 = 6 , ( U ) ∫ g dμ = 1 ⋅ μ ( x3 ) = 1 × 1 = 1 ,
and the lower integral may violate the similar equality. Example 5 Let X = {x1 , x2 } and F = P(X). Set function
and
( U ) ∫ ( f + g ) dμ = 1 ⋅ μ ( x1 ) + 1 ⋅ μ ({x2 , x3 }) = 1× 3 + 1× 5 = 8
( U) ∫ ( f + g ) dμ > ( U) ∫ f dμ +( U) ∫ g dμ .
(L) ∫ f dμ = 1 ⋅ μ ({x1 , x2 }) = 1 × 5 = 5 , (L) ∫ g dμ = 1 ⋅ μ ( x3 ) = 1 × 1 = 1 ,
⎧0 if A = ∅ . ⎩1 otherwise
μ ( A) = ⎨
.
So, in this example, we have
Similarly,
μ is defined as
It is a fuzzy measure. We have
( U ) ∫ 1 dμ = 1 ⋅ μ ({x1}) + 1 ⋅ μ ({x2 }) = 1 × 1 + 1 × 1 = 2 ≠ μ(X )
.
However, we still have the following inequalities. Theorem 6 For any generalized fuzzy measure μ,
0 ≤ (L) ∫A1 dμ ≤ μ ( A) ≤ ( U) ∫A1 dμ .
let A and B be measurable sets. If f ≤ g on A, then
(1)
Moreover, if X = {x1 , x2 , L, xn } and μ is a
(L) ∫A f dμ ≤ (L) ∫A g dμ .
fuzzy measure, then
( U) ∫A1 dμ ≤ n ⋅ μ ( X ) .
(2)
If A ⊂ B , then
(L) ∫A f dμ ≤ (L) ∫B f dμ .
Beyond the above inequalities, the upper integral and the lower integral possess most common
Finally, showing in the following theorem, the
properties that the Lebesgue-like integral and the
upper
Choquet integral have.
Lebesgue-like integral has.
Theorem 7 Let f and g be nonnegative measurable
Theorem 9 Let f be a nonnegative measurable
functions on generalized fuzzy
function on fuzzy measure space (X, F, μ ). If
measure space
(X, F, μ), A and B be measurable sets, and a be a nonnegative real constant.
integral
Conversely,
(L) ∫A f dμ = (L) ∫ f ⋅ χ A dμ .
continuous
property
that
the
( U) ∫A f dμ = 0
if
from
μ
and
below,
is then
μ ({x | f ( x) > 0} ∩ A) = 0 .
If f ≤ g on A, then
( U) ∫A f dμ ≤ ( U) ∫A g dμ . (3)
a
μ ({x | f ( x) > 0} ∩ A) = 0 , then ( U) ∫A f dμ = 0 .
(1) ( U) ∫A f dμ = ( U) ∫ f ⋅ χ A dμ and
(2)
has
If A ⊂ B , then
( U) ∫A f dμ ≤ ( U) ∫B f dμ . (4) ( U) ∫A af dμ = a ⋅ ( U) ∫A f dμ and
(L) ∫A af dμ = a ⋅ (L) ∫A f dμ .
The condition of the lower continuity of
second conclusion of the above theorem is essential. We have the following counterexample for that conclusion if the condition of the continuity from below is violated. Example 6 Let X = (0, 1], F be the class of all Borel sets in X, and fuzzy measure
μ on F be
defined as
⎧1 if A = X ∀A ∈ F. ⎩0 otherwise
μ ( A) = ⎨
Note that in (2) and (3) of Theorem 7, there is no conclusions on the lower integral. Indeed, to have
Fuzzy measure
similar properties for lower integral, we need more
Taking
condition showing in the next theorem.
μ (E) = 0
μ is not continuous from below.
f ( x) = x for
for
every
x∈ X , E∈ F
λ ⋅ χ E ≤ f with λ > 0 . So, Theorem 8 Let f and g be nonnegative measurable functions on fuzzy measure space (X, F, μ), and
μ in the
( U) ∫ f dμ = ( W ) ∫ f dμ = 0.
we
have
satisfying
is defined by
However,
μ ({x | f ( x) > 0}) = 1 ≠ 0 .
γμ =
( U ) ∫ 1 dμ −(L) ∫ 1 dμ
μ( X )
.
5. Uncertainty carried by fuzzy measures
It is evident that when μ is a classical measure,
To simplify our discussion, we restrict the
the upper integral coincides with the lower integral and, therefore, γ μ = 0 .
universal set X to be finite. This is enough for most information fusion problems since the
For a given fuzzy measure, the range of the
number of considered information sources, such
degree of the uncertainty can be estimated as
as the attributes in a data base, is usually finite.
follows.
Let X = {x1 , x2 , L, xn } . Moreover, we assume that set function μ is a fuzzy measure on (X, P(X)). In such a case, the upper integral and the
Theorem 10 For any fuzzy measure μ on (X, P(X)), 0 ≤ γ μ ≤ n .
lower integral of nonnegative function f is reduced to be
The conclusion in the next theorem can be used to 2 −1 n
( U ) ∫ f dμ = sup{ ∑ a j ⋅ μ ( A j ) | j =1
2 −1 n
∑ajχA j =1
j
= f}
calculate
the
uncertainty
carried
by
measures
in
aggregation
processes
fuzzy if
the
coordination manner is unknown.
and 2 −1
2 −1
j =1
j =1
n
(L) ∫ f dμ = inf{ ∑ a j ⋅ μ ( A j ) |
n
∑ajχA
j
respectively, where a j ≥ 0 and A j =
= f}
U{xi } ji =1
if j is expressed in binary digits as jn jn −1 ⋅ ⋅ ⋅ j1
Theorem 11 Given fuzzy measure μ on (X,
P(X)), for any nonnegative function f on X,
0 ≤ ( U) ∫ f dμ −(L) ∫ f dμ ≤ γ μ ⋅ μ ( X ) ⋅ max f ( x) x∈ X
for every j = 1, 2, ..., 2 n − 1 . Their calculation is just LP problems [8, 9].
Example 7 The data and some results in
Examples 1 and 3 are used here. Since We have seen that, due to the nonadditivity of μ, for a given nonnegative function f, different
( U) ∫ 1 dμ = 21 and (L) ∫ 1 dμ = 14 , we have
types of integral may result different integration
γμ =
values. This is just the uncertainty carried by fuzzy measure μ. Since the upper integral and
From
the lower integral are two extreme in regard to the integration value, we may estimate the uncertainty by their difference.
21 − 14 7 = . 17 17
( U) ∫ f dμ −(L) ∫ f dμ = 88 − 64 = 22
max f ( x) = 6 , and μ ( X ) = 17 , we may verify x∈ X
the conclusion showing in Theorem 11 by
Definition 3 Given a fuzzy measure μ on (X,
P(X)), the degree of the uncertainty carried by μ
,
22 ≤
7 × 6 × 17 = 42 . 17
6.
and
Conclusions
the
fuzzy
integral,
Journal
of
Mathematical Analysis and Applications 99 In information fusion, the received numerical information from various sources can be regarded as observations of attributes. Given a fuzzy measure representing the individual as well as the joint importance of the attributes, if the coordination manner is unknown, the aggregation amount is still uncertain within an interval. The upper integral and the lower integral
are
two
extremes
of
nonlinear
(1984) 195-218. [5] Z. Wang, Pan integral and Choquet integral, Proc. IFSA’93, Soeul (1993) 316-317. [6] Z. Wang and G. J. Klir, Fuzzy Measure Theory, Plenum, New York, (1992). [7] Z. Wang, K. S. Leung, and G. J. Klir, Integration on finite sets, to appear in IJIS. [8] Z. Wang, W. Li, K. H. Lee, and K. S. Leung, Lower integrals and upper integrals with
aggregation tools, by which the degree of the
respect
uncertainty carried by the fuzzy measure can be
submitted to FSS.
calculated and the aggregation value can be estimated.
to
nonadditive
Acknowledgements
The work described in this paper was supported by a direct grant of CUHK.
References
[1] P. R. Halmos, Measure Theory, Van Nostrand, New York (1967). [2] T. Murofushi, M. Sugeno, and M. Machida, Non-monotonic fuzzy measures and the Choquet integral, Fuzzy Sets and Systems 64 (1994) 73-86. [3] M. Sugeno, Theory of Fuzzy Integral and its Applications, Ph. D. dissertation, Tokyo Institute of Technology (1974). [4] Z. Wang, The autocontinuity of set function
functions,
[9] W. L. Winston, Operations Research— Applications and Algorithms, fourth edition, Duxbury Press, 2004.
It is hopeful that the inequality in Theorem 10 can be improved as 0 ≤ γ μ ≤ n − 1 .
set
