From: FLAIRS-01 Proceedings. Copyright © 2001, AAAI (www.aaai.org). All rights reserved.
A General Approach to Uncertainty Representation
Measures
Using Fuzzy
Ronald R. Yager
Machine
Intelligence Institute. IonaCollege
NewRochelle, NY10801
yager@panix.com
AssumeV is a variable which attains its value in
the space X = {x1, x2 ...... Xn}. In situations in which
the exact value of the variable V is unknownthe best
we can do is to try to formulate our knowledgeabout V
in a useful mathematicalstructure. One such structure
is a fuzzy measure [1, 2]. One useful feature of the
fuzzy measureis its ability to interact with integrals,
such as the Choquet [3] and fuzzy integral [4], to
provide the necessary tools to add in decision making
under uncertainty. A second desirable feature of this
measure its ability to represent in a unified manner
different types of characterizations of uncertainty.
Formally a fuzzy measure It on a space X is a
mappingfrom subsets of X into the unit interval, It:
Introduction
2X ~ [0, 1] such that 1.it(X) = 1, 2. It(O) = 0 and
c B then It(A) < It(B)
Within the frameworkof using the fuzzy measures
The multiplicity of modalities associated with the
to represent information about an uncertain variable,
kinds of information that can appear in intelligent
It(E) can be interpreted as a measureassociated with our
systems requires a wide spectrum of uncertainty
representation calculi. Amongthose that are commonly belief that the value of V is contained in the subset E.
Weshall generically refer to this measure as the
used are probability theory, possibility theory, fuzzy
"confidence" we have that V ~ E.
sets, rough sets, Dempster-Shafer and random set
It is clear that the properties of B are in accord with
theory. Close connections exist between some of these
this interpretation. Since we are completely confident
as is the case with Dempster-Shafer and random set
that V lies in the space X, we have It(X) = 1. Since
theory as well as between fuzzy set theory and
are also certain that the value of Vdoesn’t lie in the null
possibility theory. These calculi rather then being
set, we have kt(O) = 0. The third condition, called the
competingare neededto represent the different types of
monotonicityproperty, is a reflection of the fact that we
uncertainties such as randomness,lack of specificity and
can’t be less confidentthat V lies in a smaller set rather
imprecision. As a result of this situation there arises a
than a larger one.
need for a unified framework in which to model the
Other than satisfying these three requirements, we
above mentioned and any other required type of
are free to define the fuzzy measure. Thus the fuzzy
uncertainty representations A promising unifying
frameworkfor this is a class of non-additive measures measure provides a framework for expressing a large
class of structures useful for expressing our knowledge
called fuzzy measures [1, 2] whichhave properties very
of uncertain variables. Classes of fuzzy measureswhich
suitable for the representation and managementof
involve a simplification of the process of obtaining the
uncertain information. In the following we shall shall
value of It(E) are of considerable interest. In some
briefly discuss our ideas in this direction.
cases the measurehas been given a very specific name.
In the following we shall describe someof these special
Fuzzy Measures and the Representation
situations.
Abstract
Weintroduce the fuzzy measureas a unifying structure
for modelingknowledgeabout an uncertain variable.
Weshowthat a large class of well established types of
uncertainty representation can be modeled in this
framework. A measure of entropy associated with a
fuzzy measureis introducedand its manifestation for
different fuzzy measuresis described. The problemof
uncertain decision makingfor the case in whichthe
uncertainty represented by a fuzzy measure is
considered. The Choquetintegral is introduced as
providing a generalization of the expected value to
this environment.
of Uncertainty
Copyright
©2001,
AAAI.
Allrights
reserved.
UNCERTAINTY 619
The case in which our knowledge about the
unknown variable is of the form of probabilistic
uncertainty can be represented by a fuzzy measureg in
whichB({xi}) =Pi andwhereix(E)= ~ Ix({xi} ). It
xieE
n
=
is required that ~ Pi 1. In this situation g(E)
i=l
referred to as the measureof probability of E. It should
be emphasized that probabilistic
uncertainty is
effectively characterized by the simple additive nature of
the associated measure. Here then the Pi, the
probability distribution, distinguishes betweendifferent
manifestations of probability measures.
Another type of uncertainty is possibilistic
uncertainty [5]. Possibilistic uncertainty is represented
by a fuzzy measure in which Ix(E) = Max[Ix({xi})].
xie E
possibility measure[6] can be generated by a possibility
distribution It({ i }) =oq, where at least oneof t he i
has ai =1. Possibilistic uncertainty is often generated
from fuzzy subsets used to represent linguistic
information [7]. For example if V is the variable
corresponding to John’s height and our knowledgeof
John’s height is that he is tall then the fuzzy subset
tall is seen to inducea possibility distribution.
Another class of fuzzy measuresare those in which
our confidence that the value of a variable lies in a
subset is based upon the number of elements in the
subset. Weshall refer to these as cardinality based
measures. Let 0=D0<D1 <D2 ....
< Dn = 1 bea
set of monotonically non-decreasing values. For the
cardinality based measure we define IX(E) = DCard(E
).
Onespecial case of this class is whereDj = j/n. In this
case It(E) - Card(E).
These cardinality based measures
n
can also be expressed in terms of a set of weights wj, j
11
= 1 to n, wherewj~ [0, 1] andj=l ~ wj = 1. Using
Card(E)
these weights IX(E)=
wj. These weights are
j=l
related to the Dj by wj = Dj- Dj_ 1"
Completeuncertainty and complete certainty about
the value of a variable are notable special cases with
respect to our knowledge about a variable.
One
characterization of completeuncertainty about the value
of a variable is the fuzzy measureIX, in whichIt,(X)
1 and Ix,(E) = 0 for E ~ X. Here we are expressing our
62O
FLAIRS-2001
complete uncertainty in very conservative way, we are
only confident
the value is in X. Another
characterization of complete uncertainty is the fuzzy
measureIt* in which g*(O)= 0 and Ix*(E) = 1 for all
~: 0. Here we are expressing our complete uncertainty
in very optimistic way, we are completely confident it
lies in every set. It is worth noting these two fuzzy
measuresare special cases of cardinality based measures.
In the first case, g,, Dj = 0 for j ~ n, and Dn = 1, and in
the secondcase, it*, DO = 0 and Dj = I for j > 0.
At the opposite extreme is the case of complete
certainty, the value of the variable is knownexactly, V
= x. In this case our fuzzy measureis g(E) = 1 if x ~
and It(E) = 0 if x ~ E. Weshall denote a measure
whichis representative of completecertainty of x, Ixx"
Another frameworkused for the representation of
knowledgeabout an uncertain variable is the DempsterShafer structure [8, 9]. Werecall that the DempsterShafer structure is a mappingm: 2X ---> [0, 1], called a
basic assignment function, having the following
properties m(~) = 0 and ~ m(B) = 1. The subsets
X
Be 2
Bj of X for which m(Bj) > 0 are called focal elements.
It should be noted that the Dempster-Shaferstructure is
not a fuzzy measure. In [ 10] Yagerindicated that a D-S
structure can be viewedas representing a situation in
which we are uncertain as to the the actual underlying
fuzzy measure. That is a D-S belief can be used to
represent a situation in which there are a numberof
possible fuzzy measures. This additional level of
uncertainty can be seen as a lack of knowledgeabout the
parameters associated with the measure. Given a D-S
belief the measureof plausibility, defined as PI(E)
m(Bj) and the measureof belief, defined
j s.t. BjnE~O
as BeI(E) =
~
m(Bj) provide examples of
j s.t. Bj~E
possible fuzzy measuresthat are consistent with a D-S
structure. It can be easily shownthat these are fuzzy
measures. It is worth noting that in the DempsterShafer frameworkcomplete ignorance is represented by
the basic assignment function where m(X) = 1. For
this basic assignment function we get as the
plausibility measure It* and as the belief measure we
get Ix,.
The Information in a Fuzzy Measure
In this section we shall suggest such a measure of
information associated with a fuzzy measure in the
spirit of the Shannonentropy. In order accomplishthis
we shall use an concept introduced into the framework
of fuzzy measures by Murofishi [11]. Let It be a fuzzy
measureof X = {Xl, x2 ......
Xn}. For any xi in X its
Shapleyindex [12], Vi, is defined by
n-1
Vi=~Tk ~’~ It(Ku{xi})-Ix(K)
k =0
KG_F
i
II~=k
where[
KIis the cardinality of the set K, Fi = X - { xi }
It can be shown that it is
and Tk - (n-k-l)!k!
n!
always the case that Vi ¯ [0, 1] and ~i Vi = 1. The
Shapley index Vi can be seen as indicating the average
increment in confidence obtained by adding xi to a
subset that doesn’t contain it. Weuse this Shapley
index to help indicate the amountof information in a
fuzzy measure whenit is being used to represent our
knowledgeabout an uncertain variable.
In [13] Yager suggested the use of the Shapley
index in the calculation of the amountof information
contained in a fuzzy measureassociated with a variable.
Let It be a fuzzy measure on the space X = { x 1,, x2,
..... xn } representing our knowledgeof a variable. Let
Vi be the Shapley index of xi, then define H(it)
= - ~’i Vi in Vi and this the Shapley entropy of It.
It is well established that this formulation of
entropy provides a measure of uncertainty [14]
associated with an uncertain variable. It is recalled that
in the situation as is the case here where Vi ¯ [0, 1]
and ~iVi = 1, H(it) attains its maximalvalue of In(n)
whenVi = l/n for all i and its minimal value of zero,
whenVi = 1 for somei.
One interesting faculty provided the entropy
measure over the space of fuzzy measures is the
principle of maximal entropy. Assume we have a
variable about whichall that our available information
allows us to conclude is that one of q fuzzy measures,
Iti for i = 1 to q, is the appropriate fuzzy measure.Here
then there is some uncertainty as to which is the
appropriate fuzzy measureassociated with the variable.
Weare indifferent between these fuzzy measures. The
principle of maximalentropy suggests that we select as
the measurethe one of these that maximizesthe
entropy. That is, we select ktj* where H(Isj*)
Maxj[H(p.j)]. This is a conservative approach, we arre
choosing the measure with the most uncertainty and
thus introduce the least unjustified information.
Some Cases of Entropy
Let us look at this entropy for some particular
cases of fuzzy measures to see that it performs
appropriately. Consider first the special case of
complete certainty. This corresponds to a measure in
which ~(E) = 1 if 1 ¯ E and It (E) = 0 if 1 ~E. In
this case we see that if x1 ~ E, then kt(E u{x1 }) - It(E)
= I. On the other hand for any xj~ x1 and E
kt(E u {xj}) - It(E) = 0. From this we see that
j~l, Vj =0and v1 =1. Thus in this case we get
H(It) = 0, the case of minimaluncertainty.
Consider nowthe case of a probability measure. If
pj denotes the probability of xj, then It(E) = ~ pj.
xj~ E
In this case if xi ~ E then It(E ~) {xi}) - It(E)
n-I
Based upon this we get that Vi=pi ~’~ Tk ~1
k=0
Ec_F
i
I~=k
Using some algebraic manipulations it can be shown
n-I
that ~ Tk ~ 1 = 1. Thus for the probabilistic
k=0
Ec_F
i
Ivt=k
measurewe get Vi = Pi and hence H(it) -)-’.j In Vj =
-~j pj In pj. This is the classical Shannonentropy.
Consider the case of a cardinality based on fuzzy
IEI
measure, la(E)= ~ wj. Here for all E and i not i n
j=l
E, It(E u {xi}) - It(E) = WlEl+
1. Fromthis we get
n-I
Vi= E ~’k ~ Wk+l _1n
k=0
K_Fi
I~=k
This correspond to the case of maximalentropy, H(la)
In(n). The implication of this result is very significant
for our uncertainty framework.Essentially what this is
saying is that every cardinality based fuzzy measureis a
manifestation of a maximaluncertainty situation. Thus
with this class of cardinality based fuzzy measures we
have a whole family of potential representations of
complete uncertainty. Wehave already seen some
UNCERTAINTY 621
manifestation of this. Wehave already indicated three
examplesof cardinality measures, lA*, lA. and the case
where Dj = j/n. In the probabilistic frameworkwe saw
that the case where pj = j/n gives us the maximal
uncertainty. Wenote that this is an example of a
cardinality based measurewhere Dj =j/n.
A class of fuzzy measures can be seen to be a
collection of fuzzy measures sharing some properties
but distinguished by the value of some parameters. A
prototypical example of this are the probability
measures. Here while each memberof this class obeys
the simple additivity property, however they are
distinguished by the parameters corresponding to the
probabilities of the singleton sets. Onebenefit of using
families is that once having decided upon a family the
process of selecting the actual fuzzy measure becomes
simply a process of determining (learning) the values
the associated parameters. Weobserve that any useful
class of fuzzy measuresshould be able to represent the
states of complete uncertainty and complete knowledge
about a variable. Thus it wouldappear that any useful
class should have a cardinality based measureas one of
its members,to represent complete uncertainty, and lax
to represent completecertainty.
Wenow turn to the expression of the Shapley
value and the associated entropy for the measuresbased
upon a Dempster-Shaferstructure mwith focal elements
Bj. Werecall that the measuresof plausibility, PI, and
belief, Bel, are two fuzzy measuresconsistent with this
belief structure. In [13] Yager showedthese two fuzzy
measures have the same Shapley value
Theorem:Ifla is a plausibility
or belief measure
obtained from a D-S belief structure with basic
assignment function m and focal elements Bj then the
m(Bj)
Shapleyvalue for each xi is Vi =
~
j s.t. xi ~ Bj Card(Bj)
q
Wecan express this as Vi = ~ m(Bj)
where
j= 1 Card(Bj) Bj(xi)
Bj(xi) is the membership of i i n Bj. T his a s t he
Shapley value for a Dempster-Shafer belief structure.
The
i) entropy associated with this index - ~iVi In(V
Wenowobtain the Shapley index for possibilistic
uncertainty wherelA(E) = Max[i.t({xj}). Let us denote
xj~E
ctj = la({xj}).
Any possibility
measure can
represented as the plausibility measureof someD-S
622 FLAIRS-2001
structure which has consonant focal elements. A D-S
belief structure is called a consonantbelief structure if
the focal elements are nested, B I D B2 D ...... D Bq.
Let us see the relationship betweena given possibility
measureand the associated consonant belief structure.
Without loss of generality we assume the xi have been
indexedsuch that cti > txj if i > j. In this case ct n = 1.
Considernowa D-Sbelief structure whereBj = { xi t i =
j to n}, Bj c Bk ifk <j and therefore it is a consonant
belief structure. Let the basic assignment function m
be such that m(Bj) = ctj - ctj_ 1 forj = 1 to n whereby
definition we let cc0 = 0. For this D-Sstructure
n
J
Pl({xj})= ~ m(Bi) Bi(xj) ~ ct i -cci. 1 = ct j,
i=l
i=l
the plausibility measureis the same as the possibility
measure. From the results of the preceding section
using the plausibility representation of the possibility
measurewe obtain the Shapley index of the possibility
measure,
1
aj
Vi = ~ m(Bj) .=
j s.t. Card(Bj) j=l n + 1-j
xi~ Bj
Decision Making with Fuzzy Measures
Onesituation in which we must use an uncertainty
representation is in decision makingunder uncertainty.
The issues involved in decision making under
incertitude can best xbe understood using the decision
matrix shownin figure1 #1.
x
x
j
A.
l
C
n
ij
A
m
Figure #1. Decision Matrix
The Ai are alternatives available to the decision
maker,the xj are possible values for a relevant variable
V and Cij is the payoff to the DMif he selects Ai and
V assumes the value xj. The DMmust select one
alternative with the objective of getting the best payoff.
In decision makingunder uncertainty the value of V is
unknown before the DMselects his alternative,
knowledgeabout the value of V is expressed by some
uncertainty representation.
In uncertain decision makingeach course of action
is a multi-dimensional object consisting of the set of
possible payoffs that can result from the selection of
this action. The difficulty in makingdecisions in the
face of uncertainty is rooted in the difficulty humans
have in comparing multi-dimensional objects. One
approach is to associate with each alternative a single
value and then order the alternatives with respect to
these scalar values. To do this we need a valuation
function to mapthe alternatives into a single value. In
the case of probabilistic uncertainty the expected value
provides such a valuation function. An approach to
providing a generalization of the expected value to
uncertainties represented by a fuzzy measureis the [3]
Choquet integral. Consider alternative A. and let
1
Val(Ai) indicate its valuation. Let V be our uncertain
variable which can assumea value xj ~ X and Cij is the
payoff if we select outcomeAi and V assumes the value
xj. Weshall assumethe indexing has been done so that
Cil > Ci2 > ......
> Cin. Assume our knowledge
about the value of this variable is represented by a fuzzy
measureg. If we let Hj = {x1, x2 .........
xj} using the
n
Choquetintegral[2],
Val(Ai) = ~ (g(Hj) -g(Hj_ 1))Cij.
j=l
Theintuition for this as a generalization of the expected
value becomesclear if we consider the probabilistic
situation and begin with the expected value, E(Ai)=
j
Y~j
pj
Cij.
Let
pj = (Sum(j)-Sum(j-I))
Sum(j)
= ~ Pk then
k=l
we can rewrite E(V)
n
(Sum(j) - Sum(j-I)) Cij. Under our indexing,
j=l
Cik > Cii if k < 1, Sum(j) is the probability that
receive at least Cij. Whenour uncertainty is represented
by It, p.(Hj) indicates our belief that we receive at least
Cij then by replacing Sum(j)with ILt(Hj) in the expected
value formulation we obtain the Choquetformula.
A more general expression of the preceding which
doesn’t rely on the special indexing, Cik >_ Cil if k < 1
is the following. Welet c-index(k) indicate the index
the kth largest Cij. Welet Hk = {Xc_index(l), c_
index(2)......... Xc_index(k)},
the states of nature
the k highest payoffs. Using this we have
n
Val(Ai)
(l’t(Hk) -g(Hk-1)Cic-index(k)
k=l
References
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UNCERTAINTY 623