Generalization of the dempster~shafer theory: a fuzzy-valued ......THE Dempster–Shafer theory (DST) could be considered as a generalization of the probability theory [1]–[3]. Consider

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999 255

Generalization of the Dempster–ShaferTheory: A Fuzzy-Valued Measure

Caro Lucas and Babak Nadjar Araabi,Student Member, IEEE

Abstract—The Dempster–Shafer theory (DST) may be consid-ered as a generalization of the probability theory, which assignsmass values to the subsets of the referential set and suggestsan interval-valued probability measure. There have been severalattempts for fuzzy generalization of the DST by assigning mass(probability) values to the fuzzy subsets of the referential set.The interval-valued probability measures thus obtained are notequivalent to the original fuzzy body of evidence. In this paper,a new generalization of the DST is put forward that gives afuzzy-valued definition for the belief, plausibility, and probabilityfunctions over a finite referential set. These functions are allequivalent to one another and to the original fuzzy body ofevidence. The advantage of the proposed model is shown inthree application examples. It can be seen that the proposedgeneralization is capable of modeling the uncertainties in thereal world and eliminate the need for extra preassumptions andpreprocessing.

Index Terms—Dempster–Shafer theory, fuzzy body of evidence,fuzzy generalization of the Dempster–Shafer theory, fuzzy setof consistent probability measures, fuzzy valued belief function,fuzzy valued plausibility function.

I. INTRODUCTION

A. Dempster–Shafer Theory

T HE Dempster–Shafer theory (DST) could be consideredas a generalization of the probability theory [1]–[3].

Consider a set-valued mapping: , where isthe set of all nonfuzzy subsets of. The referential set isa finite set through this paper. Assume a probability measure

over ; now, what can be said about a probability measureover that is induced by ? This is the basic question in [1],where Dempster shows that for each , belongsto the following interval:

(1)

Manuscript received January 30, 1997; revised Septembeer 15, 1998.C. Lucas is with the Department of Electrical and Computer Engineering,

University of Tehran, Tehran 14395, Iran.B. Nadjar Araabi was with the Department of Electrical and Computer

Engineering, University of Tehran, Tehran, 14395 Iran. He is now with theDepartment of Electrical Engineering, Texas A&M University, TX 77843USA.

Publisher Item Identifier S 1063-6706(99)04940-1.

in which is any nonempty member of the range ofand

Range (2)

About ten years later Shafer introduced his evidence theoryand defined and functions. Consider a referential set

; a body of evidenceis defined asfollows [2]:

(3)

in which each is a focal element,and is the corre-spondingmass value. Evidence theory could be considered asa direct generalization of Bayesian statistics [3]. One may thinkof mass values as probability density values; but in evidencetheory, mass values are assigned to the subsets ofinstead ofthe elements of ; so, it conveys a higher level of uncertaintyand is capable of modeling both ignorance and indeterminism.Shafer defined the concepts ofbelief and plausibility as twomeasures over the subsets ofin an axiomatic manner andthen he showed that and with the following definitionswere belief and plausibility functions

(4)

(5)

Using the concept of Mobius inversion, Shafer proved a one-to-one correspondence between (3)–(5); i.e., if we have eachone of function, function, or body of evidence, then wemay build up the others two. Although Shafer’s approach wastotally different from Dempster’s, he obtained the same results.So, when we refer to the DST we mean both approaches withtheir corresponding results and interpretations.

In his capital study [1], Dempster had also introduced theset of consistent probability measures.

Definition 1—Set of Consistent Probability Measures:Consider a body of evidence (3) over a referential set

, is defined as

1063–6706/99$10.00 1999 IEEE

256 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999

if

(6)

is a set of probability density values over( , ), which defines a probability measure:

over , accordingly.Remark: The nameset of consistent probability measures

is a bit confusing since it really consists of ( )’swhich are probability density values not probability measures( ’s). However, we saved this name from the original workby Dempster [1] to make a clear reference to the well-known concept of Definition 1. There is an obvious one-to-one correspondence between ( )’s and ’s, i.e.,basically they are equivalent.

In [1], Dempster showed that

(7)

(8)

So, the set of consistent probability measures also has aone to one correspondence with function and functionas well as the body of evidence.

B. Generalizing the DST to Fuzzy Sets

While DST assigns mass values to the subsets, ratherthan the elements of , one may still wish to build up theprobability concept over the fuzzy subsets of, [4] and [5].Let us define afuzzy body of evidencelike (3), the onlydifference being that all the ’s, and , can be normal fuzzysubsets of as well as nonfuzzy ones.

The pioneering work in this connection was due to Zadeh[4]. Based on his works on information granularity [4] andpossibility theory [6], he extended to a fuzzy set-valuedmapping, and defined expected certainty () and expectedpossibility ( ) and then defined generalized andfunctions as follows:

(9)

(10)

where is the set of all normal fuzzy subsets of. Theexpected certainty and possibility functions (9) and (10) reduce

to (4) and (5) using two-valued logic definition for and .For two-valued logic, implication ( ) would be

. Ishizuka [7] and Yager [8] used alternative fuzzydefinitions for and and got different and functions.These definitions had the following structure:

(11)

only with different interpretations for . One mightthink of as a degree of inclusion. Ogawa [9] usedrelative sigma count instead of . Some other possibledefinitions for [following the structure of (11)] wereintroduced in [10].

Yen counted three deficiencies with the existing fuzzygeneralizations of the DST [5]:

1) the belief functions are not sensitive enough to thesignificant changes in focal elements;

2) the definition of fuzzy implication is not unique;3) there is no reasonable interpretation for and

functions as lower and upper probabilities.

To resolve these problems, Yen extended different parts ofthe problem based on the Dempster’s approach and thenadded them up to obtain a satisfactory fuzzy generalizationof the DST. Yen’s generalization was based on (7), wherehe extended the definition of using Zadeh’s probabilitymeasure of fuzzy events [11]

(12)

and he approximated the fuzzy body of evidence with anonfuzzy one, in a reasonable way. Where in (12) referredto the corresponding set of consistent probability measures forthe nonfuzzy approximation. Yen’s generalization saves thestructure of (11) with the following function:

(13)

where and, .

Although Yen’s generalization seems more reasonable thanthe others, it cannot resolve the three deficiencies which wereposed by himself, completely. While (13) is more sensitive tothe changes in the focal elements, it was shown in [10] that itstill suffers from probabilistic meaningless statements.

Example 2—Restated from [5, Section V(E)]:Consider abody of evidence in the DST over

with the following focal elements:

LUCAS AND ARAABI: GENERALIZATION OF THE DEMPSTER–SHAFER THEORY 257

Suppose and ; thenusing (13).

In Example 2, we face with a case in which ,which does not have a meaningful probabilistic interpretation.We don’t have such a problem in the DST itself or in any of theprevious generalizations. Elsewhere, we have suggested novelfuzzy generalizations of the DST [10]. However, neither oursuggested generalizations nor any of the previously suggestedgeneralizations were completely satisfactory or advantageous,and neither have a completely proper probabilistic interpreta-tion.

Another recent fuzzy generalization of the DST is due toMahler [12]. This generalization is restricted to finite levelfuzzy sets, i.e., all fuzzy membership functions take theirvalues only from a fixed finite list , , , ,

, . function in this generalizationagain has the structure of (11), where is substitutedwith the crisp definition of degree of inclusion, i.e.,is equal to one if and zero, otherwise.

Mahler’s method is based on the algebra of the lattice ofthe finite-level fuzzy subsets of the referential set. Therefore,it has a firm mathematical foundation. However, stilland functions in this extension suffer from some of thedeficiencies counted by Yen [5]. function with a crispinterpretation of the degree of inclusion of fuzzy sets in

, is not sensitive enough to the significant changesin focal elements. Also, we can have and, in this caseespecially, values, which cannot be well interpreted aslower and upper probabilities. For instance, thevalue of afinite-level fuzzy set which has a small nonempty intersectionwith all focal elements is one. This result is independent fromthe height of intersection. The problem comes from the factthat in (11) accepts only values in Mahler’sgeneralization.

C. Proposal: A Fuzzy Valued Measure

All the existing fuzzy generalizations of the DST, includingYen’s and Mahler’s generalizations, preserve the structure of(11). Therefore, they all result in an interval-valued probabilityfor a fuzzy event

(14)

We conjectured in [10] that a fuzzy-valued interpretation ofthe probability might resolve all the problems

(15)

The advantage of a fuzzy-valued probability measure in afuzzy environment has already been mentioned in the literature[13], [14]. In the following sections, we will first look atfuzzyextension principle[15] from a viewpoint that has already beenintroduced by the authors [16]. Then in Section III, we build,step by step, a fuzzy generalization of the DST, which resultsin the probability of a fuzzy event as a fuzzy number ratherthan an interval. This generalization is based on the resultsof Section II. We will show the probabilistic meaning of thisgeneralization. The existing generalizations will be interval-valued approximations of this fuzzy valued measure, in the

sense that all previous and values are among thepossible fuzzy and values of our method. Besides,unlike the previous generalizations, we will prove a one-to-one correspondence between our and functions and thefuzzy body of evidence, i.e., one can reconstruct the originalfuzzy body of evidence if one knows or function.

It should be noted that using the fuzzy generalization ofMobius inversion for lattice, Mahler also introduces a one toone correspondence between function of his generalizationand the fuzzy body of evidence. However, that result is onlytrue for finite-level fuzzy sets.

In Section IV, the proposed generalization will be evaluatedusing three application examples, among them an eviden-tial reasoning and a diabetes diagnosis problem. Finally, theconclusion is presented in Section V.

II. A GENERALIZATION SCHEME

A. Fuzzy Extension Principle

The fuzzy extension principle was introduced by Zadeh as auseful tool to extend the point-valued functions and relationsto fuzzy set-valued ones [15]. Consider a function

(16)

or more generally a relation

(17)

Fuzzy extension principle extends to as follows:

(18)

(19)A large part of fuzzy arithmetic stems from (18) and (19).

B. Fuzzy Extension of Set-Valued Functions

While (18) and (19) can be used to extend point-valuedfunctions to fuzzy-valued ones, one may ask if it is possibleto extend set-valued functions as well. Consider a set-valuedfunction

(20)

A fuzzy set-valued function

(21)

is a fuzzy extension of if

(22)

Zadeh has already suggested a way to extend (20) to (21) in[15]

(23)where is the cut of fuzzy set ,

. Definition (23) deeply depends on the resolution


Fig. 1. SP = Set of possible probability measures over =f!1; !2; !3g. SA = fuzzy set of consistent probability measuresfor a typical fuzzy body of evidence.P (B) = probability ofthe fuzzy event B; objective function in the optimization process.SA � SP = f(p1; p2; p3)jpi � 0,

ipi = 1g.

identity [15]. It was shown in [16] that (23) has a reasonableinterpretation only if had the following property:

(24)

Theorem 3—Extension of Set-Valued Functions to FuzzySet-Valued Ones:Consider as defined in (20). If satisfies(24), then (23) is equivalent to

(25)

and

(26)

Proof: See [16, Appendix II].So, if a set-valued function preserves the inclusion order, we

may extend it to a fuzzy set-valued function in a consistentway. The extended function again preserves the inclusionorder. We will use the idea of (25) in Section III to introducea fuzzy generalization of the DST, which will lead us to afuzzy-valued probability instead of an interval-valued one.

III. GENERALIZATION OF THE DEMPSTER–SHAFER THEORY

A. A Fundamental Observation

Theorem 4: Assume two (nonfuzzy) body of evidence over

(27)

(28)

if

and (29)

then

(30)

where and are sets of consistent probability measuresfor (27) and (28), respectively.

Proof: See Appendix A.Now, consider a fuzzy body of evidence

(31)

We define the cut of a fuzzy body of evidence as follows.Definition 5— Cut of a Fuzzy Body of Evidence:Consider

a fuzzy body of evidence (31), the cut of the fuzzy bodyof evidence is defined as a nonfuzzy body of evidence, whichconsists of the cuts of the focal elements with no changein the mass values

(32)

To make sure ; , all the focal elements’s are assumed normal.The corresponding set of consistent probability measures for

(32) may be denoted by and based on Theorem 4 we have

(33)

So, we face with a nested family of the sets of consistentprobability measures. If we have thecuts of a fuzzy bodyof evidence , then we may reconstruct the fuzzybody of evidence and vice versa. Therefore, these two areequivalent.

B. Fuzzy Set of Consistent Probability Measures

Extension of set-valued functions to fuzzy set-valued func-tions has been discussed in Section II-B. If the original set-valued function preserves the inclusion order, then the fuzzyextension will be a consistent one. We may describe thisverbally as:the cut of the mapping the mapping of thecut. Now, if we look at (33) we have a family of cuts ofthe fuzzy body of evidence and a family of sets of consistentprobability measures correspondingly; and this correspondencepreserves an inclusion order. This property yields the idea forthe following definition.

Definition 6—Fuzzy Set of Consistent Probability Measures:Consider a fuzzy body of evidence (31), and its consecutivefamily of cuts as defined in (32), where for eachcut wehave the corresponding set of consistent probability measures,

. The fuzzy set of consistent probability measuresfor thefuzzy body of evidence is defined as follows:

(34)

where is a fuzzy subset of ,, and is the set of all possible probability

measures (probability density values) over.We may describe Definition 6 verbally as follows:The

cut of the fuzzy set of consistent probability measures whichcorresponds to the fuzzy body of evidenceThe set of con-sistent probability measures which corresponds to thecut ofthe fuzzy body of evidence.

Theorem 7: The fuzzy set of consistent probability mea-sures [ in (34)] is equivalent to the fuzzy body of evidence(31).


(a) (b)

Fig. 2. (a) Pr(C): A fuzzy subset of [0, 1];(Pr(C))�=0 = [0:32; 1] and (Pr(C))�=1 = [0:80;0:90]. (b) Pr(B): A fuzzy subset of [0, 1];(Pr(B))�=0 = [0:24; 1] and (Pr(B))�=1 = [0:92;0:92].

Proof: Based on the definition of , and cut ofa fuzzy body of evidence is trivial.

Theorem 8: The fuzzy set of consistent probability mea-sures [ in (34)] is a normal convex fuzzy set.

Proof: A fuzzy set is convex if and only if all of its cutsare convex [17]. So is convex if and only if

is convex. is the set of consistent probabilitymeasures for the cut of the fuzzy body of evidence; soit could be represented by a set of linear constraints, (6) or(71), which leads to a convex feasible space [18]. So, ;

is convex. is also normal because all the focalelements has already been assumed normal, so .

C. Fuzzy Valued , , and Functions

Once we define the fuzzy set of consistent probabilitymeasures, which corresponds with the fuzzy body of evidence,it is easy to define , , and functions. Referring to (7)and (8), we can define and as the minimumand maximum values for over ; but here we havea fuzzy feasible space so we are faced with a fuzzy decisionmaking problem [19].

Definition 9—Fuzzy Valued and Functions: Considera fuzzy body of evidence (31) with its corresponding fuzzy setof consistent probability measures . and functionsare defined as follows:

(35)

(36)

If we use Zadeh’s probability of fuzzy events [like (12)] thenwe may also define and over

(37)

(38)

Fuzzy and functions are defined based on theircutshere.

Maximization and minimization processes in (35)–(38) areoperations on real numbers since we break up the fuzzydecision making problem into a sequence of nonfuzzy decisionmaking problems using the concept ofcuts. A typical fuzzydecision making problem is shown in Fig. 1. As noted in [19],it is nothing but fusion of goals and constraints. Here we havea nonfuzzy goal and fuzzy constraint . If one thinksa fuzzy valued definition of is more reasonable, e.g.,[20], basically there is no problem. In that case, we have afuzzy goal. The decision will be fuzzy not only because ofbut also because of .

Definition 10—Fuzzy Valued Function: Consider a fuz-zy body of evidence (31) with its corresponding fuzzy set ofconsistent probability measures, . For each ,function will be defined, based on (35) and (36), as follows:

(39)

and more generally for each based on (37) and (38)as follows:

(40)

So, is a fuzzy subset of unit interval, [0, 1].


Fig. 3. Mean ~E of the fuzzy body of evidence in Example 13.~E: � ~E(r) = max(p ; p ; ��; p )f�S (p1; p2; � � � ; p9)j

9i=1 ipi = r,

pi = 1; pi � 0’s.

Theorem 11: function (35) or (37), function (36) or(38), function (39) or (40). The fuzzy set of consistentprobability measures in (34) and the fuzzy body ofevidence (31) are all equivalent.

Proof: See Appendix B.So, if we have or function, we can reconstruct the

fuzzy body of evidence. Our and functions convey allthe knowledge which exists in the fuzzy body of evidence.There is this form of equivalence in the DST, but none ofthe suggested fuzzy generalizations of the DST preserves itexcept for Mahler generalization [12], which preserves theequivalence in finite-level fuzzy sets domain.

Theorem 12: , , and are all normalconvex fuzzy subsets of unit interval, [0, 1], for each fuzzyevent .

Proof: A fuzzy set is convex if and only if all of itscuts are convex. All of the cuts of , , and

are convex due to their definitions. We assumed allthe focal elements of the fuzzy body of evidence are normalso for ; so , , and arenormal.

D. Mean, Mode, and Entropy

We can introduce mean for a fuzzy body of evidence,as well as mode, entropy, or any other similar probabilisticfunction. Consider a fuzzy body of evidence (31) overandits corresponding fuzzy set of consistent probability measures

, which was introduced in (34). We may representwithits membership function

and for

(41)

Now, consider a point-valued function over the possiblevalues of

(42)

Fig. 4. Possible belief values in our generalization. Possible plausibilityvalues in our generalization. A typical [bel(B); pls(B)] interval ofprevious generalizations.

Based on the extension principle—Section II—we can definethe fuzzy value of for as follows:

(43)

where is a fuzzy subset of the range of. Let’sconsider the probability of a fuzzy event

(44)

The fuzzy probability is defined as

(45)

it is easy to show that (45) is equivalent to (40).A weighted mean for the fuzzy body of evidence is the

general case of the above function

(46)

(47)


(a)

(b) (c)

(d) (e)

Fig. 5. (a) Fuzzy representation of YOUNG, OLD, and MIDDLE (linguistic labels for age). (b)Pr(YOUNG). (c) Pr(OLD). (d) Pr(MIDDLE). (e)Mean of the body of evidence.

We can introduce mode and entropy of the fuzzy body ofevidence, as well

Mode:

(48)

Entropy:

(49)

, Mode, and Entropy in the above definitions are fuzzysubsets of their range. andEntropyare convex and normal,but Mode is not necessarily convex.

Example 13—Continued from Example 2:Suppose, , and

. Using the Yen’s method

however, our method gives two fuzzy subsets of [0, 1] asand (Fig. 2). The mean of this fuzzy body of

evidence based on (47) is shown in Fig. 3.If we use Yen’s nonfuzzy approximation of the fuzzy body

of evidence [ in (12)], then we get as aninterval-valued mean.


Fig. 6. Fuzzy representation ofGvl, Gl, Gm, Gh, andGvh as subsets ofG = f0; 0:1; 0:2; 0:3; 0:4; 0:5; 0:6; 0:7; 0:8; 0:9; 1g.

E. Comparison with the Existing Generalizations

All the previous fuzzy generalizations of the DST preservethe structure of (11) and give an interval-valued interpretationfor the probability

(50)

but our generalization gives a fuzzy valued interpretation forthe probability—(40) or equivalently, (45). Consider a fuzzybody of evidence (31). For every fuzzy set,

so

(51)

If we can prove

(52)then we have

– – –(53)

(53) means, in (50) lies among possible belief valuesof our generalization. The same thing is correct for ;and Fig. 4 explains this relation.

Theorem 14:Equation (52) is held by Zadeh [4], Ishizaka[7], Yager [8], Yen [5], and Mahler [12] generalizations of theDST.

Proof: See Appendix C.So the belief and plausibility values of these five existing

generalizations of the DST stay within possible belief andplausibility values of our generalization. One may considera typical [ , ] interval, in one of the previousgeneralizations, as an interval-valued approximation of ourfuzzy-valued probability. Now, the vagueness in the point-valued definition of the and seems natural sincethere are many possible values for and . We candefuzzify belief and plausibility values in different ways so wemay obtain different point-valued approximations. The result

of Theorem 14 cannot be proven for Ogawa’s generalization.His generalization does not reduce to the DST for nonfuzzybody of evidence, so it doesn’t have a logical correspondencewith the DST.

If we think of previous generalizations as interval-valuedapproximations of a fuzzy-valued probability measure, Yen’smethod seems more reasonable. Yen approximates the fuzzybody of evidence—or based on our view point he approxi-mates the fuzzy set of consistent probability measures—witha nonfuzzy one and then the point-valued approximations ofbelief and plausibility functions will be defined naturally, whileothers have tried to define the belief and plausibility valuesdirectly.

As explained in Example 2, for a focal element,, in Yen’sgeneralization we may face with , which doesn’thave a proper probabilistic interpretation. This observation hasmotivated us to develop a new generalization for the DST.Now, for our generalization, we will show that for every focalelement in a fuzzy body of evidence there is at least onepossible belief value which is greater than or equal to. Thepossibility of this belief value would be one.

Theorem 15:Consider a fuzzy body of evidence (31) andthe corresponding fuzzy function as defined in (36). Foreach focal element in the body of evidence there is at leastone such that

and (54)

Proof: See Appendix D.

IV. A PPLICATIONS

The applicability of proposed generalization of the DST isshown in three different examples in this section. It shouldbe noted that the aim is not to compare our method with theprevious ones based on its numerical accuracy, but to demon-strate its generality and match with the reality. Our modelhas the capability of handling a higher level of uncertainty,so we claim it is more compatible with the real world. Whenthe behavior of the phenomenon under study is determinate orwithout vagueness, our model is not necessary, simpler modelsor approximate ones can be used. One should pay the cost ofour more complicated model when previous simpler modelscan not reflect the different levels of uncertainty inherent inthe underlying system. Note that in all parts of this section wewill denote the mass value of a focal element withinstead of .

A. Evidential Reasoning

The following typical example of evidential reasoning hasbeen adopted from [21]:

Suppose and are two variables that take their possiblevalues from two spaces,and , respectively. Spacesandare referred to as the evidence space and the hypothesis space.A body of evidence for the hypothesis space is constituted by1) a set of rules that associate values of the two variables inthe form of

IF THEN is (55)


(a) (b)

(c) (d)

(e) (f)

Fig. 7. (a)Pr(Gvhjd) for m1. (b) Pr(Gvhjd) for m2. (c) Pr(Gvhj�d) for m1. (d) Pr(Gvhj�d) for m2. (e)Pr(Gljd) for m1. (f) Pr(Gljd) for m2.

where and is a fuzzy subset of and 2) a probabilitydistribution of the evidence space».

Now, consider the following set of rules:

IF the person is bold

THEN his age is OLD (56)

IF the person is not bold

THEN his age is UNKNOWN (57)

IF the person likes punk rock

THEN his age is YOUNG (58)

IF the person does not like punk rock

THEN his age is UNKNOWN (59)

where OLD and YOUNG are fuzzy subsets of [14, 80]. Onemay consider UNKNOWN as ignorance, which will be shownwith referential set (here [14, 80]) as a focal element. Supposewe have the following probabilistic judgments about a personnamed John

bold

likes punk

not bold

does not like punk

(60)

(61)

from (56), (57), and (60) we get a body of evidence

OLD, [14, 80]

OLD (62)

and from (58), (59), and (61) we get another body of evidence

YOUNG, [14, 80]

YOUNG (63)

Now, to have a judgment based on all the existing evidencesthese two bodies of evidences should be combined. To thisend, we will use a modified version ofDempster’s ruleof combination[1] besides a normalization process [5]; thefollowing combined body of evidence will be obtained:

YOUNG, OLD, YOUNG OLDYOUNG

YOUNG OLD

OLD

(64)

where YOUNG OLD is the normalized form of YOUNGOLD.


(g) (h)

(i) (j)

(k) (l)

Fig. 7. (Continued.) (g) Pr(Glj�d) for m1. (h) Pr(Glj�d) for m2. (i) Pr(bjd) for m1. (j) Pr(bjd) for m2. (k) Pr(Gvl or Glj�d) for m1. (l)Pr(Gvh or Ghjd) for m1.

Fig. 5(a) shows the fuzzy representation of OLD, YOUNG,and MIDDLE as fuzzy subsets of [14, 80]; these are linguisticlabels which may assign to our linguistic variable “age.”Fig. 5(b)–(d) shows the probability of these three labels, asnormal convex fuzzy subsets of [0, 1] using (40) as thedefinition of probability for a fuzzy body of evidence. Fig. 5(e)shows the mean of the fuzzy body of evidence (64), which isa normal convex fuzzy subsets of [14, 80], and is calculatedusing (47).

Let us compare the results with what we obtain from Yen’sgeneralization:

YOUNG (65)

OLD (66)

MIDDLE (67)

(YOUNG) in (65) is an interval approximation of whatdefined in Fig. 5(b) as the probability. Equations (66) and (67)are also interval approximations for the fuzzy-valued proba-bilities in parts (b) and (c) of Fig. 5, respectively. As noted inSection III, while Yen’s generalization may be considered asthe most rational approximation, sometimes we may need more

accurate—in adaptation with reality—results. For example, ifwe have a mechanism to assign a combination of linguisticlabels—out of a term set—to each fuzzy probability, then theinterval approximation of the probability is not enough. Meanof the fuzzy body of evidence, Fig. 5(e), may be considered asone of the best indexes that shows our overall judgment basedon the existing evidence. Here we can say “John is middleaged or somehow old.”

B. Diabetes Diagnosis

Shenoy [22] suggests a Bayesian method to represent,make a decision and solve a diabetes diagnosis problem: “Amedical intern is trying to decide a policy for treating patientssuspected of suffering from diabetes. The intern first observeswhether a patient exhibits two symptoms of diabetes—bluetoe and glucose in the urine. After she observes the presenceor absence of these symptoms, she then either prescribes atreatment for diabetes or does not.”

Now we face two questions: 1) what if a binary modelcould not describe the observation properly and 2) are thesetwo symptoms still useful in the presence of fuzzy uncertainty?


TABLE ITWO TYPICAL FUZZY BODIES OF EVIDENCES.

Gti = G AND Bti = B WHERE ti REFERS TOTOTAL IGNORANCE

DIVIDE EACH ENTRY BY 100 000TO HAVE A NORMALIZED MASS DISTRIBUTION

Consider the following model:

(68)

where , , and represents glucose, blue toe, and diabetes,respectively. means the person has diabetes, andmeansthe person has blue toe; for the glucose (68) suggests anonbinary per unit model. We may have fuzzy observationson the amount of glucose. Lets consider five possible fuzzyobservations over (Fig. 6) named , , , , and

; represented verbally byvery low, low, moderate, high,and very high.

The medical intern may have lots of observations over thespace. In our fuzzy generalization of the DST, we

can model different observations directly by focal elements.The frequency of each observation may be considered asits mass value. Table I shows two typical fuzzy bodies ofevidences with same focal elements and different mass values.The first column consists of focal elements, all of which aresubsets of . They all are real observations and

TABLE IICONDITIONAL PROBABILITY OF fdg WITH DIFFERENT

CONDITIONS FOR TWO BODIES OF EVIDENCES.ALL THE SETS ARE SUBSETS OFG EXCEPT fbg AND f�bg

two typical frequencies of observation have been shown inthe second and third column. Table II shows the conditionalprobability of for different conditions; we used theDemp-ster’s rule of conditioningto this end, which is a special case ofDempster’s rule of combination [1], [5]. Although we considera fuzzy body of evidence in the frame of our generalizationof the DST, the conditional probability of in thisspecial case would be a number rather than a fuzzy valuedprobability or an interval. This is due to the special structureof our example. Fig. 7, on the other hand, shows some of fuzzyvalued conditional probabilities.

So we built up an empirical statistical model for diabetes di-agnosis problem, which is capable of handling the uncertaintyin the observation itself. We considered fuzzy uncertaintyfor the amount of glucose in the urine and total ignorancein the observation for the blue toe; while no uncertaintywas assumed on . It may be seen that it is very easy tocombine different levels of uncertainty in our model. Carefullyexamining Table II, one finds

for (69)

for (70)

So, in the presence of ignorance, blue toe does not seem asuitable symptom for the diabetes anymore.


TABLE IIIRESULTS OF 2000 SAMPLES EXPERIMENT: “H OW MANY CIGARETTES DO YOU SMOKE A DAY?”

C. Empirical PDF

In a random sampling experiment, the members of a statis-tical society were asked about the number of cigarettes theysmoked a day. Table III shows the results of a 2000 samplesexperiment.

There is no modification or preprocessing over the answers.Our model is capable of handling all kinds of uncertainobservations directly, so there is no need for preprocessing toget rid of the uncertainty in the observation itself. Fig. 8 showssome fuzzy valued mean and probabilities for our model. Soinstead of a probability density function (pdf), we have afuzzy body of evidence that was empirically derived and itis equivalent to a fuzzy set of consistent probability measures.So, we have a fuzzy set of possible pdf’s instead of a singleapproximate one.

V. CONCLUDING REMARKS

In this paper, we tried to establish a firm foundation for afuzzy generalization of the Dempster–Shafer theory. Modelswhich are capable of handling two well-known types of uncer-tainty—randomness and fuzziness—would play an importantrole in the future of systems modeling. We started with theDST because it dealt with sets so it seemed more naturaland much easier for building a stochastic fuzzy model. Theexisting fuzzy generalizations of the DST [4], [5], [7]–[9], [12]were carefully studied. It seemed that there was an essentialcontradiction in these generalizations. While each generaliza-tion seemed natural and reasonable from a viewpoint, othergeneralizations seemed to have equal claim to being naturaland reasonable from other viewpoints.


(a) (b)

(c) (d)

Fig. 8. (a) Mean of cigarette consumption. (b) Mean of cigarette consumption among smokers. (c) Mean of cigarette consumption among those who smoketen or more a day. (d) Probability to smoke ten or more a dayPr(f10; 11;12; � � � ; 50g).

In our suggested generalization a fuzzy valued probabilitymeasure resolved the problem. Our model yielded fuzzy valued

and functions. These functions were fuzzy becauseof the fuzziness of the body of evidence itself. Belief andplausibility values of the previous generalizations are withinthe possible belief and plausibility values of our model. Ourmodel will reduce to the DST for nonfuzzy body of evidence.Our and functions are equivalent to one another and tothe fuzzy body of evidence. So, we may reconstruct the fuzzybody of evidence from , or , function.

The applicability of our generalization was demonstratedin three benchmark applications. The model suggested in thispaper is general and flexible. It has a better correspondence tothe real world problems. It is remarkable that few assumptionsand preprocessing were made in each application.

APPENDIX APROOF OF THEOREM 4

We prove a lemma first.Lemma 16—A Representation of the Set of Consistent Prob-

ability Measures: Consider the body of evidence (3) andconsider the set of consistent probability measures as definedin Definition 1. Let’s define as follows:

if (71)

then .

Proof:

1) If then

(72)

and


(73)

(74)

so .2) If , Let’s define and for

each as follows:

and

(75)

so

and

(76)

Since , based on (71) we have

and

(77)

and obviously we have

and

(78)

Now, comparing (76)–(78) we get

(79)

so .

Now we use the representation of Lemma 16 to prove Theorem4. From (29) we can conclude

(80)

So

(81)

and if we consider (80) for then we have

(82)If we consider the representation of Lemma 16 for bothand based on (81) and (82) we have

APPENDIX BPROOF OF THEOREM 11

Consider as defined in (35):

(83)

(84)

Comparing (84) with (7), the following function

(85)

is equivalent with and if we consider this equivalence, the function would be equivalent to . We

can prove the equivalence of function to considering, and is equivalent with both and

functions based on the definition. Therefore, all these threefunctions are equivalent to one another to and to the fuzzybody of evidence. While we can reconstruct from (35), weobviously can reconstruct it from (37), which is more general.On the other hand, we can build up both (35) and (37) from

. So no matter whether we use (35) or (37) as the definitionof . The same is correct for and functions.

APPENDIX CPROOF OF THEOREM 14

For the first three generalizations (only with different im-plication ( ) operators) we have

(86)


(87)

(88)

(89)

(90)

(91)

for Yen’s generalization we have

(92)

and ; , there-fore, is aweighted mean of ’s, so (86)

(93)

and for Mahler’s generalization we have

o.w.(94)

So,

(95)

APPENDIX DPROOF OF THEOREM 15

Consider a focal element and its cut for ,, where . Now, if we

consider the cut of the fuzzy body of evidence forand then calculate the belief value for in this nonfuzzybody of evidence we have

(96)

where denotes the belief function for this cut of thefuzzy body of evidence for . We know

(97)

where is the fuzzy belief value as defined in (37). So,we have

(98)

REFERENCES

[1] A. P. Dempster, “Upper and lower probabilities induced by a multi-valued mapping,”Ann. Math. Stat.,vol. 38, no. 2, pp. 325–339, 1967.

[2] G. Shafer,A Mathematical Theory of Evidence.Princeton, NJ: Prince-ton Univ. Press, 1976.

[3] J. Guan and D. A. Bell,Evidence Theory and Its Applications.Ams-terdam, The Netherlands: North-Holland, 1991.

[4] L. A. Zadeh, “Fuzzy sets and information granularity,” inAdvances inFuzzy Set Theory and Applications, M. M. Gupta, Ed. Amsterdam, TheNetherlands: North-Holland, 1979, pp. 3–18.

[5] J. Yen, “Generalizing the Dempster–Shafer theory to fuzzy sets,”IEEETrans. Syst., Man, Cybern.,vol. 20, pp. 559–570, May/June 1990.

[6] L. A. Zadeh, “Fuzzy sets as a base for a theory of possibility,”FuzzySets Syst.,vol. 1, pp. 3–28, 1978.

[7] M. Ishizuka, K. S. Fu, and T. P. Yao, “Inference procedure anduncertainty for the problem-reduction method,”Inform. Sci.,vol. 28,pp. 179–206, 1982.

[8] R. R. Yager, “Generalized probabilities of fuzzy events from fuzzy beliefstructures,”Inform. Sci.,vol. 28, pp. 45–62, 1982.

[9] H. Ogawa and K. S. Fu, “An inexact inference for damage assessment ofexisting structures,”Int. J. Man-Machine Studies,vol. 22, pp. 295–306,1985.

[10] B. N. Araabi and C. Lucas, “On the fuzzy generalization of Demp-ster–Shafer theory,” inProc. Int. Conf. Intell. Cogn. Syst. (ICICS’96),Tehran, Iran, 1996, vol. 1, pp. 161–166.

[11] L. A. Zadeh, “Probability measure of fuzzy events,”J. Math. Anal.Appl., vol. 23, pp. 423–427, 1968.

[12] R. P. S. Mahler, “Combining ambiguous evidence with respect toambiguousa priori knowledge—Part II: Fuzzy logic,”Fuzzy Sets Syst.,vol. 75, pp. 319–254, 1995.

[13] L. A. Zadeh, “Fuzzy probabilities,”Inform. Processing Mgmt., vol. 20,no. 3, pp. 363–372, 1984.

[14] R. R. Yager, “A representation of the probability of a fuzzy subset,”Fuzzy Sets Syst., vol. 13, pp. 273–283, 1984.

[15] L. A. Zadeh, “The concept of linguistic variable and its application toapproximate reasoning—Part I,”Inform. Sci.,vol. 8, pp. 199–249, 1975.

[16] B. N. Araabi and C. Lucas, “On the fuzzy extension principle,” inProc.Int. Conf. Intell. Cogn. Syst. (ICICS’96), Tehran, Iran, 1996, vol. 2, pp.68–72.

[17] G. J. Klir and T. Folger,Fuzzy Sets, Uncertainty, and Information.Englewood Cliffs, NJ: Prentice-Hall, 1988.

[18] H. Taha,Operations Research: An Introduction,2nd ed. New York:MacMillan, 1976.

[19] R. E. Bellman and L. A. Zadeh, “Decision-making in a fuzzy environ-ment,” Mgmt. Sci.,vol. 17, no. 4, pp. B141–B164, Dec. 1970.

[20] R. R. Yager, ”A representation of the probability of a fuzzy subset,”Fuzzy Sets Syst., vol. 13, pp. 273–283, 1984.

[21] J. Yen, “Computing generalized belief function for continuous fuzzysets,” Int. J. Approx. Reasoning,vol. 6, no. 1, pp. 1–31, 1992.

[22] P. P. Shenoy, “A new method for representing and solving Bayesiandecision problems,” inArtificial Intelligence Frontiers in Statistics,D.J. Hand, Ed. London, U.K.: Chapman-Hall, 1993, pp. 119–138.


Caro Lucas received the M.S. degree from theUniversity of Tehran, Iran, in 1973, and the Ph.D.degree from the University of California, Berkeley,in 1976.

He is a Professor in the Department of Electricaland Computer Engineering, University of Tehran,Iran, as well as a Researcher at the IntelligentSystems Research Faculty (ISRF), Institute for Stud-ies in Theoretical Physics and Mathematics (IPM),Tehran, Iran. He has served as the Director of ISRF(1993–1997), Chairman of the ECE Department at

the University of Tehran (1986–1988), Managing Editor of theMemoriesof the Engineering Faculty, University of Tehran (1979–1991), Reviewer ofMathematical Reviews(since 1987), and Chairman of the IEEE, Iran Section(1990–1992). He was also a Visiting Associate Professor at the Univer-sity of Toronto (Summer, 1989–1990), University of California, Berkeley(1988–1989), an Assistant Professor at Garyounis University (1984–1985),University of California, Los Angeles (1975–1976), a Senior Researcher atthe International Center for Theoretical Physics and the International Centerfor Genetic Engineering and Biotechnology, both in Trieste, Italy, the Instituteof Applied Mathematics, Chinese Academy of Sciences, Harbin Instituteof Electrical Technology, a Research Associate at Manufacturing ResearchCorporation of Ontario, and a Research Assistant at the Electronic ResearchLaboratory, University of California, Berkeley. He is currently an associateeditor ofJournal of Intelligent and Fuzzy Systems. He is the holder of a patenton speaker independent Farsi isolated word neurorecognizer. His researchinterests include biological computing, computational intelligence, uncertainsystems, intelligent control, neural networks, multiagent systems, data mining,business intelligence, financial modeling, and knowledge management.

Dr. Lucas has served as the Chairman of several international conferences.He was the founder of the ISRF, and has assisted in founding several newresearch organizations and engineering disciplines in Iran. He is the recipientof several research grants at the University of Tehran and ISRF.

Babak Nadjar Araabi (S’98) received the B.S.degree in electrical engineering from Sharif Uni-versity of Technology, Tehran, Iran, in 1992, andthe M.S. degree in electrical engineering-controlsystems from the University of Tehran, Iran, in1996. He is currently working toward the Ph.D.degree in electrical engineering at Texas A&MUniversity, College Station, TX.

He has been working in electrical and system en-gineering industries in 1996 and 1997. His researchinterests include information processing, signal and

image analysis, fuzzy and statistical modeling, pattern recognition, neuralnetworks, and biologically motivated algorithms.

Documents

Generalization of the dempster~shafer theory: a fuzzy-valued ......THE Dempster–Shafer theory (DST) could be considered as a generalization of the probability theory [1]–[3]. Consider