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Fast Fusion of Basic Belief Assignments
Defined on a Dichotomous Frame of Discernment
Jean Dezert Florentin Smarandache Albena Tchamova Deqiang Han
ONERA - DTIS Department of Mathematics Inst. of I&C Tech. Inst. of Integrated Automation
The French Aerospace Lab Univ. of New Mexico Bulgarian Academy of Sciences Xi’an Jiaotong Univ.
Palaiseau, France. Gallup, NM, USA. Sofia, Bulgaria. Xi’an, China.
[email protected] [email protected] [email protected] [email protected]
Abstract—In this paper, we propose a new fusion approach associativityproperty2 whichmakeitnotsoappealingbecause
to combine basic belief assignments (BBAs) defined on a di- the fusion order matters when sequential PCR5 fusion is
chotomous frame of discernment based on their canonical de-
appliedinsteadofglobalcombinationofthesourcesaltogether.
composition. In a companion paper, we have already proved
In this work, we show how the fusion of many sources
that the canonical decomposition of this type of BBA (called
dichotomous BBA) is always possible and unique thanks to of evidences represented by BBAs defined on a same di-
the proportional conflict redistribution rule No 5 (PCR5). More chotomous frame of discernment can be easily done based on
precisely,anydichotomousBBAisalwaysthePCR5combination the PCR5-based canonical decomposition of the BBAs. Such
of two simpler basic belief assignments named respectively the
decomposition of BBA has been proposed recently in [7].
pro-evidence, and the contra-evidence. From this interesting
We recall that another canonical decomposition based on
canonical decomposition, we present a new way of combining
many dichotomous BBAs and we show that the computational conjunctive rule (but involving improper3 BBA) had been
timeforfusingthesedichotomousBBAsbasedontheircanonical proposedin1995bySmets[8],andextendedlaterbyDenœux
decomposition is quasi-linear with the number of sources to [11] to develop the so-called cautious rule of combination.
combine, contrary to the direct fusion of the dichotomous BBAs
In this new approach we use our well justified canonical
altogether.
decomposition based on PCR5 which is strictly based on a
Keywords: Information fusion, canonical decomposition, be-
proper (i.e. normal) BBAs as defined by Shafer himself. We
lief functions, PCR5 rule, PCR6 rule.
have shown that any dichotomous BBA is always the result
of the PCR5 fusion of a simple proper pro-evidence BBA
I. INTRODUCTION
mp with a simple proper contra-evidence BBA mc, and that
The belief functions (BF) introduced by Shafer in the this decomposition is unique. Based on this important result,
mid of 1970’s [1] from Dempster’s works are well known we address in this work the problem of combination of many
and used in the artificial intelligence community to model dichotomous BBAs based on their canonical decomposition.
epistemic uncertainty and to reason with it for informa- This paper is organized as follows. After a brief recall of
tion fusion and decision-making support. Dempster’s rule to basics of belief functions in section II, we present briefly the
combine distinct sources of evidence characterized by their canonical decomposition for any dichotomous BBA based on
basic belief assignments (BBAs) defined on the same frame PCR5 rule of combination in section III which is explained
of discernment (FoD) is the historical and emblematic rule in more details with proofs, and examples in [7]. The fusion
of combination in Dempster-Shafer Theory (DST). Unfortu- of dichotomous BBAs based on the principle of canonical
nately, Dempster’s rule (denoted by DS rule for short) suffers decompositions is detailed in section IV. Concluding remarks
of serious drawbacks in high conflict evidences as pointed with perspectives are given in the last section.
out by Zadeh [2], [3], but more importantly also in some
very low conflict situations [4] as well. That is why many II. BASICSOFBELIEFFUNCTIONS
rules have been proposed in the literature [5] (Vol.2), among
Belieffunctions(BF)havebeenintroducedbyShaferin[1]
them the combination of two sources of evidence based on tomodelepistemicuncertainty.Weassumethattheanswer4 of
theproportionalconflictredistributionprincipleNo.51 (PCR5
theproblemunderconcernbelongstoaknown(orgiven)finite
rule) justified theoretically in [6], which has been shown discrete frame of discernment (FoD) Θ = {θ1,θ2,...,θn},
successful in applications. However its complexity remains with n > 1, and where all elements of Θ are mutually
oneofitslimitationswhichpreventsitsuseinfusionproblems exclusiveandexhaustive5.TheFoDissaiddichotomouswhen
involving many sources of evidence to combine, and its non
2PCR5isonlyquasi-associative.
1ActuallyPCR6ruleispreferentiallyusedforthecombinationofmorethan 3We call a BBA improper when it does not satisfy Shafer’s original
twosourcesaltogether.Fortwosources,PCR5andPCR6rulescoincideand definition.SmetscalleditageneralizedsimpleBBA(GSBBA).
becausecanonicaldecompositioninvolvedonlytwosources,weonlyneedto 4i.e.thesolution,orthedecisiontotake.
workwithPCR5ruletocombinethepro-evidencewithitscontra-evidence. 5Thisisso-calledShafer’smodelofFoD[5].
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itinvolvesonlytwoelements(onesubsetanditscomplement), ThejustificationandbehaviorofDempster’srulehavebeen
that is Θ ={A, A¯} where A¯ is the complement of A in disputedovertheyearsfrommanycounter-examplesinvolving
Θ. The set of all subsets of Θ (including empty set ∅ and high and low conflicting sources (from both theoretical and
Θ) is the power-set of Θ denoted by 2Θ. A proper Basic practicalstandpoints)asreportedin[4],[12]–[14].Manyrules
Belief Assignment (BBA) associated with a given source of ofcombinationexistintheliterature7,amongthemwerecom-
evidence is defined [1] as(cid:2)a mapping m(·) : 2Θ → [0,1] mendtherulebasedontheproportionalconflictredistribution
satisfying m(∅) = 0 and m(A) = 1. In some BF principle No. 5 (PCR5 rule) [6] which has been shown to
A∈2Θ
related frameworks, like in Smets Transferable Belief Model be successful in applications and well justified theoretically.
(TBM) [8], m(∅) is allowed to take a positive value. In this That is why we analyze it in details for solving the BF
case,m(·)issaidimproperbecauseitdoesnotsatisfyShafer’s canonical decomposition problem (BF-CDP). PCR5 transfers
definition [1]. The quantity m(A) is called the mass of A the conflicting mass only to the elements involved in the
committed by the source of evidence. Belief and plausibility conflict and proportionally to their individual masses, so that
functionsarerespectivelydefinedfromaproperBBAm(·)by the specificity of the information is entirely preserved in this
(cid:3)
fusion process (see [5], Vol. 2 and Vol. 3 for full justification
Bel(A)= m(B) (1) and examples): mPCR5(∅)=0 and ∀X ∈2Θ\{∅}
B∈2Θ|B⊆A
(cid:3)
and (cid:3) m (X)= m (X )m (X )+
PCR5 1 1 2 2
Pl(A)= m(B)=1−Bel(A¯). (2) X1,X2∈2Θ
B∈2Θ|A∩B(cid:5)=∅ X1∩X2=X
(cid:3) m (X)2m (X ) m (X)2m (X )
where A¯ is the complement of A in Θ. [m 1(X)+m2 (X2 ) + m 2(X)+m1 (X2 )] (5)
1 2 2 2 1 2
Bel(A)andPl(A)areinterpretedrespectivelyaslowerand XX22∩∈X2=Θ∅
upper bounds of an unknown (subjective) probability measure
P(A) in original Dempster’s works [9], [10]. The quantities where all denominators in (5) are different from zero. If a
m(·) and Bel(·) are one-to-one and the following Mo¨bius denominator is zero, that fraction is discarded. The properties
of PCR5 can be found in [15]. Extension of PCR5 for
inverse formula holds (see [1], p. 39)
(cid:3) combining qualitative BBA’s can be found in [5], Vol. 2 & 3.
m(A)= (−1)|A−B|Bel(B) (3) AvariantofPCR5,calledPCR6hasbeenproposedbyMartin
B⊆A⊆Θ and Osswald in [5], Vol. 2, for combining s>2 sources. The
A is called a Focal Element (FE) of m(·) if m(A) > 0. general formulas for PCR5 and PCR6 rules are also given in
When all focal elements are singletons, m(·) is called a [5], Vol. 2. PCR6 coincides with PCR5 when one combines
Bayesian BBA [1] and its corresponding Bel(·) function is two sources. The difference between PCR5 and PCR6 lies
equal to Pl(·) and they are homogeneous to a (subjective) in the way the proportional conflict redistribution is done as
probability measure P(·). The vacuous BBA, or VBBA for soon as three (or more) sources are involved in the fusion.
short, representing a totally ignorant source is defined as6 From the implementation point of view, PCR6 is simpler
mv(Θ) = 1. A dichotomous BBA is a BBA defined on to implement than PCR5. For convenience, very basic (not
optimized) MatlabTMcodes of PCR5 and PCR6 fusion rules
a dichotomous FoD. A dogmatic BBA is a BBA such that
m(Θ) = 0. If m(Θ) > 0 the BBA m(·) is nondogmatic. A can be found in [5], [16] and from the toolboxes repository
on the web [17]. The main drawback of PCR5 and PCR6
simple BBA is a BBA that has at most two focal sets and one
of them is Θ. A dichotomous non dogmatic mass of belief is rules is their combinatorial complexity when the number of
a BBA having three focal elements A, A¯ and A∪A¯ with A source is big. Even for combining BBAs defined on a simple
and A¯ subsets of Θ. dichotomousframeofdiscernment,thecomputationaltimefor
combining more than 20 sources can take several hours8.
In his Mathematical Theory of Evidence [1], Shafer pro-
posed to combine s ≥ 2 distinct sources of evidence rep- Our main motivation and contribution is to propose a faster
resented by BBAs m1(.),...,ms(.) over the same FoD Θ fusion method to combine many dichotomous BBAs in order
with Dempster’s rule (i.e. the normalized conjunctive rule). to overcome the combinatorial complexity problem by estab-
For the combination of two BBAs, Dempster’s rule formula lishing a new effective (approximating) fusion method based
[1] is given by mDS(∅)=0 and ∀X ∈2Θ\{∅} on the new PCR5-based canonical decomposition principle.
It is worth noting that our new method is very different
1 (cid:3)
mDS(X) = K m1(X1)m2(X2) (4) of the method based on the clustering of non conflicting
12 BBAsfollowedbyadiscountingstepandtheconjunctiverule
X1,X2∈2Θ
X1∩X2=X presented in [18].
(cid:3)
with K12 =1− m1(X1)m2(X2).
7see[5],Vol.2foradetailedlistoffusionrules.
X1,X2∈2Θ|X1∩X2=∅ 8duetotheexponentialcomplexityofthePCR6rule(asshowninFigure
4). For our simulations, we did use a MacBook Pro 2.8 GHz Intel Core i7
6ThecompleteignoranceisdenotedΘinShafer’sbook[1]. with16Go1600MHzDDR3memoryrunningMatlabTMR2018a.
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III. CANONICALDECOMPOSITIONOFDICHOTOMOUSBBA of combination (see Lemma in [7]). In [7], we have shown
A FoDΘ= {A,A¯} is called dichotomous if it consists of that our canonical decomposition cannot be achieved from
only two elements A and A¯ with A∪A¯=Θ and A∩A¯=∅. conjunctive, disjunctive, Yager’s [19] or Dubois-Prade [20]
A is different from Θ and from Empty-Set because we want rules of combination, neither from averaging rule. However,
such type of decomposition is unique and is always possible
to work with informative FoD. Indeed, the very special frame
{Θ,∅} does not bring any useful information since the only in all cases of dichotomous BBA m(·) using the PCR5 rule
of combination. In [7], we did prove the following Theorem.
possible BBA with such frame is the vacuous BBA. So, we
consider a given proper9 BBA m(·):2Θ →[0,1] of the form Theorem 1: Consider a dichotomous FoD Θ = {A,A¯} with
m(A)=a, m(A¯)=b, m(A∪A¯)=1−a−b (6) Adefi(cid:9)=neΘdaonndAΘ(cid:9)=by∅man(dAa)n=onad,ogmm(aA¯ti)cB=Bb,Aamnd(·)m:2(AΘ →∪A¯[0),1=]
with 0<a<1, 0<b<1 and a+b<1. 1−a−b, where a,b ∈ [0,1] and a+b < 1. Then the BBA
The conditions 0<a<1 and 0<b<1 mean that A and m(·) has a unique canonical decomposition using PCR5 rule
A¯ are focal elements of the BBA. The restriction a+b < 1 of combination of the form m = PCR5(mp,mc) with pro-
means that the BBA is nondogmatic. evidencemp(A)=x,mp(A∪A¯)=1−xandcontra-evidence
This assumption of nondogmaticity of the BBA m(·) is mc(A¯)=y, mc(A∪A¯)=1−y where x,y ∈[0,1].
necessary for Smets canonical decomposition [8], but it is
Moreover, we also proved in [7] that the canonical decom-
notessentialforourPCR5-basedcanonicaldecomposition(as
position also exists even if the dichotomous BBA is dogmatic
we will show in the sequel) because our decomposition also
(i.e. Bayesian) and the following theorem also holds.
worksdirectlywithadogmaticBBA.Ofcourseanydogmatic
BBA can always be modified as a non-dogmatic BBA by Theorem 2: Any dogmatic BBA defined by m(A) = a and
using a very small discounting number ((cid:3) > 0) so that, in m(A¯)=b, where a,b∈[0,1] and a+b=1, has a canonical
practice, Smets’ decomposition can always be applied, but decomposition using PCR5 rule of combination of the form
this is not sufficient to prove that Smets approach always m=PCR5(mp,mc) with mp(A)=x, mp(A∪A¯)=1−x
providesrelevantresults.Why?justbecauseweknow(andwe and mc(A¯)=y, mc(A∪A¯)=1−y where x,y ∈[0,1].
have proved) that Dempster’s (normalized conjunctive rule)
Theorems 1 & 2 prove that the decomposition based on
and even the conjunctive rule in Smets’ TBM suffers of
PCR5 always exists and it is unique for any dichotomous
serious drawbacks - see justifications in our aforementioned
(nondogmatic, or dogmatic) BBA.
references.Thatiswhyweexploreinthisworkanotherwayof
making a canonical decomposition, which is, for now, limited For the case of dichotomous dogmatic BBA considered
to dichotomous BBA. in Theorem 2, the expression of solutions x and y can be
Our canonical decomposition problem consists in finding established explicitly as follows - see [7] for details
the two following simple proper BBAs mp and mc of the • If a√ < b then x < y, and we have y = 1 and x =
form a+ a2+4a
.
mp(A)=x, mp(A∪A¯)=1−x (7) • If √a2> b then x > y, and we have x = 1 and y =
b+ b2+4b
mc(A¯)=y, mc(A∪A¯)=1−y (8) • If a2=b a.nd a+b=1 then a=b=0.5 and x=y =1.
with (x,y)∈[0,1]×[0,1], such that m=Fusion(mp,mc), For the case of dichotomous nondogmatic BBA considered
forachosenruleofcombinationdenotedbyFusion(·,·).The in Theorem 1, one has to find x and y solutions of the system
simple BBA mp(·) is called the pro-BBA (or pro-evidence) x2y x2+xy−xy2
of A, and the simple BBA mc(·) the contra-BBA (or contra- a=x(1−y)+ = (9)
x+y x+y
evidence) of A. The BBA mp(·) is interpreted as a source
of evidence providing an uncertain evidence in favor of A, b=(1−x)y+ xy2 = y2+xy−x2y (10)
whereasmc(·)isinterpretedasasourceofevidenceproviding x+y x+y
an uncertain contrary evidence about A. under the constraints (a,b) ∈ [0,1]2, and 0 < a+b < 1. In
This decomposition is possible with Dempster’s rule only fact, it has been proved in [7] that x ∈ [a,a + b] ⊂ [0,1]
if 0 < a < 1, 0 < b < 1 and a + b < 1, and in and y ∈ [b,a+b] ⊂ [0,1], but the explicit expression of x
this case we have x = 1−ab and y = 1−ba. However, any and y are very complicated to obtain analytically (even with
dogmatic BBA m(A) =a , m(A¯) =b with a + b = 1 is modern symbolic computing systems like MathematicaTM, or
not decomposable from Dempster’s rule for the case when MapleTM) because after algebraic calculation, and for x (cid:9)= 1,
(a,b) (cid:9)= (1,0) and (a,b) (cid:9)= (0,1) (see Theorem 4 in [7]),
one has to solve the following quartic equation which has at
and the dogmatic BBAs m(A) = 1, m(A¯) = 0 (case most four real solutions with only a valid one in [a,a+b]
(a,b)=(1,0)),orm(A)=0,m(A¯)=1(case(a,b)=(0,1))
haveinfinitelymanydecompositionsbasedonDempster’srule x4+(−a−2)x3+(2a+b)x2
+(a+b−ab−b2)x+(−a2−ab)=0 (11)
9whichmeansthatm(∅)=0.
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and then compute y as y = a+b−x. 2) Case when a = 0, or b = 0: In the case a = 0 and
1−x
Fortunately, the solutions can be easily calculated nu- 0<b≤1, then for conjunctive rule, Yager’s, Dubois-Prade’s,
merically by these computing systems, and even with Dempster’sandPCR5rulesonehasx=0andy =b(conflict
MatlabTMsystem10 as soon as the numerical values are com- between canonical masses is zero) and m(·) corresponds to
mittedtoaandtob,andthisiswhatwedoinoursimulations thefusionofvacuouspro-evidencemp =mv withthecontra-
in the sequel. evidence mc = m. In the case 0 < a ≤ 1 and b = 0, then
for conjunctive rule, Yager’s, Dubois-Prade’s, Dempster’s and
Example 1: Let consider Θ = {A,A¯} and m(A) = 0.6, PCR5 rules one has x = a and y = 0 (conflict between
m(A¯)=0.3andm(A∪A¯)=1−m(A)−m(A¯)=0.1.Hence, canonical masses is zero) and m(·) corresponds to the fusion
a=0.6 and b=0.3. The quartic equation (11) becomes ofthepro-evidencemp =mwiththevacuouscontra-evidence
x4−2.6x3+1.5x2+0.63x−0.54=0 (12) mc =mv.Thesecaseshavenoparticularinterestbecausethey
can be seen just as the combination of pro (or contra) BBA
with the vacuous BBA.
The four solutions of this quartic equation provided by the
computing system11 are approximately 3) Case when a = b ∈ (0,0.5): In this case, the BBA
m(A)=m(A¯)=aandm(A∪A¯)=1−2acanbecanonically
x1 ≈0.7774780438 x2 ≈ 0.9297589637 d√ecomposed from PCR5√rule with the BBAs mp(A√) = 1−
x ≈1.4191515820 x ≈−0.5263885898 1−2a, mp(A√∪A¯)= 1−2a and mc(A¯)=1− 1−2a,
3 4 mc(A∪A¯)= 1−2a - see details and proof in [7].
Clearlyx3 andx4 arenotacceptablesolutionsbecausethey B. Benefits of canonical decomposition
don’t belong to [0,1]. If we take x1 ≈ 0.7774780438, then
we will get y1 = (a+b−x1)/(1−x1) = (0.9−x1)/(1− The canonical decomposition based on PCR5 offers several
x1)≈0.5506061437. The pair (x1,y1)∈[0,1]2 is a solution interests and advantages that are briefly listed.
of the decomposition problem of the BBA m(·). If we take 1) This canonical decomposition of m(·) into the pro-
x2 ≈0.9297589637, then we will get y2 =(a+b−x2)/(1− evidencemp(·) andthe contra-evidencemc(·) allowsto
x2) = (0.9−x2)/(1−x2) ≈ −0.4236692006. We see that define the notion of internal conflict of a dichotomous
y2 ∈/ [0,1],andthereforethepair(x2,y2)cannotbeasolution source of evidence, denoted by Kint(m), as
ocafnthoenidceaclommapsosessitimonp(p·r)obanledmmocf(t·h)eaBreBgAivmen(·b)y.Thereforethe Kint(m)(cid:2)mp(A)mc(A¯) (13)
m (A)≈0.7774780438, m (A∪A¯)≈0.2225219562 where mp(A) =x and mc(A¯) =y are the canonical
p p factors of the BBA m(·) based on PCR5 rule of combi-
m (A¯)≈0.5506061437, m (A∪A¯)≈0.4493938563
c c nation.
2) Thecanonicaldecompositionalsoallowstoadjust/revise
ItcanbeverifiedthatthePCR5combinationofBBAsmp and
easily a dichotomous source of evidence (if needed)
mc, denoted by PCR5(mp,mc), is equal to the BBA m(·).
according to the knowledge one has on it. For instance,
Of course there are necessarily numerical approximations
if one knows that a source over (or under) estimate the
involved by the proposed decomposition because this decom- hypothesisA,thenonecouldapplyanadjustment(based
positionisobtainedbynumericalsolvers.Thismayhavesome
on some discounting or reinforcing factors) on the pro
little impact in the PCR5 fusion result but because PCR5 rule
(or contra) evidence to de-bias this source of evidence.
is numerically robust to small input changes (contrariwise to
3) This canonical decomposition can help to develop
Dempster’srule)thePCR5resultwillnotchangesubstantially
new fast rules of combination for the fusion of
with small changes (due to small numerical imprecisions) in S ≥ 2 (dichotomous) distinct12 BBAs ms(·) =
the values of BBAs to combine. (ms(A),ms(A¯),ms(A ∪ A¯)) = (as,bs,1 − as − bs),
s=1,2,...,S. This is presented next.
A. Particular cases
1) Case (a,b) = (0,0) (i.e. m is the vacuous BBA): This IV. FASTFUSIONOFDICHOTOMOUSBBAS
is the most degenerate case where the BBA m(·) corresponds Inthissection,weshowhowtocombinemanydichotomous
to the vacuous BBA. For averaging rule, conjunctive rule, BBAs defined on the same FoD Θ thanks to their canonical
Yager’s, Dubois-Prade’s, Dempster’s and PCR5 rules one has decompositions.
x = 0 and y = 0 (conflict between canonical masses is
A. Principle of the fast fusion of dichotomous BBAs
zero).InfactthevacuousBBAm(·)canalwaysbeinterpreted
as the fusion of mp and mc, where mp and mc are also The main idea for making the fast fusion of dichotomous
vacuousBBAs.Thisdegeneratecasehasnoparticularinterest BBAs is, at first, to decompose canonically each dichoto-
in practice but to model the total ignorant state of knowledge. mous BBA ms(.), for s = 1,2,...,S into their pro and
contra evidences mp,s =(mp,s(A),mp,s(A¯),mp,s(A∪A¯))=
10thankstothefsolvecommand.
11WedidgetsamesolutionswithMapleTM,andwithMatlabTM. 12i.e.cognitivelyindependent.
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(xs,0,1−xs)andmc,s =(mc,s(A),mc,s(A¯),mc,s(A∪A¯))= fusion of pro-evidences and contra-evidences with respect to
(0,ys,1−ys), and then to combine the pro-evidences mp,s the direct fusion of the BBAs with PCR5 (or PCR6 when
for s = 1,2,...,S altogether on one hand to get a global considering more than two sources to combine). To answer to
pro-evidence mp, and to combine the contra-evidences mc,s this important question we make a statistical analysis of the
for s = 1,2,...,S altogether on the other hand to get a quality of the combined result m, with respect to the direct
global contra-evidence mc. The fusion step of pro and contra PCR5, or PCR6 fusion of all BBAs altogether.
evidences is discussed in section IV-D. Once mp and mc Themeasureofthegoodnessisobtainedbythenormalized
are calculated, then one combines them with PCR5 fusion (Euclidean) Belief Interval distance dBI(mPCR5,m) (for the
rule to get the final result. This general principle of the new case of two BBAs only), or by dBI(mPCR6,m) if more
fusion method is represented by the diagram of figure 1 for than two sources are considered in the fusion process, where
convenience. m is the result of the fusion principle based on canonical
decompositions, and mPCR5 (resp. mPCR6) is the result
of the combination of original BBAs altogether with PCR5
(resp. PCR6) rule. The dBI distance between two BBAs
m1(·) and m2(·) defined on the powerset of a given FoD
Θ = {θ1,...,θn} has been proposed and justified in [21],
[22]. It is defined by
(cid:4)
(cid:3)
dBI(m1,m2)(cid:2) Nc· d2W(BI1(X),BI2(X)) (14)
X∈2Θ
where Nc = 1/2n−1 is a normalization factor to
make dBI(m1,m2) ∈ [0,1], and dW(BI1(X),BI2(X))
is Wassertein’s distance [23] between belief intervals
BI1(X) (cid:2) [Bel1(X),Pl1(X)] = [a1,b1] and BI2(X) (cid:2)
[Bel2(X),Pl2(X)] = [a2,b2]. Here, d2W(BI1(X),BI2(X))
entering in (14) is given by
(cid:5) (cid:6)
Figure 1. General principle of the fusion of dichotomous BBAs from their a +b a +b 2
canonicaldecompositions. d2 ([a ,b ],[a ,b ])(cid:2) 1 1 − 2 2
W 1 1 2 2 2 2
(cid:5) (cid:6)
ofTthheisprnoe-wevfiduesinocneamppp,rso(arcehspi.scionntetrrae-setivnigdebnecceasumsec,tsh)eisfuqsuiiotne + 31 b1−2 a1 − b2−2 a2 2 (15)
simple because there is non conflict between mp,s (resp.
Figure 2 shows the normalized histogram (i.e. the
between mc,s), so that their fusion can be done quite easily
empirical probability distribution) of the distance values
andalargenumberofsourcescanbecombinedwithoutahigh
computational burden. In fact, with this fusion approach, only dEBI(mPCR5,m) based on 20000 random15 generations of
dichotomous BBAs m1 and m2. One observes that the new
one PCR5 fusion step of simple (combined) canonical BBAs
fusion approach based on the canonical decompositions of
is needed at the very end of the fusion process. It is worth
BBAs (with the conjunctive fusion of pro-evidences, and the
noting that in this work there is no link with the canonical
conjunctive fusion of contra-evidences) provides a solution
decomposition proposed by Shafer and then extended by
which is very close to what we obtain from the direct ap-
Smets because here we use another fusion rule based on the
plication of PCR5 rule, with a mean of 0.0287 and a standard
proportional conflict redistribution principle.
deviation of 0.0289. In 98.20% of cases, the final decision
B. Analysis of the effectiveness of this new fusion approach (based on the min of dEBI decision-making strategy explained
in [22]) based on mPCR5, or on m are in agreement. This
Because the PCR5 rule13 of combination is not associative,
means that the decision agreement (DA) rate is 98.20%.
the fusion14 of the canonical BBAs followed by their PCR5
Figures 3 show the normalized histograms of the
fusion will not provide in general the same result as the dEBI(mPCR6,m) values based also on 20000 random runs
direct fusion of the dichotomous BBAs altogether but only
for the fusion of 6 dichotomous BBAs respectively. We use
an approximate result, which is normal.
PCR6ruleinsteadofPCR5ruletocombinethe6dichotomous
The main question is to know how good is the approx-
BBAs altogether because PCR6 rule has been recognized to
imation obtained by this new fusion method based on the
bemoreeffectivethanPCR5inapplications[5](Vol.2-Chap.
2). As we can observe, the shape of the histograms is a bit
13ThesameremarkholdsforPCR6rulewithmorethantwoBBAs.
different from the histogram of fig. 2, but what matters is that
14We assume here that the fusion of all the pro-evidences (resp. contra-
evidences) is done with PCR5 rule which coincides in this case with the
conjunctiverulebecausethereisnoconflictbetweenthepro-evidences(resp. 15Forthis,wegeneratethreerandomnumbersuniformlydistributedin[0,1]
thecontra-evidences). andwenormalizethemtogeneraterandomlyadichotomousBBA.
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#ofBBAs mean(dEBI) std(dEBI) DecisionAgreement(%)
2 0.0287 0.0289 98.20
3 0.0578 0.0373 97.52
4 0.0838 0.0394 96.69
5 0.1008 0.0397 96.05
6 0.1119 0.0392 95.84
7 0.1169 0.0385 95.40
8 0.1200 0.0374 94.89
9 0.1211 0.0365 94.25
10 0.1204 0.0348 94.21
TableI
COMPARATIVEEVALUATIONOFCANONICALDECOMPOSITIONFUSION
METHODW.R.T.THEDIRECTPCR-BASEDFUSIONMETHOD.
decomposition based) method. As we can see, the DA factors
areveryhighwhichmeansthatmostofthetimethedecisions
Figure 2. Normalized histogram of dEBI(mPCR5,m) for 2 dichotomous taken from the direct fusion method and from the indirect
BBAscase(20000runs). fusion method are the same.
After a deep analysis of our simulation results, one can
attest that the decision-making disagreement occurs when the
themeanvalueandthestandarddeviationofthedEBI distance numericalvaluesofthemassofAandthemassofA¯arevery
are still low (0.1119 and 0.0392 respectively) indicating that close. This indicates a very high ambiguity in the decision to
the approximation obtained by this new fusion method is take in such situation which can be easily tracked in practice
globally very good. Also the decision based on this new by evaluating the quality indicator of the decision-making -
fusion approach is globally coherent with the decision taken see [22] for details.
by the direct PCR6 fusion of the BBAs (95.84% of decision
In this paper we did not investigate the quality of the
coherence).
approximation of the fusion result based on this canonical
decomposition when replacing the PCR5 fusion step of mp
and mc by other rules of combination because the core of the
canonical decomposition is based on PCR5.
C. Computational time of the new fusion method
Because of very high combinatorial complexity (and thus
high computational time) required for applying the direct
PCR6 fusion of many BBAs, we did only make the perfor-
mance evaluation up to the fusion of ten BBAs only with
PCR6.Weconjecturethattheperformancesofthisnewfusion
method based on canonical decomposition will very slowly
degrade with the increase of the number of BBAs involved in
thefusionprocess.Ofcoursethenewfusionmethodbasedon
this canonical decomposition does not suffer of combinatorial
Figure 3. Normalized histogram of dEBI(mPCR6,m) for 6 dichotomous complexity limitation which is of great interest in some
BBAscase(20000runs). applications (like in multi-spectral imagery for detection and
classification) because many (hundreds or even thousands) of
Several Monte Carlo simulations have been done with dichotomous BBAs could be easily combined very quickly.
different numbers of dichotomous BBAs to combine. The Actually with this method what takes a bit time is only the
resultsobtainedbasedon20000runsMonteCarlosimulations canonical decomposition done by the numerical solver16.
are summarized in the Table I. Figure 4 shows the average (based on 50 random runs)
The second column of Table I indicates the mean value, computational time (in seconds) of the direct PCR6 fusion
denoted by mean(dEBI), of the normalized Euclidean belief of the BBAs altogether (red plot), and the average compu-
Interval distance between the direct fusion of the BBAs by tational time of the new fusion method based on canonical
the PCR5 (when combining 2 BBAs only), or PCR6 rule decomposition (blue plot). It is clear that the computational
(when combining more than two BBAs) and the new fusion time of the direct PCR6 fusion method (the red curve) grows
rulebasedontheircanonicaldecomposition.Thethirdcolumn exponentially with the number of sources, whereas the com-
of the Table I shows the corresponding standard deviation putational time grows only slowly and quasi-linearly with the
values denoted by std(dEBI). The last column indicates the new method proposed in this work.
decision agreement (DA) factor between the decision taken
from the direct fusion method, and the indirect (canonical 16WediduseMatlabTMfsolvefunctionforthis.
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FUSION2020 Virtual conference, Pretoria, South Africa 6 - 9 July 2020
also a subtraction because mp(A ∪ A¯) = 1 − mp(A). The
Comparison of computational times (based on 50 random runs) complexityofthisfastsuboptimalPCRfusionapproach(once
thecanonicaldecompositionisavailable)istherefore2(n−1)
with the direct PCR6 fusion of all BBAs
12 with the fusion based on the canonical decompositions of BBAs multiplications and 2 subtractions for making the conjunctive
fusion of mp,i and the conjunctive fusion of mc,i, and 7
additionsand5multiplicationsformakingthePCR5fusionof
10
mpwithmc.Thereisnoneedtousethecommonalityfunction
Averaged Time (in seconds)68 otacthhdaraenvttsoaheinnestidacwSigacmhlehyoedottewfsocmetochmaocenauponfsounssBsiiacitBoyaionAltnhdsosa.efticTndothhimcteeehsprneomoetsfiowiomgtifupoorrncueoossptmooBsphsBmeuodAtwaaksmtiecobelnetathhaaseorellddytfiumiotsshnieero,etnharaelenolaidyfrl
4 afast fusion method with respect to the direct PCR5 or PCR6
rule of combination when working with a dichotomous frame
2 of discernment.
D. On the fusion of pro and contra evidences
0
2 3 4 5 6 7 8 9 10 11 12
Number of dichotomous BBAs to combine Inthepreviousanalysis,wedidusetheconjunctiverulefor
Figure4. ComputingtimeversusnumberofBBAstocombine.
theintermediaryfusionstepofpro-evidencesinonehand,and
the intermediary fusion step of contra-evidences in the other
Based on a set of 1000 random dichotomous BBAs, figure hand.Itisworthnotingthattheintermediarystepoffusionof
5 shows that the computational time (in seconds) of the pro-evidences, and the intermediary step of fusion of contra-
fusionbasedonthecanonicaldecompositionisaquasi-linear17 evidencescanbedoneinparallelwhichoffersacomputational
function of the number of BBAs to combine. advantage with respect to the direct fusion method (if one
has many sources to combine in a specific application). This
parallelization cannot be achieved in general with the other
Computational time of the fusion based on canonical decomposition
2
existing rules of combination of evidences.
1.8 Because of the fusion principle depicted in Figure 1, this
1.6 new fusion method offers also the possibility (if one prefers
1.4 for some own specific reasons) of selecting other fusion rules
Time (in seconds)01..182 fetthhoveeridfcetuhonsemcioebnsiin,nrtauaentlriemdosnetuhdsoeieafdrcycofononftrurtrtasha-i-eeoevnvciodidsmeetnenbcpciensesas.tfiOoiomrnfpcocaoofcumtpsrrbsoteih-neetivhnifiegdnecatnhhlcoeeriescspeaurnooltd-f,
0.6 but depending on the type of rules chosen we can obtain an
0.4 associative rule, an idempotent rule, and even a new cautious
rule. For example, let’s consider the same type of fusion rule
0.2
for combining the pro-evidences mp,s s = 1,...,S, and for
0
0 100 200 Nu3m00ber of 4d0ic0hotom5o0u0s BBA6s0 t0o com7b0in0e 800 900 1000 combining contra-evidences mc,s s = 1,...,S and consider
the following cases:
Figure5. ComputingtimeversusnumberofBBAstocombine.
1) If we use the conjunctive rule [5] (Vol. 1), denoted by
Figures 4 and 5 show the computational times including Conj(.,...,.)(aswedidpreviouslyinourMonteCarlo
the canonical decomposition itself done on the fly. Of course, simulations for histogram plots), then
the canonical decomposition could be done off-line once for
m =Conj(m ,...,m )
p p,1 p,S
all and stored in the computer memory (if necessary) - see
(cid:7)
fIofrwiensthaanvceenthedi(cxh,oyto)mvoauluseBsBgAivsentoinco[7m]bfinoer,cwonevehnaiveencteo. 1an−dmonpe(Aha∪sAm¯)p.(BAec∪aAu¯s)e=thecSso=n1j(u1n−ctixvse)ruanledimsaps(sAoc)i=a-
make their canonical decomposition at first and because the n
tivethefusionofpro-evidencescanbedonesequentially.
pro-evidence BBAs to combine (resp. contra evidence BBAs)
Similarly, for the fusion of contra-evi(cid:7)dences using the
hnisauvmoebbatearisvn,eerdtyhasvtiemriyspleqmuspitcr(ukAcly)turb=eytht(cid:7)ehierni=cd1oimnrejpuc,tni(cpAtirv)oe,dufaucntsdioonwf emnpn(reeAead)l cbmoepntw(juAe¯ne)cnt=ivthe1er−uplremo-ocen(vAeidhe∪ansAc¯me)s.c((BrAeesc∪paA.u¯sc)eo=nthtrear-eSse=vi1isd(e1nno−cyecsos))n,flatinhcdet
fusion result of the pro-evidences (resp. the contra-
17It is not strictly linear because the time for the numerical fsolve search
evidences)byPCR5(orPCR6)rulesisequivalenttothe
ofpro-evidenceandcontra-evidencefactorsformakingthecanonicaldecom-
positionisnotconstant. conjunctive fusion result. The conjunctive rule however
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FUSION2020 Virtual conference, Pretoria, South Africa 6 - 9 July 2020
is not idempotent in general but in very specific cases [10] A.P. Dempster, The generalization of bayesian inference, Technical
whereonlyonefocalelementgetsallthemassofbelief. report No. 20, Dept. of Statistics, Havard University, November 15,
1967.
2) Ihfavweemppre(Afe)r(cid:2)t=o uS1se(cid:2)thSs=e1axvseraagnidngmrpu(lAe,∪thAe¯n)(cid:2)w=e 1wi−ll [11] Tin.dDuecnedœubxy,n“Conodnijsutnincctitvbeoadnidesdiosfjuenvcitdievnecceo,”mAbirntiaft.ioInntoelfl.be1l7ie2f,f2u0n0c8ti.ons
amnpd(Am)c(=AS1∪ A¯Ss)=1=(11−−xsm),c(aAn¯d) m=c(SA1¯)(cid:2)=Ss=S11(1 −Ss=y1sy)s. [12] AcIno.tm.TCbcionhnaaftm.iooonnvaaI,nndtJ..tShDeyseftoz.,uerSntdo,afi“taiOo,nnBsutolhgfeaDribeaem,hSpaesvptietor.r-6S–oh8fa,feD2r0et1mh2ep.osrtye,r”’s6trhulIeEEoEf
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itsinterpretationasageneralizationofbayesianfusionrule,”Int.J.of
quential fusion of pro-evidences (and contra-evidences)
IntelligentSyst.,Vol.29,Issue3,pages223–252,March2014.
is not recommended, however the averaging rule allows [14] F. Smarandache, J. Dezert, “On the consistency of PCR6 with the
to get an idempotent fusion rule based on canonical averaging rule and its application to probability estimation,” Proc. of
Fusion2013,Istanbul,Turkey,July2013.
decompositions if needed.
[15] J. Dezert, F. Smarandache, “Non bayesian conditioning and decondi-
3) We could also prefer to use the min rule to build a new tioning,”Int.WorkshoponBeliefFunctions,Brest,France,April2010.
cautious rule of combination which will be associative [16] F.Smarandache,J.Dezert,J.-M.Tacnet,“Fusionofsourcesofevidence
manidnsid=e1m,..p.,oSt(exnst).Fanodrtmhips(,Aw∪eAj¯u)st=ha1v−emtop(taAk)e.mSimp(iAla)rl=y, [17] w2h0titt1ph0s:d/c/ibofffnaefsreeornectniecitemy,.opErogdr/tiannbcuergsha,nUdKre,lJiaublyili2ti0e1s0,”.in Proceedings of Fusion
mc(A¯)=mins=1,...,S(ys)andmc(A∪A¯)=1−mc(A¯). [18] K.Zhou,A.Martin,Q.Pan,Abeliefcombinationruleforalargenumber
ofsources,InfiniteStudy,2018.
V. CONCLUSIONS [19] R. Yager, “On the Dempster-Shafer framework and new combination
rules,”InformationSciences,Vol.41,pp.93–138,1987.
In this research paper, we did propose a new fusion [20] D. Dubois, H. Prade, “Representation and combination of uncertainty
method to combine very quickly many BBAs defined on a withbelieffunctionsandpossibilitymeasures,”Comput.Intell.,4,1988.
[21] D. Han, J. Dezert, Y. Yang, “Belief interval based distances measures
dichotomous frame of discernment thanks to their unique
in the theory of belief functions,” IEEE Trans on SMC: Systems, Vol.
canonical decompositions. This new interesting method can 48,No.6,pp.833-850,June2018.
be parallelized and offers the advantage to have a quasi-linear [22] J. Dezert, D. Han, J.-M. Tacnet, S. Carladous, Y. Yang, “Decision-
makingwithbeliefintervaldistance,”Proc.ofBelief2016,Prague,CZ,
computational time with the number of sources. For now, this
2016.
methodislimitedtothefusionofmanyBBAsthataredefined [23] A. Irpino, R. Verde, “Dynamic clustering of interval data using a
on a simple (dichotomous) frame of discernment. After some Wasserstein-baseddistance,”PatternRec.Letters,Vol.29,2008.
unsuccessfulattempts,itappearsthatthedevelopmentofafast
fusionmethodbasedonthecanonicaldecompositionprinciple
for working with non-dichotomous frames of discernment is
actuallyaverydifficultproblemthatwewanttoaddresstothe
scientificcommunityworkingwithbelieffunctionsasafuture
research challenge. This very new method brings already a
significant benefit for real application involving inter-criteria
analysisfortheevaluationofmultiple-objectiveantcolonyop-
timization algorithm for wireless sensor networks deployment
that should be reported in a forthcoming publication.
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