Toward a combination rule to deal with partial conflict and specificity in belief functions theory

T o wa rd a combination rule to dea l with partial conflict and specificity in bel ief functions theory Arnaud Martin E 3 I 2 EA3876 ENSIET A 2 rue Franc ¸ ois V erny , 29806 Brest Cedex 09, France Email: Arn aud.Martin@ensieta.fr Christophe Osswald E 3 I 2 EA3876 ENSIET A 2 ru e Franc ¸ ois V erny , 29806 Brest Cedex 09, Franc e Email: Christophe.Osswald@ensieta.fr Abstract — W e present and discuss a mixed conju nctiv e and disjunctive rule, a generalization of conflict r epartition ru les, and a combination of th ese two rules. In the belief functions t heory one of the major problem is the conflict repartition enlightened by the famous Zadeh’s exam ple. T o date, ma ny combination rules hav e been proposed in order to solve a solution to this p r oblem. Moreo ver , it can b e important to consider the specifi city of the responses of the experts. Since few yea r some u nification rules ar e proposed. W e hav e shown in our p r evious works the i nteres t of the p r oportional confl ict redistribution rule. W e propose here a mixed combination rule f ollowing the proportional confli ct redistribution rule modifi ed by a discoun ting procedure. This rule generalizes many combination rul es. Keywords: belief functions theory , conflict repartition, combination rules, proportional conflict redistrib utio n rules. I . I N T RO D U C T I O N Many fusion th eories hav e b een studied for the co mbination of the exper ts opinions such as v oting rules [1] , [2], pos- sibility the ory [3], [4], and belief functions theory [5], [6 ]. W e can divide all these f usion approac hes into f our steps: modelization , parameters estimation depen ding on th e m odel (not always necessary) , comb ination , and decision . The most difficult step is p resumably the first one. Howe ver, it is only at the comb ination step that we can add in formation such as the conflict between expert or the specificity o f the exp ert’ s response. The voting rule s are not adap ted to the m odelization of conflict between experts. If bo th po ssibility and pr obability- based theories can m odel imprecise and uncertain data at the same time, in a lot of applications exper ts can express their certainty on their per ception o f th e reality . As a result, probab ilities-based theor y such as the belief function s theor y is mo re adapted . The belief fun ctions theory , also called evidence theory or Dem pster -Shafer theo ry [5], [6 ] is based on the u se of function s defined on the power set 2 Θ (the set of all the subsets of Θ ), where Θ is the set o f e lements. Theses b elief functions or basic belief assignmen ts , m are defined by the map ping of the power set 2 Θ onto [0 , 1] with: m ( ∅ ) = 0 , (1) and X X ∈ 2 Θ m ( X ) = 1 . (2) The equ ation (1) is the hypo thesis of a closed world [6 ]. W e can d efine the b elief functio n o nly with: m ( ∅ ) ≥ 0 , (3) and the world is ope n [7] . In order to chan ge an open world to a closed world, we can add on e elemen t in the discriminan t space. These simple conditio ns in e quation (1) and ( 2), give a large panel of definitio ns of the belief f unctions, which is one the difficulties of the theo ry . Fro m these basic be lief assignments, other belief functions can be defin ed such a s the credibility and the plausibility . T o keep a maximu m o f informa tion, it is prefer able to co mbine inform ation given by the basic belief assignmen ts in to a ne w basic belief assignment and take the decision o n the ob tained belief fu nctions. If the credibility function provide s a p essimistic decision, th e plausibility f unction is often too optimistic. The pignistic probab ility [7 ] is gen erally con sidered as a compro mise. I t is given for all X ∈ 2 Θ , with X 6 = ∅ b y: betP ( X ) = X Y ∈ 2 Θ ,Y 6 = ∅ | X ∩ Y | | Y | m ( Y ) 1 − m ( ∅ ) . (4) The normalized conjunctive combinatio n rule is the first rule propo sed in the be lief theor y by [5]. In th e b elief fu nctions theory one of the major problem is the conflict rep artition enlightene d by the famous Zadeh’ s example. T o date, m any combinatio n rules have been pro posed, building a solution to this p roblem [8]– [17]. Last y ears some unification rules have been proposed [1 8]–[20]. The remainde r of the paper is organized as follows. Sectio n II highlights the importa nce o f the conflict in the classical combinatio n rules. An histor ical p oint o f view of the comb i- nation ru les and the p roportiona l conflict redistribution ru les are recalled. A ge neral fo rmulation for co mbination rules is presented in Sectio n III. First we p ropose a mixed rule between the conjunc ti ve and disjunctive rules in subsection III -A, and a pr oportional conflict rep artition rules with a discou nting proced ure in su bsection II I-B. From these two new rules we propo se a m ore gener al ru le in subsection III- C . Sec tion IV presents a discussion for a more gener al rule, and finally th e section V outlines the conclusion s of th e pap er . An algorithm implementatio n is pro posed in section VI. I I . T H E C L A S S I C A L C O M B I N A T I O N R U L E S A N D T H E C O N FL I C T R E PA RT I T I O N A. An historical poin t of view The first combination ru le pr oposed b y De mpster and Shafer is the n ormalized co njunctive co mbination rule given fo r two basic belief assignm ents m 1 and m 2 and for all X ∈ 2 Θ , X 6 = ∅ by: m DS ( X ) = 1 1 − k X A ∩ B = X m 1 ( A ) m 2 ( B ) , (5) where k = X A ∩ B = ∅ m 1 ( A ) m 2 ( B ) is the glob al co nflict of the com bination. The prob lem en lightened by the famous Zadeh’ s example is the repar tition of th e g lobal co nflict. Indeed , con sider Θ = { A, B , C } and two experts opinion s giv en by m 1 ( A ) = 0 . 9 , m 1 ( C ) = 0 . 1 , and m 2 ( B ) = 0 . 9 , m 1 ( C ) = 0 . 1 , th e mass giv en by th e com bination is m ( C ) = 1 . So as to resolve this pro blem Smets [10] prop oses to consider an open world, therefore the conju ncti ve rule is non- normalized and we have for two basic belief assignmen ts m 1 and m 2 and f or all X ∈ 2 Θ by: m Conj ( X ) = X A ∩ B = X m 1 ( A ) m 2 ( B ) . (6) m Conj ( ∅ ) can be in terpreted as a non -expected solution. I n the T ransferab le Belief Model of Smets, the rep artition o f the global conflict is do ne in the dec ision step by the pig nisitic probab ility (4). Y age r [8 ] pr oposes to tran sfer th e glo bal co nflict on the ignoran ce Θ : m Y ( X ) = m Conj ( X ) , ∀ X ∈ 2 Θ r { ∅ , Θ } m Y (Θ) = m Conj (Θ) + m Conj ( ∅ ) m Y ( ∅ ) = 0 . (7) These three b ased-conjunctive rules red uce th e imp recision and uncertainty but can be used only if all th e experts ar e reliable. I n the o ther case a disjunc ti ve combin ation r ule can be used [12 ] given for two basic belief assignm ents m 1 and m 2 and f or all X ∈ 2 Θ by: m Dis ( X ) = X A ∪ B = X m 1 ( A ) m 2 ( B ) . (8) Of course with this rule we hav e a loss of sp ecificity . When we can quantify the reliability of each expert, we can weaken the basic b elief assign ment befo re the combin ation by the discountin g proced ure:  m ′ j ( X ) = α j m j ( X ) , ∀ X ∈ 2 Θ r { Θ } m ′ j (Θ) = 1 − α j (1 − m j (Θ)) . (9) α j ∈ [0 , 1] is the discou nting factor of the expert j that is in this case the r eliability of th e expert j , eventually as a fu nction of X ∈ 2 Θ . Dubois and Pra de [ 9] propose a mixed ru le with a repartition of the par tial co nflict on the par tial igno rance. Conseq uently , the conflict is consider ed m ore p recisely than p re v iously . T his rule is given for two basic b elief assignments m 1 and m 2 and for all X ∈ 2 Θ , X 6 = ∅ b y: m DP ( X ) = X A ∩ B = X m 1 ( A ) m 2 ( B ) + X A ∪ B = X A ∩ B = ∅ m 1 ( A ) m 2 ( B ) . (10) The r epartition o f the co nflict is im portant because o f the non-id empotency of th e r ules ( except the rule of [17] that can be applied when the d ependency between exper ts is high) and due to th e respon ses of th e exper ts that can be con flicting. Hence, we hav e d efine the auto- conflict [21] in order to quantify the in trinsic con flict of a mass an d the distribution of th e conflict accordin g to the number o f experts. B. The pr opo rtional conflict r ed istri bution rules Dezert and Smaran dache propo sed a list of proportio nal conflict redistribution ( PCR ) methods [14], [22] to redistribute the partial conflict on the elements imp lied in the partial conflict. The most ef ficien t is the PCR rule gi ven for two basic b elief assignments m 1 and m 2 and f or all X ∈ 2 Θ , X 6 = ∅ by: m PCR ( X ) = m Conj ( X ) + X Y ∈ 2 Θ , X ∩ Y = ∅  m 1 ( X ) 2 m 2 ( Y ) m 1 ( X ) + m 2 ( Y ) + m 2 ( X ) 2 m 1 ( Y ) m 2 ( X ) + m 1 ( Y )  , (11) where m Conj ( . ) is the conjun cti ve rule given by th e equa tion (6). W e h a ve studied and formulate d this rule fo r more than two experts in [1 6], [21] X ∈ 2 Θ , X 6 = ∅ : m PCR6 ( X ) = m Conj ( X ) + M X i =1 m i ( X ) 2 X M − 1 ∩ k =1 Y σ i ( k ) ∩ X = ∅ ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ (2 Θ ) M − 1        M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) ) m i ( X ) + M − 1 X j =1 m σ i ( j ) ( Y σ i ( j ) )        , (12) where Y j ∈ 2 Θ is the response of the expert j , m j ( Y j ) the associated b elief fun ction and σ i counts fro m 1 to M av oiding i :  σ i ( j ) = j if j < i , σ i ( j ) = j + 1 if j ≥ i , (13) The idea is h ere to redistribute the masses of the foc al elements giving a p artial conflict prop ortionally to the initial masses o n these elements. I I I . A G E N E R A L F O R M U L A T I O N F O R C O M B I N A T I O N R U L E S In [1 8], [20] we can find two propositions of a general formu lation of the co mbination rules. In the first one, Smets considers th e combin ation r ules from a matrix notation and find the shape of this matrix according to some assumptions on the rule, such as linear ity , commutativity , associativity , etc . In the second on e, a gener ic operator is d efined from the plausibility function s. A gener al formu lation of the global con flict repartition have been proposed in [1 1], [ 19] for all X ∈ 2 Θ by: m c ( X ) = m Conj ( X ) + w ( X ) m Conj ( ∅ ) , (14) where X X ∈ 2 Θ w ( X ) = 1 . The problem is the choice of the weights w ( X ) . W e h a ve prop osed also a param etrized PCR to decr ease or increase th e influence o f many small values tow ar d one large one. T he first way is given by PCR6f , apply ing a function on ea ch belief value im plied in the partial con flict. Any non - decreasing p ositi ve function f defined o n ]0 , 1 ] can be u sed. m PCR6f ( X ) = m Conj ( X ) + M X i =1 m i ( X ) f ( m i ( X )) X M − 1 ∩ k =1 Y σ i ( k ) ∩ X = ∅ ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ (2 Θ ) M − 1        M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) ) f ( m i ( X )) + M − 1 X j =1 m σ i ( j ) f ( Y σ i ( j ) )        The second way , gi ven b y PC R6 g is to app ly a similar function g on the sum of b elief fu nctions given to a f ocal element. m PCR6g ( X ) = m Conj ( X ) + M X i =1 m i ( X ) X M − 1 ∩ k =1 Y σ i ( k ) ∩ X = ∅ ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ (2 Θ ) M − 1 M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) ) ! Y Y σ i ( j ) = X 1 l j >i ! g m i ( X ) + X Y σ i ( j ) = X m σ i ( j ) ( Y σ i ( j ) ) ! X Z ∈{ X,Y σ i (1) ,...,Y σ i ( M − 1) } g   X Y σ i ( j ) = Z m σ i ( j ) ( Y σ i ( j ) ) + m i ( X ) 1 l X = Z   where 1 l is the ch aracteristic functio n ( 1 l X = Z is 1 if X = Z and 0 elsewhere). Nev ertheless, here also the pro blem is the choice of the func tions f and g . A. How to choose con junctive an d disjunctive rules? W e h a ve seen that conjun cti ve r ule reduc es the imp recision and uncer tainty but can be used only if one of the experts is reliable, wh ereas the d isjuncti ve rule can be used when the experts are not reliable, but allows a loss of specificity . Hence, Florea [15] p roposes a weigh ted sum of these two rules accord ing to the global co nflict k = m Conj ( ∅ ) given for X ∈ 2 Θ , X 6 = ∅ by: m Flo ( X ) = β 1 ( k ) m Dis ( X ) + β 2 ( k ) m Conj ( X ) , (15) where β 1 and β 2 can ad mit k = 1 2 as sym metric weight: β 1 ( k ) = k 1 − k + k 2 , β 2 ( k ) = 1 − k 1 − k + k 2 . (16) Consequently , if the glo bal co nflict is h igh ( k near 1) th e b e- havior of this rule will give more impo rtance to the disjun cti ve rule. Thus, this rule considers the global con flict co ming fro m the no n-reliability of the exper ts. In ord er to take into account the weights more p recisely in eac h partial combination, we propose the following n e w rule. For two basic belief assignments m 1 and m 2 and for all X ∈ 2 Θ we hav e: m Mix ( X ) = X A ∪ B = X δ 1 m 1 ( A ) m 2 ( B ) + X A ∩ B = X δ 2 m 1 ( A ) m 2 ( B ) . (17) Of cou rse, if δ 1 = β 1 ( k ) an d δ 2 = β 2 ( k ) we obtain the Florea’ s rule. In the same mann er , if δ 1 = 1 − δ 2 = 0 we obtain the conjunctive r ule and if δ 1 = 1 − δ 2 = 1 the disjunctive rule . If δ 1 ( A, B ) = 1 − δ 2 ( A, B ) = 1 l A ∩ B = ∅ we retrieve the Dubo is and Prad e’ s ru le and the partial conflict c an b e consider ed, whereas th e rule ( 15). The choice o f δ 1 = 1 − δ 2 can be done by a dissimilarity such as: δ ( A, B ) = 1 − | A ∩ B | min( | A | , | B | ) , (18) where | A | is the cardinality of A . Note th at is n ot a distance nor a p roper dissimilarity ( e.g. δ ( A, B ) = 0 does not imply A = B ). W e can also take for δ 2 , the Jaccar d distance given by: d ( A, B ) = | A ∩ B | | A ∪ B | , (19) used b y [23 ] on the belief func tions. Thus, if we have a p artial conflict between A and B , | A ∩ B | = 0 and the rule transfer s the mass o n A ∪ B . In the case A ⊂ B (or the contrary), A ∩ B = A an d A ∪ B = B , so with δ the r ule transfer s the mass on A and with d on A and B accord ing to the rate | A | / | B | of the card inalities. In the case A ∩ B 6 = A, B and ∅ , the rule tran sfers the m ass on A ∩ B an d A ∪ B accor ding to δ and d . Consider the following example for two experts on Θ = { A, B , C } : ∅ A B A ∪ B A ∪ C Θ Expert 1 0 0.3 0 0.4 0 0.3 Expert 2 0 0 0.2 0 0.5 0.3 m Conj 0.06 0.44 0.14 0.12 0.15 0.09 According ly , we o btain for δ 1 = δ : δ A A ∪ B Θ B 1 0 0 A ∪ C 0 1/2 0 Θ 0 0 0 where th e column s are the focal elemen ts of the basic be lief assignment given by the expert 1 and the lin es are the focal elements of the basic belief assignmen t giv en by expe rt 2. The mass 0.06 on ∅ is transfered on A ∪ B an d the mass 0.2 on A given b y the responses A ∪ B a nd A ∪ C is transfered on A with a value of 0.1 and on Θ with the same value. For δ 1 = 1 − d we have: 1 − d A A ∪ B Θ B 1 1 /2 2/3 A ∪ C 1 /2 2/3 1 /3 Θ 2 /3 1/3 0 Note that δ can be used when the experts a re co nsidered reli- able. In th is case we consider th e more precise r esponse. With d , we h a ve the conjunctive rule only wh en the experts give the same response, else we con sider the doub tful responses an d we transfer th e m asses in propo rtion o f th e im precision of the responses (given by the cardin ality of the re sponses) o n the part in ag reement and on th e partial ig norance. For m ore than two experts, if the intersection of th e re- sponses o f the experts is not empty , we can still transfer on the inter section and the un ion, and the equ ation (18) becomes: δ ( Y 1 , ..., Y M ) = 1 − | Y 1 ∩ ... ∩ Y M | min 1 ≤ i ≤ M | Y i | . (20) From th e equation (19), we can de fine δ by: δ ( Y 1 , ..., Y M ) = 1 − | Y 1 ∩ ... ∩ Y M | | Y 1 ∪ ... ∪ Y M | . (21) Finally , the rule is given by: m Mix ( X ) = X Y 1 ∪ ... ∪ Y M = X δ ( Y 1 , ..., Y M ) M Y j =1 m j ( Y j ) + X Y 1 ∩ ... ∩ Y M = X (1 − δ ( Y 1 , ..., Y M )) M Y j =1 m j ( Y j ) . (22) This formu lation can be inter esting accord ing to the co her- ence of the respo nses of the experts. Ho wev er , this f ormulation does no t allow the repartition o f the p artial co nflict in an oth er way than the Dub ois and Prade’ s rule. In the later (3 1) and (3 2) equations, we will simply write δ instead o f δ ( Y 1 , ..., Y M ) . B. A discou nting pr oportiona l con flict repartition rule The PCR6 re distrib utes the masses of th e conflicting focal elements proportio nally to the initial masses o n th ese elemen ts. For instance, consider three experts expressing th eir opinion on Θ = { A, B , C, D } : A B A ∪ C Θ Expert 1 0.7 0 0 0.3 Expert 2 0 0.5 0 0.5 Expert 3 0 0 0.6 0.4 The glo bal conflict is given her e by 0.21+0. 14+0.09=0.44, coming from A , B and A ∪ C for 0.2 1, A , B and Θ for 0.14 and Θ , B and A ∪ C fo r 0.0 9. W ith the gener alized PCR6 rule ( 12) we obtain: m PCR6 ( A ) = 0 . 1 4 + 0 . 2 1 + 0 . 21 7 18 + 0 . 14 7 16 ≃ 0 . 493 , m PCR6 ( B ) = 0 . 06 + 0 . 21 5 18 + 0 . 14 5 16 + 0 . 09 5 14 ≃ 0 . 1 94 , m PCR6 ( A ∪ C ) = 0 . 09 + 0 . 2 1 6 18 + 0 . 09 6 14 ≃ 0 . 1 99 , m PCR6 (Θ) = 0 . 06 + 0 . 14 4 16 + 0 . 09 3 14 ≃ 0 . 114 . First of all, the repartition is on ly on the elements g i ven the partial co nflict. W e can apply a discounting p rocedure in the comb ination r ule in order to tran sfer a part of the partial c onflict on the partial ignoran ce. This new discoun ting PCR ( noted DP C R ) can be expressed for two basic belief assignments m 1 and m 2 and f or all X ∈ 2 Θ , X 6 = ∅ by: m DPCR ( X ) = m Conj ( X ) + X Y ∈ 2 Θ , X ∩ Y = ∅ α  m 1 ( X ) 2 m 2 ( Y ) m 1 ( X ) + m 2 ( Y ) + m 2 ( X ) 2 m 1 ( Y ) m 2 ( X ) + m 1 ( Y )  + X Y 1 ∪ Y 2 = X Y 1 ∩ Y 2 = ∅ (1 − α ) m 1 ( Y 1 ) m 2 ( Y 2 ) , (23) with α ∈ [0 , 1] , the discoun ting factor . In a general ca se for M experts, we co uld write this rule as: m DPCR ( X ) = m Conj ( X ) + M X i =1 m i ( X ) 2 X M − 1 ∩ k =1 Y σ i ( k ) ∩ X = ∅ ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ (2 Θ ) M − 1 α        M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) ) m i ( X ) + M − 1 X j =1 m σ i ( j ) ( Y σ i ( j ) )        + X Y 1 ∪ ... ∪ Y M = X Y 1 ∩ ... ∩ Y M = ∅ (1 − α ) M Y j =1 m j ( Y j ) , (24) where Y j ∈ 2 Θ is a respo nse o f the expert j , m j ( Y j ) its assigned m ass and σ i is given by (13). Hence, if we choo se α = 0 . 9 in the p re vious exam ple, we obtain: m DPCR ( A ) = 0 . 14 + 0 . 21 + 0 . 21 7 18 0 . 9 + 0 . 1 4 7 16 0 . 9 ≃ 0 . 4 79 , m DPCR ( B ) = 0 . 06 + 0 . 21 5 18 0 . 9 + 0 . 14 5 16 0 . 9 +0 . 09 5 14 0 . 9 ≃ 0 . 181 , m DPCR ( A ∪ C ) = 0 . 09 + 0 . 2 1 6 18 0 . 9 + 0 . 09 6 14 0 . 9 ≃ 0 . 187 , m DPCR ( A ∪ B ∪ C ) = 0 . 21 × 0 . 1 = 0 . 021 , m DPCR (Θ) = 0 . 06 + 0 . 14 4 16 0 . 9 + 0 . 0 9 3 14 0 . 9 + 0 . 1 4 × 0 . 1 +0 . 09 × 0 . 1 ≃ 0 . 13 2 . Howe ver, in this example, the partial conflict due to the experts 1, 2 and 3 saying respectively A , B , and A ∪ C , the conflict is 0 .21; noneth eless o nly th e experts 1 and 2 and the expert 2 an d 3 are in conflict. The experts 1 and 3 are no t in con flict. Now , co nsider another ca se where the experts 1 , 2 and 3 say respectively A , B , and C with the same conflict 0.21. In these both cases, the DPCR rule transfer s th e masses with the same weight α . Altho ugh, we co uld pref er tr ansfer more mass o n Θ in the seco nd th an in the fir st case. Consequently , the transfer o f mass can depe nd on the existence o f conflict between eac h pair of expe rts. W e define the con flict fu nction giving th e n umber of exper ts in conflict two by two fo r each response Y i ∈ 2 Θ of the expert i as the number of responses of the o ther experts in conflict with i . A function f i is defined by the map ping of (2 Θ ) M onto  0 , 1 M  with: f i ( Y 1 , ..., Y M ) = M X j =1 1 l { Y j ∩ Y i = ∅} M ( M − 1) . (2 5) Hence, we can cho ose α depe nding o n the r esponse of the experts such as: α ( Y 1 , ..., Y M ) = 1 − M X i =1 f i ( Y 1 , ..., Y M ) . (26) In this case α ∈ [0 , 1] , likewise we do not transfer the mass on elemen ts that c an be written as the union of th e respo nses of th e experts. Therefo re, if we take again ou r pre vious example we obtain: α ( A, B , A ∪ C ) = 1 − 2 3 = 1 3 , α ( A, B , Θ) = 1 − 1 3 = 2 3 , α (Θ , B , A ∪ C ) = 1 − 1 3 = 2 3 . Thus th e provided mass by the DPCR is: m DPCR ( A ) = 0 . 14 + 0 . 21 + 0 . 21 7 18 1 3 + 0 . 14 7 16 2 3 ≃ 0 . 418 , m DPCR ( B ) = 0 . 06 + 0 . 21 5 18 1 3 + 0 . 14 5 16 2 3 + 0 . 09 5 14 2 3 ≃ 0 . 130 , m DPCR ( A ∪ C ) = 0 . 09 + 0 . 2 1 6 18 1 3 + 0 . 09 6 14 2 3 ≃ 0 . 1 39 , m DPCR ( A ∪ B ∪ C ) = 0 . 21 2 3 = 0 . 1 40 , m DPCR (Θ) = 0 . 06 + 0 . 14 4 16 2 3 + 0 . 09 3 14 2 3 + 0 . 14 1 3 +0 . 09 1 3 ≃ 0 . 1 73 . W e want to take account of th e degree o f conflict (or non-co nflict) within eac h pair o f expert d if f erently for each element. W e can conside r th e non-co nflict function given fo r each expert i by the nu mber of experts not in co nflict with i . Hence, we can choo se α i ( Y 1 , ..., Y M ) defined by th e mapp ing of (2 Θ ) M onto  0 , 1 M  with: α i ( Y 1 , ..., Y M ) = 1 M − f i ( Y 1 , ..., Y M ) = M X j =1 ,j 6 = i 1 l { Y j ∩ Y i 6 = ∅} M ( M − 1) . (27) The discountin g PCR rule (equatio n (24)) can be written for M experts as: m DPCR ( X ) = m Conj ( X ) + M X i =1 m i ( X ) 2 X M − 1 ∩ k =1 Y σ i ( k ) ∩ X = ∅ ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ (2 Θ ) M − 1 α i λ        M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) ) m i ( X ) + M − 1 X j =1 m σ i ( j ) ( Y σ i ( j ) )        + X Y 1 ∪ ... ∪ Y M = X Y 1 ∩ ... ∩ Y M = ∅ (1 − M X i =1 α i ) M Y j =1 m j ( Y j ) , (28) where α i ( X, Y σ i (1) , ..., Y σ i ( M − 1) ) is noted α i for notations and λ d epending on ( X , Y σ i (1) , ..., Y σ i ( M − 1) ) , is chosen to obtain the no rmalization giv en by the equation (2). λ is given when α i 6 = 0 , ∀ i ∈ { 1 , ..., M } by: λ = M X i =1 α i < α , γ > , (29) where < α , γ > is the scalar produ ct of α = ( α i ) i ∈{ 1 ,...,M } and γ = ( γ i ) i ∈{ 1 ,...,M } with: γ i = m i ( X ) m i ( X ) + M − 1 X j =1 m σ i ( j ) ( Y σ i ( j ) ) , (30) where γ i ( X, Y σ i (1) , ..., Y σ i ( M − 1) ) is n oted γ i for notations. W ith this last version of the rule, for α i giv en by the equation (27), we obtain on o ur illustrative example λ = 36 13 when the experts 1, 2 and 3 say re specti vely A , B , and A ∪ C (the conflict is 0.21 ), λ = 16 5 when the conflict is 0.1 4 and λ = 56 17 when the con flict is 0 .09. Thus, th e masses are g i ven by: m DPCR ( A ) = 0 . 1 4 + 0 . 2 1 + 0 . 21 7 18 1 6 36 13 + 0 . 14 7 16 1 6 16 5 ≃ 0 . 420 , m DPCR ( B ) = 0 . 06 + 0 . 14 5 16 1 6 16 5 + 0 . 09 5 14 1 6 56 17 ≃ 0 . 101 , m DPCR ( A ∪ C ) = 0 . 09 + 0 . 2 1 6 18 1 6 36 13 + 0 . 09 6 14 1 6 56 17 ≃ 0 . 143 , m DPCR ( A ∪ B ∪ C ) = 0 . 21 2 3 = 0 . 1 4 m DPCR (Θ) = 0 . 06 + 0 . 14 4 16 1 3 16 5 + 0 . 09 3 14 1 3 56 17 +0 . 14 1 3 + 0 . 09 1 3 ≃ 0 . 1 96 . This last version allows to consider a kind of de gree of conflict (a degree of pair of n on-conflict), b ut th is degree is not so easy to introduc e in the co mbination rule. C. A mixed discoun ting con flict r ep artition rule From both new rules, the mixed rule (22) an d th e dis- counting PCR (24), we pr opose a combin ation o f these rules, giv en for two basic belief assignm ents m 1 and m 2 and for all X ∈ 2 Θ , X 6 = ∅ by: m MDPCR ( X ) = X Y 1 ∪ Y 2 = X, Y 1 ∩ Y 2 6 = ∅ δ m 1 ( Y 1 ) m 2 ( Y 2 ) + X Y 1 ∩ Y 2 = X, Y 1 ∩ Y 2 6 = ∅ (1 − δ ) m 1 ( Y 1 ) m 2 ( Y 2 ) + X Y ∈ 2 Θ , X ∩ Y = ∅ α  m 1 ( X ) 2 m 2 ( Y ) m 1 ( X ) + m 2 ( Y ) + m 2 ( X ) 2 m 1 ( Y ) m 2 ( X ) + m 1 ( Y )  , + X Y 1 ∪ Y 2 = X Y 1 ∩ Y 2 = ∅ (1 − α ) m 1 ( Y 1 ) m 2 ( Y 2 ) . (31) α ca n be given by the equ ation (26) an d δ by the equ ation (20) or (21). The weigh ts ar e taken to g et a kind o f contin uity between the mixed and DPCR ru les. Actually , when the intersection of th e respon ses is almost emp ty but not empty we use th e m ixed rule and when th is in tersection is empty we ch ose the DPCR rule. In the first case all the m ass is transfered on th e un ion an d in the secon d one it will be the same accor ding to the p artial conflict. Indee d, α = 0 if the intersection is not empty and δ = 1 if the intersection is empty . W e c an also intro duce α i giv en by the eq uation (27), an d this continuity is con served. This r ule is given in a gen eral case for M experts, fo r all X ∈ 2 Θ , X 6 = ∅ by: m MDPCR ( X ) = X Y 1 ∪ ... ∪ Y M = X, Y 1 ∩ ... ∩ Y M 6 = ∅ δ M Y j =1 m j ( Y j ) + X Y 1 ∩ ... ∩ Y M = X, Y 1 ∩ ... ∩ Y M 6 = ∅ (1 − δ ) M Y j =1 m j ( Y j ) + M X i =1 m i ( X ) 2 X M − 1 ∩ k =1 Y σ i ( k ) ∩ X = ∅ ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ (2 Θ ) M − 1 α        M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) ) m i ( X ) + M − 1 X j =1 m σ i ( j ) ( Y σ i ( j ) )        + X Y 1 ∪ ... ∪ Y M = X Y 1 ∩ ... ∩ Y M = ∅ (1 − α ) M Y j =1 m j ( Y j ) , (32) where Y j ∈ 2 Θ is the response of the expert j , m j ( Y j ) the associated belief function an d σ i is giv en by (1 3). This form ula could seem difficult to u nderstand, but we can implemen t it easily as shown in appen dix VI. If we take again the p re v ious examp le with δ given by the equation (2 0), there is n o difference with the DPCR . If δ is calculated by the equation (21), the only difference per tains to the mass 0.09 coming fr om th e respo nses o f the three experts: Θ , Θ an d A ∪ C . This m ass is tr ansfered on A ∪ C ( 0.06) an d on Θ (0.0 3). I V . D I S C U S S I O N : T O W A R D A M O R E G E N E R A L R U L E The rules presented in the previous section , pro pose a repartition of th e masses g i v ing a partial con flict on ly (wh en at m ost two exp erts are in discord ) and d o no t take heed of the le vel of imprecision of the responses of the exper ts (the non specificity of th e resp onses). Th e imp recision o f th e responses of each expert is only con sidered by the mixed and MDPCR rules wh en there is no conflict between the experts. In the mixed rule, if the in tersection o f the r esponses of the experts is em pty , the best way is not necessarily to transfer the mass on th e u nion. For example, if three experts say A ∪ B , A ∪ C , D , two experts agr ee on A . So, it co uld be better to transfer the mass on A and A ∪ B ∪ C ∪ D . Consider M experts, we define the set o f subsets of the responses of the exper ts that are not in co nflict: ε k ( Y 1 , ..., Y M ) = {{ Y i 1 , ..., Y i k } , i j ∈ I : I ⊂ { 1 , ..., M } , | I | = k , ∩ k j =1 Y i j 6 = ∅} , (33) where Y i is the re sponse of the expert i . Additionally , we define k = argmax k { ε k 6 = ∅} . In the pr e v ious example, ε k = ε 3 = {{ A ∪ B , A ∪ C, A ∪ D }} . In the M case experts, we d efine δ ( Z ) for all Z ∈ ε k with Z = { Y i 1 , ..., Y i k } as: δ ( Z ) = 1 − | ∩ k j =1 Y i j | min j ∈{ 1 ,..., k } | Y i j | . (34) An extended mixed rule for M exper ts can be written: m EMix ( X ) = X Y 1 ∪ ... ∪ Y M = X X Z ∈ ε k ( Y 1 ,...,Y M ) δ ( Z ) M Y j =1 m j ( Y j ) + X { Y i 1 ,...,Y i k } = Z ∈ ǫ k ( Y 1 ,...,Y M ) Y i 1 ∩ ... ∩ Y i k = X, (1 − δ ( Z )) | ε k ( Y 1 , ..., Y M ) | M Y j =1 m j ( Y j ) . (35) This r ule keep the spirit of the mixed ru le. Nevertheless, imagine a very high m ass on D com pared to the masses on A ∪ B and A ∪ C in the previous example. Therefore, we w o uld prefer tr ansfer the mass prop ortionally o n D and o n the oth er connected elements A ∪ B and A ∪ C in th e spirit of DPCR . For the mass allocated on these con nected elements, we can apply the extended mixed rule EMix . Consequently , in the case of conflict between all the exper ts, we must find which experts are in co nflict together, e.g. the c onnected responses of the experts. Th is pa rtial con flict is mor e precise than the partial co nflict provided consid ering all the respo nses of the experts. Thus, we obtain an extende d MDPCR . T o compute ε k taking into accou nt M f ocal classes having at most a size | Θ | = n , we h a ve to read all the foca l c lasses, an d count how often each singleton app ears in the focal classes: O ( nM ) o perations. For each of these singletons, we m ight have to distribute a par t of the loca l conflict over k focal classes. Each M -u ple of focal elements can req uest a treatme nt of O ( n 2 M 2 ) ope rations, as k 6 M and | ε k | 6 n . If each belief fu nction h as p focal eleme nts, global complexity is bound ed by O ( n 2 M 2 p M ) . Figure 1 shows two sets of fou r focal eleme nts with an empty inter section. In th e left situation , each singleton is a n intersection of two focal elem ents, and ev ery inte rsection of three focal elemen ts is ∅ . So k is 2, and ε 2 is {{ A ∪ B , B ∪ C } , { B ∪ C, C ∪ D } , { C ∪ D , A ∪ D } , { A ∪ B , A ∪ D } } . In the r ight situation , A app ears in three focal elements, B in two, and the other singleto ns appear only in one f ocal element. So k is 3, an d ε 3 is {{ A ∪ B , A ∪ C A ∪ D }} . The singleton E do es no t r ecei ve any part o f the conflict du e to its p resence A B C D A B C D E Fig. 1. T wo conflicti ng focal element s sets in the f ocal elem ent B ∪ E , but only th rough its presence in the d isjunction of all the fo cal elements: A ∪ B ∪ C ∪ D ∪ E . V . C O N C L U S I O N S In this p aper , we p ropose so me solutions to d eal with the conflict and to weigh the imp recision o f the respon ses of the exper ts, from the classical co mbination rules. Th us, we first con sider a mixed rule p rovided b y a weighted sum of the co njunctiv e and disjunctive r ules. The weig hts are defined from a measure of n onspecifity calculated by th e cardinality of th e respon ses o f the experts. Th is rule transfers the partial conflict on p artial ignoran ce. Again, the prop ortional conflict distribution ru le redistrib utes the partial con flict on the eleme nt implied in this conflict. W e pr opose an extension of this rule by a discou nting proced ure, thereb y , a par t of th e partial conflict is also redistributed o n th e par tial ig norance. So as to quantify this part, we introdu ce a measure of conflict between pair of experts and anoth er measur e of no n-conflict between pair of experts. In or der to take heed of th e nonsp ecifity and to redistributed the par tial co nflict, we p ropose a fused rule of these tw o ne w rules. This ne w rule is ma de in such way tha t we retain a kind of con tinuity of the mass on the partial igno rance, between both cases with and witho ut pa rtial conflict. Finally , we propose to d iscuss a more general ru le that can deal with the n onspecifity of each respon se of the expert also in the ca se with par tial conflict between some p artition of th e experts. The com ments of th ese new rules show that the classical combinatio n r ules in th e belief function s theory cannot take precisely into account the nonsp ecifity of the experts an d the partial conflict of the experts. W e can introd uce m ore and more artificial -or not- measures of imperf ections (conflict, nonspecificity , and so on ) in the conjunctive a nd disjunctive combinatio n r ules. Ano ther p oint to trea t in a futu r work is how these ru les perfo rm in pratical application s. V I . A P P E N D I X – MDPCR A L G O R I T H M Formula (32), like m ost of th e formu la of this ar ticle, seems simpler wh en expressed throu gh an algo rithm instead of a direct expression of m ( X ) . W e list all th e M -u ples of fo cal elements o f the M belief functio ns. An input belief function e is an association of a list of focal elemen ts an d th eir masses. W e write size( e ) the num ber of its focal elements. The focal classes are e [1] , e [2] , . . . , e [size( e )] . The mass associated to a class c is e ( c ) , wr itten with parenth esis. T he cardinality of a focal element e [ i ] is also written size( e [ i ]) . The princip le of the algo rithm 1 is to use the variable ind to build all the n -uples of fo cal elements o f the n input belief functions. Then, if the in tersection of these is n ot ∅ or equ i valent to ∅ , the cor responding conjun cti ve m ass (the produ ct o f all the masses of the fo cal elemen ts in the n - uple) is pu t on the in tersection; other wise, it is distributed over the input focal elem ents and their disjun ction. Algorithm 1: Fusion by the MDPCR combinatio n rule Data : M experts e x : ex [1] . . . ex [ M ] Result : Fu sion of ex by M DPCR rule : ep for i = 1 to M do foreach c in e x [ i ] do Append c to cl [ i ] ; foreach ind in [1 , size ( cl [1] )] × [1 , size ( cl [2] )] × . . . × [1, size ( cl [ M ] )] do δ = 1 - size( s ) / min 1 6 i 6 M (size( cl [ i ][ ind [ i ]])) ; s ← Θ ; lpro d ← 1; lsum ← 0; l u ← ∅ ; for i = 1 to M do s ← s ∩ cl [ i ][ ind [ i ]] ; lpro d ← lpro d × ex [ i ]( cl [ i ][ ind [ i ]]) ; lsum ← lsum + ex [ i ]( cl [ i ][ ind [ i ]]) ; l u ← lu ∪ cl [ i ] ; if s = ∅ then nc ← 0 ; for i = 1 to M do for j = 1 to M , j 6 = i do if cl [ i ] ∩ cl [ j ] = ∅ then nc ← nc + 1 ; α ← 1 - nc/ ( M ( M − 1 )) ; for i = 1 to M do ep ( ex [ i ][ ind [ i ]]) ← α.ep ( ex [ i ][ ind [ i ]]) + ex [ i ]( cl [ i ][ ind [ i ]]) ∗ lpro d / lsum ; ep ( l u ) ← ep ( l u ) + (1 − α ) ∗ lpro d ; else ep ( s ) ← e p ( s ) + (1 − δ ) ∗ lpr od ; ep ( l u ) ← ep ( l u ) + δ ∗ lpro d ; A C K N OW L E D G M E N T This work is supporte d by the Eur opean Union (FEDER), the French state (FN ADT), th e Brittany region and the Fren ch Departmen t of Finistere. R E F E R E N C E S [1] L. Xu, A. Krzyzak and C.Y . Suen, “Methods of Combinin g Multiple Classifiers and Their Application to Handwritin g Recogniti on, ” IEEE T ransactio ns on Systems, Man Cybernetic s , vol. 22, no. 3, pp. 418-435, May 1992. [2] L. Lam and C.Y . Suen, “ Applicati on of Majority V oting to Pattern Recog- nition: An Analysis of Its Beha vior and Performance, ” IEE E T ransacti ons on Systems, Man Cybernetic s - P art A: Systems and Humans , vol. 27, no. 5, pp. 553-568, September 1997. [3] L. Zadeh, “Fuzzy sets as a basis for a theory of possibility , ” Fuzzy Sets and Systems , vol. 1, no. 3, pp. 3-28, 1978. [4] D. Dubois and H. Prade, P ossibility Theory: An Approa ch to Computer- ized Proc essing of Uncertainty , Location: Plenum Press, New Y ork, 1988. [5] A.P . Dempster , “Upper and Lower probabilitie s induced by a multi va lued mapping, ” Annals of Mathe matical Statistic s , vol. 83, pp. 325-339, 1967. [6] G. Shafer , A mathematical theory of ev idence . L oca tion: Princet on Uni versity Press, 1976. [7] Ph. Smets, “Constructing the pignistic probabilit y function in a conte xt of uncertainty , ” Uncertainty in A rtific ial Intellig ence , vol. 5, pp. 29-39, 1990. [8] R.R. Y ager , “On the Dempster-Shafe r Framew ork and Ne w Combination Rules, ” Informations Sciences , vol. 41, pp. 93-137, 1987. [9] D. Dubois and H. Prade, “Represent ation and Combination of uncertainty with belief functio ns and possibili ty measures, ” Computational Intelli - genc e , vol. 4, pp. 244-264, 1988. [10] Ph. Smets, “The Combinati on of E vide nce in the Transfe rable Belief Model, ” IEEE T ransaction s on P attern Analysis and Machine Intel lig ence , vol. 12, no. 5, pp. 447-458, 1990. [11] T . Inagaki, “Indepe ndence between safety-control policy and multiple - sensors schemes via Dempster-Shafer theory , ” IEEE T ransaction on re liabil ity , vol. 40, no. 2, pp. 182-188, 1991. [12] Ph. Smets, “Belie f functions: the Disjuncti ve Rule of Combinat ion and the Generalized Bayesia n Theorem, ” International Journa l of A ppr oxi- mate Reasoning , vol. 9, pp. 1-35, 1993. [13] A. Josang, M. Daniel and P . V annoorenbe rghe, “Strate gies for Combin- ing Conflicting Dogmatic Belief, ” International Confer ence on Informa- tion Fusion , Cairns, Australia , 7-10 July 2003. [14] F . Smarandache and J. Dezert, “Informati on Fusion Based on New Proportion al Conflict Redistri buti on Rules, ” Internation al Confer ence on Informatio n Fusion , Philadelphia , USA, 25-29 July 2005. [15] M.C. Florea, J. Dezert, P . V alin, F . Sm ara ndache and A .L. Jousselme, “ Adaptati ve combination rule and proportional conflict redistrib ution rule for information fusion, ” COGni tive systems with Interactiv e Sensors , Paris, France , March 2006. [16] A. Martin and C. Osswald, “ A new generali zation of the proportiona l conflict redistrib ution rule stable in terms of decision , ” Applications and Advances of DSmT for Information Fusion, Book 2 , American Research Press Rehoboth, F . Smarandac he and J. Dezert, pp. 69-88 2006. [17] T . Denœux, “The cautious rule of combinat ion for belief functions and some exte nsions, ” Internatio nal Confer ence on Information Fusion , Florence , Italy , 10-13 July 2006. [18] Ph. Sm et s, “The α -junctions: the commutati ve and associati ve non inter - acti ve combinat ion operato rs applic able to belie f functio ns, ” Qualitati ve and quanti tative pratical reason ing , Springer V erlag, D. Gabbay and R. Kruse and A. Nonnengart and H.J . Ohlba cg, Berlin , pp. 131-153, 1997. [19] E. Lefe vre, O. Colot and P . V annoore nbergh e, “Belief functio n combina- tion and conflict management, ” Information F usion , vol. 3, pp. 149-162, 2002. [20] A. Appriou, “ Approche g ´ en ´ erique de la gestion de l’incertain dans les processus de fusion multisenseur , ” T raitement du Signal , vol. 22, no. 4, pp. 307-319, 2005. [21] C. Osswald and A. Ma rtin, “Underst anding the lar ge family of De mpster- Shafer theory’ s fusion operators - a decisio n-based measure, ” Interna- tional Confer ence on Information Fusion , Florence, Italy , 10-13 July 2006. [22] F . Smarandache and J. Dezert, “Proportion al Conflict Redistribut ion Rules for Information Fusion, ” A ppli cations and A dvan ces of DSmT for Informati on F usion, Book 2 , American Research Press Rehoboth, F . Smarandac he and J. Dezert, pp. 3-68, 2006. [23] A.-L. Jousselme, D. Grenier and E. Boss ´ e, “ A new distance between two bodies of eviden ce, ” Information Fusion , vol. 2, pp. 91-101, 2001.