AN ALGORITHM FOR QUASI-ASSOCIATIVE AND QUASI- MARKOVIAN RULES OF COMBINATION IN INFORMATION FUSION

Transcription

1 ehnial Sienes and Applied Mathematis AN ALGORIHM FOR QUASI-ASSOCIAIVE AND QUASI- MARKOVIAN RULES OF COMBINAION IN INFORMAION FUSION Florentin SMARANDACHE*, Jean DEZER** *University of New Mexio, Gallup, USA, **ONERA/DIM/IED, Châtillon, Frane Abstrat: In this paper one proposes a simple algorithm of ombining the fusion rules, those rules whih first use the onjuntive rule and then the transfer of onfliting mass to the non-empty sets, in suh a way that they gain the property of assoiativity and fulfill the Markovian requirement for dynami fusion. Also, a new fusion rule, SDL-improved, is presented. Keywords: Conjuntive rule, partial and total onflits, Dempster s rule, Yager s rule, BM, Dubois-Prade s rule, Dezert-Smarandahe lassi and hybrid rules, SDL-improved rule, quasiassoiative, quasi-markovian, fusion algorithm. 1. INRODUCION We first present the formulas for the onjuntive rule and total onflit, then try to unify some theories using an adequate notation. Afterwards, we propose an easy fusion algorithm in order to transform a quasiassoiative rule into an assoiative rule, and a quasi-markovian rule into a Markovian rule. One gives examples using the DSm lassi and hybrid rules and SDL-improved rule within DSm. One studies the impat of the VBF on SDLi and one makes a short disussion on the degree of the fusion rules ad-ho-ity. 2. HE CONJUNCIVE RULE For n 2 let = {t1, t 2,, t n } be the frame of disernment of the fusion problem under onsideration. We need to make the remark that in the ase when these n elementary hypotheses t 1, t 2,, t n are exhaustive and exlusive one an use the Dempster-Shafer heory, Yager s, BM. Dubois-Prade heory, while for the ase when the hypotheses are not exlusive one an use Dezert-Smarandahe heory, while for nonexhaustivity one uses BM. Let m: 2 [0,1] be a basi belief assignment or mass. he onjuntive rule works in any of these theories, and it is the following in the first theories: for A 2, (A) = m (X)m (X) (1) m 1 2 X,Y 2 X Y= A while in DSm the formula is similar, but instead of the power set 2 one uses the hyperpower set D, and similarly m: D [0,1] be a basi belief assignment or mass: for A D, (A) = m (X)m (X) (2) m 1 2 X,Y D X Y= A he power set is losed under, while the hyper-power set is losed under both and. Formula (2) allows the use of intersetion of sets (for the non-exlusive hypotheses) and it is alled DSm lassi rule. he onjuntive rule (1) and its extension (2) to DSm are assoiative, whih is a nie property needed in fusion ombination that we need to extend to other rules derived from it. Unfortunately, only three fusion rules derived from the onjuntive rule are known as assoiative, i.e. Dempster s rule, Smets s 5

2 An algorithm for quasi-assoiative and quasi-markovian rules of ombination in information fusion BM s rule, and Dezert-Smarandahe lassi rule, the others are not. For unifiation of theories let s note by G either 2 or D depending on theories. he onfliting mass k12 is omputed similarly: = m ( θ ) = m (X)m (X) (3) k X,Y G X Y= θ Formulas (1), (2), (3) an be generalized for any number of masses s ASSOCIAIVIY he propose of this artile is to show a simple method to ombine the masses in order to keep the assoiativity and the Markovian requirement, important properties for information fusion. Let m 1, m 2, m 3 : G [0,1] be any three masses, and a fusion rule denoted by operating on these masses. One says that this fusion rule is assoiative if: ((m1 m2) m3)(a) = (m1 (m2 m3))(a) f or all A G, (4) whih is also equal to (m1 m2 m3)(a) for all A G. (5) 4. MARKOVIAN REQUIREMEN Let m 1, m 2,, m k : G [ 0,1] and k 2 masses, and a fusion rule denoted by operating on these masses. One says that this fusion rule satisfies Markovian requirement if: (m1 m2... mn )(A) = ((m1 m2... mn 1) mn )(A) for all A G. (6) Similarly, only three fusion rules derived from the onjuntive rule are known satisfying the Markovian Requirement, i.e. Dempster s rule, Smets s BM s rule, and Dezert- Smarandahe lassi rule. he below algorithm will help transform a rule into a Markovian rule. 5. FUSION ALGORIHM A trivial algorithm is proposed below in order to restore the assoiativity and 6 Markovian properties to any rule derived from the onjuntive rule. Let s onsider a rule formed by using: first the onjuntive rule, noted by, and seond the transfer of the onfliting mass to non-empty sets, noted by operator O (no matter how the transfer is done, either proportionally with some parameters, or transferred to partial or total ignoranes and/or to the empty set; if all onfliting mass is transferred to the empty set, as in Smets s rule, there is no need for transformation into an assoiative or Markovian rule sine Smets s rule has already these properties). Clearly = O( ). he idea is simple, we store the onjuntive rule s result (before doing the transfer) and, when a new mass arises, one ombines this new mass with the onjuntive rule s result, not with the result after the transfer of onfliting mass. Let s have two masses m 1, m 2 defined as above. a) One applies the onjuntive rule to m 1 and m 2 and one stores the result: m 1 m2 = m C(1,2) (by notation). b) One applies the operator O of transferring onfliting mass to the non-empty sets, i.e. O(m C(1,2) ). his alulation ompletely does the work of our fusion rule, i.e. m 1 m 2 = O(m C(1,2) ) that we ompute for deision-making proposes. ) When a new mass, m3, arises, we ombine using the onjuntive rule this mass m3 with the previous onjuntive rule s result m(12), not with O(m C(1,2) ). herefore: m C(1,2) m 3 = m C(C(1,2)3) (by notation). One stores this results, while deleting the previous one stored. d) Now again we apply the operator O to transfer the onfliting mass, i.e. ompute ô m C(C(1,2)3) needed for deision-making. e) And so one the algorithm is ontinued for any number n 3of masses. he properties of the onjuntive rule, i.e. assoiativity and satisfation of the Markovian requirement, are transmitted to the fusion rule too. his is the algorithm we use in DSm in order to onserve the assoiativity and Markovian requirement for DSm hybrid rule and SDL improved rule for n 3.

3 ehnial Sienes and Applied Mathematis Depending on the type of problem to be solved we an use in DSm either the hybrid rule, or the SDL rule, or a ombination of both (i.e., partial onfliting mass is transferred using DSm hybrid, other onfliting mass is transferred using SDL improved rule). Yet, this easy fusion algorithm an be extended to any rule whih is omposed from a onjuntive rule first and a transfer of onfliting mass seond, returning the assoiativity and Markovian properties to that rule. One an remark that the algorithm gives the same result if one applies the rule to n 3masses together, and then one does the transfer of onfliting mass. Within DSm we designed fusion rules that an transfer a part of the onfliting mass to partial or total ignorane and the other part of the onfliting mass to non-empty initial sets, depending on the type of appliation. A non-assoiative rule that an be transformed through this algorithm into an assoiative rule is alled quasi-assoiative rule. And similarly, a non-markovian rule than an be transformed through this algorithm into a Markovian rule is alled quasi-markovian rule. 6. SDL-IMPROVED RULE Let = {t 1, t 2,, t n } be the frame of disernment and two masses m 1, m 2 : G [0,1]. One applies the onjuntive rule (1) or (2) depending on theory, then one alulates the onfliting mass (3). In SDL improved rule one transfers partial onfliting masses, instead of the total onfliting mass. If an intersetion is empty, say A B = Ø, then the mass m(a B) is transferred to A and B proportionally with respet to the non-zero sum of masses assigned to A and respetively B by the masses m1, m 2. Similarly, if another intersetion, say A C D = Ø, then again the mass m( A C D) is transferred to A, C, and D proportionally with respet to the nonzero sum of masses assigned to A, C and respetively D by the masses m1, m 2. And so on til all onfliting mass is distributed. hen one umulates the orresponding masses to eah non-empty set. For two masses one has the formula: For Ø A D, m ( A) = m ( X ) m ( X ) + 12 SDLi ( A) X G X A=Φ 1 X, Y G X Y = A m1( X ) m ( A) + m1( A) m ( A) + ( X ) 12 2 ( X ) (7) where 12 (A) is the non-zero sum of the mass matrix olumn orresponding to the set A, i.e. 12 (A) = m 1 (A) + m 2 (A) 0. (8) For more masses one applies the algorithm to formulas (7) and (8). 7. AD-HOC-ICIY OF FUSION RULES Eah fusion rule is more or less ad-ho. Same thing for SDL improved. here is up to the present no rule that fully satisfies everybody. Let s analyze some of them. Dempster s rule transfers the onfliting mass to non-empty sets proportionally with their resulting masses. What is the reasoning for doing this? Just to swallow the masses of nonempty sets in order to sum up to 1? Smets s rule transfers the onfliting mass to the empty set. Why? Beause, he says, we onsider on open world where unknown hypotheses might be. Not onvining. Yager s rule transfers the onfliting mass to the total ignorane. Should the onfliting mass be ignored? Dubois-Prade s rule and DSm hybrid rule transfers the onfliting mass to the partial and total ignoranes. Not ompletely justified either. SDL improved rule is based on partial onfliting masses, transferred to the orresponding sets proportionally with respet to the non-zero sums of their assigned masses. But other weighting oeffiients an be found. Inagaki (1991), Lefevre-Colot- Vannoorenberghe (2002) proved that there are infinitely many fusion rules based on the onjuntive rule and then on the transfer of the onfliting mass, all of them depending on the weighting oeffiients that transfer that onfliting mass. How to hoose them, what parameters should they rely on that s the question! here is not a measure for this. 7

4 An algorithm for quasi-assoiative and quasi-markovian rules of ombination in information fusion In my opinion, neither DSm hybrid rule nor SDLi rule are not more ad-ho than other fusion rules. No matter how you do, people will have objetions (Wu Li). 8. NUMERICAL EXAMPLES We show how it is possible to use the above fusion algorithm in order to transform a quasiassoiative and quasi-markovian rule into an assoiative and Markovian one. Let = {A, B, C}, all hypotheses exlusive, and two masses m1, m2 that form the orresponding mass matrix: A B A C m m Let s take the DSm hybride rule: Let s hek the assoiativity: a) First we use the DSm lassi rule and we get at time t 1 : m DSmC12 (A) = 0.38, m DSmC12 (B) = 0.10, m DSmC12 (A C) = 0.02, mdsmc12(a B) = 0,38, mdsmc12(b (A C)) = 0.12, and one stores this result. (S1) b) One uses the DSm hybrid rule and we get: m DSmH12 (A) = 0.38, m DSmH12 (B) = 0.010, m DSmH12 (A C) = 0.02, m DSmH12 (A B) = 0,38, m DSmH12 (A B C) = his result was omputed beause it is needed for deision making on two soures/masses only. (R1) ) A new masses, m 3, arise at time t 2, and has to be taken into onsideration, where m 3 (A) = 0.7, m 3 (B) = 0.2, m 3 (A C) = 0.1. Now one ombines the result stored at (S1) with m 3, using DSm lassi rule, and we get: m DSmC(12)3 (A) = 0.318, m DSmC(12)3 (B) = 0.020, m DSmC(12)3 (A C) = 0.002, m DSmC(12)3 (A B) = 0,610, m DSmC(12)3 (B (A C)) = 0.050, and one stores this result, while deleting (S1) (S2) d) One uses the DSm hybrid rule and we get: m DSmH(12)3 (A) = 0.318, m DSmH(12)3 (B) = 0.020, m DSmH(12)3 (A C) = 0.002, mdsmh(12)3 (A B) = 0,610, mdsmh(12)3(a B C) = his result was also omputed beause it is needed for deision making on three soures/masses only. (R2) e) And so on for as many masses as needed. First ombining the last masses, m 2, m 3, one gets: m DSmC23 (A) = 0.62, m DSmC23 (B) = 0.04, m DSmC23 (A C) = 0.02, m DSmC23 (A B) = 0,26, mdsmc23(b (A C)) = 0.06, and one stores this result. (S3) Using DSm hybrid one gets: m DSmH23 (A) = 0.62, m DSmH23 (B) = 0.04, m DSmH23 (A C) = 0.02, m DSmH23 (A B) = 0,26, mdsmh23 (A B C) = hen, ombining m1 with m DSmC23 {stored at (S3)} using DSm lassi and then using DSm hybrid one obtain the same result (R2). If one applies the DSm hybride rule to all three masses together one gets the same result (R2). We showed on this example that DSm hybrid applied within the algorithm is assoiative (i.e. using the notation DSmHa one has): DSmHa((m 1, m 2 ), m 3 ) = DSmHa (m 1, (m 2, m 3 )) = DSmHa (m 1, m 2, m 3 ) Let s hek the Markov requirement: a) Combining three masses together using DSm lassi: A B A C (M1) m m M one gets as before: m DSmC123 (A) = 0.318, m DSmC123 (B) = 0.020, m DSmC123 (A C) = 0.002, m DSmC123 (A B) = 0,610, m DSmC123 (B (A C)) = 0.050, and one stores this result in (S2). b) One uses the DSm hybrid rule to transfer the onfliting mass and we get: m DSmH123 (A) = 0.318, m DSmH123 (B) = 0.010, m DSmH123 (A C) = 0.002, m DSmH123 (A B) = 0,610, mdsmh123(a B C) = ) Suppose a new mass m 4 arises m 4 (A) = 0.5, m 4 (B) = 0.5, m 4 (A C) = 0. Use DSm lassi to ombine m 4 with m DSmC123 and one gets: m 4 (A) = 0.5, m 4 (B) = 0.5, m 4 (A C) = 0. Use DSm lassi to ombine m4 and m DSmC123 and one gets: m DSmC(123)4 (A) = 0.160, m DSmC(123)4 (B) = 0.010, m DSmC(123)4 (A C) = 0, mdsmc(123)4(a B) = 0,804, mdsmc(123)4(b (A C)) = 0.026, and one stores this result in (S3). d) Use DSm hybrid rule: m DSmH(123)4 (A) = 0.160, m DSmH(123)4 (B) = 0.010, m DSmH(123)4 (A C) = 0, mdsmh(123)4(a B) = 0,804, m DSmH(123)4 (A B C) = (R4) 8

5 ehnial Sienes and Applied Mathematis Now, if one ombines all previous four masses, m1, m2, m3, m4, together using first the DSm lassi then the DSm hybrid one still get (R4). Whene the Markovian requirement. We didn t take into aount any disounting of masses Let s use the SDL improved rule on the same example. a) One onsiders the above mass matrix (M1) and one ombines m 1 and m 2 using DSm lassi and one gets as before: m DSmC12 (A) = 0.38, m DSmC12 (B) = 0.10, m DSmC12 (A C) = 0.02, m DSmC12 (A B) = 0,38, m DSmC12 (B (A C)) = 0.12, and one stores this result in (S1). b) One transfers the partial onfliting mass 0.38 to A and B respetively: x/1 = y/0.7 = 0.38/1.8; whene x= , y= One transfers the other onfliting mass 0.12 to B and A C respetively: z/0.7 = w/0.3 = 0.12/1; whene z=0.084, w= One umulates them to the orresponding sets and one gets: msdli12(a) = = ; m SDLi12 (B) = = ; m SDLi12 (A C) = = ) One uses the DSm lassi rule to ombine the above m 3 and the result in (S1) and one gets again: m DSmH(12)3 (A) = 0.318, m DSmH(12)3 (B) = 0.020, m DSmH(12)3 (A C) = 0.002, mdsmh(12)3 (A B) = 0,610, mdsmh(12)3(a B C) = 0.050, and and one stores this result in (S2) while deleting (S1). d) One transfers the partial onfliting masses to A and B respetively, and to B and A C respetively. hen one umulates the orresponding masses and one gets: msdli(12)3(a) = ; m SDLi(12)3 (B) = ; m SDLi(12)3 (A) = Same result we obtain if one ombine first m 2 and m 3, and the result ombine with m 1, or if we ombine all three masses m 1, m 2, m 3 together. 9. VACUOUS BELIEF FUNCION SDLi seems to satisfy Smets s impat of VBF (Vauum Belief Funtion. i.e. m()=1), beause there is no partial onflit ever between the total ignorane and any of the sets of G. Sine in SDLi the transfer is done after eah partial onflit, will reeive no mass, not being involved in any partial onflit. hus VBF ats as a neutral elements with respet with the omposition of masses using SDLi. he end ombination does not depend on the number of VBF s inluded in the ombination. Let s hek this on the previous example. Considering the first two masses m 1 and m 2 in (M1) and using SDLi one got: m SDLi12 (A) = ; m SDLi12 (B) = ; m SDLi12 (A C) = Now let s ombine the VBF too: A B A C A B C (M2) VFB m m a) One uses the DSm lassi rule to ombine all three of them and one gets again: m DSmC(VFB) (A) = 0.38, m DSmC(VFB) (B) = 0.10, m DSmC(VFB) (A C) = 0.02, mdsmc(vfb) (A B) = 0,38, m DSmC(VFB) (B (A C)) = 0.12, mdsmc(vfb)( A B C) = 0 and one stores this result in (S1). b) One transfers the partial onfliting mass 0.38 to A and B respetively: x/1 = y/0.7 = 0.38/1.8; whene x= , y= One transfers the other onfliting mass 0.12 to B and A4C respetively: z/0.7 = w/0.3 = 0.12/1; whene z=0.084, w= herefore nothing is transferred to the mass of A B C, then the results is the same as above: m SDLi12 (A) = ; m SDLi12 (B) = ; m SDLi12 (A C) = CONLUSIONS We propose an elementary fusion algorithm that transforms any fusion rule (whih first uses the onjuntive rule and then the transfer of onfliting masses to non-empty sets, exept for Smets s rule) to an assoiative and Markovian rule. his is very important in information fusion sine the order of ombination of masses should not matter, and for the Markovian requirement the algorithm allows the storage of information of all previous masses into the last result (therefore not neessarily to store all the masses), whih later will be ombined with the new mass. 9

6 An algorithm for quasi-assoiative and quasi-markovian rules of ombination in information fusion In DSm, using this fusion algorithm for n 3 soures, the DSm hybrid rule and SDLi are ommutative, assoiative, Markovian, and SDLi also satisfies the impat of vauous belief funtion. BIBLIOGRAPHY 1. Daniel, M. (2003). Assoiativity in Combination of belief funtions; a derivation of minc ombination. Soft Computing, 7(5): Dubois, D., Prade, H. (1992). On the ombination of evidene in various mathematial frameworks. In J. Flamm and. Luisi. Reliability Data Colletion and Analysis. Brussels: ECSC, EEC, EAFC, Inagaki,. (1991). Interdependene between safety-ontrol poliy and multiple-senor shemes via Dempster- Shafer theory. IEEE rans. on reliability, Vol. 40, no. 2: Lefevre, E., Colot, O., Vannoorenberghe, P. (2002). Belief funtions ombination and onflit management. Information Fusion Journal. Vol. 3, No. 2: Murphy, C. K. (2000). Combining belief funtions when evidene onflits. Deision Support Systems. Vol. 29: Shafer, G. (1976). A Mathematial heory of Evidene. Prineton, NJ: Prineton Univ. Press. 7. Smarandahe, F., Dezert, J. (Eds). (2004). Appliations and Advanes of DSm for Information Fusion. Rehoboth: Am. Res. Press. 8. Smets, P. (2000). Quantified Epistemi Possibility heory seen as an Hyper Cautious ransferable Belief Model Voorbraak, F. (1991). On the justifiation of Dempster's rule of ombination. Artifiial Intelligene, 48: Yager, R. R. (1983). Hedging in the ombination of evidene. Journal of Information &Optimization Sienes, Vol. 4, No. 1: Yager, R. R. (1985). On the relationships of methods of aggregation of evidene in expert systems. Cybernetis and Systems, Vol. 16: Zadeh, L. (1984). Review of Mathematial theory of evidene, by Glenn Shafer. AI Magazine. Vol. 5, No. 3: Zadeh, L. (1984). A simple view of the Dempster-Shafer theory of evidene and its impliation for the rule of ombination. AI Magazine 7, No.2: