New Measures of Specificity and Bayesianity of Basic Belief Assignments and of Fusion Rules

Transcription

1 New Measures of Specificity and Bayesianity of Basic Belief Assignments and of Fusion Rules Flentin Smarandache, Arnaud Martin, Christophe Osswald ENSIETA E 3 I -EA3876 Labaty rue François Verny 9806 Brest Cedex 9, France Flentin.Smarandache@ensieta.fr, Arnaud.Martin@ensieta.fr, Christophe.Osswald@ensieta.fr Abstract. In this paper we introduce the degree of specificity of a mass, which is the distance between a mass and its most specific associated mass, and we measure the specificity of a fusion rule. Also, we determine the Bayesianity of a mass. We propose certain new distances between masses as well. Keywds: belief function, uncertainty, conflict, auto-conflict, specificity of a mass, specificity of a fusion rule, distance between masses, Bayesianity. Introduction. In der f the paper to be self-contained we list the main distances between masses and also the three pignistic transfmations. We list some specificity measures from the known literature and we also propose some new ways of measuring the degree of specificity of a bba. We also list the known distances and propose some new approaches. The most specific mass associated to a given mass is defined. The degree of uncertainty of a set and the degree of Bayesianity/non-Bayesianity is also defined at the end of the paper.. Specificity. Yager [] has defined the specificity measure of a mass m(.) defined on X as): S m ma ( ) () A A X A One has /n Sm, where n is the cardinality of X, minimum value occurs f the vacuous belief function mvbf(x) =, and maximum value occurs f any Bayesian mass.

2 In our opinion this fmula should be adjusted in der to get the minimum specificity value 0 f the vacuous belief function and f unifmly distributed masses (i.e. m(ai) = /p f each A, A,, Ap in the fusion space, while the specificity should be f m(a)= where A has the cardinality. Because, f example, if we have three Bayesian bba s defined on ={A, B, C}, where all A, B, C are singletons, and all their intersections are empty: A B C m /3 /3 /3 m ½ ½ 0 () m3 0 0 we get the same specificity f all three of them, i.e. Sm = Sm = Sm3 =, while intuitively there should be 0 Sm < Sm < Sm3 = since m is me specific than m, and m3 is the most specific than all of them. Uncertainty results from randomness and non-specificity. Non-specificity is related to vagueness imprecision (Ristic and Smets, []).. Degree of Specificity. We define a new degree of specificity measure of a bba m(.) in the following way: SSMO(m) = -d(m, ms) (3) where ms(.) is the most specific mass associated with m(.), and d(.) is a distance function whose values are in the interval [0, ] between the masses m(.) and ms(.). As distance between masses we prefer to use Jousselme distance which is the most accurate one, but other mass distance can be used as well: Euclidean distance, Bhattacharyya s distance, Tessem s distance, GPT distance, DSmPε distance, etc. The restriction is that each mass distance should have the values in the closed interval [0, ]. We recall some mass distance fmulas: 3.) Jousselme Distance: T dj( m, m) ( m m) D( m m) (4)

3 where D is an n n matrix whose elements are D( A, B) AB A B. (5) It is easier to take the Jousselme matrix cresponding to the union of all focal elements of bba s m and respectively m, instead of considering the large size matrix defined on the whole fusion space Θ. The result is the same. 3.) Euclidian Distance: de m, m [ m( A) m( A)]. (6) A 3.3) Bhattacharya Distance: db m, m [ m( A) m( A)]. (7) A 3.4) Tessem s Distance between pignistic probabilities BetP and BetP associated to the bba s m, and respectively m (distance between betting commitments from two pignistic transfmations) of the two bba s is defined as [7, 8]: dt(m,m) = max BetP(A)-BetP(A) (8) A, A = singleton 3.5) This distance extended to DSmT gives the GPT distance, between generalized pignistic probabilities: ddsmt(m,m) = max GPT(A)-GPT(A) (9) A, c(a) = where c(a) is the DSm cardinality of A. 3.6) A DSmPε Distance, extension of the above two distances should be: ddsmt ε (m,m) = max DSmP(ε)(A)-DSmP(ε)(A). (0) A, c(a) = 3.7) Another idea would be to take, instead of max, the average arithmetic operat in the above three fmulas of distances between pignistic probabilities, and we get pseudo-distances (the axiom of triangular inequality is in general not verified). Other way to define the distance between two masses is the following: ) Use the pignistic transfmation (BetP), generalized pignistic transfmation (GPT), DSmPε to transfm a given bba into a Bayesian mass, and then make a city-block distance between the masses of singletons:

4 dpt(m,m) = BetP( A) BetP( A) () A c( A) respectively in DSmT: dgpt(m,m) = GPT ( A) GPT ( A) () A c( A) DSmP ( A) DSmP ( A). (3) ddsmpε(m,m) = A c( A) In a me general way one can construct the above fmulas by inserting a lambda parameter λ : dpt(m,m) = ( BetP( A) BetP( A) λ ) / λ (4) A c( A) dgpt(m,m) = ( GPT ( A) GPT ( A) λ ) / λ (5) A c( A) ddsmpε(m,m) = ( DSmP ( A) DSmP ( A) λ ) / λ (6) A c( A) Remark that d(m, ms) represents the quantity that misses to m(.) in der f m(.) to becoming the most specific mass (closest to it). 4. The most specific mass associated with a given mass. We define the most specific mass ms(.) associated to a given mass m(.) as follows: ms(amax) =, where AmaxG. The problem is how to find Amax? a) If m(.) is Bayesian, then we compute Amax = max{m(x), X ={,, }} (7) and therefe ms(.) is considered to be ms(amax) = ; if there exist many Amax s (i.e. having the same maximal mass), we take any of them because the distance between m(.) and any of these most specific masses ms (Amax ) = associated with m(.) will be the same.

5 b) If m(.) is non-bayesian we can compute Amax in a similar way: Amax = {max{m(x)/card(x), X ={,, }} }, (8) but if there exist me maximal masses we take the element with the smallest cardinality. c) Another method f the case when m(.) is non-bayesian is to use the Smets s pignistic transfmation, GPT, DSmPε in der to transfm the non-bayesian mass m(.) into a Bayesian mass m (.) cresponding to m(.). And then use case a). We recall below the three fmulas of the pignistic transfmations: 4.. The pignistic probability, which transfms a basic belief assignment (bba) to a Bayesian probability, was introduced by Smets [5] and defined as: f all X Θ. The following double inequality holds: Bel(A) BetP(A) Pl(A). (0) 4.. An extension of the pignistic probability from DST and TBM to DSmT is the following (Dezert, Smarandache, Daniel, [9]): cm( X Y) GPT ( X ) m( Y) () cm( Y) YD^ Y (9) where cm(y) is the DSm cardinality of Y in the given model M, which means the number of disjoint parts of Y in the Venn diagram A generalization of the pignistic probability is DSmPε (Dezert & Smarandache, [6]): YG^ Z X Y c( Z) ZY c( Z) m( Z) c( X Y ) DSmP ( X ) m( Y) m( Z) c( Y) () where ε 0 is a tuning parameter and G^ cresponds to the generic set (power set Θ, hyperpower set D Θ, super-power set S Θ -- including eventually all integrity constraints (if any) of the model M); c(z) denote the DSm cardinals of the set Z;

6 ε allows to reach the maximum specificity value of the approximation of m(.) into a subjective probability measure; the smaller ε is the greater the specificity of the mass m(.) is. DSmP provides a better specificity than the pignistic probability and other transfmations that map the belief masses to Bayesian probabilities (Sudano s, Cuzzolin s). Remark: the most specific mass m s(.) can also be defined in a different user s need f solving a particular application. F DSmT, Amax may be, f example, a non-empty intersection, let s say A B if m(a B) is maximal and Card(A B) =. F UFT (Unification of Fusion Theies) (Smarandache, [3] ) Amax may be, f example, a difference of elements, let s say A-B if m(a-b) is maximal and Card(A-B) =. Examples: A A B m (3) m The most specific mass associated with both of them, m and m, is ms(a) =. dj(m, ms) = 0.88 where dj(.,.) is Jousselme distance, whence the specificity of m is S(m) = = dj(m, ms) = , whence S(m) = = So, m is me specific than m. If we take m3(a) = 0.5, m3(b) = 0.5, with A B =, the specificity of m3 is smaller than m s specificity although the masses are the same. The above ms(a) = can serve as the most specific mass associated with m3, then dj(m3, ms) = 0.50, whence S(m3) = = 0.50 < Me Bayesian examples in der to observe the convergence of specificity:

7 A B C dj(mi, m0) S(mi) mo m m m m m m9 /3 /3 / m The smallest specificity of a Bayesian mass is when the bba has a unifm distribution (in these examples m9), and the largest specificity is of course when the mass of a singleton is (m0). The specificity increases when the differences between the mass of the largest singleton and the masses of other singletons are getting bigger: S(m5) < S(m6) < S(m7) < S(m8). In the case when one has three disjoint singletons and the largest mass of one of them is 0.45, we have the minimum specificity when the masses of B and C are getting further from the mass of A (m8). The problem becomes me complex f non-bayesian masses about how to find the most specific mass. Let s consider these examples where we apply method b): A B C A B A C B C A B C msi dj(mi, m0) S(mi) m ms(a B)= {if one decides on ignances}

8 ms (A)= {if one decides on singletons} m ms(b)= m ms3(a)= Table F m (second case), m and m3 we applied method c). 5. Contradiction of an element with respect to a mass. The contradiction of an element A with respect to a mass mi(.) is the distance between the masses mi(.) and ma(.), where ma(a)=: c(a) = d(mi, ma) (4) We define the weighted contradiction ci of a mass mi(.) as: ci mi( A) d( mi, ma ) (5) AG^ and the contradiction between two masses m(.) and m(.) as: c(m, m) = c+c-d(m, m) (6) 6. Measure of specificity of a fusion rule. Let s consider two masses m and m. One applies the rules R, R,, Rp m(r)m, m(r)m,, m(rp)m, and then one computes the specificity measure of each result and compare the results: what specificity (cresponding to what fusion rule) is bigger. This can be generalized to s bba s in the following way: If we combine me bba s, m, m,, ms, in the same way we compute the specificity of each mass mi, and then the arithmetic average of these specificity. Afterwards, we fusion all these s masses simultaneously with a fusion rule, and then compute the specificity of the resulted mass. Let s take the masses m4 and m0 that have the same most specific mass m0(a)=. {It is true that the most specific mass of m4 could equally be mb(b)= too.}

9 6.. Bayesian Example. A B C A B A C B C A B C m S of m i d J(m i, m S) S(m i) m m S(A)= m m S(A)= mconj 0. 4 mds msmets mdp 0. 4 myage r mpcr5 6 mfle a m S(A)= m S(A)= m S(A)= m S(A)= m S(A)= m S(A B)= m S(A)= m S(A)= mdis m S(A)= m S(A B)= mmean m S(A)= mdpmixt m S(A)= m S(A B)= m Martin m S(A)=

10 Osswald- mixt m DPCR 0. 4 m MDPC R mzhan g 4 m S(A B)= m S(A)= m S(A B)= m S(A)= m S(A B)= m S(A)= Table. 6.. Non-Bayesian Example. A B A B m S of m i d J(m i, m S) S(m i) m ms(a)= m ms(b)= mconj ms(a)= mds ms(a)= msmets ms(a)= myager ms(a)= mdisjunctive ms(a)= mflea ms(a)= m PCR5/m PCR ms(a)= m mean ms(a)= m DuboisPrade ms(a)= m DPmixt ms(a)= m MartinOsswald ms(a)= mixt mdpcr ms(a)= mmdpcr ms(a)= mzhang ms(a)= Table 3.

11 In [4], Osswald and Martin computed the distance between fusion operat results using a class of random belief functions. Roussilhe [0] compared the fusion rules from a statistical point of view. Randomly generating masses and fusing them 000 times using various fusion rules, then with the classical pignistic transfmation each non-bayesian result was converted into a Bayesian result, which was interpreted as a random variable. Then one calculated the crelation coefficients (similarities) as in statistics between the decision vects Continuous Frame Example. [, ] [3, 5] [, ] [3, 5] m S of m i d J(m i, m S) S(m i) m ms([3, m ms([3, mconj ms([3, m ms([3, m ms([3, mconj ms([3, mds ms([3, msmets ms([3, myager ms([3, mdisjunctive ms([3, mflea ms([3, m PCR5/m PCR ms([3, m mean ms([3, m DuboisPrade ms([3, m DPmixt ms([3, m MartinOsswaldmixt ms([3, mdpcr ms([3,

12 mmdpcr ms([3, mzhang ms([3, Table In der to define the Bayesianity (non-bayesianity) we present the measures of uncertainty of a set and of a mass. 7. Measure of Uncertainty of a Set. In DST (Dempster-Shafer s They), Hartley defined the measure of uncertainty of a set A by: I( A) log A, f A \, (7) where A is the cardinality of the set A. We can extend it to DSmT in the same way: I( A) log A, f AG \ (8) where G is the super-power set, and A means the DSm cardinality of the set A ; in the case of Shafer s model (i.e. all intersections of the sets in the frame of discernment are empty), DSm cardinality coincides with classical cardinality in DST. We even improve it to a degree of uncertainty of a set: s d : G \ 0,, UdS (A) = log(a)/ log It (9) s If A is a singleton, i.e. A, then d A 0 (minimum degree of uncertainty of a set), s F the total ignance I t, since I t is the maximum cardinality, we get d It (maximum degree of uncertainty of a set). F all other sets X from G \, whose cardinality is in between and I t, we have s 0 X. d We consider our degree of uncertainty of a set wks better than Hartley Measure since it is referred to the frame of discernment. Let s see an Example. AB, and A B, we have the model If A B Fig. I( A) log A log

13 s While A d log A log log A B log Example. A, B, C, and A B, but A C, B C, we have the model If A B C Fig. s While A I( A) log A as in Example. d log A log log A B C log It is nmal to have a smaller degree of uncertainty of set A when the frame of discernment is larger, since herein the total ignance has a bigger cardinality. There are two types of uncertainty: nonspecificity and discd. 8. Generalized Hartley Measure of uncertainty f masses is defined as: GH ( m) m( A)log A A \ (30) This is also called non-specificity. From DST we simply extend the GM(.) to DSmT as follows: GH ( m) m( A)log A AG \ (3) Degree of Uncertainty ( Degree of non-bayesianity) of a mass. We go further and define a degree of uncertainty of a mass m as log A M d ( m) m( A) m( A)log It A log I AG \ t AG \ where I t is the total ignance. M If m() is a mass whose focal elements are only singletons then d ( m) 0 (minimum uncertainty degree of a mass). M m I, then ( m) (maximum uncertainty degree of a mass). If t d (3)

14 M F all other masses m() we have 0 ( m). d Whence we can define a Degree of Bayesianity of a mass, which means how close is a bba to a Bayesian (probability) measure: log B( m) ( ) log AG A m A. (33) \ It If m(.) is Bayesian, then B(m) =. F the vacuous belief assignment mvba(it) = we have B(mVBS) = 0 If m(.) is non-bayesian, with m(.) mvba(.), then 0 < B(m) <. Examples: A B C A B A C B C A B C NB(m i) B(m i) m m m m m m m m m m m Table 5

15 Remark. The degree of uncertainty ( non-bayesianity) and respectively the degree of Bayesianity depend on the cardinality of the frame of discernment. F example B(m) = 0.75 in the frame = {A, B, C} where the cardinality of the total ignance It = A B C is 3, but if we consider m in the frame = {A, B, C, D} where the cardinality of the total ignance is 4, its Bayesianity increases: B(m ) = The large is the frame, the larger becomes the Bayesianity. 9. Conclusion. In this paper we introduced a new degree of specificity measure of a belief function, a degree of specificity measure of a set, and a degree of Bayesianity/non-Bayesianity of a belief function. We also introduced new distances between masses based on GPT and DSmP pignistic transfmations from DSmP. References [] R. Yager, Entropy and Specificity in the Mathematical They of Evidence, in R. Yager, L. Liu, A.P. Dempster, G. Shafer edits, Classic Wks of the Dempster-Shafer They of Belief Functions, Springer, 9-308, 008. [] Branko Ristic, Philippe Smets, The TBM global distance measure f the association of uncertain combat ID declarations, Infmation Fusion, 7 (006), [3] F. Smarandache, Unification of Fusion Theies (UFT), International Journal of Applied Mathematics & Statistics, Rokee, India, Vol., -4, 004. [4] C. Osswald, A. Martin, Understanding the large family of Dempster-Shafer they s fusion operats a decision-based measure, Fusion 006 International Conference, Flence, Italy, July 006. [5] P. Smets, Constructing the pignistic probability function in a context of uncertainty, Uncertainty in Artificial Intelligence 5 (990), pp [6]J. Dezert, F. Smarandache, A new probabilistic transfmation of belief mass assignment, in Proceedings of Fusion 008 Conference, Cologne, Germany, July 008. [7] B. Tessem, Approximations f efficient computations in the they of evidence, Artificial Intelligence, Vol. 6, No., June 993, pages [8] Mihai Cristian Flea, Éloi Bossé, Crisis Management using Dempster Shafer They: Using dissimilarity measures to characterize sources reliability, NATO-OTAN, RTO-MP-IST-086.

16 [9] Dezert J., Smarandache F., Daniel M., The Generalized Pignistic Transfmation, Proc. of the 7-th International Conference on Infmation Fusion, Fusion 004, Stockholm, Sweden, July 004. [0] Laurence Roussilhe, Comparaison d opérateurs de fusion, Master Professionel, Maître de stage Christophe Osswald, ENSIETA, Brest, France, 4 June 00.