J. Abellan and A. Masegosa Harmanec and Klir(1996)proposed the measure S"(m)which is equal to the maximum of the entropy (upper entropy) of the probability distributions verifying Bel(A)s AP(r)s Pl(A), VA CX. This set of probability distributions is the credal set associated with a b p a. m, and will be denoted as K Maeda and Ichihashi (1993)proposed a total uncertainty measure using the measures which quantifies the conflict and non-specificity contained in a b p a. on X following way MI(m)=I(m)+S(m) where 1(m)is used as a non-specificity function and S"(m)is used as a measure of conflict. 二Ez∞N一 This measure was analysed in Abellan and Moral (1999) Harmanec and Klir(1996)proposed S as a total uncertainty measure in DST, i.e.as a neasure that quantifies conflict and non-specificity, but they do not separate this into parts that quantify these two types of uncertainty on DST. More recently, abellan et al.(2006 proposed upper entropy as an aggregate measure on more general theories than dst, coherently separating conflict and non-specificity. These parts can also be obtained in DSt in a similar way. In DSt, we can consider where S(m) represents maximum entropy and S(m) represents minimum entropy on the credal Km associated to a b p a. m, with S(m) coherently quantifying the conflict part and (S-S)(m)its non-specificity part Quite recently, Jousselme et al.(2006)presented a measure to quantify ambiguity (discord or conflict and non-specificity)in DST, i.e. a total uncertainty measure on dst. This measure is based on the pignistic distribution on DST: let m be a b p a. on a finite set X, then the pignistic probability distribution BetPm, on all the subsets A in X is defined by A∩B BetPm(A)=>m(B For a singleton set A=[x], we have BetPm([))=CreB[m(B)/Bl]. Therefore, the ambiguity measure for a b p a. m on a finite set X is defined as M(m)=->BetPm(x)log(BetPm(x) 3. Basic properties of total uncertainty measures in DST In Klir and wierman(1998), we can find five requirements for a total uncertainty measure (TU in DST, i.e. for a measure which captures both conflict and non-specificity. Using the above notation, these requirements can be expressed in the following way Pl) Probabilistic consistency all the focal elements of a b P a. m are singletons hen a total uncertainty measure must be equal to the Shannon entropyHarmanec and Klir (1996) proposed the measure S* (m) which is equal to the maximum of the entropy (upper entropy) of the probability distributions verifying P BelðAÞ # x[ApðxÞ # PlðAÞ; ;A # X: This set of probability distributions is the credal set associated with a b.p.a. m, and will be denoted as Km. Maeda and Ichihashi (1993) proposed a total uncertainty measure using the above measures which quantifies the conflict and non-specificity contained in a b.p.a. on X in the following way: MIðmÞ ¼ IðmÞ þ S*ðmÞ; where I(m) is used as a non-specificity function and S* (m) is used as a measure of conflict. This measure was analysed in Abella´n and Moral (1999). Harmanec and Klir (1996) proposed S* as a total uncertainty measure in DST, i.e. as a measure that quantifies conflict and non-specificity, but they do not separate this into parts that quantify these two types of uncertainty on DST. More recently, Abella´n et al. (2006) proposed upper entropy as an aggregate measure on more general theories than DST, coherently separating conflict and non-specificity. These parts can also be obtained in DST in a similar way. In DST, we can consider S*ðmÞ ¼ S*ðmÞþðS* 2 S*ÞðmÞ; where S* (m) represents maximum entropy and S*ðmÞ represents minimum entropy on the credal Km associated to a b.p.a. m, with S*ðmÞ coherently quantifying the conflict part and ðS* 2 S*ÞðmÞ its non-specificity part. Quite recently, Jousselme et al. (2006) presented a measure to quantify ambiguity (discord or conflict and non-specificity) in DST, i.e. a total uncertainty measure on DST. This measure is based on the pignistic distribution on DST: let m be a b.p.a. on a finite set X, then the pignistic probability distribution BetPm, on all the subsets A in X is defined by BetPmðAÞ ¼ X B#X mðBÞ jA > Bj jBj : For a singleton set A ¼ {x}, we have BetPmð{x}Þ ¼ P x[B ½mðBÞ=jBj
. Therefore, the ambiguity measure for a b.p.a. m on a finite set X is defined as AMðmÞ ¼ 2 X x[X BetPmðxÞ log ðBetPmðxÞÞ: 3. Basic properties of total uncertainty measures in DST In Klir and Wierman (1998), we can find five requirements for a total uncertainty measure (TU) in DST, i.e. for a measure which captures both conflict and non-specificity. Using the above notation, these requirements can be expressed in the following way: (P1) Probabilistic consistency: when all the focal elements of a b.p.a. m are singletons, then a total uncertainty measure must be equal to the Shannon entropy: TUðmÞ ¼ X x[X mðxÞ log mðxÞ: 736 J. Abella´n and A. Masegosa Downloaded by [New York University] at 12:09 08 November 2011