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An alternative approach to the handling of subnormal possibility distributions
Fuzzy Sets and Systems 24 (1987) 123-126 North-Holland
123
SHORT COMMUNICATION
AN ALTERNATIVE APPROACH TO THE HANDLING SUBNORMAL POSSIBILITY DISTRIBUTIONS - A Critical Comment on a Proposal by Yager Didier DUBOIS Universitt! Narbonne,
OF
and Henri PRADE
Paul Sabatier Laboratoire 31062 Toulouse, France
Langages
et Systhes
Informatiques,
118, route
de
Received January 1987 In this short note a definition of the necessity measure in case of a subnormal possibility distribution already proposed by the authors is recalled. This definition both satisfies the axiomatics of necessity measures and preserves the desirable inequality between possibility and necessity measures. The handling of subnormal basic probability assignments in the more general framework of Shafer’s belief functions is also discussed. Keywords:
Possibility distribution,
Necessity measure, Normalization,
Belief function.
1. Introduction A normal possibility distribution such that
it, defined on U, is a function from U to [0, l]
s,llpun(u) = 1.
(1)
Then, a so-called possibility measure 17 based on n, is defined by [ll]
(2) where A is an ordinary or a fuzzy subset of U. Note that choices other than min are possible in (2); see [7] for instance. A measure of necessity N [l, 41, is naturally associated with n by the duality relation VA c U,
A’(A) = 1 - n(A),
where A denotes the complement have VA c U,
(3)
of A defined by ~2 = 1 - Pi. Consequently
N(A) = hf, max(pA(u),
1 - n(u)).
0165-0114/87/$3.50 @ 1987, Elsevier Science Publishers B.V. (North-Holland)
we (4)
124
D. Dubois,
H. Prade
The set functions nand N defined by (2) and (4) satisfy the properties, which are characteristic ones when A and B are not fuzzy: VA c U, VB G U, VA E U, VB E U,
17(AU B) = max(n(A), 17(B)), N(A 17B) = min(N(A), N(B)).
(5) (6)
Then it can be easily checked that VAS
U, 17(A)sN(A)
(7)
which is a satisfactory inequality since an event must be possible before being certain (i.e. occurring necessarily). It has been noticed for a long time [S], that the inequality (7) no longer holds when n is not normal when keeping (3) as the definition of the necessity measure.
2. Yager’s proposal In a recent paper, Yager [lo] proposed to define a new quantity defined by
Cert(A)=min(n(A),
N(A))
(8)
which obviously always satisfies the inequality VA5
U, Cert(A)sfl(A)
(9)
even if n is subnormal. However if using Cert instead of N preserves the desirable inequality in any case, Cert has a severe drawback; namely it does no longer satisfy the axiom (6). Thus Cert appears to be somewhat ad hoc, without a clear interpretation, and even arbitrary. Indeed, we might think as well of keeping N and defining a new measure Pos as VA c
U, Pas(A) = max(II(A),
N(A)) a N(A)
(10)
and use it in place of n!
3. Alternative
approach
The authors have proposed in various papers [6,2] (see also [9]) to modify (3) under the form VAc
U, N(A)=fl(U)-17(A),
(11)
where II(U) is defined by (2) (with p&u) = 1, VU). Obviously (11) reduces to (3) when n is normal. The greatest degree of possibility attached to an element of U plays the role of a reference point with respect to which the degree of certainty is estimated. It is easy to check that (11) is still compatible with the axioms (5), (6) and the inequality (7).
Handling
subnormal
possibility
distributions
125
4. Extension to Shafer’s framework Belief and plausibility functions [8] whose necessity and possibility measures are particular cases respectively, are built from a so-called basic probability assignment m from 2O to [0, 11, such that 2
m(A)=l.
(12)
AEU
A subset A of U such that m(A) > 0 is called a focal subset. When the sum of the weights m(A) is less than 1, we may attach the remaining weight to the empty set, i.e. m(0)=
l-
2
m(A).
(13)
AGU
Then belief and plausibility Bel(A)=
c
functions are defined for ordinary subsets as
m(B),
(14)
0fBsA PI(A)
=
2 BrlA#B
m(B),
(1%
which remain identical to the classical definitions when m(0) = 0. Then it is easy to see that b’A c_ U,
Bel(A) + Pi(A) = 1 - m(0) = PI(U),
(16)
which clearly generalizes (11). When the focal elements and the events are fuzzy sets, (14) and (15) can be generalized by [12] Bel(A) =
2
m(B).
N,(A),
(17)
0fBEU
WA)
= 2
m(B).
n,(A),
(18)
BSU
where NB and IT, are the necessity and possibility measures defined with n = p9, which may be subnormal. Note that in (17), N,(A) is calculated via (11). Moreover the empty set is a possible focal element, i.e. m(0) 2 0. The inequality PI(A) 2 Bel(A) is obviously preserved. As proved in Dubois and Prade [3], the subadditivity of the function Pl in (17) is preserved in the setting of evaluation of fuzzy events, in the presence of fuzzy focal elements, i.e. vn 2 2J Pl(;=~,,,
Ai) s ,GtlT,,, n) (-1Y”+1 P1(UAi) id iG0
(19)
’
where n and U are expressed by minimum and maximum operations respectively. (19) is obvious because the possibility of a fuzzy event n,(A) is itself a
126
D. Dubois,
H. Prade
subadditive function, and subadditivity is preserved through convex combination. is a superadditive function. Hence As a consequence N,(A) = Sup pB - II&i) the dual of (19) (changing n into U and conversely, and G into 5) is also valid for Bel, as defined by (17). Hence (17) and (18) are axiomatically valid extensions of belief and plausibility functions. This is probably not the case when N,(A) is changed into CertB(A) in (17).
5. Conclusion The proposed technique for handling subnormal distributions in uncertainty evaluation preserves the basic axioms of the uncertainty theory it refers to. Subnormal distributions are useful to express the fact that a variable whose value is ill-known may actually carry no value at all. For instance, when we want to represent in a data base the tentative departure date for a trip which is not completely decided yet. 1 - n(U) represent the degree of possibility that the trip will be cancelled.
References [l] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [2] D. Dubois and H. Prade, On several representations of an uncertain body of evidence, in: M.M. Gupta, E. Sanchez, Eds., Fuzzy Information and Decision Processes (North-Holland, Amsterdam, 1982) 167-181. [3] D. Dubois and H. Prade, Evidence measures based on fuzzy information, Automatica 21 (1985) 547-562. [4] H. Prade, Nomenclature of fuzzy measures, in: E.P. KIement, Ed., Proc. 1st Inter. Seminar on Theory of Fuzzy Sets, Johannes Kepler Univ., Linz, Austria (Sept. 1979) 9-25. [5] H. Prade, Possibilite - NCcessitC- Principe d’extension, Busefal 4 (1980) 60-63. [6] H. Prade, On the link between Dempster’s rule of combination of evidence and fuzzy set intersection, Busefal 8 (1981) 60-64. (71 H. Prade, Modal semantics and fuzzy set theory, in: R.R. Yager, Ed., Fuzzy Set and Possibility Theory: Recent Developments (Pergamon Press, New York, 1982) 232-246. [8] G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, NJ, 1976). [9] C. Testemale, Book review of “Fuzzy relational databases - A key to expert systems” (M. Zemankova-Leech, A. Kandel), Fuzzy Sets and Systems 17 (1985) 106-108. [lo] R.R. Yager, A modification of the certainty measure to handle subnormal distributions, Fuzzy Sets and Systems 20 (1986) 317-324. [ll] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28. [12] L.A. Zadeh, Fuzzy sets and information granularity, in: M.M. Gupta, R.K. Ragade, R.R. Yager, Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 3-18.
123
SHORT COMMUNICATION
AN ALTERNATIVE APPROACH TO THE HANDLING SUBNORMAL POSSIBILITY DISTRIBUTIONS - A Critical Comment on a Proposal by Yager Didier DUBOIS Universitt! Narbonne,
OF
and Henri PRADE
Paul Sabatier Laboratoire 31062 Toulouse, France
Langages
et Systhes
Informatiques,
118, route
de
Received January 1987 In this short note a definition of the necessity measure in case of a subnormal possibility distribution already proposed by the authors is recalled. This definition both satisfies the axiomatics of necessity measures and preserves the desirable inequality between possibility and necessity measures. The handling of subnormal basic probability assignments in the more general framework of Shafer’s belief functions is also discussed. Keywords:
Possibility distribution,
Necessity measure, Normalization,
Belief function.
1. Introduction A normal possibility distribution such that
it, defined on U, is a function from U to [0, l]
s,llpun(u) = 1.
(1)
Then, a so-called possibility measure 17 based on n, is defined by [ll]
(2) where A is an ordinary or a fuzzy subset of U. Note that choices other than min are possible in (2); see [7] for instance. A measure of necessity N [l, 41, is naturally associated with n by the duality relation VA c U,
A’(A) = 1 - n(A),
where A denotes the complement have VA c U,
(3)
of A defined by ~2 = 1 - Pi. Consequently
N(A) = hf, max(pA(u),
1 - n(u)).
0165-0114/87/$3.50 @ 1987, Elsevier Science Publishers B.V. (North-Holland)
we (4)
124
D. Dubois,
H. Prade
The set functions nand N defined by (2) and (4) satisfy the properties, which are characteristic ones when A and B are not fuzzy: VA c U, VB G U, VA E U, VB E U,
17(AU B) = max(n(A), 17(B)), N(A 17B) = min(N(A), N(B)).
(5) (6)
Then it can be easily checked that VAS
U, 17(A)sN(A)
(7)
which is a satisfactory inequality since an event must be possible before being certain (i.e. occurring necessarily). It has been noticed for a long time [S], that the inequality (7) no longer holds when n is not normal when keeping (3) as the definition of the necessity measure.
2. Yager’s proposal In a recent paper, Yager [lo] proposed to define a new quantity defined by
Cert(A)=min(n(A),
N(A))
(8)
which obviously always satisfies the inequality VA5
U, Cert(A)sfl(A)
(9)
even if n is subnormal. However if using Cert instead of N preserves the desirable inequality in any case, Cert has a severe drawback; namely it does no longer satisfy the axiom (6). Thus Cert appears to be somewhat ad hoc, without a clear interpretation, and even arbitrary. Indeed, we might think as well of keeping N and defining a new measure Pos as VA c
U, Pas(A) = max(II(A),
N(A)) a N(A)
(10)
and use it in place of n!
3. Alternative
approach
The authors have proposed in various papers [6,2] (see also [9]) to modify (3) under the form VAc
U, N(A)=fl(U)-17(A),
(11)
where II(U) is defined by (2) (with p&u) = 1, VU). Obviously (11) reduces to (3) when n is normal. The greatest degree of possibility attached to an element of U plays the role of a reference point with respect to which the degree of certainty is estimated. It is easy to check that (11) is still compatible with the axioms (5), (6) and the inequality (7).
Handling
subnormal
possibility
distributions
125
4. Extension to Shafer’s framework Belief and plausibility functions [8] whose necessity and possibility measures are particular cases respectively, are built from a so-called basic probability assignment m from 2O to [0, 11, such that 2
m(A)=l.
(12)
AEU
A subset A of U such that m(A) > 0 is called a focal subset. When the sum of the weights m(A) is less than 1, we may attach the remaining weight to the empty set, i.e. m(0)=
l-
2
m(A).
(13)
AGU
Then belief and plausibility Bel(A)=
c
functions are defined for ordinary subsets as
m(B),
(14)
0fBsA PI(A)
=
2 BrlA#B
m(B),
(1%
which remain identical to the classical definitions when m(0) = 0. Then it is easy to see that b’A c_ U,
Bel(A) + Pi(A) = 1 - m(0) = PI(U),
(16)
which clearly generalizes (11). When the focal elements and the events are fuzzy sets, (14) and (15) can be generalized by [12] Bel(A) =
2
m(B).
N,(A),
(17)
0fBEU
WA)
= 2
m(B).
n,(A),
(18)
BSU
where NB and IT, are the necessity and possibility measures defined with n = p9, which may be subnormal. Note that in (17), N,(A) is calculated via (11). Moreover the empty set is a possible focal element, i.e. m(0) 2 0. The inequality PI(A) 2 Bel(A) is obviously preserved. As proved in Dubois and Prade [3], the subadditivity of the function Pl in (17) is preserved in the setting of evaluation of fuzzy events, in the presence of fuzzy focal elements, i.e. vn 2 2J Pl(;=~,,,
Ai) s ,GtlT,,, n) (-1Y”+1 P1(UAi) id iG0
(19)
’
where n and U are expressed by minimum and maximum operations respectively. (19) is obvious because the possibility of a fuzzy event n,(A) is itself a
126
D. Dubois,
H. Prade
subadditive function, and subadditivity is preserved through convex combination. is a superadditive function. Hence As a consequence N,(A) = Sup pB - II&i) the dual of (19) (changing n into U and conversely, and G into 5) is also valid for Bel, as defined by (17). Hence (17) and (18) are axiomatically valid extensions of belief and plausibility functions. This is probably not the case when N,(A) is changed into CertB(A) in (17).
5. Conclusion The proposed technique for handling subnormal distributions in uncertainty evaluation preserves the basic axioms of the uncertainty theory it refers to. Subnormal distributions are useful to express the fact that a variable whose value is ill-known may actually carry no value at all. For instance, when we want to represent in a data base the tentative departure date for a trip which is not completely decided yet. 1 - n(U) represent the degree of possibility that the trip will be cancelled.
References [l] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [2] D. Dubois and H. Prade, On several representations of an uncertain body of evidence, in: M.M. Gupta, E. Sanchez, Eds., Fuzzy Information and Decision Processes (North-Holland, Amsterdam, 1982) 167-181. [3] D. Dubois and H. Prade, Evidence measures based on fuzzy information, Automatica 21 (1985) 547-562. [4] H. Prade, Nomenclature of fuzzy measures, in: E.P. KIement, Ed., Proc. 1st Inter. Seminar on Theory of Fuzzy Sets, Johannes Kepler Univ., Linz, Austria (Sept. 1979) 9-25. [5] H. Prade, Possibilite - NCcessitC- Principe d’extension, Busefal 4 (1980) 60-63. [6] H. Prade, On the link between Dempster’s rule of combination of evidence and fuzzy set intersection, Busefal 8 (1981) 60-64. (71 H. Prade, Modal semantics and fuzzy set theory, in: R.R. Yager, Ed., Fuzzy Set and Possibility Theory: Recent Developments (Pergamon Press, New York, 1982) 232-246. [8] G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, NJ, 1976). [9] C. Testemale, Book review of “Fuzzy relational databases - A key to expert systems” (M. Zemankova-Leech, A. Kandel), Fuzzy Sets and Systems 17 (1985) 106-108. [lo] R.R. Yager, A modification of the certainty measure to handle subnormal distributions, Fuzzy Sets and Systems 20 (1986) 317-324. [ll] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28. [12] L.A. Zadeh, Fuzzy sets and information granularity, in: M.M. Gupta, R.K. Ragade, R.R. Yager, Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 3-18.