An Approach to Interval Fuzzy Probabilities - Atlantis Press
16th World Congress of the International Fuzzy Systems Association (IFSA) 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT)
An Approach to Interval Fuzzy Probabilities Tiago da Cruz Asmus1 Graçaliz Pereira Dimuro2 Benjamín Bedregal3 1
2
IMEF, Universidade Federal do Rio Grande, Rio Grande, Brazil PPGCOMP, C3, Universidade Federal do Rio Grande, Rio Grande, Brazil 3 DIMAP, Universidade Federal do Rio Grande do Norte, Natal, Brazil
Abstract
mated and where the modeling of fuzzy numbers is not trivial. To elaborate the calculation of interval fuzzy probabilities, we introduce an approach based on the one used by Buckley and Eslami [10], where the probabilities respect an arithmetic restriction on the interval [0, 1]. We discuss several properties of the proposed approach. In the literature, we can find other approaches to fuzzy probabilities. For example, the approach proposed by da Costa [11], without the use of an arithmetic restriction, facilitates the calculation of each probability, avoiding the interdependence between them, but it produces fuzzy numbers with greater dispersion than the ones produced by our approach. As the probabilities defined in this paper are calculated to be utilized in problems of decision making, we prefer to work with fuzzy values with the least possible dispersion to avoid ambiguous cases in the ordering of imprecise values. The paper is organized as follows. In Section 2, we provide the notion of interval fuzzy numbers, their arithmetic and a total order for symmetric triangular and positive interval fuzzy numbers. In Section 3, preliminary aspects and the aforementioned approach for fuzzy probabilities is presented, as the definition of fuzzy mean1 . In Section 4, the focus is on development of the approach to interval fuzzy probabilities, joint with some propositions and properties, and the definition of interval fuzzy mean. Section 5 brings the final considerations of this paper and some ideas for future work.
Fuzzy sets and logic have been largely used for the treatment of the uncertainty, vagueness and ambiguity found in the modeling of real problems. However, there may exist also the case when there is uncertainty related to the membership functions to be used in the modeling of fuzzy sets and fuzzy numbers, as there are many ways to define the shape of this kind of number. Thus, one can use, for example, the theory of interval fuzzy sets to address this uncertainty, considering different modelings of fuzzy numbers into a single interval fuzzy number. In this paper, we use interval fuzzy numbers to represent probabilities that are difficult to be estimated and where the modeling of fuzzy numbers is not trivial. To elaborate the calculation of interval fuzzy probabilities, we introduce an approach based on the one used by Buckley and Eslami for fuzzy probabilities, where the probabilities respect an arithmetic restriction. We discuss several properties of the proposed approach. Keywords: interval fuzzy numbers, ranking interval fuzzy numbers, interval fuzzy probabilities, interval fuzzy mean 1. Introduction In problems of decision making in an environment with uncertainty, the decision maker must estimate the probabilities of different actions, which may lead to different outcomes. However, sometimes these probabilities are difficult to estimate precisely, such as problems in agent-based social simulation [1, 2, 3], which often have linguistic variables to define some parameters of the agents involved in the modeling [4, 5, 6] under vagueness, ambiguity and uncertainty. However, one can find in the theory of fuzzy sets an alternative to model this imprecision, as fuzzy numbers are ideal for representing linguistic variables and modeling imprecise values [7, 8]. But still there may be uncertainty in how to model this fuzzy numbers, as there are many ways to define the shape of this kind of number. Thus, one can use the theory of interval fuzzy sets to address this uncertainty, considering different modelings of fuzzy numbers into a single interval fuzzy number [9]. In this paper, we use interval fuzzy numbers to represent probabilities that are difficult to be esti© 2015. The authors - Published by Atlantis Press
2. Interval Fuzzy Numbers The fuzzy set theory is a useful tool for modeling uncertainty. However, sometimes it is difficult to determine the membership degree to be used for certain problems. To work around this situation, several authors (e.g., as discussed in [14, 15, 16, 17]) represent the membership degrees through real intervals, thus extending fuzzy sets to interval fuzzy sets. Considering IR as the set of all real intervals, let U I = [0, 1] ∈ IR be the real unit interval, and define U = {[a, b] | 0 ≤ a ≤ b ≤ 1} as the set of all sub-intervals of U I. Thus, an interval fuzzy subset A of a universe X 1 For the lack of space, we omit some proofs of propositions and properties, which can be found in [12, 13].
1520
usuppNˆ = (au , bu ) and core coreNˆ = [u1 , u2 ], and is denoted by the tuple
is defined as the set of ordered pairs A = {(x, µA (x)) | x ∈ X},
([au , al ]/[u1 , u2 ]/[bl , bu ]). where µA : R → U is the interval membership function of A. If the interval membership function µA is continuous2 , then there are continuous functions µAl , µAu : X → U I called respectively as lower membership function (LMF) and upper membership function (UMF), such that, for every x ∈ X: µA (x) = [µAl (x), µAu (x)],
ˆ is a triangular (linear) If u1 = u2 = u, then N interval fuzzy number. The set of all interval fuzzy ˆ numbers is denoted by F(R). One can observe that the functions LMF and ¯l e UMF describe, respectively, the fuzzy numbers N ¯ Nu , which can represent the interval fuzzy number ˆ . Assuming that N ¯l and N ¯u are both symmetN ric triangular fuzzy numbers described respectively by the functions µNˆi (LMF) and µNˆs (UMF), and represented respectively by the tuples (al /u/bl ) and ˆ is a symmetric triangular inter(au /u/bu ), then N val fuzzy number. Thus, an interval fuzzy number may be represented as an ordered pair of fuzzy numbers. Thereˆ = (N ¯l , N ¯u ), where N ¯l and N ¯u are called fore, N lower generator number and upper generator numˆ. ber, respectively, as they constitute N The addition of two triangular interval fuzzy numbers
(1)
where µAl (x) ≤ µAu (x). The inner and outer supports of an interval fuzzy set A of X are defined, respectively, by: lsuppA = {x ∈ X | µAl (x) > 0}; usuppA = {x ∈ X | µAu (x) > 0}.
(2) (3)
For the same A, the core of this interval fuzzy subset is defined by: coreA = {x ∈ X | µA (x) = [1; 1]}.
(4) ˆ1 = ([au , al ]/[u1 , u2 ]/[bu , bl ]) N
In other words: and coreA = {x ∈ X | µAl (x) = µAu (x) = 1}.
ˆ2 = ([cu , cl ]/[v1 , v2 ]/[du , dl ]) N
(5)
is the triangular interval fuzzy number
For α = [α1 , α2 ] ∈ U, we define the [α1 , α2 ]-cuts of A as:
A[α1 , α2 ]=
Clearly, in this case, one has that
{x ∈ X | µA (x) ≥K α},
if α1 6= 0
Al [0] ∩ Au [0],
if α1 =α2 =0.
Al [0]∩{x∈X | µA (x)≥K α},
ˆ = ([au +cu , al +cl ]/[u1 +v1 , u2 +v2 ]/[bu +du , bl +dl ]). N
ˆ = (N ¯1l + N ¯2l , N ¯1u + N ¯2u ) N
if α1 =06=α2
where the addition is the usual adition among triangular fuzzy numbers. Analogously, the pseudoinverse additive of a triangular interval fuzzy numˆ = ([au , al ]/[u1 , u2 ]/[bu , bl ]) is the triangular ber N interval fuzzy number
(6)
where Al [0] is the closure of the support of Al and Au [0] is the closure of the support of Au . As in the classical fuzzy theory, an interval fuzzy set is completely determined by its [α1 , α2 ]-cuts. An interval fuzzy number is defined as an interval extension of the usual definition of the fuzzy number, considering the approach of interval fuzzy sets presented here. Therefore, an interval fuzzy numˆ is defined as a interval fuzzy set of R with ber N the following characteristics: [9]
ˆ = ([−bl , −bu ]/[−u2 , −u1 ]/[−al , −au ]) −N ˆ = (−N ¯u , −N ¯l ). or equivalently, −N Based on the concept that two symmetric triangular fuzzy generator numbers with the same core produce a symmetric triangular interval fuzzy number, an order relation was defined, based on the AD-order for comparing the fuzzy generator numbers, and analyzing those comparisons to determine the ordering of symmetric triangular interval fuzzy numbers. [21] Let F¯1l and F¯1u be two symmetric triangular interval fuzzy numbers obtained through the functions LMF and UMF, respectively, of an interval fuzzy number Fˆ1 . Analogously, let F¯2l e F¯2u be, respectively, the lower and upper fuzzy generator numbers of another symmetric triangular interval ˆ is defined fuzzy number, Fˆ2 . Then, the relation < by:
ˆ are real in(a) the [α1 , α2 ]-cuts and the core of N ˆ tervals, that is, N [α1 , α2 ], coreNˆ ∈ IR ; (b) lsuppNˆ and usuppNˆ are bounded. If ear, ber, ship
the functions LMF and UMF are both linˆ is called linear interval fuzzy numthen N which can be defined by an interval memberfunction µNˆ , with supports lsuppNˆ = (al , bl ),
2 The continuity of interval functions was defined by Moore as an extension of the continuity of real functions. More information on this subject can be seen in [18, 19, 20].
1521
ˆ Fˆ2 ⇔(F¯1u < ¯ F¯2u )∨(F¯1u = F¯2u ∧F¯1l < ¯ F¯2l ) Fˆ1
An Approach to Interval Fuzzy Probabilities Tiago da Cruz Asmus1 Graçaliz Pereira Dimuro2 Benjamín Bedregal3 1
2
IMEF, Universidade Federal do Rio Grande, Rio Grande, Brazil PPGCOMP, C3, Universidade Federal do Rio Grande, Rio Grande, Brazil 3 DIMAP, Universidade Federal do Rio Grande do Norte, Natal, Brazil
Abstract
mated and where the modeling of fuzzy numbers is not trivial. To elaborate the calculation of interval fuzzy probabilities, we introduce an approach based on the one used by Buckley and Eslami [10], where the probabilities respect an arithmetic restriction on the interval [0, 1]. We discuss several properties of the proposed approach. In the literature, we can find other approaches to fuzzy probabilities. For example, the approach proposed by da Costa [11], without the use of an arithmetic restriction, facilitates the calculation of each probability, avoiding the interdependence between them, but it produces fuzzy numbers with greater dispersion than the ones produced by our approach. As the probabilities defined in this paper are calculated to be utilized in problems of decision making, we prefer to work with fuzzy values with the least possible dispersion to avoid ambiguous cases in the ordering of imprecise values. The paper is organized as follows. In Section 2, we provide the notion of interval fuzzy numbers, their arithmetic and a total order for symmetric triangular and positive interval fuzzy numbers. In Section 3, preliminary aspects and the aforementioned approach for fuzzy probabilities is presented, as the definition of fuzzy mean1 . In Section 4, the focus is on development of the approach to interval fuzzy probabilities, joint with some propositions and properties, and the definition of interval fuzzy mean. Section 5 brings the final considerations of this paper and some ideas for future work.
Fuzzy sets and logic have been largely used for the treatment of the uncertainty, vagueness and ambiguity found in the modeling of real problems. However, there may exist also the case when there is uncertainty related to the membership functions to be used in the modeling of fuzzy sets and fuzzy numbers, as there are many ways to define the shape of this kind of number. Thus, one can use, for example, the theory of interval fuzzy sets to address this uncertainty, considering different modelings of fuzzy numbers into a single interval fuzzy number. In this paper, we use interval fuzzy numbers to represent probabilities that are difficult to be estimated and where the modeling of fuzzy numbers is not trivial. To elaborate the calculation of interval fuzzy probabilities, we introduce an approach based on the one used by Buckley and Eslami for fuzzy probabilities, where the probabilities respect an arithmetic restriction. We discuss several properties of the proposed approach. Keywords: interval fuzzy numbers, ranking interval fuzzy numbers, interval fuzzy probabilities, interval fuzzy mean 1. Introduction In problems of decision making in an environment with uncertainty, the decision maker must estimate the probabilities of different actions, which may lead to different outcomes. However, sometimes these probabilities are difficult to estimate precisely, such as problems in agent-based social simulation [1, 2, 3], which often have linguistic variables to define some parameters of the agents involved in the modeling [4, 5, 6] under vagueness, ambiguity and uncertainty. However, one can find in the theory of fuzzy sets an alternative to model this imprecision, as fuzzy numbers are ideal for representing linguistic variables and modeling imprecise values [7, 8]. But still there may be uncertainty in how to model this fuzzy numbers, as there are many ways to define the shape of this kind of number. Thus, one can use the theory of interval fuzzy sets to address this uncertainty, considering different modelings of fuzzy numbers into a single interval fuzzy number [9]. In this paper, we use interval fuzzy numbers to represent probabilities that are difficult to be esti© 2015. The authors - Published by Atlantis Press
2. Interval Fuzzy Numbers The fuzzy set theory is a useful tool for modeling uncertainty. However, sometimes it is difficult to determine the membership degree to be used for certain problems. To work around this situation, several authors (e.g., as discussed in [14, 15, 16, 17]) represent the membership degrees through real intervals, thus extending fuzzy sets to interval fuzzy sets. Considering IR as the set of all real intervals, let U I = [0, 1] ∈ IR be the real unit interval, and define U = {[a, b] | 0 ≤ a ≤ b ≤ 1} as the set of all sub-intervals of U I. Thus, an interval fuzzy subset A of a universe X 1 For the lack of space, we omit some proofs of propositions and properties, which can be found in [12, 13].
1520
usuppNˆ = (au , bu ) and core coreNˆ = [u1 , u2 ], and is denoted by the tuple
is defined as the set of ordered pairs A = {(x, µA (x)) | x ∈ X},
([au , al ]/[u1 , u2 ]/[bl , bu ]). where µA : R → U is the interval membership function of A. If the interval membership function µA is continuous2 , then there are continuous functions µAl , µAu : X → U I called respectively as lower membership function (LMF) and upper membership function (UMF), such that, for every x ∈ X: µA (x) = [µAl (x), µAu (x)],
ˆ is a triangular (linear) If u1 = u2 = u, then N interval fuzzy number. The set of all interval fuzzy ˆ numbers is denoted by F(R). One can observe that the functions LMF and ¯l e UMF describe, respectively, the fuzzy numbers N ¯ Nu , which can represent the interval fuzzy number ˆ . Assuming that N ¯l and N ¯u are both symmetN ric triangular fuzzy numbers described respectively by the functions µNˆi (LMF) and µNˆs (UMF), and represented respectively by the tuples (al /u/bl ) and ˆ is a symmetric triangular inter(au /u/bu ), then N val fuzzy number. Thus, an interval fuzzy number may be represented as an ordered pair of fuzzy numbers. Thereˆ = (N ¯l , N ¯u ), where N ¯l and N ¯u are called fore, N lower generator number and upper generator numˆ. ber, respectively, as they constitute N The addition of two triangular interval fuzzy numbers
(1)
where µAl (x) ≤ µAu (x). The inner and outer supports of an interval fuzzy set A of X are defined, respectively, by: lsuppA = {x ∈ X | µAl (x) > 0}; usuppA = {x ∈ X | µAu (x) > 0}.
(2) (3)
For the same A, the core of this interval fuzzy subset is defined by: coreA = {x ∈ X | µA (x) = [1; 1]}.
(4) ˆ1 = ([au , al ]/[u1 , u2 ]/[bu , bl ]) N
In other words: and coreA = {x ∈ X | µAl (x) = µAu (x) = 1}.
ˆ2 = ([cu , cl ]/[v1 , v2 ]/[du , dl ]) N
(5)
is the triangular interval fuzzy number
For α = [α1 , α2 ] ∈ U, we define the [α1 , α2 ]-cuts of A as:
A[α1 , α2 ]=
Clearly, in this case, one has that
{x ∈ X | µA (x) ≥K α},
if α1 6= 0
Al [0] ∩ Au [0],
if α1 =α2 =0.
Al [0]∩{x∈X | µA (x)≥K α},
ˆ = ([au +cu , al +cl ]/[u1 +v1 , u2 +v2 ]/[bu +du , bl +dl ]). N
ˆ = (N ¯1l + N ¯2l , N ¯1u + N ¯2u ) N
if α1 =06=α2
where the addition is the usual adition among triangular fuzzy numbers. Analogously, the pseudoinverse additive of a triangular interval fuzzy numˆ = ([au , al ]/[u1 , u2 ]/[bu , bl ]) is the triangular ber N interval fuzzy number
(6)
where Al [0] is the closure of the support of Al and Au [0] is the closure of the support of Au . As in the classical fuzzy theory, an interval fuzzy set is completely determined by its [α1 , α2 ]-cuts. An interval fuzzy number is defined as an interval extension of the usual definition of the fuzzy number, considering the approach of interval fuzzy sets presented here. Therefore, an interval fuzzy numˆ is defined as a interval fuzzy set of R with ber N the following characteristics: [9]
ˆ = ([−bl , −bu ]/[−u2 , −u1 ]/[−al , −au ]) −N ˆ = (−N ¯u , −N ¯l ). or equivalently, −N Based on the concept that two symmetric triangular fuzzy generator numbers with the same core produce a symmetric triangular interval fuzzy number, an order relation was defined, based on the AD-order for comparing the fuzzy generator numbers, and analyzing those comparisons to determine the ordering of symmetric triangular interval fuzzy numbers. [21] Let F¯1l and F¯1u be two symmetric triangular interval fuzzy numbers obtained through the functions LMF and UMF, respectively, of an interval fuzzy number Fˆ1 . Analogously, let F¯2l e F¯2u be, respectively, the lower and upper fuzzy generator numbers of another symmetric triangular interval ˆ is defined fuzzy number, Fˆ2 . Then, the relation < by:
ˆ are real in(a) the [α1 , α2 ]-cuts and the core of N ˆ tervals, that is, N [α1 , α2 ], coreNˆ ∈ IR ; (b) lsuppNˆ and usuppNˆ are bounded. If ear, ber, ship
the functions LMF and UMF are both linˆ is called linear interval fuzzy numthen N which can be defined by an interval memberfunction µNˆ , with supports lsuppNˆ = (al , bl ),
2 The continuity of interval functions was defined by Moore as an extension of the continuity of real functions. More information on this subject can be seen in [18, 19, 20].
1521
ˆ Fˆ2 ⇔(F¯1u < ¯ F¯2u )∨(F¯1u = F¯2u ∧F¯1l < ¯ F¯2l ) Fˆ1
