Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Fuzzy Dissimilarity and Distance Functions

Fuzzy Dissimilarity and Distance Functions Fuzzy Inf. Eng. (2009)2:205-217 DOI 10.1007/s12543-009-0016-y ORIGINAL ARTICLE Ismat Beg · Samina Ashraf Received: 5 December 2008/ Revised: 17 May 2009/ Accepted: 20 May 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract −fuzzy dissimilarity relation is defined by using the concept of− fuzzy equivalence relation and a strong negator. It is proved that the − fuzzy dissimilarity relation so defined satisfies inequalities resembling to generalized triangle inequality. Keywords Fuzzy dissimilarity · Distance axioms · Fuzzy equivalence relation 1. Introduction When it seems that indistinguishability should be an equivalence relation and thus, in particular, transitive, there are many examples in literature that suggest otherwise (see [3], [7], [16], [17]). The phenomenon called “Poincare paradox” was pointed out in history by Poincare [22]. Later on Menger [20] also tried to address it. Fuzzy logic [32] seems a natural approach to deal with vagueness, since it does not require a predicate to be necessarily true or false; rather, it can be true to a certain degree. In the framework of fuzzy set theory, the notion of T−transitive relations was developed with hope to resolve the paradox of approximate equality [25] but unfortunately it did not work (see [3], [7], [16], [17]). In this scenario, the notion of −fuzzy transitivity to get a better picture of approximate equality was introduced by Beg and Ashraf [4]. A fundamental property that accompanies every new definition of indistinguishabil- ity relations is that their complements satisfy the properties of distance. This does not only hold for crisp cases, but for every T−transitive equivalence relations as was observed by Zadeh [33] in case of his max-min transitive similarity relations. The key step in this direction was taken by representation theorem of Valverde [30], which proves that the application of an order reversing bijection always converts a T−transitive fuzzy equivalence relation to an S−pseudo metric. These developments provide sound reason for believing that the notion of dissimilarity is nothing else than distance. This is why it is customary to take dissimilarity as distance (see [1], Ismat Beg () · Samina Ashraf Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan e-mail: ibeg@lums.edu.pk 206 Ismat Beg · Samina Ashraf (2009) [12], [24], [31], [23]). It is now widely suspected that the standard distance-based similarity measures do not provide an adequate account of perceived similarity (see [13], [19], [24] and [29]). Because of their questionable empirical validity, it is de- sirable to investigate theories of perceived dissimilarity not constrained by distance axioms. The axioms of self dissimilarity and symmetry face less criticism but the tri- angle inequality has been the hardest of the metric axioms to be tested experimentally. Therefore depending upon the context in which these notions are going to be used for people state different sets of axioms for similarity and dissimilarity. In this paper, we present a unified approach to quantifying dissimilarity within the framework of fuzzy set theory. For this purpose, we use axiom of−transitivity [4] to develop inequalities that replace triangle inequality, when dissimilarity is taken as a concept different from distance. Throughout this paper, X represents a crisp universe of generic elements and F(X) stands for the set of all fuzzy subsets of X. Whereas, a fuzzy relation onagiven universe X is a fuzzy subset of X × X. Definition 1.1 [32] For A, B ∈ F(X), if for all x ∈ X, A(x)  B(x), then A is said to be subset of B in Zadeh’s sense, where, A(x) and B(x) represent the membership grades of x in A and B respectively. In this case we write A ⊆ B. The triangular norms and conorms are considered as the standard models for in- tersecting and unifying fuzzy sets respectively [18]. Definition 1.2 [18] The triangular norm (t-norm) T and triangular conorm (t- conorm) S are increasing, associative, commutative and [0, 1] → [0, 1] mappings satisfying: T (1, x) = x and S (x, 0) = x, for all x ∈ [0, 1]. Some choices for t-norms are: (i) The minimum operator M : M(x, y) = min(x, y); (ii) The Lukasiewicz’s t-norm W : W(x, y) = max(x+ y− 1, 0); (iii) The product operator P : P(x, y) = xy. The corresponding dual t-conorms are: ∗ ∗ (i) The maximum operator M : M (x, y) = max(x, y); ∗ ∗ (ii) The bounded sum W : W (x, y) = min(x+ y, 1); ∗ ∗ (iii) The probabilistic sum P : P (x, y) = x+ y− xy. Definition 1.3 [11] A t-norm T is said to be: (i) continuous, if T is continuous as a function on the unit interval; (n) (ii) Archimedean, if lim x = 0 for all x ∈]0, 1[, n→∞ (n) where, for any x ∈ [0, 1] and any associative binary operation K on [0, 1], x denotes (n) the nth power of x defined as: x = K(x,..., x) (n.times). Theorem 1.4 [11] A t-norm T is Archimedean if and only if it can be represented in the following form: T (x, y) = g( f (x)+ f (y)), where (a) f :[0, 1] → R = [0,∞] is a continuous, strictly decreasing function such that f (1) = 0; + −1 (b) g is a continuous function from R onto [0, 1] such that g(x) = f (x) on [0, f (0)] and g(x) = 0 for x ∈ R \[0, f (0)]. Fuzzy Inf. Eng. (2009) 2: 205-217 207 In this case, f is said to be an additive generator of T. Definition 1.5 [11] A negator N is an order-reversing [0, 1] → [0, 1] mapping such that N(0) = 1 and N(1) = 0. A negator is called strict if it is continuous and strictly decreasing. A strict negator is said to be strong if it is involutive too i.e., N(N(x)) = x for all x ∈ [0, 1]. The standard negator is defined as N (x) = 1− x, it was proposed by Zadeh [32] to define the complement of a fuzzy set. Definition 1.6 [11] A triplet (T, S, N) of a t-norm T, a t-conorm S and a strong negator N is called a De Morgan triplet, if and only if S (x, y) = N(T (N(x), N(y))), for all x, y ∈ [0, 1]. S and T are said to be dual to each other in this case. Definition 1.7 [2] A fuzzy implicator is a binary operation on [0, 1] with order reversing first partial mappings and order preserving second partial mappings such that I(0, 1) = I(0, 0) = I(1, 1) = 1, I(1, 0) = 0. An implicator I is said to have contrapositive symmetry with respect to a negator N if it satisfies I(x, y) = I(N(y), N(x)), for all x, y ∈ [0, 1]. Definition 1.8 [10] Let T be a t-norm, a mapping I :[0, 1] → [0, 1] defined as I (x, y) = sup{u ∈ [0, 1] : T (u, x)  y} is called an R−implicator with respect to T . R−implicators are defined by using the concept of residuation (for more details on this notion see [18]). R−implicators are often (in fact, for left-continuous t-norms) partial orderings on the unit interval: I(x, y) = 1 if and only if x  y. Let T be a t-norm, I (the R−implicator associated with T ) is said to satisfy resid- uation condition if T (x, y)  z ⇔ x  I (y, z). (1) Obviously, taking into account that T is nondecreasing in both places, condition (1) is equivalent to the following one: T (x,·) is a left-continuous function for any fixed x ∈ [0, 1]. Remark 1.9 [18] If T is a continuous Archimedean t-norm and f :[0, 1] → [0,∞] an additive generator of T , then the R−implicator can be obtained by the formula (−1) I (x, y) = f (max(T (y)− T (x), 0)). Theorem 1.10 [10] Suppose that T is a t-norm such that (1) is satisfied and N is a strong negator. If I has contrapositive symmetry with respect to N, then (i) I (x, y) = N(T (x, N(y))) for all x, y ∈ [0, 1]. (ii) N(x) = I (x, 0), for all x ∈ [0, 1] (iii) T (x, y) = 0 if and only if x  N(y), for any x, y ∈ [0, 1]. Definition 1.11 [6] Following properties are defined for a fuzzy relation R defined on a universe X and a t-norm T : 208 Ismat Beg · Samina Ashraf (2009) (i) Reflexivity: R(x, x) = 1, for all x ∈ X; (ii) Symmetry: R(x, y) = R(y, x), for all x, y ∈ X; (iii) T−transitivity: R(x, z)  T (R(x, y), R(y, z)), for all x, y, z ∈ X. A reflexive, symmetric and T−transitive fuzzy relation is called T−equivalence relation (or T−indistinguishability relation). Definition 1.12 [30] Let S be t-conorm. An S -pseudometric m is a map from X× X into [0, 1] such that (i) m(x, x) = 0, for all x ∈ X; (ii) m(x, y) = m(y, x), for all x, y ∈ X; (iii) S (m(x, y), m(y, z))  m(x, y), for all x, y, z ∈ X, (S− triangle inequality). An S−metric is defined in the usual way i.e., replacing (i) by m(x, y) = 0 if and only if x = y. Theorem 1.13 [30] Let E be a T−indistinguishability relation on X and let φ be a continuous and order-reversing bijection from [0, 1] into itself, then m (x, y) = φ(E(x, y)), for all x, y ∈ X −1 is a S -pseudometric and vice-versa, where S (x, y) = φ (T (φ(x),φ(y)). 2.−Dissimilarity Relations The concepts introduced in this section are based on notions of −transitivity and −equivalence relation (for details see [4]). We first recall following definitions and remarks to explain notions and notations to be used in this section. Definition 2.1 [4] For a given fuzzy relation R, the measure of transitivity of R is given by Tr(R) = inf (I(T (R(x, y), R(y, z)), R(x, z))). (2) x,y,z∈X A fuzzy relation R is called − fuzzy transitive if  = Tr(R), R is non-transitive if = 0, strong fuzzy transitive if   0.5 and weak fuzzy transitive if  ∈]0, 0.5[. Remark 2.2 [4] If an R−implicator is used in calculating, degree of transitivity, then the T−transitive fuzzy relations are 1− fuzzy transitive. This is due to the fact that R−implicators possess the property that a  b implies I(a, b) = 1. So the T−transitivity i.e., T (R(x, y), R(y, z))  R(x, z) for all x, y, z ∈ X implies I(T (R(x, y), R(y, z)), R(x, z)) = 1 for all x, y, z ∈ X, hence the result. Definition 2.3 [4] A fuzzy relation E on X is called an−fuzzy equivalence relation if, (i) E(x, x) = 1, for all x ∈ X; (ii) E(x, y) = E(y, x), for all x, y ∈ X; (iii) T r(E) = , for all x, y, z ∈ X. In this case, E will be called an −equivalence relation. In general, if> 0, then R will be called an−fuzzy equivalence relation. Now we develop the notion of fuzzy dissimilarity relation as complement of a fuzzy equivalence relation on any universe X. Definition 2.4 Let E be a fuzzy equivalence relation defined on a crisp universe X and N be a strong negator. A fuzzy dissimilarity relation d associated with E, is a E Fuzzy Inf. Eng. (2009) 2: 205-217 209 fuzzy relation on the same universe X defined as d (x, y) = N(E(x, y)), for all x, y ∈ X. If E is an −equivalence relation instead of just being fuzzy equivalence relation, then d is called−dissimilarity relation. Remark 2.5 In (2), if we select an implicator I satisfying: I(x, y) = 0 implies x = 1 and y = 0, then the fuzzy dissimilarity relation d possesses following properties as consequence of its definition : (D1) d (x, x) = 0, (Self dissimilarity), (D2) d (x, y) = d (y, x), (Symmetry), E E (D3) At least one of the following holds: (a) d (x, y) ∈]0, 1]; (b) d (y, z) ∈]0, 1]; (c) d (x, z) ∈ [0, 1[. From the Remark 2.5, it is obvious that the first two conditions are the ones sat- isfied by any distance function, whereas, the dissimilarity relation defined in this manner satisfies (D3) instead of triangle inequality. If the dissimilarity between x and y is zero and between y and z is also zero, then x and z can not have dissimilarity of 1 degree. This alternative of triangle inequality is a very soft condition because it is established for the fuzzy relations with nonzero degrees of transitivity i.e.,  ∈]0, 1]. Next we develop another form of condition (D3) for negator of an −dissimilarity relation. Remark 2.6 Due to Theorem 1.13, it is obvious that if E is a T−transitive equiva- lence relation, then d always satisfies the identity as follows, d (x, z)  S (d (x, y), d (y, z)), E E E where, S is the conorm dual to to the t-norm T. Hence in case of T−transitive fuzzy relations we do not need to seek for an alternative for triangle inequality. This is why, in what follows, we shall restrict to the establishment of results for only those points x, y, z ∈ X, for which T (E(x, y), E(y, z)) > E(x, z). Theorem 2.7 Let d be the fuzzy dissimilarity relation obtained by applying a strong negator N to an −equivalence relation E defined on a given universe X, T be a left continuous Archimedean t-norm with differentiable generators and I is a contrapositive R−implicator associated with T , then there exists a strictly decreasing continuous function f and a positive real number k such that d (x, z)  S (d (x, y), E E f () d (y, z))+ = S (d (x, y), E E f´(θ) d (y, z))− kf (), where, (T, S, N) form a De Morgan’s triplet. Proof Since E is an −fuzzy equivalence relation, inf (I(T (R(x, y), R(y, z)), R(x, z))) = . x,y,z∈X 210 Ismat Beg · Samina Ashraf (2009) It implies that I(T (E(x, y), E(y, z)), E(x, z)) > 0. (3) Using Remark 1.9 of R−implicators, associated with continuous Archimedean t-norm T (A left continuous Archimedean t-norm is also continuous see [18, Proposition 2.16, page 30] or [8]), there exist a continuous and strictly decreasing function f : [0, 1] → [0,∞] such that −1 I(x, y) = f (max{ f (y)− f (x), 0}). Thus, there exist a continuous and strictly decreasing function f :[0, 1] → [0,∞], such that −1 f (max{ f (E(x, z))− f (T (E(x, y), E(y, z)), 0}). Applying f to both sides and using its strictly decreasing nature, we obtain max{ f (E(x, z))− f (T (E(x, y), E(y, z))), 0}  f (). (4) Since E(x, z) < T (E(x, y), E(y, z)) by the Remark 2.6 and f is a strictly decreasing function, it implies that f (E(x, z)) > f (T (E(x, y), E(y, z))). Hence f (E(x, z))− f (T (E(x, y), E(y, z))) > 0. By using this fact in (4) we get f (E(x, z))− f (T (E(x, y), E(y, z)))  f (). By use of the involutive negator N in d (x, y) = N(E(x, y)) we get E(x, y) = N(d (x, y)) E E and it further implies f (N(d (x, z)))− f (T (N(d (x, y)), N(d (y, z))))  f (). E E E Since (T, S, N) is a De Morgan triplet, therefore, we have f (N(d (x, z)))− f (N(S (d (x, y), d (y, z)))  f (). (5) E E E Now by applying the representation Theorem 1.10 for implicator to I (x, 0) = N(x), we get, −1 N(x) = f (max{ f (0)− f (x), 0}). We have f (N(x)) = max{ f (0)− f (x), 0}. by applying f to both sides. Since both S (d (x, y), d (y, z))  0, and d (x, z)  0, so f (0)  f (S (d (x, y), d (y, z)) E E E E E and f (0)  f (d (x, z)) by the decreasing nature of the function f . Thus in this partic- ular case we have f (N(x)) = f (0)− f (x). (6) Fuzzy Inf. Eng. (2009) 2: 205-217 211 Using (6) in the equation (5), we get f (0)− f (d (x, z))− f (0)+ f (S (d (x, y), d (y, z))  f (), E E E it implies that f (S (d (x, y), d (y, z)))− f (d (x, z))  f (). E E E Applying the Cauchy mean value theorem, there exist a number θ satisfying d (x, z)  θ  S (d (x, y), d (y, z)), E E E such that f (θ)(S (d (x, y), d (y, z))− d (x, z)))  f (). E E E Dividing both sides by f (θ)(< 0) f () S (d (x, y), d (y, z))− d (x, z)  . E E E f (θ) Hence f () d (x, z)  S (d (x, y), d (y, z))+ . E E E f (θ) Letting k = − . We get the result. f (θ) Remark 2.8 Though the Theorem 2.7 has been proved R−implicators, but the results are expected to hold for S−implicators as well. Next we prove the theorem for Reichenbach implicator which is not an R−implicator. Theorem 2.9 If the Reichenbach implicator defined as I(a, b) = 1− a+ ab, for all a, b ∈ [0, 1] is used along with P and P as t-norm and t-conorm respectively in the definition of an −equivalence relation E and standard negator is used to obtain the corresponding fuzzy dissimilarity relation, then d satisfies the following inequality: d (x, z)  P (d (x, y), d (y, z))+ (1−). E E E Proof Since E is an −equivalence relation so, for any x, y, z ∈ X, I(P(E(x, y), E(y, z)), E(x, z))  . Putting value of E(x, y) = N(d (x, y)) in the above equation, we get I(P(N(d (x, y)), N(d (E(y, z))), N(d (x, z)))  . E E E Using value of I 1− P(N(d (x, y)), N(d (y, z)))+ P(N(d (x, y)), N(d (y, z)))N(d (x, z))  . E E E E E Using the value of P(a, b) = a.b and N(a) = 1− a. 1− (1− d (x, y))(1− d (y, z))+ (1− d (x, y))(1− d (y, z))(1− d (x, z))  . E E E E E 212 Ismat Beg · Samina Ashraf (2009) On re-arranging expressions, we get 1−  (1− d (x, y))(1− d (y, z))(1− (1− d (x, z))). E E E So, 1−  d (x, z)(1− d (y, z)− d (x, y)+ d (x, y)d (y, z)). E E E E E On simplifying, d (x, z)  d (x, z)(d (y, z)+ d (x, y)− d (x, y)d (y, z))+ 1−. E E E E E E Using d (x, z)  1, d (x, z)  d (y, z)+ d (x, y)− d (x, y)d (y, z)+ 1−. (7) E E E E E Hence by using the value of corresponding P , we get d (x, z)  P (d (x, y), d (y, z))+ 1−. E E E Corollary 2.10 In Equation (7) if the negative term is omitted (which does not dis- turb the inequality) instead of adjusting in disjunction operator of the inequality is even strengthened and we get the form of triangular inequality used in definition of metric d (x, z)  d (y, z)+ d (x, y)+ 1−. E E E Theorem 2.11 Let d be the corresponding dissimilarity relation of an−equivalence relation on X, T a left-continuous t-norm and N be a strong negator. If I is a contra- positive implicator associated with T , then d (x, z)  S (d (y, z), d (x, y)) if and only if tr(x, y, z) = 1. E E E where, (T, S, N) form a De Morgan triplet and tr(x, y, z) denotes pointwise transitivity defined as tr(x, y, z) = I(T (E(x, y), E(y, z)), E(x, z)). Proof Since E is an −equivalence relation, inf I(T (E(x, y), E(y, z)), E(x, z)) = . x,y,z∈X For any x, y, z ∈ X, tr(x, y, z) = I(T (E(x, y), E(y, z)), E(x, z))  . Let tr(x, y, z) = I(T (E(x, y), E(y, z)), E(x, z)) =  . Using E(x, y) = N(d (x, y)), we get I(T (N(d (x, y)), N(d (y, z))), N(d (x, z))) =  . E E E 1 By properties of De Morgan triplet, we have I(N(S (d (y, z), d (x, y))), N(d (x, z))) =  . E E E 1 Since Theorem 1.10 (i) further implies that I(N(S (d (y, z), d (x, y))), N(d (x, z))) E E E = N(T (N(S (d (y, z), d (x, y))), N(N(d (x, z))) =  . E E E 1 Fuzzy Inf. Eng. (2009) 2: 205-217 213 Since N is an involutive, we have N(T (N(S (d (y, z), d (x, y))), d (x, z))) =  . E E E 1 Applying negator operator on both the sides, we get T (N(S (d (y, z), d (x, y))), d (x, z)) = N( ). (8) E E E 1 Now if d (x, z)  S (d (y, z), d (x, y)), then N(S (d (y, z), d (x, y)))  N(d (x, z)) and E E E E E E applying Theorem 1.10, we get T (N(S (d (y, z), d (x, y))), d (x, z)) = 0 = N( ). E E E 1 This implies that  = 1. Conversely letting = 1 in Equation (8), one can easily obtain d (x, z)  S (d (y, z), d (x, y)). E E E 3. Examples of Dissimilarity Measures In this section, we present some formulae which calculates degree of dissimilarity between elements of F(X)(F(X) in itself is a crips universe). In all the following examples we will use implicators with property stated in Remark 2.5. Example 3.1 Let X be a given universe and let A, B ∈ F(X), and let I be an R− implicator. If we define fuzzy set of dissimilarity Dis between A and B as: A,B Dis (x) = 1− T (I(A(x), B(x)), I(B(x), A(x))), for all x ∈ X, (9) A,B Dis A, B then Dis(A, B) = is a dissimilarity measure on F(X) where, |A| denotes the |X| scalar cardinality of a fuzzy subset A of a finite universe X, defined as|A| = A(x). x∈X Proof Self dissimilarity: For all A ∈ F(X), 1− T (I(A(x), A(x)), I(A(x), A(x))) Dis A,A x∈X Dis(A, A)= = |X| |X| 1− T (1, 1) 1− 1 x∈X x∈X = = = = 0. |X| |X| |X| (D2) Symmetry is straightforward due to commutative nature of T . (D3) Suppose on contrary that there exist A, B, C ∈ F(X) such that Dis(A, B) = 0 = Dis(B, C) and Dis(A, C) = 1. By putting values of these functions, we get Dis A,B Dis(A, B)= |X| 1− T (I(A(x), B(x)), I(B(x), A(x))) x∈X = = 0 (10) |X| 214 Ismat Beg · Samina Ashraf (2009) and 1− T (I(B(x), C(x)), I(C(x), B(x))) Dis B,C x∈X Dis(B, C) = = = 0 (11) |X| |X| and 1− T (I(A(x), C(x)), I(C(x), A(x))) Dis A,C x∈X Dis(A, C) = = = 1. (12) |X| |X| Solving Equation (10), we get A(x) = B(x), for all x ∈ X. Similarly solving equation (11), we get B(x) = C(x) for all x ∈ X. But Equation (12) can hold only if there exist some x ∈ X, such that T (I(A(x), C(x)), I(C(x), A(x)))  1. So, either I(A(x), C(x))  1 or I(C(x), A(x)))  1. In every case A and C are not equal fuzzy sets, a contradiction. Hence the result. Definition 3.2 [27] A fuzzy inclusion is a fuzzy relation Inc on F(X) defined as Inc(A, B) = inf I(A(x), B(x)), for all A, B ∈ F(X), x∈X where, A(·) refers to the membership function of the fuzzy set A etc. Theorem 3.3 Let X be a given universe. The fuzzy relation Dis on F(X) defined by Dis(A, B) = N(T (Inc(A, B), Inc(B, A))) (13) is a fuzzy dissimilarity measure on F(X) for any t-norm T , and an implicator I satis- fying I(x, x) = 1. Proof For all A, B, C ∈ F(X)wehave (D1) Self dissimilarity Dis(A, A)= N(T (Inc(A, A), Inc(A, A))) = N(T (1, 1)) = N(1) = 0. (D2) Symmetry is obvious. (D3) Assume that the dissimilarity so defined does not satisfy (D3), so there exist fuzzy sets A, B,C ∈ F(X), such that Dis(A, B) = Dis(B, C) = 0 and Dis(A, C) = 1. Applying negator, it implies that T (Inc(A, B), Inc(B, A)) = T (Inc(B, C), Inc(C, B)) = 1 (14) and T (Inc(A, C), Inc(C, A)) = 0. (15) Using properties of t-norm in Equations (14), we have Inc(A, B) = Inc(B, A) = Inc(B, C) = Inc(C, B) = 1, (16) Using the value of inclusion operator in Equations (16), we get inf I(A(x), B(x)) = inf I(B(x), A(x)) = 1, (17) x∈X x∈X Fuzzy Inf. Eng. (2009) 2: 205-217 215 inf I(B(x), C(x)) = inf I(C(x), B(x)) = 1, (18) x∈X x∈X Equations (17) and (18) imply that A = B and B = C, i.e., A = C, where = stands for equality of fuzzy sets. From here it follows that Inc(A, C) = Inc(C, A) = 1, which contradicts (15) hence the result. |A∩ B| Example 3.4 For all A, B ∈ F(X), d (A, B) = 1 − is a fuzzy dissimilarity |A∪ B| relation on F(X), provided∩ and∪ are modelled by max and min, respectively. Proof For all A, B, C ∈ F(X), we have: (D1) Self dissimilarity |A∩ A| d (A, A) = 1− = 0. |A∪ A| (D2) Symmetry: |A∩ B| d (A, B) = 1− = d (B, A). E E |A∪ B| (D3) Suppose on contrary that d (A, B) = 0 = d (B, C) and d (A, C) = 1, so, E E E min(A(x), B(x)) min(B(x), C(x)) x∈X x∈X 1−  = 0 and 1−  = 0, (19) max(A(x), B(x)) max(B(x), C(x)) x∈X x∈X together with min(A(x), C(x)) x∈X 1−  = 1. (20) max(A(x), C(x)) x∈X Using properties of t-norms and t-conorms, Equations (19) lead us to the conclusion that A, B and C are equal fuzzy sets while Equation (20) implies that T (A(x), C(x)) = 0. x∈X It further implies that for all x ∈ X either A(x) = 0or C(x) = 0, a contradiction to the above conclusion that A and C are equal (nonzero) fuzzy sets. Hence the result. 4. Conclusions In this paper we have explored the relationship between the concepts of dissimilarity and distance. The conclusion reached is that fuzzy transitivity of a fuzzy equivalence relation behaves as a controlling factor for the triangle inequality satisfied by the nega- tor of the fuzzy equivalence relation. The greater the degree of fuzzy equivalence, the closer is the inequality satisfied by the dissimilarity function to the triangle inequality. The negator of a fuzzy equivalence relation satisfies the triangular inequality or the inequality for being an ultra-metric only if it is a fuzzy equivalence relation of degree Acknowledgments The present version of the paper owes much to the precise and kind remarks of the learned referees. 216 Ismat Beg · Samina Ashraf (2009) References 1. Balopoulos V, Hatzimichailidis AG, Papadopoulos BK (2007) Distance and similarity measures for fuzzy operators. Information Sciences 177:2336-2348 2. Bandler W, Kohout L (1980) Semantics of implicators and fuzzy relational products. Man-Machine Studies 12:89-116 3. Bazu K (1984) Fuzzy revealed preference theory. Economics Theory 32:212-227 4. Beg I, Ashraf S (2008) Fuzzy equivalence relations. Kuwait J. Sci. and Eng 35(1A):191-206 5. Bezdek JC, Harris JO (1978) Fuzzy partitions and relations. An axiomatic basis for clustering. Fuzzy Sets and Systems 1:111-126 6. Bodenhofer U, De Baets B, Fodor JC (2007) A compendium on fuzzy weak orders: Representations and constructions. Fuzzy Sets and Systems 158:811-829 7. De Cock M, Kerre EE (2003) On (un)suitable fuzzy relations to model approximate equality. Fuzzy Sets and Systems 133(2):137-153 8. De Baets B, Mesier R (1996) Residual implicators of continuous t-norms, Proceedings EUFIT’96 1:22-26 9. De Baets B, Mesier R (1997) Pseudo-metrics and T-equivalences. Fuzzy Math 5(2) 10. Fodor J (1995) Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems 69:141-156 11. Fodor J, Yager RR (2000) “Fuzzy set-theoretic operators and quantifiers”, in Fundamentals of Fuzzy Sets, (edited by: D. Dubois and H. Prade), Kluwer Academic Publishers: Norwell, Ma:125-193 12. Getty DJ, Swets JA, Swets JB, Green DM (1979) On the prediction of confusion matrices form similarity judgements. Perception and Psycophysics 26:1-19 13. Gregory Ashby F, Perrin NA (1988) Towards a unified theory of similarity and recognition. Psycho- logical Review 95: 124-150 14. Jain R, Murthy S, Tran P, Chattarjee S (1995) Similarity measures for image database, IEEE Tran- sections on Knowledge and Data Engineering 7:1247-1254 15. James W (1890) Principle of psychology. New York Holt 16. Klawonn F (1999) The role of similarity in the fuzzy reasoning. Fuzzy sets, logic and reasoning about knowledge, Appl. Log. Ser,15, Kluwer Acad.Pub, Dordrecht:243-253 17. Klawonn F (2003) Should fuzzy equality and similarity satisfy transitivity? Comments on the paper by M. De Cock and E. Kerre. Fuzzy Sets and Systems 133 2:175-180 18. Klement EP, Mesiar R, Pap E (2000) Triangular norms, Trends in Logic. Kluwer Academic Publish- ers, Dordrecht,Vol. 8 19. Krumhansi CL (1978) Concerning the applicability of geometric models to similarity data: The in- terrelationship between similarity and spatial density. Psycological Review 85:445-463 20. Menger K (1951) Probabilistic theories of relations. Proceedings of the National Academy of Sci- ences 37:178-180 21. Nguyen HT, Walker EA (2006) A first course in fuzzy logic. (Third Edition) CRC Press 22. Poincar’e H (1902) La Science et 1’Hypothese. Flammarion, Paris 23. Ruspini EH (1992) On truth, utility, and similarity. Fuzzy engineering toward human friendly systems (Yokohama, 1991), 42-50, Ohm, Tokyo 24. Santini S, Jain R (1999) Similarity measures. IEEE Transactions on Pattern Analysis and Machine Intelligence 21: 871-883 25. Trillas E (1982) Assaig sobre les relacions d’indistingibilitate. Actes del Primer Congress Catala de Logica Matematica. Barcelona:51-59 26. Trillas E, Valverde L (1984) An essay on indistinguishability, in aspects of vagueness. Reidel, Eds.H.J.Skala et al 27. Tsiporkova-Hristoskova E, De Baets B, Kerre EE (1997) A fuzzy inclusion based approach to upper inverse images under fuzzy multivalued mappings. Fuzzy Sets and Systems 85:93-108 28. Tversky A (1977) Features of similarity. Psychological Review 84(4):327-352 29. Tverski A, Gati I (1982) Similarity, separability, and the triangle inequality. Psychological Review 89(2):123-154 30. Valverde L (1985) On the structure of F-indistinguishability operators. Fuzzy Sets and Systems Fuzzy Inf. Eng. (2009) 2: 205-217 217 17:313-328 31. Williams J, Steele N (2002) Difference, distance and similarity as a basis for fuzzy decision support based on prototypical decision classes. Fuzzy Sets and Systems 131:35-46 32. Zadeh LA (1965) Fuzzy sets. Information and Control 8(3):338-353 33. Zadeh LA (1971) Similarity relations and fuzzy orderings. Inf. Sci.,3:177-200 34. Zwick R, Carlstein E, Budesco DV (1987) Measures of similarity amongst fuzzy concepts: a com- parative analysis. Int.J. of Approximate Reasoning 1:221-242 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Fuzzy Dissimilarity and Distance Functions

Fuzzy Information and Engineering , Volume 1 (2): 13 – Jun 1, 2009

Fuzzy Dissimilarity and Distance Functions

Abstract

Abstract∈-fuzzy dissimilarity relation is defined by using the concept of ∈-fuzzy equivalence relation and a strong negator. It is proved that the ∈-fuzzy dissimilarity relation so defined satisfies inequalities resembling to generalized triangle inequality.
Loading next page...
 
/lp/taylor-francis/fuzzy-dissimilarity-and-distance-functions-lU03FjWj49
Publisher
Taylor & Francis
Copyright
© 2009 Taylor and Francis Group, LLC
ISSN
1616-8666
eISSN
1616-8658
DOI
10.1007/s12543-009-0016-y
Publisher site
See Article on Publisher Site

Abstract

Fuzzy Inf. Eng. (2009)2:205-217 DOI 10.1007/s12543-009-0016-y ORIGINAL ARTICLE Ismat Beg · Samina Ashraf Received: 5 December 2008/ Revised: 17 May 2009/ Accepted: 20 May 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract −fuzzy dissimilarity relation is defined by using the concept of− fuzzy equivalence relation and a strong negator. It is proved that the − fuzzy dissimilarity relation so defined satisfies inequalities resembling to generalized triangle inequality. Keywords Fuzzy dissimilarity · Distance axioms · Fuzzy equivalence relation 1. Introduction When it seems that indistinguishability should be an equivalence relation and thus, in particular, transitive, there are many examples in literature that suggest otherwise (see [3], [7], [16], [17]). The phenomenon called “Poincare paradox” was pointed out in history by Poincare [22]. Later on Menger [20] also tried to address it. Fuzzy logic [32] seems a natural approach to deal with vagueness, since it does not require a predicate to be necessarily true or false; rather, it can be true to a certain degree. In the framework of fuzzy set theory, the notion of T−transitive relations was developed with hope to resolve the paradox of approximate equality [25] but unfortunately it did not work (see [3], [7], [16], [17]). In this scenario, the notion of −fuzzy transitivity to get a better picture of approximate equality was introduced by Beg and Ashraf [4]. A fundamental property that accompanies every new definition of indistinguishabil- ity relations is that their complements satisfy the properties of distance. This does not only hold for crisp cases, but for every T−transitive equivalence relations as was observed by Zadeh [33] in case of his max-min transitive similarity relations. The key step in this direction was taken by representation theorem of Valverde [30], which proves that the application of an order reversing bijection always converts a T−transitive fuzzy equivalence relation to an S−pseudo metric. These developments provide sound reason for believing that the notion of dissimilarity is nothing else than distance. This is why it is customary to take dissimilarity as distance (see [1], Ismat Beg () · Samina Ashraf Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan e-mail: ibeg@lums.edu.pk 206 Ismat Beg · Samina Ashraf (2009) [12], [24], [31], [23]). It is now widely suspected that the standard distance-based similarity measures do not provide an adequate account of perceived similarity (see [13], [19], [24] and [29]). Because of their questionable empirical validity, it is de- sirable to investigate theories of perceived dissimilarity not constrained by distance axioms. The axioms of self dissimilarity and symmetry face less criticism but the tri- angle inequality has been the hardest of the metric axioms to be tested experimentally. Therefore depending upon the context in which these notions are going to be used for people state different sets of axioms for similarity and dissimilarity. In this paper, we present a unified approach to quantifying dissimilarity within the framework of fuzzy set theory. For this purpose, we use axiom of−transitivity [4] to develop inequalities that replace triangle inequality, when dissimilarity is taken as a concept different from distance. Throughout this paper, X represents a crisp universe of generic elements and F(X) stands for the set of all fuzzy subsets of X. Whereas, a fuzzy relation onagiven universe X is a fuzzy subset of X × X. Definition 1.1 [32] For A, B ∈ F(X), if for all x ∈ X, A(x)  B(x), then A is said to be subset of B in Zadeh’s sense, where, A(x) and B(x) represent the membership grades of x in A and B respectively. In this case we write A ⊆ B. The triangular norms and conorms are considered as the standard models for in- tersecting and unifying fuzzy sets respectively [18]. Definition 1.2 [18] The triangular norm (t-norm) T and triangular conorm (t- conorm) S are increasing, associative, commutative and [0, 1] → [0, 1] mappings satisfying: T (1, x) = x and S (x, 0) = x, for all x ∈ [0, 1]. Some choices for t-norms are: (i) The minimum operator M : M(x, y) = min(x, y); (ii) The Lukasiewicz’s t-norm W : W(x, y) = max(x+ y− 1, 0); (iii) The product operator P : P(x, y) = xy. The corresponding dual t-conorms are: ∗ ∗ (i) The maximum operator M : M (x, y) = max(x, y); ∗ ∗ (ii) The bounded sum W : W (x, y) = min(x+ y, 1); ∗ ∗ (iii) The probabilistic sum P : P (x, y) = x+ y− xy. Definition 1.3 [11] A t-norm T is said to be: (i) continuous, if T is continuous as a function on the unit interval; (n) (ii) Archimedean, if lim x = 0 for all x ∈]0, 1[, n→∞ (n) where, for any x ∈ [0, 1] and any associative binary operation K on [0, 1], x denotes (n) the nth power of x defined as: x = K(x,..., x) (n.times). Theorem 1.4 [11] A t-norm T is Archimedean if and only if it can be represented in the following form: T (x, y) = g( f (x)+ f (y)), where (a) f :[0, 1] → R = [0,∞] is a continuous, strictly decreasing function such that f (1) = 0; + −1 (b) g is a continuous function from R onto [0, 1] such that g(x) = f (x) on [0, f (0)] and g(x) = 0 for x ∈ R \[0, f (0)]. Fuzzy Inf. Eng. (2009) 2: 205-217 207 In this case, f is said to be an additive generator of T. Definition 1.5 [11] A negator N is an order-reversing [0, 1] → [0, 1] mapping such that N(0) = 1 and N(1) = 0. A negator is called strict if it is continuous and strictly decreasing. A strict negator is said to be strong if it is involutive too i.e., N(N(x)) = x for all x ∈ [0, 1]. The standard negator is defined as N (x) = 1− x, it was proposed by Zadeh [32] to define the complement of a fuzzy set. Definition 1.6 [11] A triplet (T, S, N) of a t-norm T, a t-conorm S and a strong negator N is called a De Morgan triplet, if and only if S (x, y) = N(T (N(x), N(y))), for all x, y ∈ [0, 1]. S and T are said to be dual to each other in this case. Definition 1.7 [2] A fuzzy implicator is a binary operation on [0, 1] with order reversing first partial mappings and order preserving second partial mappings such that I(0, 1) = I(0, 0) = I(1, 1) = 1, I(1, 0) = 0. An implicator I is said to have contrapositive symmetry with respect to a negator N if it satisfies I(x, y) = I(N(y), N(x)), for all x, y ∈ [0, 1]. Definition 1.8 [10] Let T be a t-norm, a mapping I :[0, 1] → [0, 1] defined as I (x, y) = sup{u ∈ [0, 1] : T (u, x)  y} is called an R−implicator with respect to T . R−implicators are defined by using the concept of residuation (for more details on this notion see [18]). R−implicators are often (in fact, for left-continuous t-norms) partial orderings on the unit interval: I(x, y) = 1 if and only if x  y. Let T be a t-norm, I (the R−implicator associated with T ) is said to satisfy resid- uation condition if T (x, y)  z ⇔ x  I (y, z). (1) Obviously, taking into account that T is nondecreasing in both places, condition (1) is equivalent to the following one: T (x,·) is a left-continuous function for any fixed x ∈ [0, 1]. Remark 1.9 [18] If T is a continuous Archimedean t-norm and f :[0, 1] → [0,∞] an additive generator of T , then the R−implicator can be obtained by the formula (−1) I (x, y) = f (max(T (y)− T (x), 0)). Theorem 1.10 [10] Suppose that T is a t-norm such that (1) is satisfied and N is a strong negator. If I has contrapositive symmetry with respect to N, then (i) I (x, y) = N(T (x, N(y))) for all x, y ∈ [0, 1]. (ii) N(x) = I (x, 0), for all x ∈ [0, 1] (iii) T (x, y) = 0 if and only if x  N(y), for any x, y ∈ [0, 1]. Definition 1.11 [6] Following properties are defined for a fuzzy relation R defined on a universe X and a t-norm T : 208 Ismat Beg · Samina Ashraf (2009) (i) Reflexivity: R(x, x) = 1, for all x ∈ X; (ii) Symmetry: R(x, y) = R(y, x), for all x, y ∈ X; (iii) T−transitivity: R(x, z)  T (R(x, y), R(y, z)), for all x, y, z ∈ X. A reflexive, symmetric and T−transitive fuzzy relation is called T−equivalence relation (or T−indistinguishability relation). Definition 1.12 [30] Let S be t-conorm. An S -pseudometric m is a map from X× X into [0, 1] such that (i) m(x, x) = 0, for all x ∈ X; (ii) m(x, y) = m(y, x), for all x, y ∈ X; (iii) S (m(x, y), m(y, z))  m(x, y), for all x, y, z ∈ X, (S− triangle inequality). An S−metric is defined in the usual way i.e., replacing (i) by m(x, y) = 0 if and only if x = y. Theorem 1.13 [30] Let E be a T−indistinguishability relation on X and let φ be a continuous and order-reversing bijection from [0, 1] into itself, then m (x, y) = φ(E(x, y)), for all x, y ∈ X −1 is a S -pseudometric and vice-versa, where S (x, y) = φ (T (φ(x),φ(y)). 2.−Dissimilarity Relations The concepts introduced in this section are based on notions of −transitivity and −equivalence relation (for details see [4]). We first recall following definitions and remarks to explain notions and notations to be used in this section. Definition 2.1 [4] For a given fuzzy relation R, the measure of transitivity of R is given by Tr(R) = inf (I(T (R(x, y), R(y, z)), R(x, z))). (2) x,y,z∈X A fuzzy relation R is called − fuzzy transitive if  = Tr(R), R is non-transitive if = 0, strong fuzzy transitive if   0.5 and weak fuzzy transitive if  ∈]0, 0.5[. Remark 2.2 [4] If an R−implicator is used in calculating, degree of transitivity, then the T−transitive fuzzy relations are 1− fuzzy transitive. This is due to the fact that R−implicators possess the property that a  b implies I(a, b) = 1. So the T−transitivity i.e., T (R(x, y), R(y, z))  R(x, z) for all x, y, z ∈ X implies I(T (R(x, y), R(y, z)), R(x, z)) = 1 for all x, y, z ∈ X, hence the result. Definition 2.3 [4] A fuzzy relation E on X is called an−fuzzy equivalence relation if, (i) E(x, x) = 1, for all x ∈ X; (ii) E(x, y) = E(y, x), for all x, y ∈ X; (iii) T r(E) = , for all x, y, z ∈ X. In this case, E will be called an −equivalence relation. In general, if> 0, then R will be called an−fuzzy equivalence relation. Now we develop the notion of fuzzy dissimilarity relation as complement of a fuzzy equivalence relation on any universe X. Definition 2.4 Let E be a fuzzy equivalence relation defined on a crisp universe X and N be a strong negator. A fuzzy dissimilarity relation d associated with E, is a E Fuzzy Inf. Eng. (2009) 2: 205-217 209 fuzzy relation on the same universe X defined as d (x, y) = N(E(x, y)), for all x, y ∈ X. If E is an −equivalence relation instead of just being fuzzy equivalence relation, then d is called−dissimilarity relation. Remark 2.5 In (2), if we select an implicator I satisfying: I(x, y) = 0 implies x = 1 and y = 0, then the fuzzy dissimilarity relation d possesses following properties as consequence of its definition : (D1) d (x, x) = 0, (Self dissimilarity), (D2) d (x, y) = d (y, x), (Symmetry), E E (D3) At least one of the following holds: (a) d (x, y) ∈]0, 1]; (b) d (y, z) ∈]0, 1]; (c) d (x, z) ∈ [0, 1[. From the Remark 2.5, it is obvious that the first two conditions are the ones sat- isfied by any distance function, whereas, the dissimilarity relation defined in this manner satisfies (D3) instead of triangle inequality. If the dissimilarity between x and y is zero and between y and z is also zero, then x and z can not have dissimilarity of 1 degree. This alternative of triangle inequality is a very soft condition because it is established for the fuzzy relations with nonzero degrees of transitivity i.e.,  ∈]0, 1]. Next we develop another form of condition (D3) for negator of an −dissimilarity relation. Remark 2.6 Due to Theorem 1.13, it is obvious that if E is a T−transitive equiva- lence relation, then d always satisfies the identity as follows, d (x, z)  S (d (x, y), d (y, z)), E E E where, S is the conorm dual to to the t-norm T. Hence in case of T−transitive fuzzy relations we do not need to seek for an alternative for triangle inequality. This is why, in what follows, we shall restrict to the establishment of results for only those points x, y, z ∈ X, for which T (E(x, y), E(y, z)) > E(x, z). Theorem 2.7 Let d be the fuzzy dissimilarity relation obtained by applying a strong negator N to an −equivalence relation E defined on a given universe X, T be a left continuous Archimedean t-norm with differentiable generators and I is a contrapositive R−implicator associated with T , then there exists a strictly decreasing continuous function f and a positive real number k such that d (x, z)  S (d (x, y), E E f () d (y, z))+ = S (d (x, y), E E f´(θ) d (y, z))− kf (), where, (T, S, N) form a De Morgan’s triplet. Proof Since E is an −fuzzy equivalence relation, inf (I(T (R(x, y), R(y, z)), R(x, z))) = . x,y,z∈X 210 Ismat Beg · Samina Ashraf (2009) It implies that I(T (E(x, y), E(y, z)), E(x, z)) > 0. (3) Using Remark 1.9 of R−implicators, associated with continuous Archimedean t-norm T (A left continuous Archimedean t-norm is also continuous see [18, Proposition 2.16, page 30] or [8]), there exist a continuous and strictly decreasing function f : [0, 1] → [0,∞] such that −1 I(x, y) = f (max{ f (y)− f (x), 0}). Thus, there exist a continuous and strictly decreasing function f :[0, 1] → [0,∞], such that −1 f (max{ f (E(x, z))− f (T (E(x, y), E(y, z)), 0}). Applying f to both sides and using its strictly decreasing nature, we obtain max{ f (E(x, z))− f (T (E(x, y), E(y, z))), 0}  f (). (4) Since E(x, z) < T (E(x, y), E(y, z)) by the Remark 2.6 and f is a strictly decreasing function, it implies that f (E(x, z)) > f (T (E(x, y), E(y, z))). Hence f (E(x, z))− f (T (E(x, y), E(y, z))) > 0. By using this fact in (4) we get f (E(x, z))− f (T (E(x, y), E(y, z)))  f (). By use of the involutive negator N in d (x, y) = N(E(x, y)) we get E(x, y) = N(d (x, y)) E E and it further implies f (N(d (x, z)))− f (T (N(d (x, y)), N(d (y, z))))  f (). E E E Since (T, S, N) is a De Morgan triplet, therefore, we have f (N(d (x, z)))− f (N(S (d (x, y), d (y, z)))  f (). (5) E E E Now by applying the representation Theorem 1.10 for implicator to I (x, 0) = N(x), we get, −1 N(x) = f (max{ f (0)− f (x), 0}). We have f (N(x)) = max{ f (0)− f (x), 0}. by applying f to both sides. Since both S (d (x, y), d (y, z))  0, and d (x, z)  0, so f (0)  f (S (d (x, y), d (y, z)) E E E E E and f (0)  f (d (x, z)) by the decreasing nature of the function f . Thus in this partic- ular case we have f (N(x)) = f (0)− f (x). (6) Fuzzy Inf. Eng. (2009) 2: 205-217 211 Using (6) in the equation (5), we get f (0)− f (d (x, z))− f (0)+ f (S (d (x, y), d (y, z))  f (), E E E it implies that f (S (d (x, y), d (y, z)))− f (d (x, z))  f (). E E E Applying the Cauchy mean value theorem, there exist a number θ satisfying d (x, z)  θ  S (d (x, y), d (y, z)), E E E such that f (θ)(S (d (x, y), d (y, z))− d (x, z)))  f (). E E E Dividing both sides by f (θ)(< 0) f () S (d (x, y), d (y, z))− d (x, z)  . E E E f (θ) Hence f () d (x, z)  S (d (x, y), d (y, z))+ . E E E f (θ) Letting k = − . We get the result. f (θ) Remark 2.8 Though the Theorem 2.7 has been proved R−implicators, but the results are expected to hold for S−implicators as well. Next we prove the theorem for Reichenbach implicator which is not an R−implicator. Theorem 2.9 If the Reichenbach implicator defined as I(a, b) = 1− a+ ab, for all a, b ∈ [0, 1] is used along with P and P as t-norm and t-conorm respectively in the definition of an −equivalence relation E and standard negator is used to obtain the corresponding fuzzy dissimilarity relation, then d satisfies the following inequality: d (x, z)  P (d (x, y), d (y, z))+ (1−). E E E Proof Since E is an −equivalence relation so, for any x, y, z ∈ X, I(P(E(x, y), E(y, z)), E(x, z))  . Putting value of E(x, y) = N(d (x, y)) in the above equation, we get I(P(N(d (x, y)), N(d (E(y, z))), N(d (x, z)))  . E E E Using value of I 1− P(N(d (x, y)), N(d (y, z)))+ P(N(d (x, y)), N(d (y, z)))N(d (x, z))  . E E E E E Using the value of P(a, b) = a.b and N(a) = 1− a. 1− (1− d (x, y))(1− d (y, z))+ (1− d (x, y))(1− d (y, z))(1− d (x, z))  . E E E E E 212 Ismat Beg · Samina Ashraf (2009) On re-arranging expressions, we get 1−  (1− d (x, y))(1− d (y, z))(1− (1− d (x, z))). E E E So, 1−  d (x, z)(1− d (y, z)− d (x, y)+ d (x, y)d (y, z)). E E E E E On simplifying, d (x, z)  d (x, z)(d (y, z)+ d (x, y)− d (x, y)d (y, z))+ 1−. E E E E E E Using d (x, z)  1, d (x, z)  d (y, z)+ d (x, y)− d (x, y)d (y, z)+ 1−. (7) E E E E E Hence by using the value of corresponding P , we get d (x, z)  P (d (x, y), d (y, z))+ 1−. E E E Corollary 2.10 In Equation (7) if the negative term is omitted (which does not dis- turb the inequality) instead of adjusting in disjunction operator of the inequality is even strengthened and we get the form of triangular inequality used in definition of metric d (x, z)  d (y, z)+ d (x, y)+ 1−. E E E Theorem 2.11 Let d be the corresponding dissimilarity relation of an−equivalence relation on X, T a left-continuous t-norm and N be a strong negator. If I is a contra- positive implicator associated with T , then d (x, z)  S (d (y, z), d (x, y)) if and only if tr(x, y, z) = 1. E E E where, (T, S, N) form a De Morgan triplet and tr(x, y, z) denotes pointwise transitivity defined as tr(x, y, z) = I(T (E(x, y), E(y, z)), E(x, z)). Proof Since E is an −equivalence relation, inf I(T (E(x, y), E(y, z)), E(x, z)) = . x,y,z∈X For any x, y, z ∈ X, tr(x, y, z) = I(T (E(x, y), E(y, z)), E(x, z))  . Let tr(x, y, z) = I(T (E(x, y), E(y, z)), E(x, z)) =  . Using E(x, y) = N(d (x, y)), we get I(T (N(d (x, y)), N(d (y, z))), N(d (x, z))) =  . E E E 1 By properties of De Morgan triplet, we have I(N(S (d (y, z), d (x, y))), N(d (x, z))) =  . E E E 1 Since Theorem 1.10 (i) further implies that I(N(S (d (y, z), d (x, y))), N(d (x, z))) E E E = N(T (N(S (d (y, z), d (x, y))), N(N(d (x, z))) =  . E E E 1 Fuzzy Inf. Eng. (2009) 2: 205-217 213 Since N is an involutive, we have N(T (N(S (d (y, z), d (x, y))), d (x, z))) =  . E E E 1 Applying negator operator on both the sides, we get T (N(S (d (y, z), d (x, y))), d (x, z)) = N( ). (8) E E E 1 Now if d (x, z)  S (d (y, z), d (x, y)), then N(S (d (y, z), d (x, y)))  N(d (x, z)) and E E E E E E applying Theorem 1.10, we get T (N(S (d (y, z), d (x, y))), d (x, z)) = 0 = N( ). E E E 1 This implies that  = 1. Conversely letting = 1 in Equation (8), one can easily obtain d (x, z)  S (d (y, z), d (x, y)). E E E 3. Examples of Dissimilarity Measures In this section, we present some formulae which calculates degree of dissimilarity between elements of F(X)(F(X) in itself is a crips universe). In all the following examples we will use implicators with property stated in Remark 2.5. Example 3.1 Let X be a given universe and let A, B ∈ F(X), and let I be an R− implicator. If we define fuzzy set of dissimilarity Dis between A and B as: A,B Dis (x) = 1− T (I(A(x), B(x)), I(B(x), A(x))), for all x ∈ X, (9) A,B Dis A, B then Dis(A, B) = is a dissimilarity measure on F(X) where, |A| denotes the |X| scalar cardinality of a fuzzy subset A of a finite universe X, defined as|A| = A(x). x∈X Proof Self dissimilarity: For all A ∈ F(X), 1− T (I(A(x), A(x)), I(A(x), A(x))) Dis A,A x∈X Dis(A, A)= = |X| |X| 1− T (1, 1) 1− 1 x∈X x∈X = = = = 0. |X| |X| |X| (D2) Symmetry is straightforward due to commutative nature of T . (D3) Suppose on contrary that there exist A, B, C ∈ F(X) such that Dis(A, B) = 0 = Dis(B, C) and Dis(A, C) = 1. By putting values of these functions, we get Dis A,B Dis(A, B)= |X| 1− T (I(A(x), B(x)), I(B(x), A(x))) x∈X = = 0 (10) |X| 214 Ismat Beg · Samina Ashraf (2009) and 1− T (I(B(x), C(x)), I(C(x), B(x))) Dis B,C x∈X Dis(B, C) = = = 0 (11) |X| |X| and 1− T (I(A(x), C(x)), I(C(x), A(x))) Dis A,C x∈X Dis(A, C) = = = 1. (12) |X| |X| Solving Equation (10), we get A(x) = B(x), for all x ∈ X. Similarly solving equation (11), we get B(x) = C(x) for all x ∈ X. But Equation (12) can hold only if there exist some x ∈ X, such that T (I(A(x), C(x)), I(C(x), A(x)))  1. So, either I(A(x), C(x))  1 or I(C(x), A(x)))  1. In every case A and C are not equal fuzzy sets, a contradiction. Hence the result. Definition 3.2 [27] A fuzzy inclusion is a fuzzy relation Inc on F(X) defined as Inc(A, B) = inf I(A(x), B(x)), for all A, B ∈ F(X), x∈X where, A(·) refers to the membership function of the fuzzy set A etc. Theorem 3.3 Let X be a given universe. The fuzzy relation Dis on F(X) defined by Dis(A, B) = N(T (Inc(A, B), Inc(B, A))) (13) is a fuzzy dissimilarity measure on F(X) for any t-norm T , and an implicator I satis- fying I(x, x) = 1. Proof For all A, B, C ∈ F(X)wehave (D1) Self dissimilarity Dis(A, A)= N(T (Inc(A, A), Inc(A, A))) = N(T (1, 1)) = N(1) = 0. (D2) Symmetry is obvious. (D3) Assume that the dissimilarity so defined does not satisfy (D3), so there exist fuzzy sets A, B,C ∈ F(X), such that Dis(A, B) = Dis(B, C) = 0 and Dis(A, C) = 1. Applying negator, it implies that T (Inc(A, B), Inc(B, A)) = T (Inc(B, C), Inc(C, B)) = 1 (14) and T (Inc(A, C), Inc(C, A)) = 0. (15) Using properties of t-norm in Equations (14), we have Inc(A, B) = Inc(B, A) = Inc(B, C) = Inc(C, B) = 1, (16) Using the value of inclusion operator in Equations (16), we get inf I(A(x), B(x)) = inf I(B(x), A(x)) = 1, (17) x∈X x∈X Fuzzy Inf. Eng. (2009) 2: 205-217 215 inf I(B(x), C(x)) = inf I(C(x), B(x)) = 1, (18) x∈X x∈X Equations (17) and (18) imply that A = B and B = C, i.e., A = C, where = stands for equality of fuzzy sets. From here it follows that Inc(A, C) = Inc(C, A) = 1, which contradicts (15) hence the result. |A∩ B| Example 3.4 For all A, B ∈ F(X), d (A, B) = 1 − is a fuzzy dissimilarity |A∪ B| relation on F(X), provided∩ and∪ are modelled by max and min, respectively. Proof For all A, B, C ∈ F(X), we have: (D1) Self dissimilarity |A∩ A| d (A, A) = 1− = 0. |A∪ A| (D2) Symmetry: |A∩ B| d (A, B) = 1− = d (B, A). E E |A∪ B| (D3) Suppose on contrary that d (A, B) = 0 = d (B, C) and d (A, C) = 1, so, E E E min(A(x), B(x)) min(B(x), C(x)) x∈X x∈X 1−  = 0 and 1−  = 0, (19) max(A(x), B(x)) max(B(x), C(x)) x∈X x∈X together with min(A(x), C(x)) x∈X 1−  = 1. (20) max(A(x), C(x)) x∈X Using properties of t-norms and t-conorms, Equations (19) lead us to the conclusion that A, B and C are equal fuzzy sets while Equation (20) implies that T (A(x), C(x)) = 0. x∈X It further implies that for all x ∈ X either A(x) = 0or C(x) = 0, a contradiction to the above conclusion that A and C are equal (nonzero) fuzzy sets. Hence the result. 4. Conclusions In this paper we have explored the relationship between the concepts of dissimilarity and distance. The conclusion reached is that fuzzy transitivity of a fuzzy equivalence relation behaves as a controlling factor for the triangle inequality satisfied by the nega- tor of the fuzzy equivalence relation. The greater the degree of fuzzy equivalence, the closer is the inequality satisfied by the dissimilarity function to the triangle inequality. The negator of a fuzzy equivalence relation satisfies the triangular inequality or the inequality for being an ultra-metric only if it is a fuzzy equivalence relation of degree Acknowledgments The present version of the paper owes much to the precise and kind remarks of the learned referees. 216 Ismat Beg · Samina Ashraf (2009) References 1. Balopoulos V, Hatzimichailidis AG, Papadopoulos BK (2007) Distance and similarity measures for fuzzy operators. Information Sciences 177:2336-2348 2. Bandler W, Kohout L (1980) Semantics of implicators and fuzzy relational products. Man-Machine Studies 12:89-116 3. Bazu K (1984) Fuzzy revealed preference theory. Economics Theory 32:212-227 4. Beg I, Ashraf S (2008) Fuzzy equivalence relations. Kuwait J. Sci. and Eng 35(1A):191-206 5. Bezdek JC, Harris JO (1978) Fuzzy partitions and relations. An axiomatic basis for clustering. Fuzzy Sets and Systems 1:111-126 6. Bodenhofer U, De Baets B, Fodor JC (2007) A compendium on fuzzy weak orders: Representations and constructions. Fuzzy Sets and Systems 158:811-829 7. De Cock M, Kerre EE (2003) On (un)suitable fuzzy relations to model approximate equality. Fuzzy Sets and Systems 133(2):137-153 8. De Baets B, Mesier R (1996) Residual implicators of continuous t-norms, Proceedings EUFIT’96 1:22-26 9. De Baets B, Mesier R (1997) Pseudo-metrics and T-equivalences. Fuzzy Math 5(2) 10. Fodor J (1995) Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems 69:141-156 11. Fodor J, Yager RR (2000) “Fuzzy set-theoretic operators and quantifiers”, in Fundamentals of Fuzzy Sets, (edited by: D. Dubois and H. Prade), Kluwer Academic Publishers: Norwell, Ma:125-193 12. Getty DJ, Swets JA, Swets JB, Green DM (1979) On the prediction of confusion matrices form similarity judgements. Perception and Psycophysics 26:1-19 13. Gregory Ashby F, Perrin NA (1988) Towards a unified theory of similarity and recognition. Psycho- logical Review 95: 124-150 14. Jain R, Murthy S, Tran P, Chattarjee S (1995) Similarity measures for image database, IEEE Tran- sections on Knowledge and Data Engineering 7:1247-1254 15. James W (1890) Principle of psychology. New York Holt 16. Klawonn F (1999) The role of similarity in the fuzzy reasoning. Fuzzy sets, logic and reasoning about knowledge, Appl. Log. Ser,15, Kluwer Acad.Pub, Dordrecht:243-253 17. Klawonn F (2003) Should fuzzy equality and similarity satisfy transitivity? Comments on the paper by M. De Cock and E. Kerre. Fuzzy Sets and Systems 133 2:175-180 18. Klement EP, Mesiar R, Pap E (2000) Triangular norms, Trends in Logic. Kluwer Academic Publish- ers, Dordrecht,Vol. 8 19. Krumhansi CL (1978) Concerning the applicability of geometric models to similarity data: The in- terrelationship between similarity and spatial density. Psycological Review 85:445-463 20. Menger K (1951) Probabilistic theories of relations. Proceedings of the National Academy of Sci- ences 37:178-180 21. Nguyen HT, Walker EA (2006) A first course in fuzzy logic. (Third Edition) CRC Press 22. Poincar’e H (1902) La Science et 1’Hypothese. Flammarion, Paris 23. Ruspini EH (1992) On truth, utility, and similarity. Fuzzy engineering toward human friendly systems (Yokohama, 1991), 42-50, Ohm, Tokyo 24. Santini S, Jain R (1999) Similarity measures. IEEE Transactions on Pattern Analysis and Machine Intelligence 21: 871-883 25. Trillas E (1982) Assaig sobre les relacions d’indistingibilitate. Actes del Primer Congress Catala de Logica Matematica. Barcelona:51-59 26. Trillas E, Valverde L (1984) An essay on indistinguishability, in aspects of vagueness. Reidel, Eds.H.J.Skala et al 27. Tsiporkova-Hristoskova E, De Baets B, Kerre EE (1997) A fuzzy inclusion based approach to upper inverse images under fuzzy multivalued mappings. Fuzzy Sets and Systems 85:93-108 28. Tversky A (1977) Features of similarity. Psychological Review 84(4):327-352 29. Tverski A, Gati I (1982) Similarity, separability, and the triangle inequality. Psychological Review 89(2):123-154 30. Valverde L (1985) On the structure of F-indistinguishability operators. Fuzzy Sets and Systems Fuzzy Inf. Eng. (2009) 2: 205-217 217 17:313-328 31. Williams J, Steele N (2002) Difference, distance and similarity as a basis for fuzzy decision support based on prototypical decision classes. Fuzzy Sets and Systems 131:35-46 32. Zadeh LA (1965) Fuzzy sets. Information and Control 8(3):338-353 33. Zadeh LA (1971) Similarity relations and fuzzy orderings. Inf. Sci.,3:177-200 34. Zwick R, Carlstein E, Budesco DV (1987) Measures of similarity amongst fuzzy concepts: a com- parative analysis. Int.J. of Approximate Reasoning 1:221-242

Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Jun 1, 2009

Keywords: Fuzzy dissimilarity; Distance axioms; Fuzzy equivalence relation

References