Generalizated local divergence measures

Abstract

The comparison of sets is an important topic with application in several fields. Divergence measures were introduced as an adequate measure of comparison of two fuzzy sets and an alternative of the dissimilarities. The particular study for local divergences is here generalized to any t-conorm instead just the sum. The concept of divergence is revisited and studied in detail. This study is complemented with the whole characterization of a new family of divergence measures, the generalized local divergence measures. Some applications of these divergences in pattern recognition and decision making illustrate their utility.

Keywords

Divergence measure T-conorm locality S-locality aggregation operator

1 Introduction

Estimation of similarity between sets (patterns or alternatives) is crucial in many aspects. This task is usually performed by applying a suitable comparison measure to fuzzy sets corresponding to given patterns or alternatives.

This can be done from different points of view. In some cases we measure the equality degree and in other ones the difference degree. A study of measures of comparison was given by Bouchon-Meunier et al. [2]. Recently, a review of the measures based on the differences was proposed by Couso et al. [4]. Among these measures, divergences appears as a good alternative [7, 8]. An important family of such measures fulfills the following natural property: if we change one of the two images only at a singleton, then the variation of divergence only depends on the changed values. This is called locality and divergence measures fulfilling it are called local. The computational advantage is obvious and so they were studied and characterized in [8]. The definition of a divergence measure is based on the union and intersection of fuzzy sets in a standard way by Zadeh [13] based on the maximum t-conorm and the minimum t-norm, respectively. Our previous studies are now completed for any t-conorm and any t-norm. Apart from that, the family of local divergences is generalized to the family of S-local divergences. They are obtained by combining the divergence at particular points of the universe with “distances” among them using a triangular conorm.

2 Basic concepts

The universal set is denoted by X. A fuzzy subset of X is a mapping from X into the unit interval [0, 1]. In this framework, we use the following notations:

$P (X)$ is the set of all subsets of X,

$F (X)$ is the set of all fuzzy subsets of X,

We identify a fuzzy set and its membership function. Thus we have that X (x) =1 for all x ∈ X and for the empty set we have ∅ (x) =0 for all x ∈ X.

Another important concepts will be the containment relation and the complement set. We consider the standard Zadeh’s negation for the complement (see [13]).

Definition 1. Let $\tilde{A}, \tilde{B} \in F (X)$ . The complement of $\tilde{A}$ is the fuzzy set ${\tilde{A}}^{c} (x) = 1 - \tilde{A} (x), x \in X$ . $\tilde{A}$ is contained in $\tilde{B}$ , denoted by $\tilde{A} \subseteq \tilde{B}$ if $\tilde{A} (x) \leq \tilde{B} (x)$ for all x ∈ X.

Apart from the previous relation of containment, we consider the concepts of intersection and union of fuzzy sets. The initial definitions were also given in [13] as minimum and maximum. However, they are not the only way to generalize the classical set operations, but there exist a broad class of functions to represent them. For the intersection, this class is referred as t-norms and for the union as t-conorm.

The triangular norm, (t-norm) is a function T : [0, 1] × [0, 1] → [0, 1] satisfying the following:

T (a, b) = T (b, a), for all a, b ∈ [0, 1],

T (T (a, b), c) = T (a, T (b, c)), for all a, b, c ∈ [0, 1],

b ≤ c ⇒ T (a, b) ≤ T (a, c), for all a, b, c ∈ [0, 1],

T (a, 1) = a, for all a ∈ [0, 1].

Some important examples of t-norms are:

Minimum: T_M (a, b) = min(a, b), for all a, b ∈ [0, 1],

Product: T_P (a, b) = a · b, for all a, b ∈ [0, 1],

Łukasiewicz t-norm: T_L (a, b) = max(a + b - 1, 0), for all a, b ∈ [0, 1],

Drastic t-norm: $T_{D} (a, b) = {\begin{matrix} min {a, b}, & if max {a, b} = 1 . \\ 0, & otherwise . \end{matrix}$

For these basic t-norms, it holds that T_D ≤ T_L ≤ T_P ≤ T_M. In fact, for any t-norm T, there is T_D ≤ T ≤ T_M.

By changing the neutral element from 1 to 0, we obtain the triangular conorm (t-conorm).

The t-norm T and t-conorm S are dual iff for each a, b ∈ [0, 1] there is T (a, b) =1 - S (1 - a, 1 - b).

For the previous examples, the duals are:

Maximum: S_M (a, b) = max(a, b), for all a, b ∈ [0, 1],

Probabilistic sum: S_P (a, b) = a + b - a · b, for all a, b ∈ [0, 1],

Łukasiewicz t-conorm: S_L (a, b) = min(a + b, 1), for all a, b ∈ [0, 1],

Drastic t-conorm: $S_{D} (a, b) = {\begin{matrix} max {a, b}, & if min {a, b} = 0 . \\ 1, & otherwise . \end{matrix}$

Using t-norms and t-conorms, we can define in general the intersection and union of two fuzzy as follows.

Definition 2. Let $\tilde{A}, \tilde{B} \in F (X)$ . Given a t-norm T and a t-conorm S,

$\tilde{A} \cap \tilde{B} (x) = T (\tilde{A} (x), \tilde{B} (x)), \forall x \in X$ ;

$\tilde{A} \cup \tilde{B} (x) = S (\tilde{A} (x), B (x)), \forall x \in X$ .

Thus, we can denote by (X, T, S) the triple formed by the universe with the t-norm and the t-conorm defining the intersection and the union, respectively.

3 Divergences

In 1996, Bouchon-Meunier [2] tried to define a general measure of comparison between fuzzy sets. Since then more measures for comparing fuzzy sets have been introduced (see [1 , 15]). A nice study about that can be found in [4]. The usual measures of comparison are dissimilarities [6].

Definition 3. A map $D : F (X) \times F (X) \to ℝ$ is a dissimilarity if it satisfies the following axioms:

$D (\tilde{A}, \tilde{A}) = 0$ , for every $\tilde{A} \in F (X)$ .

$D (\tilde{A}, \tilde{B}) = D (\tilde{B}, \tilde{A})$ , for every $\tilde{A}, \tilde{B} \in F (X)$ .

For every $\tilde{A}, \tilde{B}, \tilde{C} \in F (X)$ such that $\tilde{A} \subseteq \tilde{B} \subseteq \tilde{C}$ , there is $D (\tilde{A}, \tilde{C}) \geq max (D (\tilde{A}, \tilde{B}), D (\tilde{B}, \tilde{C}))$ .

As this definition is not too restrictive, it is possible to define a counterintuitive measure of comparison for which the above axioms hold. The restriction associated to this definition is given by the fact that the requirement in Axiom (Diss.3) is only given for sets such that $\tilde{A} \subseteq \tilde{B} \subseteq \tilde{C}$ , but there are a lot of sets which are not comparable with respect to ⊆ and therefore, nothing is required for them. Thus, we need a concept where the restriction about “proximity” are given for any set.

So, another measure of comparison was proposed in [7], the divergence, with the following properties:

It becomes zero when the two sets coincide.

It is a nonnegative and symmetric function.

It decreases when the two subsets become “more similar” in some sense.

While it is easy to formulate the first and the second conditions analytically, the third one depends on the formalization of the concept “more similar”. We base our approach on the fact that if we add a subset $\tilde{C}$ to both fuzzy subsets $\tilde{A}, \tilde{B}$ , we obtain two subsets which are closer to each other; the same with the intersection.

Definition 4. Let (X, T, S) be a triple with X a universe and T and S any t-norm and t-conorm, respectively. A map $D : F (X) \times F (X) \to ℝ$ is a divergence measure with respect to (X, T, S) iff for all $\tilde{A}, \tilde{B} \in F (X)$ , D satisfies the following conditions:

$D (\tilde{A}, \tilde{A}) = 0$ ;

$D (\tilde{A}, \tilde{B}) = D (\tilde{B}, \tilde{A})$ ;

$max {D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}), D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C})} \leq D (\tilde{A}, \tilde{B})$ , for all $\tilde{C} \in F (X)$ , where the union and intersection are defined by means of S and T, respectively.

It is clear that a divergence measure is associated to a triple (X, T, S) and a map D can be a divergence measure with respect a t-norm and it cannot be with respect to a different t-norm. However, when there is not ambiguity, we will say just a divergence measure without specifying the used t-norm and t-conorm.

We present some examples of divergence measures.

Example 1. $D (\tilde{A}, \tilde{B}) = {\begin{matrix} 0, & if \tilde{A} = \tilde{B} . \\ 1, & if \tilde{A} \neq \tilde{B} . \end{matrix}$

Clearly D is a divergence for any triple (X, T, S).

Example 2. We consider the triple (X, T_M, S_M) where X is finite. For any pair of fuzzy sets in X we put: $D (\tilde{A}, \tilde{B}) = \sum_{x \in X} α_{x} \cdot | \tilde{A} (x) - \tilde{B} (x) |$ where α_x ≥ 0 for any x ∈ X and ∑_x∈Xα_x = 1.

Again, the first and second conditions are trivial. Denote $\tilde{A} (x) = a, \tilde{B} (x) = b, \tilde{C} (x) = c$ . Without loss of generality assume that a ≥ b and then T (a, c) ≥ T (b, c) for any t-norm T, so in particular for T_M. Thus,

if c > a ≥ b, then |T_M (a, c) - T_M (b, c) | = |a - b| ≤ |a - b|,

if a ≥ c ≥ b, then |T_M (a, c) - T_M (b, c) | = |c - b| ≤ |a - b|,

if a ≥ b ≥ c, then |T_M (a, c) - T_M (b, c) | = |c - c|=0 ≤ |a - b|,

and therefore, in all the cases, this inequality holds. From here we have that

\begin{matrix} D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) \\ = \sum_{x \in X} α_{x} \cdot | (\tilde{A} \cap \tilde{C}) (x) - (\tilde{B} \cap \tilde{C}) (x) | \\ = \sum_{x \in X} α_{x} \cdot | T_{M} (\tilde{A} (x), \tilde{C} (x)) - T_{M} (\tilde{B} (x), \tilde{C} (x)) | \\ \leq \sum_{x \in X} α_{x} \cdot | \tilde{A} (x) - \tilde{B} (x) | = D (\tilde{A}, \tilde{B}) . \end{matrix}

Analogously we prove $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) \leq D (\tilde{A}, \tilde{B})$ with the maximum t-conorm. Thus, D is a divergence.

D is also a divergence if we consider the product t-norm or the Łukasiewicz t-norm and their dualt-conorms, since

|T_P (a, c) - T_P (b, c) | = |a · c - b · c| = c · |a - b| ≤ |a - b| and |S_P (a, c) - S_P (b, c) | = |a + c - a · c - (b + c - b · c) | = (1 - c) · |a - b| ≤ |a - b|;

if a + c ≥ 1 and b + c ≥ 1 then |T_L (a, c) - T_L (b, c) | = | (a + c - 1) - (b + c - 1) | = |a - b| and |S_L (a, c) - S_L (b, c) | = |1 - 1|=0 ≤ |a - b|,

if a + c < 1 and b + c ≥ 1 then |T_L (a, c) - T_L (b, c) | = |0 - (b + c - 1) | = |b - (1 - c) |,but a < 1 - c ≤ b and therefore |T_L (a, c) - T_L (b, c) | ≤ |b - a| = |a - b|. Moreover,|S_L (a, c) - S_L (b, c) | = |a + c - 1 | = |(1 - c) - a| ≤ |b - a|.

the case a + c ≥ 1 and b + c < 1 is analogous to the previous one,

if a + c < 1 and b + c < 1 then |T_L (a, c) - T_L (b, c) | = |0 - 0|=0 ≤ |a - b| and |S_L (a, c) - S_L (b, c) | = | (a + c) - (b + c) | = |a - b|.

However, this does not hold in general. For instance, if we consider the drastic t-norm, for the case a = 1, b = 0.2, c = 0.9 we have that |T_D (a, c) - T_D (b, c) | = |T_D (1, 0.9) - T_D (0.2, 0.9) | = |0.9 - 0|=0.9 > 0.8 = |a - b|. Then, $D (\tilde{A}, \tilde{B}) = \sum_{x \in X} α_{x} \cdot 0.8 = 0.8 > 0.9 = D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C})$ .

By the last axiom of the Definition 4, we could think that conditions $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) \leq D (\tilde{A}, \tilde{B})$ and $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) \leq D (\tilde{A}, \tilde{B})$ could be equivalent. However, this is not true in general, even in the case T and S are dual.

Example 3. We will show that the two conditions in Axiom (Div3) are not equivalent, even in (X, T_M, S_M).

Let $D : F (X) \times F (X) \to ℝ$ be defined by: $\begin{matrix} D (\tilde{A}, \tilde{B}) & = & \sum_{x \in X} h (\tilde{A} (x), \tilde{B} (x)), \\ where h (x, y) & = & {\begin{matrix} 0, & if x = y . \\ 1 - xy, & otherwise . \end{matrix} \end{matrix}$

Axiom (Diss2) from the definition of divergence is satisfied since the function h is symmetric. By definition h (x, x) =0 and so Axiom (Diss1) is also satisfied.

If we consider the following partition of the universal set X = X₁ ∪ X₂ ∪ X₃ ∪ X₄ ∪ X₅ ∪ X₆, where: $\begin{matrix} X_{1} & = & {x \in X, \tilde{A} (x) \leq \tilde{B} (x) \leq \tilde{C} (x)}, \\ X_{2} & = & {x \in X, \tilde{A} (x) \leq \tilde{C} (x) < \tilde{B} (x)}, \\ X_{3} & = & {x \in X, \tilde{B} (x) < \tilde{A} (x) \leq \tilde{C} (x)}, \\ X_{4} & = & {x \in X, \tilde{B} (x) \leq \tilde{C} (x) < \tilde{A} (x)}, \\ X_{5} & = & {x \in X, \tilde{C} (x) < \tilde{A} (x) \leq \tilde{B} (x)}, \\ X_{6} & = & {x \in X, \tilde{C} (x) < \tilde{B} (x) < \tilde{A} (x)}, \end{matrix}$ then $D (\tilde{A}, \tilde{B}) = \sum_{i = 1}^{6} \sum_{x \in X_{i}} h (\tilde{A} (x), \tilde{B} (x)) .$

Applying the standard maximum t-conorm we have: $\begin{matrix} (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) & = & \sum_{x \in X_{1} \cup X_{3}} h (\tilde{C} (x), \tilde{C} (x)) \\ + \sum_{x \in X_{2}} h (\tilde{C} (x), \tilde{B} (x)) \\ + \sum_{x \in X_{4}} h (\tilde{A} (x), \tilde{C} (x)) \\ + \sum_{x \in X_{5} \cup X_{6}} h (\tilde{A} (x), \tilde{B} (x)) . \end{matrix}$

Since h is, by definition, a positive and decreasing function in both components, we have that: $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) \leq \sum_{x \in X} h (\tilde{A} (x), \tilde{B} (x)) = D (\tilde{A}, \tilde{B}) .$

We have shown that $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) \leq D (\tilde{A}, \tilde{B})$ . However, the inequality for the intersection is not fulfilled in general. Thus, if we consider the universal set X = {x}, the dual minimum t-norm and the fuzzy sets $\tilde{A}, \tilde{B}, \tilde{C}$ defined as $\tilde{A} (x) = 0.2, \tilde{B} (x) = 0.8, \tilde{C} (x) = 0.5$ , we have that $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) = h (\tilde{A} (x), \tilde{C} (x)) = (1 - 0.2 \cdot 0.5) = 0.9$ and $D (\tilde{A}, \tilde{B}) = (1 - 0.2 \cdot 0.8) = 0.84$ .

In the previous example we have shown that $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) \leq D (\tilde{A}, \tilde{B})$ does not imply $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) \leq D (\tilde{A}, \tilde{B})$ . If we change the definition of D with h (x, y) = xy if x ≠ y, then we can show that even the converse implication does not hold, as it fulfills $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) \leq D (\tilde{A}, \tilde{B})$ but not $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) \leq D (\tilde{A}, \tilde{B})$ .

So, we can conclude that both conditions in Axiom 3 of Definition 4 are independent.

Proposition 1. Let D be a divergence measure for (X, T, S). Then $D (\tilde{A}, \tilde{B}) \geq 0$ for all $\tilde{A}, \tilde{B} \in F (X)$ .

Proof. For any $\tilde{A}$ and $\tilde{B}$ in $F (X)$ , we have that $D (\tilde{A}, \tilde{B}) \geq D (\tilde{A} \cap \emptyset, \tilde{B} \cap \emptyset) = D (\emptyset, \emptyset) = 0$ , for any t-norm T.

Divergences appeared as an alternative to dissimilarities, but they are not related in general. We will show a divergence which is not a dissimilarity and conversely.

Example 4. Let us consider the divergence measure D defined on the one-point set X ={ x } as follows: $\begin{matrix} D (\tilde{A}, \tilde{B}) \\ = {\begin{matrix} 0, & if \tilde{A} = \tilde{B} or \tilde{A} (x) \in {0, 1} or \tilde{B} (x) \in {0, 1}, \\ 1, & otherwise . \end{matrix} \end{matrix}$

Now we verify that the D is a divergence measure. The first two conditions are trivial. We prove the third one.

if $\tilde{A} = \tilde{B}$ then $\tilde{A} \cap \tilde{C} = \tilde{B} \cap \tilde{C}$ and $\tilde{A} \cup \tilde{C} = \tilde{B} \cup \tilde{C}$ for any $\tilde{C} \in F (X)$ . Thus, $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) = D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) = D (\tilde{A}, \tilde{B}) = 0$ .

if $\tilde{A} \neq \tilde{B}$ and $\tilde{A} (x), \tilde{B} (x) \in (0, 1)$ , then $D (\tilde{A}, \tilde{B}) = 1$ by definition and since $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}), D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) \in {0, 1}$ by definition, they are lower than or equal to $D (\tilde{A}, \tilde{B})$ .

if $\tilde{A} = \emptyset$ and $\tilde{B} \neq \emptyset$ , then $D (\tilde{A}, \tilde{B}) = 0$ .

For the intersection, we have that $\tilde{A} \cap \tilde{C} (x) = T (\tilde{A} (x), \tilde{C} (x)) = T (0, \tilde{C} (x)) = 0$ , and therefore $\tilde{A} \cap \tilde{C} = \emptyset$ . Thus, $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) = 0 \leq D (\tilde{A}, \tilde{B})$ .

For the union, we have that $\tilde{A} \cup \tilde{C} (x) = S (\tilde{A} (x), \tilde{C} (x)) = S (0, \tilde{C} (x)) = \tilde{C} (x)$ , and so $\tilde{A} \cup \tilde{C} = \tilde{C}$ . Thus,

if $\tilde{C} (x) \in {0, 1}$ , then $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) = D (\tilde{C}, \tilde{B} \cup \tilde{C}) = 0$ .

if $\tilde{C} (x) \in (0, 1)$ , then $\tilde{B} \cup \tilde{C} = X$ if we consider the union defined by means of the drastic t-conorm. Thus, $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) = D (\tilde{C}, X) = 0$ .

In all the cases, $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) \leq D (\tilde{A}, \tilde{B})$ .

the case $\tilde{B} = \emptyset$ and $\tilde{A} \neq \emptyset$ is analogicalto (iii).

if $\tilde{A} = X$ and $\tilde{B} \neq X$ then $D (\tilde{A}, \tilde{B}) = 0$ by definition. Moreover, $\tilde{A} \cap \tilde{C} = \tilde{C}$ and $\tilde{A} \cup \tilde{C} = X$ , since 1 is the neutral element of any t-norm and the annihilating element of any t-conorm. Thus, the divergence between the intersections is 0 in for any $\tilde{B}$ if we consider the drastic t-norm, since

if $\tilde{C} (x) \in {0, 1}$ , then $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) = D (\tilde{C}, \tilde{B} \cap \tilde{C}) = 0$ by definition.

if $\tilde{C} (x) \in (0, 1)$ , then $\tilde{B} \cap \tilde{C} = \emptyset$ if we consider T_D to define the intersection. Thus, $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) = D (\tilde{C}, \emptyset) = 0$ .

In both cases,

D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) \leq D (\tilde{A}, \tilde{B})

. Moreover,

D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) = D (X, \tilde{B} \cup \tilde{C}) = 0 \leq D (\tilde{A}, \tilde{B})

from the definition of the divergence D.

the case $\tilde{B} = X, \tilde{A} \neq X$ is analogical to (v).

Thus, the map D is a divergence. However, if we consider $\tilde{A} (x) = 0, \tilde{B} (x) = 0.3, \tilde{C} (x) = 0.5$ , we have that $\tilde{A} \subseteq \tilde{B} \subseteq \tilde{C}$ but $D (\tilde{B}, \tilde{C}) = 1 > 0 = D (\tilde{A}, \tilde{C})$ . Thus, D does not fulfill Axiom (Diss.3) in Definition 3.

Thus not all divergences are dissimilarities. The converse does not hold either:

Example 5. Let $D : F (X) \times F (X) \to ℝ$ be defined by: $\begin{matrix} D (\tilde{A}, \tilde{B}) \\ = {\begin{matrix} 0, & if \tilde{A} = \tilde{B}, \\ 1, & if \tilde{A} \neq \tilde{B} and either \tilde{A} = X or \tilde{B} = X, \\ 0.5, & otherwise . \end{matrix} \end{matrix}$

Let us check that D is a dissimilarity. The first and second conditions are trivial. Consider now the fuzzy set $\tilde{A}, \tilde{B}, \tilde{C} \in F (X)$ for which $\tilde{A} \subseteq \tilde{B} \subseteq \tilde{C}$ . We are going to prove that $D (\tilde{A}, \tilde{C}) \geq max {D (\tilde{A}, \tilde{B}), D (\tilde{B}, \tilde{C})}$ . $D (\tilde{A}, \tilde{C})$ only can attain three values:

if $D (\tilde{A}, \tilde{C}) = 1$ , then $D (\tilde{A}, \tilde{B}), D (\tilde{B}, \tilde{C}) \in {0, 0.5, 1} \leq 1 = D (\tilde{A}, \tilde{B})$ .

if $D (\tilde{A}, \tilde{C}) = 0.5$ , then $\tilde{A} \neq \tilde{C}, \tilde{A} \neq X, \tilde{C} \neq X$ . Since $\tilde{A} \subseteq \tilde{B} \subseteq \tilde{C}$ we take $\tilde{B} \neq X$ and hence $D (\tilde{A}, \tilde{B}), D (\tilde{B}, \tilde{C}) \in {0, 0.5} \leq 0.5 = D (\tilde{A}, \tilde{C})$ .

if $D (\tilde{A}, \tilde{C}) = 0$ , then $\tilde{A} = \tilde{C}$ by definition. Since $\tilde{A} \subseteq \tilde{B} \subseteq \tilde{C}$ we have $\tilde{A} = \tilde{B} = \tilde{C}$ and hence $D (\tilde{A}, \tilde{B}) = D (\tilde{B}, \tilde{C}) = 0 \leq D (\tilde{A}, \tilde{B})$ .

Thus, D is a dissimilarity, but it is not a divergence. To prove that it is enough to consider the fuzzy sets $\tilde{A}, \tilde{B}, \tilde{C}$ defined on X ={ x₁, x₂ } as follows: $\tilde{A} (x_{1}) = 1, \tilde{A} (x_{2}) = 0, \tilde{B} (x_{1}) = 0, \tilde{B} (x_{2}) = 1, \tilde{C} = \tilde{B}$ . Thus, in this case, we have that $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) = D (X, \tilde{B}) = 1 > 0.5 = D (\tilde{A}, \tilde{B})$ for any t-conorm. Therefore, (Div.3) from Definition 4 is not satisfied.

We see that divergences and dissimilarities are not related in general. However, there are some maps in both families, as we can see in the next example.

Example 6. For the map from Example 1, we proved that D is a divergence measure for any t-norm and t-conorm. Moreover, if $\tilde{A} \subseteq \tilde{B} \subseteq \tilde{C}$ , we have:

$D (\tilde{A}, \tilde{C}) = 1 \Rightarrow D (\tilde{A}, \tilde{B}), D (\tilde{B}, \tilde{C}) \in {0, 1} \leq D (\tilde{A}, \tilde{B})$ ,

$D (\tilde{A}, \tilde{C}) = 0 \Rightarrow \tilde{A} = \tilde{C} \Rightarrow \tilde{A} = \tilde{B} = \tilde{C} \Rightarrow D (\tilde{A}, \tilde{B}) = D (\tilde{B}, \tilde{C}) = 0$ .

Thus, it is also a dissimilarity.

Although in general both concept are not related, they are some functions which are both divergences and dissimilarities. In fact, in the particular case we reduce our focus on the standard fuzzy set operations using only minimum t-norm and its dual t-conorm, we can consider the divergences as a subset of dissimilarities (see Proposition 2.4 in [8]). Their relationship is even stronger for a particular kind of families, which have the local property, as we will see in the next section.

4 Local divergences

Comparing two fuzzy sets it seems natural to suppose that if we only change their values at one point, the divergence should only depend on this change.

Definition 5. Let D be a divergence measure for a triple (X, T, S). Than D has the local property (is local), if for all $\tilde{A}, \tilde{B} \in F (X)$ and for all x ∈ X, there exists a map $h_{x} : [0, 1] \times [0, 1] \to ℝ$ such that $D (\tilde{A}, \tilde{B}) - D (\tilde{A} \cup {x}, \tilde{B} \cup {x}) = h_{x} (\tilde{A} (x), \tilde{B} (x)) .$

A particular case of locality was introduced in [7, 8], but in that case the map h_x was fixed and all the elements in the universe were of the same importance. However, based on some application of comparison of multivalued sets (see, e.g. [10, 12]), this is not always the case and different maps should be considered. Hence we introduce a more general definition.

In the case X is finite, a representation theorem for local divergences was obtained in [8]. However, this result holds only for the minimum t-norm and its dual t-norm, that is, in (X, T_M, S_M). We will present a general result for any t-norm and any t-conorm.

Theorem 1. (Representation Theorem 1)Let (X, T, S) be a triple with X a finite universe and T and S any t-norm and t-conorm, respectively. Let D be a divergence associated to X. D is local if and only if $D (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x)),$ where {h_x} _x∈X is a family of maps from [0, 1] × [0, 1] into $ℝ$ such that, for any x ∈ X, a, b, c ∈ [0, 1] there is

h_x (a, a) =0, for all a∈ [0, 1];

h_x (a, b) = h_x (b, a), for all a, b∈ [0, 1];

h_x (a, b) ≥ max(h_x (S (a, c), S (b, c)), h_x (T (a, c), T (b, c))).

Proof. If D is local then, by definition, $D (\tilde{A}, \tilde{B}) - D (\tilde{A} \cup {x}, \tilde{B} \cup {x}) = h_{x} (\tilde{A} (x), \tilde{B} (x))$ for any x ∈ X. We apply this equation for other elements in X. Therefore: $D (\tilde{A}, \tilde{B}) = D (\tilde{A} \cup {x}, \tilde{B} \cup {x}) + h_{x} (\tilde{A} (x), \tilde{B} (x)) = D (\tilde{A} \cup {x} \cup {y}, \tilde{B} \cup {x} \cup {y}) + h_{x} (\tilde{A} (x), \tilde{B} (x)) + h_{y} (\tilde{A} \cup {x} (y), \tilde{B} \cup {x} (y))$ . But, $\tilde{A} \cup {x} (y) = S (\tilde{A} (y), {x} (y)) = S (\tilde{A} (y), 0) = \tilde{A} (y)$ and analoguously, $\tilde{B} \cup {x} (y) = \tilde{B} (y)$ .

Thus, $D (\tilde{A}, \tilde{B}) = D (\tilde{A} \cup {x} \cup {y}, \tilde{B} \cup {x} \cup {y}) + h_{x} (\tilde{A} (x), \tilde{B} (x)) + h_{y} (\tilde{A} (y), \tilde{B} (y)) = D (\tilde{A} \cup X, \tilde{B} \cup X) + \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x)) = D (X, X) + \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x)) = \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x))$ , since D is a divergence measure.

We have to check if for any x ∈ X, the function $h_{x} : [0, 1] \times [0, 1] \to ℝ$ fulfills (i)-(iii). For any fixed x ∈ X, we can define the fuzzy sets $\tilde{A} (t) = a, \tilde{B} (t) = b, \tilde{C} (t) = c$ if t = x and $\tilde{A} (t) = \tilde{B} (t) = \tilde{C} (t) = 0$ otherwise.

In this case we have $0 = D (\tilde{A}, \tilde{A}) = \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{A} (x)) = h_{x} (a, a)$ .

By definition we have that $D (\tilde{A}, \tilde{B}) = D (\tilde{B}, \tilde{A})$ , but $D (\tilde{A}, \tilde{B}) = h_{x} (a, b)$ and $D (\tilde{B}, \tilde{A}) = h_{x} (b, a)$ .

Since the D be a divergence measure we have that $D (\tilde{A}, \tilde{B}) \geq max (D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}), D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}))$ , but $D (\tilde{A}, \tilde{B}) = h_{x} (a, b)$ and $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) = \sum_{x \in X} h_{x} (S (\tilde{A} (x), \tilde{C} (x)), S (\tilde{B} (x), \tilde{C} (x)))$ = h_x (S (a, c), S (b, c)) + ∑_t∈X-{x}h_t (S (0, 0), S (0, 0)) = h_x (S (a, c), S (b, c)) by property (i). Analogously we could prove that $D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) = h_{x} (T (a, c), T (b, c))$ and therefore the proof of this implication is concluded.

To show the converse implication let $D : F (X) \times F (X) \to ℝ$ be a map, which can be expressed as a sum $D (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x)),$ where {h_x} _x∈X is a family of maps from [0, 1] × [0, 1] into $ℝ$ such that, for any x ∈ X, h_x fulfills (i)-(iii) from the previous definition. We will show that D is a local divergence measure. It is immediate that D is well-defined. Apart from that, D is a divergence, since for any $\tilde{A}, \tilde{B}, \tilde{C}$ in $F (X)$ we have

$D (\tilde{A}, \tilde{A}) = \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{A} (x)) = 0,$

$D (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x)) = \sum_{x \in X} h_{x} (\tilde{B} (x), \tilde{A} (x)) = D (\tilde{B}, \tilde{A}),$

$D (\tilde{A} \cap \tilde{C}, \tilde{B} \cap \tilde{C}) = \sum_{x \in X} h_{x} (T (\tilde{A} (x), \tilde{C} (x)), T (\tilde{B} (x), \tilde{C} (x))) \leq \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x)) = D (\tilde{A}, \tilde{B}),$ and analogously, $D (\tilde{A} \cup \tilde{C}, \tilde{B} \cup \tilde{C}) \leq D (\tilde{A}, \tilde{B}) .$

Moreover, $D (\tilde{A}, \tilde{B}) - D (\tilde{A} \cup {x}, \tilde{B} \cup {x}) = \sum_{t \in X} h_{t} (\tilde{A} (t), \tilde{B} (t)) - \sum_{t \in X} h_{t} (\tilde{A} \cup {x} (t), \tilde{B} \cup {x} (t)) = \sum_{t \in X - {x}} (h_{t} (\tilde{A} (t), \tilde{B} (t)) - h_{t} (S (\tilde{A} (t), 0), S (\tilde{B} (t), 0))) + (h_{x} (\tilde{A} (x), \tilde{B} (x)) - h_{x} (S (\tilde{A} (t), 1), S (\tilde{B} (t), 1))) = \sum_{t \in X - {x}} (h_{t} (\tilde{A} (t), \tilde{B} (t)) - h_{t} (\tilde{A} (t), \tilde{B} (t))) + (h_{x} (\tilde{A} (x), \tilde{B} (x)) - h_{x} (1, 1)) = h_{x} (\tilde{A} (x), \tilde{B} (x))$ for any $\tilde{A}, \tilde{B} \in F (X), x \in X$ . So D is a local divergence.

We present an example of a local divergence.

Example 7. The divergence proposed in Example 2 is local. It is clear that for any x ∈ X, h_x (a, b) = α_x · |a - b|, ∀ a, b ∈ [0, 1] where α_x ≥ 0 for any x ∈ X and ∑_x∈Xα_x = 1. Thus, it suffices to prove that it fulfills the conditions in Theorem 1. The first two are trivial. For the third one follows immediately from Example 2.

In the previous example, the map h_x was based on a distance between real numbers. However, this is not true in general, as we proved in [8]. Only some specific distances can be used for generating divergence measures. In fact, h_x is itself a divergence measure on the referential {x}.

Although the divergence given in Example 2 is local, this is not true in general. Clearly there are divergences which are not local. As an example we can consider the measure proposed in Example 1. Let us suppose D is local, then there is a family of maps {h_x} _x∈X with the appropriate properties, such that $D (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x))$ for any $\tilde{A}, \tilde{B} \in F (X)$ .

In particular, if we consider three fuzzy sets $\tilde{A}, \tilde{B}, \tilde{C}$ such that $\tilde{A} (x) = a_{1}, \tilde{A} (y) = a_{2}$ for x, y ∈ X, $\begin{matrix} \tilde{B} (t) & = & {\begin{matrix} b_{1} & if t = x, \\ \tilde{A} (x) & otherwise, \end{matrix} and \\ \tilde{C} (t) & = & {\begin{matrix} c_{2} & if t = y, \\ \tilde{A} (x) & otherwise, \end{matrix} \end{matrix}$ with a₁ ≠ b₁ and a₂ ≠ c₂, we have $1 = D (\tilde{A}, \tilde{B}) = h (a_{1}, b_{1})$ and $1 = D (\tilde{A}, \tilde{C}) = h (a_{2}, c_{2})$ .

On the other hand, the fuzzy sets $\tilde{B}, \tilde{C}$ are also different, then $D (\tilde{B}, \tilde{C}) = 1$ . But if D is local, we also have that $D (\tilde{B}, \tilde{C}) = h (b_{1}, a_{1}) + h (a_{2}, c_{2}) = 1 + 1 = 2 \neq 1$ which contradicts the definition of D. So, D is a divergence measure but it has not the local property.

Thus, the family of local divergences is a proper subset of the family of divergences. This subfamily has some specific properties as we will see in the following results. The first one is an equivalent definition for locality based on the intersection instead of the union.

Theorem 2. Let (X, T, S) be a triple with X finite and T, S any t-norm and t-conorm, respectively. Let D be a divergence. D is local if and only if for all $\tilde{A}, \tilde{B} \in F (X), x \in X$ , there is $h_{x} : [0, 1] \times [0, 1] \to ℝ$ such that $\begin{matrix} D (\tilde{A}, \tilde{B}) - D (\tilde{A} \cap {0_{x}}, \tilde{B} \cap {0_{x}}) = h_{x} (\tilde{A} (x), \tilde{B} (x)), \\ where 0_{x} (y) = {\begin{matrix} 0, & if y \in X and y = x . \\ 1, & if y \in X and y \neq x . \end{matrix} \end{matrix}$

Proof. If D is local, then by Theorem 1, $D (\tilde{A}, \tilde{B}) - D (\tilde{A} \cap {0_{x}}, \tilde{B} \cap {0_{x}}) = [\sum_{y \neq x} h_{y} (\tilde{A} (y), \tilde{B} (y)) - \sum_{y \neq x} h_{y} ((\tilde{A} \cap {0_{x}}) (y), (\tilde{B} \cap {0_{x}}) (y))] + [h_{x} (\tilde{A} (x), \tilde{B} (x)) - h_{x} ((\tilde{A} \cap {0_{x}}) (x), (\tilde{B} \cap {0_{x}}) (x))] = h_{x} (\tilde{A} (x), \tilde{B} (x)) - h_{x} (0, 0) = h_{x} (\tilde{A} (x), \tilde{B} (x))$ , since $\tilde{A} \cap {0_{x}} (y) = T (\tilde{A} (y), 1) = \tilde{A} (y)$ for any y ≠ x and $\tilde{A} \cap {0_{x}} (x) = T (\tilde{A} (x), 0) = 0$ . Analogously, $\tilde{B} \cap {0_{x}} (y) = \tilde{B} (y)$ for y ≠ x and $\tilde{B} \cap {0_{x}} (x) = 0$ .

Conversely, if D fulfills the condition for the intersection, we can apply recursively the equation for all the elements in X. Therefore:

$D (\tilde{A}, \tilde{B}) = D (\tilde{A} \cap {0_{1}}, \tilde{B} \cap {0_{1}}) + h_{x_{1}} (\tilde{A} (x_{1}), \tilde{B} (x_{1})) = D (\tilde{A} \cap {0_{1}} \cap {0_{2}}, \tilde{B} \cap {0_{1}} \cap {0_{2}}) + \sum_{i = 1}^{2} h_{x_{i}} (\tilde{A} (x_{i}), \tilde{B} (x_{i})) = \dots = D (\tilde{A} \cap \emptyset, \tilde{B} \cap \emptyset) + \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x)) = \underset{0}{\underset{︸}{D (\emptyset, \emptyset)}} + \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x))$ , applying again that $\tilde{A} \cap 0_{x} (y) = \tilde{A} (y)$ and $\tilde{B} \cap {0_{x}} (y) = \tilde{B} (y)$ for y ≠ x.

Moreover, it is easy to see that h_x fulfills (i)–(iii) in Theorem 1 for any x ∈ X. Thus D is a local divergence.

Apart from the definition for local divergences, we can consider a different one. Its advantage is that it allows us to generalize this notion later.

Proposition 2.Let (X, T, S) be a triple with X finite and T, S any t-norm and t-conorm, respectively. Let D be a divergence. D has the local property if and only if $D (\tilde{A}, \tilde{B}) = D (\tilde{A} \cup Z, \tilde{B} \cup Z) + D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})$ for any $\tilde{A}, \tilde{B} \in F (X)$ and any $Z \in P (X)$ .

Proof. If D is local, then there is a family of function {h_x} _x∈X such that for any $\tilde{A}, \tilde{B} \in F (X)$ and $Z \in P (X)$ , $\begin{matrix} D (\tilde{A}, \tilde{B}) & = & \sum_{x \in Z} h_{x} (\tilde{A} (x), \tilde{B} (x)) \\ + \sum_{x \in Z^{c}} h_{x} (\tilde{A} (x), \tilde{B} (x)), \\ D (\tilde{A} \cup Z, \tilde{B} \cup Z) & = & \sum_{x \in Z} h_{x} (1, 1) \\ + \sum_{x \in Z^{c}} h_{x} (\tilde{A} (x), \tilde{B} (x)) \\ = & \sum_{x \in Z^{c}} h_{x} (\tilde{A} (x), \tilde{B} (x)) \end{matrix}$ and $\begin{matrix} D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c}) \\ = \sum_{x \in Z} h_{x} (\tilde{A} (x), \tilde{B} (x)) + \sum_{x \in Z^{c}} h_{x} (1, 1) \\ = \sum_{x \in Z} h_{x} (\tilde{A} (x), \tilde{B} (x)) \end{matrix}$ by the properties of the t-norms and t-conorms. Therefore, $D (\tilde{A}, \tilde{B}) = D (\tilde{A} \cup Z, \tilde{B} \cup Z) + D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})$ .

Conversely, if D fulfills $D (\tilde{A}, \tilde{B}) = D (\tilde{A} \cup Z, \tilde{B} \cup Z) + D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})$ for any $\tilde{A}, \tilde{B} \in F (X)$ and any $Z \in P (X)$ , for any x ∈ X we can consider Z = {x} and so $D (\tilde{A}, \tilde{B}) - D (\tilde{A} \cup {x}, \tilde{B} \cup {x}) = D (\tilde{A} \cup {x}^{c}, \tilde{B} \cup {x}^{c})$ . Thus, we have to prove $h_{x} (u, v) = D ({\tilde{A}}_{x, u}, {\tilde{A}}_{x, v})$ fulfills the conditions in Theorem 1 for any x ∈ X, where ${\tilde{A}}_{x, u} (x) = u$ and ${\tilde{A}}_{x, u} (y) = 1$ for any y ≠ x.

The first two conditions are very easy: $h_{x} (u, u) = D ({\tilde{A}}_{x, u}, {\tilde{A}}_{x, u}) = 0$ and $h_{x} (u, v) = D ({\tilde{A}}_{x, u}, {\tilde{A}}_{x, v}) = D ({\tilde{A}}_{x, v}, {\tilde{A}}_{x, u}) = h_{x} (v, u)$ . For the third one, let us consider any u, v, z ∈ [0, 1], then $\begin{matrix} h_{x} (u, v) \\ = D ({\tilde{A}}_{x, u}, {\tilde{A}}_{x, v}) \\ \geq max (D ({\tilde{A}}_{x, u} \cup {\tilde{A}}_{x, z}, {\tilde{A}}_{x, v} \cup {\tilde{A}}_{x, z})), \\ D ({\tilde{A}}_{x, u} \cap {\tilde{A}}_{x, z}, {\tilde{A}}_{x, v} \cap {\tilde{A}}_{x, z}) \\ = max (D ({\tilde{A}}_{x, S (u, z)}, {\tilde{A}}_{x, S (v, z)}), \\ D ({\tilde{A}}_{x, T (u, z)}, {\tilde{A}}_{x, T (v, z)})) \\ = max (h_{x} (S (u, z), S (v, z)), \\ h_{x} (T (u, z), T (v, z))) . \end{matrix}$

So, D is a local divergence from Theorem 1.

The previous properties were proven in the general case of any t-norm and t-conorm. In the particular case we consider the standard union or intersection, S_M and T_M, an interesting property can be proven. It will allow us to relate divergences and dissimilarities. First we characterize the dissimilarities with the local property.

Theorem 3. (Representation Theorem 2)Let (X, T, S) be a triple with X finite and T, S any t-norm and t-conorm, respectively. Let D be a dissimilarity on X. D is local, that is, for all $\tilde{A}, \tilde{B} \in F (X)$ and for all x ∈ X, there exists a map $h_{x} : [0, 1] \times [0, 1] \to ℝ$ such that $D (\tilde{A}, \tilde{B}) - D (\tilde{A} \cup {x}, \tilde{B} \cup {x}) = h_{x} (\tilde{A} (x), \tilde{B} (x))$ if and only if $D (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x)),$ where, for any x ∈ X, h_x has the following properties:

h_x (a, a) =0, for all a∈ [0, 1];

h_x (a, b) = h_x (b, a), for all a, b∈ [0, 1];

h_x (a, c) ≥ max {h_x (a, b), h_x (b, c)}, for all a, b, c ∈ [0, 1] with a < b < c.

Proof. Let D be a dissimilarity with the local property. From locality we can prove that $D (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (\tilde{A} (x), \tilde{B} (x))$ and as the first axioms of dissimilarity are the same as the axioms for divergences, we proved in Theorem 1 that h_x fulfills that h_x (a, a) =0 and h_x (a, b) = h_x (b, a), for all a, b ∈ [0, 1].

Moreover, for any a, b, c ∈ [0, 1] such that a ≤ b ≤ c, if we apply Axiom (Diss.3) in Definition 3 to $\tilde{A}, \tilde{B}, \tilde{C}$ with $\tilde{A} (x) = a$ , $\tilde{B} (x) = b$ , $\tilde{C} (x) = c$ for a fixed x ∈ X and $\tilde{A} (y) = \tilde{B} (y) = \tilde{C} (y) = 0$ if y ≠ x. Thus, we have that $h_{x} (a, c) = D (\tilde{A}, \tilde{C}) \geq max (D (\tilde{A}, \tilde{B}), D (\tilde{B}, \tilde{C})) = max (h_{x} (a, b), h_{x} (b, c))$ .

The converse implication is analogical to the proof for local divergences (Theorem 1).

In the following we will study the relationship between the locality of divergence and locality of dissimilarity for the minimum t-norm and its dual t-conorm.

Proposition 3. Let (X, T_M, S_M) be a triple with X finite. A map $D : F (X) \times F (X) ⟶ ℝ$ is a local divergence if only if D is a local dissimilarity.

Proof. It is sufficient to prove that (iii) in Theorem 1 is equivalent to (iii’) in Theorem 3.

If (iii) is fulfilled and a ≤ b ≤ c, then h_x (b, c) = h_x (S_M (a, b), S_M (c, b)) ≤ h_x (a, c) and h_x (a, b) = h_x (T_M (a, b), T_M (c, b)) ≤ h_x (a, c).

Conversely, we consider the case a ≤ b without loss of generality. Now we discuss three cases:

If c ≤ a ≤ b, then h_x (S_M (a, c), S_M (b, c)) = h_x (a, b) and h_x (T_M (a, c), T_M (b, c)) = h_x (c, c) =0 ≤ h_x (a, b).

If a ≤ c ≤ b, then h_x (S_M (a, c), S_M (b, c)) = h_x (c, b) ≤ h_x (a, b) and h_x (T_M (a, c), T_M (b, c)) = h_x (a, c) ≤ h_x (a, b), by applying (iii’) in both cases.

If a ≤ b ≤ c, then h_x (S_M (a, c), S_M (b, c)) = h_x (c, c) =0 ≤ h_x (a, b) and h_x (T_M (a, c), T_M (b, c)) = h_x (a, b).

Thus, although divergences and dissimilarities are not the same concept, we know that in the case of the minimum t-norm and its dual t-conorm, any divergence is a dissimilarity. Now we have proven that the family of the local divergences is the same as the family of the local dissimilarities.

As we could see in Proposition 1, any divergence measure has 0 as a lower bound. Moreover, if it is local and we consider T_M and S_M, it also fulfills Axiom (Diss.3) of Definition 3. Thus, $D ({\tilde{A}}^{'}, {\tilde{B}}^{'}) \leq D (\emptyset, X)$ for any ${\tilde{A}}^{'} \subset {\tilde{B}}^{'}$ . Moreover, for any $\tilde{A}, \tilde{B} \in F (X)$ , if D is local we also have that $D (\tilde{A}, \tilde{B}) = D ({\tilde{A}}^{'}, {\tilde{B}}^{'})$ with $\begin{matrix} {\tilde{A}}^{'} (x) & = & {\begin{matrix} \tilde{A} (x) & if \tilde{A} (x) \leq \tilde{B} (x), \\ \tilde{B} (x) & otherwise, \end{matrix} \\ {\tilde{B}}^{'} (x) & = & {\begin{matrix} \tilde{B} (x) & if \tilde{A} (x) \leq \tilde{B} (x), \\ \tilde{A} (x) & otherwise . \end{matrix} \end{matrix}$

Thus, $D (\tilde{A}, \tilde{B}) \leq D (\emptyset, X)$ , for any $\tilde{A}, \tilde{B} \in F (X)$ , that is, D (∅, X) is a upper bound for D.

From now on, we can consider the normalized version of a divergence measure, that is, $\frac{1}{D (\emptyset, X)} \cdot D$ .

5 Generalizated local divergences

As we could see in Proposition 2, a divergence D is local if and only if $D (\tilde{A}, \tilde{B}) = D (\tilde{A} \cup Z, \tilde{B} \cup Z) + D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})$ , for any $\tilde{A}, \tilde{B} \in F (X)$ and any $Z \in P (X)$ . In this case we are decomposing by means of the sum, but any other aggregation operator (see, for instance, [3]) could be considered. Thus, we could consider the particular family of divergence measures such that $\begin{matrix} D (\tilde{A}, \tilde{B}) \\ = Ag (D (\tilde{A} \cup Z, \tilde{B} \cup Z), D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})), \end{matrix}$ for any $\tilde{A}, \tilde{B} \in F (X)$ and any $Z \in P (X)$ and a fixed aggregation operator Ag.

Let us study the behaviour of this map Ag in detail:

If Z is a crisp set, then $D (\tilde{A}, \tilde{B}) = Ag (D (\tilde{A} \cup Z, \tilde{B} \cup Z), D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c}))$ and $D (\tilde{A}, \tilde{B}) = Ag (D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c}), D (\tilde{A} \cup Z, \tilde{B} \cup Z))$ and therefore Ag has to be commutative.

If we consider as the crisp set the referential X, then $D (\tilde{A}, \tilde{B}) = Ag (D (\tilde{A} \cup X, \tilde{B} \cup X), D (\tilde{A} \cup \emptyset, \tilde{B} \cup \emptyset)) = Ag (D (X, X), D (\tilde{A}, \tilde{B})) = Ag (0, D (\tilde{A}, \tilde{B}))$ . Thus, zero has to be the neutral element for Ag.

We could see that for local divergences, if $D (\tilde{A} \cup Z, \tilde{B} \cup Z) \leq D ({\tilde{A}}^{'} \cup Z, {\tilde{B}}^{'} \cup Z)$ and $D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c}) \leq D ({\tilde{A}}^{'} \cup Z^{c}, {\tilde{B}}^{'} \cup Z^{c})$ for a crisp set Z them $D (\tilde{A}, \tilde{B}) \leq D ({\tilde{A}}^{'}, {\tilde{B}}^{'})$ . If we require this property in a general context, then Ag is increasing in both components.

Moreover, we can apply Ag recursively and obtain $D (\tilde{A}, \tilde{B}) = Ag (D (\tilde{A} \cup Z, \tilde{B} \cup Z), D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})) = Ag (Ag (D (\tilde{A} \cup Z \cup Z, \tilde{B} \cup Z \cup Z), D (\tilde{A} \cup Z \cup Z^{c}, \tilde{B} \cup Z \cup Z^{c})), D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})) = Ag (Ag (D (\tilde{A} \cup Z \cup Z^{c}, \tilde{B} \cup Z \cup Z^{c}), D (\tilde{A} \cup Z, \tilde{B} \cup Z)), D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})),$ but also $D (\tilde{A}, \tilde{B}) = Ag (D (\tilde{A} \cup X, \tilde{B} \cup X), D (\tilde{A} \cup \emptyset, \tilde{B} \cup \emptyset)) = Ag (D (\tilde{A} \cup Z \cup Z^{c}, \tilde{B} \cup Z \cup Z^{c}), Ag (D (\tilde{A} \cup Z, \tilde{B} \cup Z), D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c}))) .$

Thus, Ag has to be associative. As Ag is associative, commutative, monotone with neutral element 0, then it is a t-conorm.

Definition 6. Let $D : F (X) \times F (X) \to [0, 1]$ be a divergence and let S be a t-conorm. Then D has the S-local property (is S-local), if for all $\tilde{A}, \tilde{B} \in F (X)$ and for all $Z \in P (X)$ x ∈ X, it is fulfilled that $\begin{matrix} D (\tilde{A}, \tilde{B}) \\ = S [D (\tilde{A} \cup Z, \tilde{B} \cup Z), D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})] . \end{matrix}$

Again we can characterize these divergences.

Theorem 4. (Representation Theorem 3)Let (X, T, S) be a triple with X finite and T, S any t-norm and t-conorm, respectively. Let S′ be another t-conorm. A map D is S′-local divergence measure if and only if $D (\tilde{A}, \tilde{B}) = \underset{x \in X}{S^{'}} h_{x} (\tilde{A} (x), \tilde{B} (x)),$ where {h_x} _x∈X is a family of maps such that, for any x ∈ X, h_x : [0, 1] × [0, 1] → [0, 1] and

h_x (a, a) =0, for all a∈ [0, 1];

h_x (a, b) = h_x (b, a), for all a, b∈ [0, 1];

h_x (a, b) ≥ max(h_x (S (a, c), S (b, c)), h_x (T (a, c), T (b, c))), for all a, b, c ∈ [0, 1].

Proof. If D is S′-local then $D (\tilde{A}, \tilde{B}) = S^{'} (D (\tilde{A} \cup {x_{1}}, \tilde{B} \cup {x_{1}}), D (\tilde{A} \cup {x_{1}}^{c}, \tilde{B} \cup {x_{1}}^{c}))$ $= S^{'} (D (\tilde{A} \cup {x_{1}}, \tilde{B} \cup {x_{1}}), S^{'} (D (\tilde{A} \cup {x_{1}}^{c} \cup {x_{2}},$ $\tilde{B} \cup {x_{1}}^{c} \cup {x_{2}})), D (\tilde{A} \cup {x_{1}}^{c} \cup {x_{2}}^{c}, \tilde{B} \cup {x_{1}}^{c} \cup {x_{2}}^{c}))) = S^{'} (D (\tilde{A} \cup {x_{1}}, \tilde{B} \cup {x_{1}}), S^{'} (D (\tilde{A} \cup {x_{2}}, \tilde{B} \cup {x_{2}})), D (X, X))$ $= \dots = S^{'} (\underset{x \in X}{S^{'}} D (\tilde{A} \cup {x}, \tilde{B} \cup {x}), D (X, X)) = S^{'} (\underset{x \in X}{S^{'}} D (\tilde{A} \cup {x}, \tilde{B} \cup {x}), 0) = \underset{x \in X}{S^{'}} D (\tilde{A} \cup {x}, \tilde{B} \cup {x}) .$

Thus, it is enough to prove that for any x ∈ X, the map h_x defined by $h_{x} (u, v) = D ({\tilde{A}}_{x, u}, {\tilde{A}}_{x, v})$ is well-defined and it fulfills Conditions (i)–(iii) for any x ∈ X, where ${\tilde{A}}_{x, u} (x) = 1$ and ${\tilde{A}}_{x, u} (y) = u$ for any y ≠ x.

The proof is analogical to the one in Proposition 2.

It is easy to prove that the map D defined in the statement is a divergence measure. For the S′-locality, $D (\tilde{A}, \tilde{B}) = S^{'} (\underset{x \in Z}{S^{'}} h_{x} (\tilde{A} (x), \tilde{B} (x)), \underset{x \in Z^{c}}{S^{'}} h_{x} (\tilde{A} (x), \tilde{B} (x)))$ and $D (\tilde{A} \cup Z, \tilde{B} \cup Z) = \underset{x \in Z}{S^{'}} h_{x} (\tilde{A} \cup Z (x), \tilde{B} \cup Z (x)) + \underset{x \in Z^{c}}{S^{'}} h_{x} (\tilde{A} \cup Z (x), \tilde{B} \cup Z (x)) = \underset{x \in Z}{S^{'}} 0 + \underset{x \in Z^{c}}{S^{'}} h_{x} (\tilde{A} (x), \tilde{B} (x)) = \underset{x \in Z^{c}}{S^{'}} h_{x} (\tilde{A} (x), \tilde{B} (x)) .$

Analogously, $D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c}) = \underset{x \in Z}{S^{'}} h_{x} (\tilde{A} (x), \tilde{B} (x))$ . Thus, $\begin{matrix} D (\tilde{A}, \tilde{B}) \\ = S^{'} (D (\tilde{A} \cup Z, \tilde{B} \cup Z), D (\tilde{A} \cup Z^{c}, \tilde{B} \cup Z^{c})) \end{matrix}$ for any $Z \in P (X)$ , that is, D is a S′-local divergence.

From the previous theorem it is immediate that any divergence measure assuming values on the interval [0, 1] is local iff it is S_L-local.

As this holds for local divergences, an equivalent definition will be given by means of the intersection instead of the union in the following proposition.

Proposition 4. A divergence measure D has theS-local property iff for all $\tilde{A}, \tilde{B} \in F (X)$ and for all $Z \in P (X)$ the following equality is fulfilled: $\begin{matrix} D (\tilde{A}, \tilde{B}) \\ = S (D (\tilde{A} \cap Z, \tilde{B} \cap Z), D (\tilde{A} \cap Z^{c}, \tilde{B} \cap Z^{c})) \end{matrix}$

Proof. If D is S-local, by the properties of thet-conorms and the t-norms, we have that $\begin{matrix} S (D (\tilde{A} \cap Z, \tilde{B} \cap Z), D (\tilde{A} \cap Z^{c}, \tilde{B} \cap Z^{c})) \\ = \underset{x \in Z}{S} S (h (\tilde{A} \cap Z (x), \tilde{B} \cap Z (x)), \\ h (\tilde{A} \cap Z^{c} (x), \tilde{B} \cap Z^{c} (x))) \\ + \underset{x \in Z^{c}}{S} S (h (\tilde{A} \cap Z (x), \tilde{B} \cap Z (x)), \\ h (\tilde{A} \cap Z^{c} (x), \tilde{B} \cap Z^{c} (x))) \\ = \underset{x \in Z}{S} S (h (\tilde{A} (x), \tilde{B} (x)), h (0, 0)) \\ + \underset{x \in Z^{c}}{S} S (h (0, 0), h (\tilde{A} (x), \tilde{B} (x))) = D (\tilde{A}, \tilde{B}) . \end{matrix}$

Conversely, if D is a divergence, it is easy to prove that D can be decomposed as in Theorem 4 for the map h_x defined by $h_{x} (u, v) = D ({\tilde{A}}_{x, u}, {\tilde{A}}_{x, v})$ , where ${\tilde{A}}_{x, u} (x) = u$ and ${\tilde{A}}_{x, u} (y) = 1$ for any y ≠ x. Thus, D is S-local.

Another equivalent definition is shown below.

Proposition 5. Let D be a divergence measure and let S be a t-conorm. D is S-local if and only if for any $\tilde{A}, \tilde{B} \in F (X)$ and for any crisp partitions ${Z_{i}}_{i = 1}^{m}$ of the reference space X, the following equationholds $D (\tilde{A}, \tilde{B}) = S_{i = 1}^{m} D (\tilde{A} \cap Z_{i}, \tilde{B} \cap Z_{i}) .$

Proof. If D is S-local, we have that $\begin{matrix} D (\tilde{A}, \tilde{B}) \\ = S (D (\tilde{A} \cap Z_{1}, \tilde{B} \cap Z_{1}), D (\tilde{A} \cap Z_{1}^{c}, \tilde{B} \cap Z_{1}^{c})) . \end{matrix}$

But if we apply again the result to the second component, we have that $D (\tilde{A} \cap Z_{1}^{c}, \tilde{B} \cap Z_{1}^{c}) = S (D (\tilde{A} \cap Z_{1}^{c} \cap Z_{2}, \tilde{B} \cap Z_{1}^{c} \cap Z_{2}), D (\tilde{A} \cap Z_{1}^{c} \cap Z_{2}^{c}, \tilde{B} \cap Z_{1}^{c} \cap Z_{2}^{c})) .$

Since $\tilde{A} \cap Z_{1}^{c} \cap Z_{2} = \tilde{A} \cap Z_{2}$ and $\tilde{B} \cap Z_{1}^{c} \cap Z_{2} = \tilde{B} \cap Z_{2}$ and accounting the associativity of the t-conorm, we obtain $D (\tilde{A}, \tilde{B}) = S (S_{i = 1}^{2} D (\tilde{A} \cap Z_{i}, \tilde{B} \cap Z_{i}), D (\tilde{A} \cap Z_{1}^{c} \cap Z_{2}^{c}, \tilde{B} \cap Z_{1}^{c} \cap Z_{2}^{c})) .$

If we apply the previous procedure recursively, we obtain that $D (\tilde{A}, \tilde{B}) = S (S_{i = 1}^{m} D (\tilde{A} \cap Z_{i}, \tilde{B} \cap Z_{i}), D (\tilde{A} \cap Z_{1}^{c} \cap \dots \cap Z_{m}^{c}, \tilde{B} \cap Z_{1}^{c} \cap \dots \cap Z_{m}^{c})) = S (S_{i = 1}^{m} D (\tilde{A} \cap Z_{i}, \tilde{B} \cap Z_{i}), D (\emptyset, \emptyset)) = S (S_{i = 1}^{m} D (\tilde{A} \cap Z_{i}, \tilde{B} \cap Z_{i}), 0) = S_{i = 1}^{m} D (\tilde{A} \cap Z_{i}, \tilde{B} \cap Z_{i}) .$

The converse implication is trivial, since we only have to consider for any $Z \in P (X)$ the partition {Z, Z^c} of X and Proposition 4.

In the following propositions we establish some important properties of the S-local divergences, which express natural characteristics of ourmeasure.

Proposition 6. Let D be a S-local divergence.Then $D (Z, Z^{c}) = D (X, \emptyset), \forall Z \in P (X) .$

Proof. Let Z be any crisp subsets of X. Then: $\begin{matrix} D (Z, Z^{c}) = \underset{x \in X}{S} h_{x} (Z (x), Z^{c} (x)) \\ = S (\underset{x \in Z}{S} h_{x} (Z (x), Z^{c} (x)), \underset{x \in Z^{c}}{S} h_{x} (Z (x), Z^{c} (x))) \\ = S (\underset{x \in Z}{S} h_{x} (1, 0), \underset{x \in Z^{c}}{S} h_{x} (0, 1)) \\ = \underset{x \in X}{S} h_{x} (1, 0) = D (X, \emptyset), \end{matrix}$ where we have applied the symmetry of h_x for any x ∈ X and the associativity of the t-conorm.

Although trivial, the following proposition allows us to change the scale factor of a divergence.

Proposition 7. Let D be a S-local divergence generated by the family of functions {h_x} _x∈X, and let φ : [0, 1] → [0, 1] be a increasing function with φ (0) =0. The maps D_φ and D^φ defined below by $\begin{matrix} D_{φ} (\tilde{A}, \tilde{B}) & = & \underset{x \in X}{S} φ (h (\tilde{A} (x), \tilde{B} (x)) \\ D^{φ} (\tilde{A}, \tilde{B}) & = & φ (\underset{x \in X}{S} h (\tilde{A} (x), \tilde{B} (x)) \end{matrix}$ are also divergence measures and D_φ is S-local.

The proof is straightforward.

Proposition 8. Let (X, T_M, S_M) be a triple with X a finite universe, let S be any t-conorm and let D be a S-local divergence measure. For any $\tilde{A}, \tilde{B}, \tilde{C} \in F (X)$ such that for any x ∈ X either $\tilde{A} (x) \leq \tilde{B} (x) \leq \tilde{C} (x)$ or $\tilde{C} (x) \leq \tilde{B} (x) \leq \tilde{A} (x)$ , we have that $D (\tilde{A}, \tilde{C}) \geq max (D (\tilde{A}, \tilde{B}), D (\tilde{B}, \tilde{C})) .$

The proof that S-local divergence D is a dissimilarity is based on Proposition 2.4 in [8] by aggregating with S.

The divergence between a set and its complement increases and it attains the maximum when $\tilde{A}$ is crisp. Moreover, this is true when we compare any pair of fuzzy sets, as it follows from Proposition 6 and the next result.

Proposition 9. Let (X, T_M, S_M) be a triple with X finite and let D be a S-local divergence. For any $Z \in P (X), \tilde{A}, \tilde{B} \in F (X)$ we have $0 \leq D (\tilde{A}, \tilde{B}) \leq D (Z, Z^{c}) .$

Proof. For any $\tilde{A}, \tilde{B} \in F (X)$ , if we consider $Z = {x \in X ∣ \tilde{A} (x) \leq \tilde{B} (x)$ then $Z^{c} (x) \leq \tilde{A} (x) \leq \tilde{B} (x) \leq Z (x)$ if x ∈ Z and $Z (x) \leq \tilde{B} (x) \leq \tilde{A} (x) \leq Z (x)$ if x ∈ Z^c. Since D is also a dissimilarity, then $D (\tilde{A}, \tilde{B}) \leq D (Z^{c}, Z)$ .

6 Applications

We present two examples of application of generalized local divergence measures for pattern recognition and decision making. In both cases we will remark the different situations depending on the selected t-conorm.

6.1 Application to pattern recognition

Let X be finite and the patterns ${\tilde{A}}_{1}, {\tilde{A}}_{2}, . . ., {\tilde{A}}_{m}$ and a sample $\tilde{B}$ be represented by fuzzy sets.

We can measure the difference between $\tilde{B}$ and ${\tilde{A}}_{i}$ for i∈ { 1, . . ., m }. We obtain the finite set of divergences: $D ({\tilde{A}}_{1}, \tilde{B}), \dots, D ({\tilde{A}}_{m}, \tilde{B})$ . Finally, the sample $\tilde{B}$ will be associated to the pattern ${\tilde{A}}_{j}$ whenever $D ({\tilde{A}}_{j}, \tilde{B}) = min_{i = 1, . . ., m} D ({\tilde{A}}_{i}, \tilde{B}) .$

So, the sample $\tilde{B}$ is classified into the pattern from which it differs least.

Let us see how to proceed by an example, based on the one proposed in [10].

Example 8. Let us consider five kinds of mineral fields, each of them featured by the content of six minerals and containing one kind of typical hybrid mineral. The five kinds of typical hybrid mineral are represented by fuzzy sets ${\tilde{A}}_{1}$ , ${\tilde{A}}_{2}$ , ${\tilde{A}}_{3}$ , ${\tilde{A}}_{4}$ and ${\tilde{A}}_{5}$ in X = {x₁, …, x₆}, respectively. Assume that there is another kind of mineral B, and we want to classify it into one of the aforementioned mineral fields. The minerals are described by means of the fuzzy sets in Table 1.

Table 1
The kinds of hybrid minerals represented by fuzzy sets

X x ₁ x ₂ x ₃ x ₄ x ₅ x ₆

${\tilde{A}}_{1}$ 0.74 0.03 0.19 0.49 0.02 0.74

${\tilde{A}}_{2}$ 0.12 0.03 0.05 0.14 0.02 0.39

${\tilde{A}}_{3}$ 0.45 0.66 1.00 1.00 1.00 1.00

${\tilde{A}}_{4}$ 0.28 0.52 0.47 0.30 0.19 0.74

${\tilde{A}}_{5}$ 0.33 1.00 0.18 0.16 0.05 0.68

$\tilde{B}$ 0.63 0.52 0.21 0.22 0.07 0.66

X	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆
${\tilde{A}}_{1}$	0.74	0.03	0.19	0.49	0.02	0.74
${\tilde{A}}_{2}$	0.12	0.03	0.05	0.14	0.02	0.39
${\tilde{A}}_{3}$	0.45	0.66	1.00	1.00	1.00	1.00
${\tilde{A}}_{4}$	0.28	0.52	0.47	0.30	0.19	0.74
${\tilde{A}}_{5}$	0.33	1.00	0.18	0.16	0.05	0.68
$\tilde{B}$	0.63	0.52	0.21	0.22	0.07	0.66

Let us also assume that experts established the weight vector α = {0.2, 0.3, 0.125, 0.125, 0.125, 0.125} on X. Let us use our method to classify B. If we consider the divergence proposed in Examples 2 and 7: $D (\tilde{A}, \tilde{B}) = \underset{x \in X}{S} α_{x} \cdot | \tilde{A} (x) - \tilde{B} (x) |$ with α_x ≥ 0 for any x ∈ X and ∑_x∈Xα_x = 1. Then $\underset{x \in X}{S_{L}} α_{x} = 1 \geq \underset{x \in X}{S_{P}} α_{x} \geq \underset{x \in X}{S_{M}} α_{x}$ and therefore, it is clear that D is a S-local divergence measure for all basic t-conorms. Thus, the values assumed by the four different divergences are obtained in Table 2.

Table 2

The S-local divergences obtained for S∈ { S_M, S_P, S_L, S_D }

	S _M	S _P	S _L	S _D
$D ({\tilde{A}}_{1}, \tilde{B})$	0.1470	0.2090	0.2215	1
$D ({\tilde{A}}_{2}, \tilde{B})$	0.1470	0.2864	0.3190	1
$D ({\tilde{A}}_{3}, \tilde{B})$	0.1163	0.3644	0.4330	1
$D ({\tilde{A}}_{4}, \tilde{B})$	0.0700	0.1314	0.1375	1
$D ({\tilde{A}}_{5}, \tilde{B})$	0.1440	0.2084	0.2203	1

We can see that in all the cases we should classify B into the hybrid mineral A₄ (in the case of the drastic t-conorm it is just an option). However, the behaviour of any divergence is different. Thus, for instance, for the maximum t-conorm, the divergence is not able to distinguish between ${\tilde{A}}_{1}$ and ${\tilde{A}}_{2}$ , since in this case only the maximum distance is considered. In the other two cases, S_P and S_L, all the points in X are essential to obtain the value of the divergence and therefore the difference between ${\tilde{A}}_{1}$ and ${\tilde{A}}_{2}$ is remarked. In the case of S_D, the information given by the divergence is null.

Apart from the t-conorm used to defined the S-local divergence, the weights also can play an interesting role. Thus, at the previous example, we will consider the weight vector α = {k, 0.5 - k, 0.125, 0.125, 0.125, 0.125} for k ∈ {0, 0.1, …, 0.5}.

In this case, if we consider the previous D for S_L, we obtain the divergences shown in Table 3.

Table 3

Variations of the S_L-local divergence based on the weights

	α = 0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5
$D ({\tilde{A}}_{1}, \tilde{B})$	0.2975	0.2595	0.2215	0.1835	0.1455	0.1075
$D ({\tilde{A}}_{2}, \tilde{B})$	0.315	0.317	0.319	0.321	0.323	0.325
$D ({\tilde{A}}_{3}, \tilde{B})$	0.425	0.429	0.433	0.437	0.441	0.445
$D ({\tilde{A}}_{4}, \tilde{B})$	0.0675	0.1025	0.1375	0.1725	0.2075	0.2425
$D ({\tilde{A}}_{5}, \tilde{B})$	0.2563	0.2383	0.2203	0.2023	0.1843	0.1663

In this case we see that $D ({\tilde{A}}_{1}, \tilde{B}) \leq D ({\tilde{A}}_{5}, \tilde{B}) \leq D ({\tilde{A}}_{4}, \tilde{B}) \leq D ({\tilde{A}}_{2}, \tilde{B}) \leq D ({\tilde{A}}_{3}, \tilde{B})$ for α = 0.5 and therefore, in this case, we should classify B into the hybrid mineral A₁. In fact, $\tilde{B}$ should be classified into

${\tilde{A}}_{1}$ for α = 0.4, 0.5,

${\tilde{A}}_{4}$ for α = 0, 0.1, 0.2, 0.3.

The obtained results are expected, since for α = 0.5, the most important point at the referential is x₁. If we look at this element, ${\tilde{A}}_{1}$ and $\tilde{B}$ are the most similar. In the case α = 0, the most important point changes to be x₂ and this is the reason of the selection of ${\tilde{A}}_{4}$ .

As we can see in the previous examples, if we consider different maps h_x for any x on defining the divergence measure we can increase the information we can obtain from the data. The same happens if we consider the concept of generalized locality, since we can aggregate the divergences among the points by different t-conorm apart from S_L (just the sum) and we can consider the most appropriate t-conorm to any problem. Thus, the information obtained by means of the class of divergence measures considered in this paper is much richer than the one considered in [8].

6.2 Application to decision making

Let $A = {{\tilde{A}}_{1}, \dots, {\tilde{A}}_{m}}$ denote a set of alternatives, let X = {x₁, …, x_n} be a set of attributes, and α = (α₁, …, α_n) be its weight vector (α_i ≥ 0,∑_iα_i = 1).

Each alternative ${\tilde{A}}_{i}$ will be expressed by a fuzzy set with the elements x_j, where ${\tilde{A}}_{i} (x_{j})$ represents the degree in which alternative ${\tilde{A}}_{i}$ agrees with attribute x_j.

The new alternatives ${\tilde{A}}^{+}$ and ${\tilde{A}}^{-}$ will be ${\tilde{A}}^{+} = ⋃_{i = 1}^{m} {\tilde{A}}_{i}, and {\tilde{A}}^{-} = ⋂_{i = 1}^{m} {\tilde{A}}_{i} .$ ${\tilde{A}}^{+}$ and ${\tilde{A}}^{-}$ canbe interpreted as the “optimal” and “least optimal”, respectively. In this sense, the preferred alternative $\tilde{A}$ would be more similar to ${\tilde{A}}^{+}$ and more different from ${\tilde{A}}^{-}$ , simultaneously.

Finally, we consider the quotient k_i defined as: $k_{i} = \frac{D ({\tilde{A}}_{i}, {\tilde{A}}^{+})}{D ({\tilde{A}}_{i}, {\tilde{A}}^{+}) + D ({\tilde{A}}_{i}, {\tilde{A}}^{-})} .$

It means that if some alternative ${\tilde{A}}_{j}$ has a quotient k_j for which k_j < k_i then the alternative ${\tilde{A}}_{j}$ is better as ${\tilde{A}}_{i}$ in the sense previous described. Thus, the optimal is the alternative whose k_i is the minimum.

The previous procedure will now be explained by means of an example based on the one from [12].

Example 9. The government has to decide among five different energy strategies: ${\tilde{A}}_{1} - {\tilde{A}}_{5}$ . Each of them is assessing four attributes: economic (x_EC), technological (x_T), environmental (x_EN) and socio-political (x_P). The weight vector of these attributes is (α_EC, α_T, α_EN, α_P) = (0.4, 0.2, 0.3, 0.1).

Assume that any alternatives ${\tilde{A}}_{i}$ is defined by the fuzzy sets given in Table 4. If we consider S_M to define the union and T_M to define the intersection, we have the corresponding fuzzy sets ${\tilde{A}}^{+}$ and ${\tilde{A}}^{-}$ shown also in Table 4.

Table 4
Definition of 5 energy strategies, the optimal and less optimal alternatives

X x _EC x _T x _EN x _P

${\tilde{A}}_{1}$ 0.2 0.7 0.6 0.5

${\tilde{A}}_{2}$ 0.4 0.5 0.8 0.6

${\tilde{A}}_{3}$ 0.5 0.6 0.9 0.7

${\tilde{A}}_{4}$ 0.3 0.8 0.7 0.5

${\tilde{A}}_{5}$ 0.8 0.7 0.1 0.3

${\tilde{A}}_{+}$ 0.8 0.8 0.9 0.7

${\tilde{A}}_{-}$ 0.2 0.5 0.1 0.3

X	x _EC	x _T	x _EN	x _P
${\tilde{A}}_{1}$	0.2	0.7	0.6	0.5
${\tilde{A}}_{2}$	0.4	0.5	0.8	0.6
${\tilde{A}}_{3}$	0.5	0.6	0.9	0.7
${\tilde{A}}_{4}$	0.3	0.8	0.7	0.5
${\tilde{A}}_{5}$	0.8	0.7	0.1	0.3
${\tilde{A}}_{+}$	0.8	0.8	0.9	0.7
${\tilde{A}}_{-}$	0.2	0.5	0.1	0.3

If we consider again the S_M-local divergence measure proposed in the previous example, that is, $D (\tilde{A}, \tilde{B}) = \underset{x \in X}{S} α_{x} \cdot | \tilde{A} (x) - \tilde{B} (x) |,$ then the values of the quotients given in Table 5.

Table 5

Comparison of five alternatives optimality

$D ({\tilde{A}}^{+}, {\tilde{A}}_{i})$	$D ({\tilde{A}}^{-}, {\tilde{A}}_{i})$	k _i
i = 1	0.24	0.15	0.62
i = 2	0.16	0.21	0.43
i = 3	0.12	0.24	0.33
i = 4	0.2	0.18	0.53
i = 5	0.24	0.24	0.50

Since k₃ < k₂ < k₅ < k₄ < k₁, we conclude that the most optimal alternative is A₃.

7 Conclusion

We have introduced a new class of divergence measures between fuzzy sets, the S-local divergences, which is constructed by using a triangular conorm S instead of the sum. In these studies we have used the previous results about divergence measures with the local property, in fact we have shown that any local divergence is a S-local divergence. Since this new class of t-conorm dependent divergences satisfies all the properties already proposed for the local ones, we think that this generalization will allow us to manage a larger class of divergence measures and use them for define new classes of fuzziness measures.

One of the advantages of the proposed attitude is, that by considering different maps h_x for the points (or ranges of points) in X we can stress the specific aspects of a particular model and thus obtain more accurate information when specifying the measure of difference.

The most immediate open problem is to generalize the S-local divergence measure to the case when the universe is infinite. Another important point is the choice of h_x in some typical cases.

Footnotes

Acknowledgments

The research in this communication has been supported in part by MINECO-TIN2014-59543-P its financial support is gratefully acknowledged.

References

Anthony

and Hammer

P.L.

, A Boolean measure of similarity, Discrete Applied Mathematics154/16 (2006), 2242–2246.

Bouchon-Meunier

, Rifqi

and Bothorel

, Towards general measures of comparison of objects, Fuzzy Sets and Systems84 (1996), 143–153.

Calvo

, Mayor

, Mesiar

, Aggregation operators: New trends and Applications, in: Physica-Verlag, 2002.

Couso

, Garrido

and Sánchez

, Similarity and dissimilarity measures between fuzzy sets: A formal relational study, Information Sciences229 (2013), 122–141.

, Qin

and He

, Some new approaches to constructing similarity measures, Fuzzy Sets and Systems234/1 (2014), 46–60.

Lui

, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems52 (1992), 305–318.

Montes

, Partitions and divergence measures in fuzzy models, Ph. D. Dissertation, University of Oviedo, Spain, 1998.

Montes

, Couso

, Gil

and Bertoluzza

, Divergence measure between fuzzy sets, International Journal of Approximate Reasoning30 (2002), 91–105.

Valverde

and Ovchinnikov

, Representations of Tsimilarity relations, Fuzzy Sets and Systems159/17 (2008), 2211–2220.

10.

Wang

and Xin

, Distance measure between intuitionistic fuzzy sets, Pattern Recognition Letters26 (2005), 2063–2069.

11.

Wilbik

and Keller

J.M.

, A fuzzy measure similarity between sets of linguistic summaries, IEEE Transactions on Fuzzy Systems21/1 (2013), 183–189.

12.

and Xia

, Distance and similarity measures for hesitant fuzzy sets, Information Sciences181 (2011), 2128–2138.

13.

Zadeh

L.A.

, Fuzzy sets, Information and Control8 (1965), 338–353.

14.

Zadeh

L.A.

, A note on similarity-based definitions of possibility and probability, Information Sciences267 (2014), 334–336.

15.

Zhang

and Fu

, Similarity measures on three kinds of fuzzy sets, Pattern Recognition Letters27/12 (2006), 1307–1317.

Generalizated local divergence measures

Abstract

Keywords

1 Introduction

2 Basic concepts

3 Divergences

4 Local divergences

5 Generalizated local divergences

6 Applications

6.1 Application to pattern recognition

Table 4 Definition of 5 energy strategies, the optimal and less optimal alternatives X x EC x T x EN x P A ˜ 1 0.2 0.7 0.6 0.5 A ˜ 2 0.4 0.5 0.8 0.6 A ˜ 3 0.5 0.6 0.9 0.7 A ˜ 4 0.3 0.8 0.7 0.5 A ˜ 5 0.8 0.7 0.1 0.3 A ˜ + 0.8 0.8 0.9 0.7 A ˜ - 0.2 0.5 0.1 0.3

Footnotes

Acknowledgments

References