Parameters of Fuzzy Bags

A revised definition for fuzzy bags is reviewed, developing the concept of bags given by Delgado et al. 2009 from which each bag has two parts, function and summary information. Then, the definitions of fuzzy bag expected value, bag entropy and bag similarity are introduced. By some examples, the new concepts are illustrated.


Introduction
The initial notion of bags, an alternative name for the multisets, was introduced by Yager [1] as an algebraic set-like structure where an element can appear more than once.So far, several works have been done using this new concept.Moreover, bags have been employed in practice, for instance: in flexible querying, representation of relational information, decision problem analysis, criminal career analysis, and even in fields such as biology.However, due to some existing drawbacks in the first definition of bags [1], the necessity of a revision of this notion revealed.The proposed definition by Delgado et al. [2] has corrected these drawbacks.By some examples, they showed that the given definition for bags by Yager has some deficiencies and it was not well suited for representing and reasoning with real-world information.Then, they proposed new definitions for bags and fuzzy bags.As it is shown in [3], the lattice of all fuzzy bags defined by Delgado et al. [2] is a complete Boolean algebra which is not compatible with the nature of fuzziness.Improving this incompatibility, in [4], we quoted a revised definition for fuzzy bags based on the proposed definition of bags in [2].In [5], we gave the concept of fuzzy bag expected value.Here, parameters of fuzzy bags are introduced and studied.

Preliminaries and notations
In this section, some basic concepts which are needed in the sequel are given.For more details, see [2], [4].In what follows, O is the set of all objects, and F (O) = {A|A : O → [0, 1]} is the set of all fuzzy subsets of O. Also, i ∈ I n = {1, 2, . . ., n}, where n ∈ N, N is the set of natural numbers, P(O) is the power set of O and card(X) is the cardinality of set X. We will use the convention here that card( / 0) = 0.
Definition 2.1.[2] Let P and O be two finite universes (sets) called "properties" and "objects", respectively.A (crisp) bag B f is a pair ( f , B f ), where f : P → P(O) is a function and B f is the following subset of P × N 0 Here, N 0 = {0, 1, . . .} = {0} ∪ N.
In this characterization, a bag B f consists of two parts.The first one is the function f that can be seen as an information source about the relation between objects and properties.The second part B f is a summary of the information in f obtained by means of the count operation card(.).This summary corresponds to the classical view of bags in the sense of [1].
Definition 2.2.[4] We define B(P, O) as the set of all bags Clearly, a crisp bag is a particular case of fuzzy bag where, for all p ∈ P, f (p) is a crisp subset of O.
Here, the concept of fuzzy bag is illustrated by an example.
where ∪ i∈I n fi : Note that by Definition 2.

Parameters of fuzzy sets
In this section, we review three parameters of fuzzy sets; expected values, entropy and similarity measures.For terminology of this section and more details see [7], [8], [9].

Expected values of fuzzy sets
Let (X, A ) be a measurable space consisting of a non-empty set X (the universe of discourse) and a σ -algebra A of subsets of X. Recall that a σ -algebra is a collection of subsets of X which contains the empty set / 0 and the universe X and which is closed under complementation and countable unions.

boundary conditions), b) m(A) ≤ m(B) whenever A ⊆ B (monotonicity).
Note that a capacity as given in Definition 3.1, is neither required to be (finitely or σ -)additive nor to be continuous in any sense.Given a specific measurable space (X, A ), the set of all capacities m : A → [0, 1] will be denoted by M(X, A ), and the set of all measurable functions f : X → [0, 1] by F (X, A ). Recall that a function f : X → [0, 1] is measurable (with respect to the σ -algebra A on X and the σ -algebra A fuzzy subset A of X with µ A ∈ F (X, A ), i.e., with a measurable membership function (which is equivalent to A α ∈ A for each α ∈ [0, 1]), will be called a fuzzy event.The set of all fuzzy events will be denoted by A .The classical expected value of a random variable Y : X → R is given by where p : A → [0, 1] is a probability measure on the measurable space (X, A ) and the integral is a Lebesgue(stieltjes) integral [9].This formula was proposed by Zadeh [10] when, introducing expected value of a fuzzy event, just replacing Y by the membership function µ A .Since then, several approaches to the expected value of fuzzy events were proposed and studied.For their overview we recommend [9].In the following text we shall briefly write EV p (A) rather than EV p (µ A ), and we shall generalize this notion as follows: Definition 3.2.[9] Let (X, A ) be a measurable space.An expected value of fuzzy events is a function EV : Hence, each expected value of fuzzy events EV : A → [0, 1] as given in Definition 3.2 can be seen as an extension of some capacity m : A → [0, 1] given by m(A) = EV (A), A ∈ A .In such a case, the notation EV m will often be used to stress the link between the capacity m and the expectation of fuzzy events.Given an integral with respect to some capacity m, the expected value of a fuzzy event A can be defined in a straightforward way as the integral of the membership function µ A with respect to m.
where the integral on the right hand side is a Riemann integral.
In the case of a finite universe X = {x 1 , x 2 , . . ., x n }, where we usually put A = 2 X , and a fuzzy event A is given by µ A = ∑ n i=1 a i .1 {x i } we can apply Formula(3.1)and obtain its expected value (Ch)EV m (A) based on the Choquet integral by where the function σ : {1, . . ., n} → {1, . . ., n} is a permutation such that a σ (1) ≤ . . .≤ a σ (n) , and {x σ (n+1) , x σ (n) } = / 0 by convention.Another expected value of fuzzy events is based on the Sugeno integral.

Entropy of fuzzy sets
In general, a measure of fuzziness H is a mapping which assigns to each fuzzy subset A of a considered universal set X a non-negative number H(A) that quantifies the degree of fuzziness present in A. All measures of fuzziness should satisfy at least two properties, a) fuzziness of crisp sets should be equal to zero, b) if a fuzzy set A is sharper than B, which expressed by membership degrees means that A(x) ≤ B(x) if B(x) < 0.5 and A(x) ≥ B(x) if B(x) > 0.5 for all x ∈ X, then H(A) should not be greater than H(B).Various types of measures of fuzziness were proposed and investigated in the literature: entropy like measures, distance like measures, or general measures of fuzziness, see [11].The concept of "entropy" in fuzzy set theory has been already mentioned by Zadeh [10].De Luca and Termini using the functional formally similar to the Shannon entropy and its generalization, defined the "entropy" of a fuzzy set A (on a finite universal set) by where k is a positive constant and A(x i ) is a membership degree of the element x i ∈ A and the convention 0. log 0 = 0 is adopted.H(A) can be regarded as an "entropy" in the sense that it measures the uncertainty about presence or absence of a certain property described by A. A deeper explanation can be found in [12,13,11].Denote by F (X) the set of fuzzy subsets of X.We have the next definition for entropy measure of fuzzy sets.

Note that from these properties it follows that E(A) = 0 whenever A is crisp.
In the following definition, we will define entropy measures by means of so-called norm functions.

Definition 3.6. [14] A continuous function h
Example 3.1.[14] The following functions are norm functions a) h

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Where, H = (h x ) x∈X , h x are norm functions for all x ∈ X, H(A)(x) = h x (A(x)) and I m is any universal integral on [0, 1], see [15].Theorem 3.2.E is symmetric if and only if H = (h x ) x∈X is a constant system and m is a symmetric capacity.Remark 3.1.a) In the case of a general universe X, some measutability constraints should be considered.However, is a well defined entropy of fuzzy events on X. b) One can consider, in general, any strong negationn on [0, 1] and modify Definition 3.5 accordingly.By [16], we know that n(x) = φ −1 (1 − φ(x)), where φ is automorphism of [0, 1] and then n−based entropy has the following properties Where, c is the fixed point of n, i.e., n(c) = c.From these properties it follows that E n (A) = 0 whenever A is crisp.

Similarity measures for fuzzy sets
In this section we present a brief review of similarity measures for fuzzy sets and their axiomatic basis.Since the concept of similarity has a wide range of applications, there are different approaches presented in literature as axioms for degree or measure of similarity.Here, we present a set of axioms formulated by Bustince [17] in a more general form for interval-valued fuzzy sets.Definition 3.7.[17] A function S : F (X) × F (X) → [0, 1] is called a similarity measure, if S satisfies the following properties for all A, B,C ∈ F (X):

then S(A, B) ≥ S(A,C) and S(B,C) ≥ S(A,C).
One method to calculate the similarity of fuzzy sets is based on their distance.then d(A, B) ≤ d(A,C) and d(B,C) ≤ d(A,C).This calculation is in two steps: in the first part the distance between two fuzzy sets is obtained by a distance function and in the second part one of the relationships between similarity and distance comes into play to reach at the degree of similarity.Various distance functions d are presented in literature.The most commonly employed distance measures are a) normalized Hamming distance, The relationship between the notions of similarity and distance is expressed in several ways.If d is the distance measure between two fuzzy sets A and B on a universe X, then the following measures of similarity are based on proposed ones in [18], [19] and [20].given by C(x) = s(A(x), B(x)) for all x ∈ X.Then, S(A, B) is given by some expected value of C, i.e.S(A, B) = E(C).
As an example, one can use (weighted) arithmetic mean and thus, e.g.
or, in general, Choquet integral, Sugeno integral etc.
For more similarity measures see [7].In the next section, the parameters of fuzzy bags will be introduced.

Parameters of fuzzy bags
Any parameters defined for sets (fuzzy sets) can be extended for bags (fuzzy bags) considering a bag (fuzzy bag) as a subset (a fuzzy subset) of the universe P × O.Alternatively, we can proceed as follows Take a monotone normed measure m (a capacity) on P, a universal integral I m on [0, 1], and parameter ψ on (fuzzy) sets or ω on couples of (fuzzy) sets, such as expected values or fuzzy entropy as ψ, and similarity or dissimilarity measures as ω.Then, for (fuzzy) bags we define these parameters by ) Where ψ( f ) : P → [0, 1] is given by ψ( f )(p) = ψ( f (p)) and similarly ψ( f ), ω( f , g) and ω( f , g).

Expected value of a fuzzy bag event
In this subsection, we determine the "size" of a (fuzzy) bag via the definition of (fuzzy) bag expected value.
Similar to the crisp case, we can have the fuzzy version.Note that equation (4.4) is Lebesgue integral based expected value.One may compare it to expectation of a fuzzy set as defined in [10].In the case of crisp bag, we have

Entropy of fuzzy bags
In this subsection, we introduce the concept of entropy of fuzzy bags.

Conclusion
Employing the proposed definition of bags by Delgado et al., which is an improved version of Yager's one, a new definition for fuzzy bags has been given.Then, the new concept of fuzzy bag expected value and of some other parameters for fuzzy bags has been introduced.

Definition 3 . 3 .
[9] Given a measurable space (X, A ) and a capacity m : A → [0, 1], the expected value (Ch)EV m (A) of a fuzzy event A ∈ A based on the Choquet integral (Choquet expectation) is given by

Definition 3 . 4 .
[9] Given a measurable space (X, A ) and a capacity m : A → [0, 1], the expected value (Su)EV m (A) of a fuzzy event A ∈ A based on the Sugeno integral (Sugeno expectation) is given by

Theorem 3 . 1 .
For any finite universe X, we have E(A) = I m (H(A)).

Definition 4 . 2 .Example 4 . 1 .
A fuzzy bag expected value is the function FBEV : B(P, O) → [0, 1] which satisfies the following conditions a) FBEV (B 0 ) = 0 and FBEV (B 1 ) = 1 (boundary conditions), b) FBEV ( B f ) ≤ FBEV ( B f ) whenever B f ⊑ B f (monotonicity).Remark 4.1.The restriction of a fuzzy bag expected value to the crisp bags is bag expected value, i.e.FBEV | B(P,O) = BEV.Employing different measures, one can have various (fuzzy) bag expected values as it can be seen in the following examples.Considering probability measure Pr on P × O, we can define the following fuzzy bag expected value FBEV Pr ( B f ) = ∑ p∈P o∈O Pr({(p, o)}).f (p)(o).(4.4)

Table 1 :
Bill, Tom, Sue, Stan, Ben} and P = {17, 21, 27, 35} be the set of objects and the set of properties, respectively.Let f : P → P(O) be the function in Table 1 with f (p) ⊆ O for all p ∈ P. Function: age-people.

Table 2 :
The degrees of memberships for Example 2.2 [4]mple 2.2.[4]Let O = {Ben, Sue, Tom, John, Stan, Bill, Kim, Ana, Sara} and P = {young, middle age, old} is the set of some linguistic descriptions of age.Let the degrees of membership of all o ∈ O in the set of each property p ∈ P are given as in Table2.

Table 3 :
The degrees of memberships for Example 4.5