The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets

: The processing of uncertain information is increasingly becoming a hot topic in the artiﬁcial intelligence ﬁeld, and the information measures of uncertainty information processing are also becoming of importance. In the process of decision-making, decision-makers make decisions mostly according to information measures such as similarity, distance, entropy, and cross-entropy in order to choose the best one. However, we found that many researchers apply cross-entropy to multi-attribute decision-making according to the minimum principle, which is in accordance with the principle of distance measures. Thus, among all the choices, we ﬁnally chose the one with the smallest cross-entropy (distance) from the ideal one. However, the relation between cross-entropy and distance measures in fuzzy sets or neutrosophic sets has not yet been veriﬁed. In this paper, we mainly consider the relation between the discrimination measure of fuzzy sets and distance measures, where we found that the fuzzy discrimination satisﬁed all the conditions of distance measure; that is to say, the fuzzy discrimination was found to be consistent with distance measures. We also found that the cross-entropy, which improved when it was based on the fuzzy discrimination, satisﬁed all the conditions of distance measure, and we ﬁnally proved that cross-entropy, including fuzzy cross-entropy and neutrosophic cross-entropy, was also a distance measure.


Introduction
In the real world, there exists much uncertain, imprecise, and incomplete information, meaning that there are also many tools to settle them. Zadeh [1] first proposed the concept of a fuzzy set, which, defined by a membership function, is used to depict the membership value of one object to a set. Atanassov [2] proposed an intuitionistic fuzzy set (IFS) described by two functions, including a membership function depicting the membership value, and a non-membership function depicting the non-membership value of one object to the intuitionistic fuzzy set. The intuitionistic fuzzy set is the extension of a fuzzy set through adding a non-membership function. It has provided a more flexible mathematical framework to process uncertainty, imprecise, and incomplete information. Smarandache [3] firstly proposed the notion of neutrosophy and a neutrosophic set in 1998. The neutrosophic set is defined by the truth-membership function, indeterminacy-membership function, and the falsity-membership function, and it is comprised of a membership value, non-membership value, and indeterminacy-membership value. The neutrosophic sets theory has been successfully applied in the image-processing field by Vlachos and Sergiadis [4]. Therefore, Wang et al. [5] put forward the definition of a single-value neutrosophic set (SVNS) and some operations for better application in real scientific and engineering fields. A single value neutrosophic set is the extension of fuzzy set, and the SVNS theory also provides us with a more convenient tool for the uncertain information processing. Recently, some researchers have devoted themselves to the and improved fuzzy cross-entropy and neutrosophic cross-entropy based on discrimination satisfying all the conditions of distance in the paper. In Section 2, we mainly introduce some relevant knowledge, and provide proof that the fuzzy discrimination measure satisfies all the conditions of a distance measure, i.e., it is actually a kind of distance measure. In Section 3, we mainly prove that the fuzzy cross-entropy satisfies all the conditions of a distance measure, and that cross-entropy in single-value neutrosophic sets is also a kind of distance; that is to say, that the cross-entropy measure is consistent with distance measures.

Fuzzy Discrimination Is Consistent with Distance Measure
Let X be a universe course, and a fuzzy set A is denoted by a membership function µ A (x) which is used to express the degree of belongingness of one point x ∈ X to the set A, and for all x ∈ X, . Let X be a universe course, and fuzzy set A defined on X be given as: 1], and every point has a membership value to express the degree of belongingness to a set.
Let FS(X) be the set of all the fuzzy sets. The following are some properties of fuzzy sets: M, N, T ∈ FS(X), Definition 2. Bhandari [21] proposed the fuzzy discrimination measure which is used to express a discrimination degree in favor of A against B. A, B ∈ FS(X) are defined as follows: From the above, the fuzzy discrimination has similar properties, such as distance measures (except for one axiom of distance measure), and they also have a similar principle corresponding to the principle of minimum cross-entropy. In other words, there exists a perfect solution, A, but because it is generally unlikely to exist in a real situation, we should aim to find a solution which is likely to exist in the real world denoted by B, C, ..., so that we can get their cross-entropy to A. We ended up choosing the smallest cross-entropy, and the solution corresponding to the smallest cross-entropy was the optimal solution. The distance measure also had the same principle as the cross-entropy. We redefined the fuzzy distance measure as follows, considering the infinity of the discrimination measure: It is obvious that the symmetry fuzzy discrimination has satisfied the first three conditions of fuzzy distance: Thus, we just needed to verify that the fuzzy discrimination satisfied condition (4) of the fuzzy distance measure.

Theorem 1. Let m, n, t be three numbers in
Firstly, we needed to prove that (3) ≥ (1).
Finally, we obtained the proof of the theorem.
From Theorem 1, we know that the Theorem has been satisfied to every single membership value, meaning that the proof can be easily obtained from Theorem 1. Theorem 3. The above-defined symmetry fuzzy discrimination is a distance measure.
Finally, the above proves that the symmetry fuzzy discrimination, defined by Definition 2, is consistent with the distance measure from the above theorems.

Fuzzy Cross-Entropy Is Consistent with Distance Measure
Bhandari and Pal pointed out that the fuzzy discrimination has a defect-when µ B (x i ) approaches 0 or 1, its value will be infinity. Thus, it has been modified on the basis of directed divergence proposed by Lin [20], and also modified by Shang et al. [22] as follows: 22]). Let M, N ∈ FS(X), where we can define a fuzzy cross-entropy as: This shows that it is well-defined and independent of every value of µ(x i ), which can express the discrimination degree of A from B.
It also has the same properties as the above discrimination measures, in that when I 2 (M, N) ≥ 0, Let E 2 (M, N) = I 2 (M, N) + I 2 (N, M); then, symmetry is satisfied. Thus, we mainly consider how the above-defined fuzzy cross-entropy E 2 (M, N) satisfies Condition (4) of the distance measure.

Theorem 4. Let m, n, t be three numbers in
Proof.
Finally, we obtain the proof of the theorem.
We can easily obtain the proof from Theorem 3.
Theorem 6. The above-defined symmetry fuzzy cross-entropy is a kind of distance measure.

Neutrosophic Cross-Entropy Is a Distance Measure
Smarandache [3,27] firstly proposed the definition of a neutrosophic set, which is an extension of an intuitionistic fuzzy set(IFS) and an interval-valued intuitionistic fuzzy set, as follows: . Let X be a universe course, where a neutrosophic set A in X is comprised of the truth-membership function T A (x), indeterminacy-membership function I A (x), and falsity-membership function There is no restriction on the sum of T A (x), Wang et al. [5] introduced the definition of single value neutrosophic set (SVNS) for better application in the engineering field. SVNS is an extension of the IFS, and also provides another way in which to express and process uncertainty, incomplete, and inconsistent information in the real world. Definition 7 ([5]). Let X be a space of points, where a single-value netrosophic set A in X is comprised of the truth-membership function T A (x), indeterminacy-membership function I A (x), and falsity-membership function F A (x). For each point x in X, T A (x), I A (x), F A (x) ∈ [0, 1]. Therefore, a SVNS A can be denoted by: There is no restriction on the sum of T A (x), The following are some properties about SVNSs M and N: Let X be a universe course, SV NS(X) be the set of all the single-value neutrosophic sets, and M, N, T ∈ SV NS(X): Then, Ye [25] first generalized the fuzzy cross-entropy measure to the SVNSs. The information measure of neutrosophic sets are composed of the information measure of the truth-membership, indeterminacy-membership, and falsity-membership in SVNSs. Let M, N ∈ SV NS(X), where Ye introduced the discrimination information of T M (x i ) from T N (x i ) for (i = 1, 2, ..., n) on the basis of the definition of fuzzy cross-entropy I 2 (M, N) as the following; We can define the following information in terms of the indeterminacy-membership function and the falsity-membership function in the same way: ) + (1 − I M (x i )) ln( 1 − I M (x i ) 1 − 1/2(I M (x i ) + I N (x i )) )  According to the proof of Theorem 4, we can easily find that E T 2 (M, T) ≥ E T 2 (M, N), and E T 2 (M, T) ≥ E T 2 (N, T). In a similar way, E I 2 (M, T) ≥ E I 2 (M, N), and E I 2 (M, T) ≥ E I 2 (N, T), E F 2 (M, T) ≥ E 2 2 (M, N), and E F 2 (M, T) ≥ E F 2 (N, T), meaning that E 3 (M, T) ≥ E 3 (M, N), E 3 (M, T) ≥ E 3 (N, T). Thus, conclusively, we were able to easily obtain the proof.