CBD: A New Divergence Measure for Complex Mass Function and its Application in Pattern Recognition

The theory of complex mass function is an effective method to deal with uncertainty information, and it is a generalized of Dempster-Shafer evidence theory. However, divergence measure is still an open issue in the realm of complex mass function theory. The main contribution of our paper is to propose a generalized divergence measure for complex mass function that is called complex belief divergence (CBD),which has the properties of symmetry, nonnegativity, nondegeneracy. When complex mass function degenerates into classical mass function, the CBD will degenerate into classical belief divergence, which has a better ability to measure uncertainty of information. Finally, a pattern recognition algorithm based on CBD is designed and applied to a medical diagnosis problem, which proves its practical prospect.

Dempster-Shafer evidence theory is one of the most effective and efficient models to deal with uncertain information [32][33][34], and evidence theory is an extension to traditional probability theory as it assigns belief to multi-element subsets [35,36]. Hence, evidence theory has better ability to deal with uncertain information than traditional probability theory [37,38].However, there are many open issues in Dempster-Shafer eviednce theory. For example, it can lead to counterintuitive results when the evidence is highly conflicting [39][40][41], how to determine the basic assignment [42][43][44],how to measure the uncertainty of the evidence [45], and so on. Many applications are developed under the framework of Dempster-Shafer evidence theory [46][47][48][49]. Dempster-Shafer theory has also promoted the development of other theories, such as evidential reasoning [50][51][52][53], D numbers theory [54,55], Pignistic belief transform [56,57], and so on. Therefore, a generalized Dempster-Shafer evidence theory is proposed [58,59], which extends the classic Dempster-Shafer evidence theory to the complex form as a way of quantum processing. There are also many other models and methods for quantum information processing [60]. In this paper, a novel divergence measure called complex belief divergence (CBD) for complex mass function is proposed.
The following is the rest of paper organised. In Section 2, preliminaries include Dempster-Shafer evidence theory, complex Dempster-Shafer evidence theory and divergence measure in Dempster-Shafer theory will be briefly introduce. In Section 3, complex divergence measure and its properties will be discussed. Some numerical examples will be given in Section 4. And an application will be given in Section 5. In Section 6, conclusions will be made in the end.

Dempster-Shafer evidence theory
Definition 2.1. The frame of discernment(FOD), a set of mutually collective and exclusive non-empty events Θ, is defined below [61,62]: The power set of Θ is denoted as 2 Θ , and it is defined as: where ∅ is an empty set.

Complex Dempster-Shafer evidence theory
Complex Dempster-Shafer evidence theory is the generalization of Dempster-Shafer evidence theory as it is defined on complex numbers [58,59].

Definition 2.4.
A set of mutually exclusive and collective non-empty events Θ is called the frame of discernment (FOD). It is defined as follows: The power set of Θ is denoted as 2 Θ , and it is defined as: where ∅ is an empty set.
The following is another expression of M.
It should be noted that when should be interpreted as 0. Also, it can be proved that KL(m 1 ||m 2 ) > 0. In this paper, a natural number e is taken for all log functions. Definition 2.9. Jensen-Shannon belief divergence is defined by Fei and Deng as a distance measure in belief theory [71], and it is presented as follows.
Inspired by belief entropy [73,74], Song and Deng improve the relative divergence by considering the uncertainty caused by subsets [75], and the improved relative divergence is given as follows.
Based on improved relative divergence, the definition of the improved Jensen-Shannon belief entropy is as follows.

Complex belief divergence
In this section, we propose a new divergence measure called CBD. The new measure extends from a real number system to a complex number system and also takes into account the cardinality of subsets.

Definition of CBD
Based on the relative belief entropy improved by Song and Deng [75], the proposed relative entropy is extended to the complex number system.
And improved Jensen-Shannon belief divergence is extended to the complex number system as complex belief divergence (CBD) as follows: There are two complex mass functions M 1 and M 2 . The defini- Proof. According to Eq.(10) we have:

Properties of CBD
Proof. According to the knowledge that we have: To prove in the same way, we can get: Then, sum the Eq.(17) and Eq. (18).
And we assume that there are two complex numbers M 1 = a + bi and So,we have|M 1 | + |M 2 | − |M 1 + M 2 | ≥ 0,and use this in Eq. (17).So, we Proof.
The calculation process of divergence are as follows:  In addition, from the two results showed in Fig.1, it can gain a conclusion that the divergence is symmetrical and nonnegative.   Fig.3(a). And from the Fig.3(b), we can see that the JS divergence is on the symmetry of a = 0.1 and x = 0.5.It once again proves that the JS divergence is symmetrical.In addition, Fig.3(c) show that when a = 0, we have the minimum JS divergence and the two imaginary parts are same.
In total, the only condition that JS divergence is equal is that the real parts and the imaginary parts are equal at the same time.

Application
Recently, the research of pattern recognition has attracted many researchers' attention. And in this part, a decision making algorithm based on CBD measure for pattern recognition is designed. The practicability of the algorithm is verified by its application in medical diagnosis.In addition, the ex-tension of the algorithm and the related work are analyzed and compared to verify the effectiveness of the algorithm.
1. The distance between a given pattern P j and sample S k measured us- 2. Choose the minimum distance between the CBBAs M S k and M P j 3. Sample S k is classified as P µ , where

Application in a medical diagnosis
Suppose a pattern recognition problem given pattern CBBAs P = {P 1 , P 2 , P 3 } and a kind of test sample CBBAs S, which relates to three kinds of attributes Table 1. The goal is to determine which of the given pattern {P 1 , P 2 , P 3 } is best suited to test sample S. The calculation process is described below.
1. The distance between a given pattern P 1 ,P 2 , andP 3 and sample S k measured using the CBD JS CBBA , as follows: 3. Sample S is classified as P 3 µ = 3; After above algorithm running, the results are presented in Table.2. As is obvious from step.2, P 3 is the smallest distance from S, so the sample S is assigned to pattern P 3 .

Extension and Comparison
In Section 6.2, the weight of each feature is evenly distributed, so an extended algorithm is proposed to include a weight ratio of each attributes.
where ∑ n i=1 ω i = 1. ω i can be considered as a subjective or objective factors. On this basis, the validity of the two methods of average weight distance measurement is further verified, so we compare them with Garg and Rani method and [76] Xiao method [63].
In the medical diagnosis application, the frame of discernment has two elements, namely, Y and N. M(Y),M(N), M(Y, N),three complex intuitionistic fuzzy sets, respectively relate to the membership, nonmembership, and hesitancy degrees. In [76], the weights ω i are set to ω 1 = 0.3,ω 2 = 0.35,ω 3 = 0.35. After the calculation is completed, the results are presented in the Table., it is obvious that JS ω (M P 3 ||M S ) ≤ JS ω (M P 2 ||M S ) ≤ JS ω (M P 1 ||M S ). The conclusion is that pattern P 3 belongs to the sample S.
In [76], < K 1 , K 2 , K 3 , K 4 > four processed data correlation coefficients with Garg and Rani method [76] and d CBBA , d ω CBBA with Xiao method [63],the distance between complex intuitionistic fuzzy sets,are presented in Table   2. It is obvious that all of them have a result that P 3 belongs to the sample S and the proposed method for pattern recognition based on JS, JS ω as effective as them. Compared with Garg and Rani method [76], the proposed method can measure the distance when the frame of discernment has two or more elements. Compared with Xiao method [63], the proposed method can take the number of elements in the subset into account. This application shows that the CBD can be used in the real applications under uncertain environments.

Conclusions
In this paper, complex belief divergence (CBD), a novel divergence measure, is proposed for complex mass function.This is the first time that divergence measure is studied in the realm of complex mass function. The properties of the CBD is analysed. The effectiveness of the CBD is illustrated by numerical examples. Finally, an application is proposed based on CBD. In the future, we will investigate on how to use CBD in many more complex applications. Also, the relationship of CBD and other divergence measures, like Rényi divergence and Tsallis divergence, will be also explored.

Conflict of interest
The authors declare that they have no conflict of interest.

Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.