Generalization of Dempster-Shafer theory: A complex belief function

TL;DR

This paper introduces a generalized Dempster-Shafer evidence theory using complex numbers to better model and combine uncertain information, especially in dynamic and imprecise data scenarios.

Contribution

It proposes a complex-valued belief function and a generalized combination rule, extending classical evidence theory to handle more complex uncertainties.

Findings

Complex basic belief assignment enhances uncertainty modeling.

Generalized combination rule relaxes conflict constraints.

Degenerates to classical theory when complex mass functions reduce to real numbers.

Abstract

Dempster-Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. However, the existing evidence theory is insufficient to consider the situations where it has no capability to express the fluctuations of data at a given phase of time during their execution, and the uncertainty and imprecision which are inevitably involved in the data occur concurrently with changes to the phase or periodicity of the data. In this paper, therefore, a generalized Dempster-Shafer evidence theory is proposed. To be specific, a mass function in the generalized Dempster-Shafer evidence theory is modeled by a complex number, called as a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster's combination rule is exploited. In contrast to the classical Dempster's combination rule, the condition in terms of the conflict coefficient between the evidences K<1 is released in the generalized Dempster's combination rule. Hence, it is more general and applicable than the classical Dempster's combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster's combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences K is less than 1. In a word, this generalized Dempster-Shafer evidence theory provides a promising way to model and handle more uncertain information.

Equations

$$z=x+yi,$$

$$z=r{e}^{i\theta },$$

$$e^{i\theta }=\cos (\theta )+i\sin (\theta ),$$

$$r=\sqrt{x^2+y^2},\text{ and }\theta =\arctan (\frac{y}{x})={\tan }^{-1}(\frac{y}{x}),$$

$$|z{|}^2=z\bar{z}=x^2+y^2,$$

$$r=|z|,\text{ and }\theta =\angle z,$$

$$z_1+z_2=(x_1+y_{1}i)+(x_2+y_{2}i)=(x_1+x_2)+(y_1+y_2)i.$$

$$z_1-z_2=(x_1+y_{1}i)-(x_2+y_{2}i)=(x_1-x_2)+(y_1-y_2)i.$$

$$(x_1+y_{1}i)(x_2+y_{2}i)=(x_1{x}_2-y_1{y}_2)+(x_1{y}_2+x_2{y}_1)i.$$

$$\frac{x_1+y_{1}i}{x_2+y_{2}i}=\frac{x_1{x}_2+y_1{y}_2}{x_2^2+y_2^2}+\frac{x_2{y}_1-x_1{y}_2}{x_2^2+y_2^2}i.$$

$$\Omega =\{F_1,F_2,\dots ,F_i,\dots ,F_N\},$$

$$2^{\Omega }=\{\varnothing ,\{F_1\},\{F_2\},\dots ,\{F_N\},\{F_1,F_2\},\dots ,\{F_1,$$

$$F_2,\dots ,F_i\},\dots ,\Omega \},$$

$$m:2^{\Omega }\to [0,1],$$

$$m(\varnothing )$$

$$Bel(A)=\sum _{B\subseteq A}m(B).$$

$$Pl(A)=\sum _{B\cap A\ne \varnothing }m(B).$$

$${m(C)=\left\{\begin{array}[]{l}\begin{array}[]{*{20}{c}}{\frac{1}{1-K}\sum\limits_{A\cap B=C}m_{1}(A)m_{2}(B),}&{{\kern 30.0pt}C\neq\emptyset,}\end{array}\\ \begin{array}[]{*{20}{c}}{0,}&{{\kern 136.0pt}C=\emptyset,}\end{array}\end{array}\right.}$$

$$K=\sum _{A\cap B=\varnothing }{m}_1(A)m_2(B),$$

$$\Omega =\{E_1,E_2,\dots ,E_i,\dots ,E_N\},$$

$$2^{\Omega }=\{\varnothing ,\{E_1\},\{E_2\},\dots ,\{E_N\},\{E_1,E_2\},\dots ,\{E_1,$$

$$E_2,\dots ,E_i\},\dots ,\Omega \},$$

$$\text{mathds}M:2^{\Omega }\to \mathbb{C},$$

$$\text{mathds}M(\varnothing )=0,$$

$$\text{mathds}M(A)=\mathbf{m}(A)e^{i\theta (A)},A\in 2^{\Omega }$$

$$\sum _{A\in 2^{\Omega }}\text{mathds}M(A)=1,$$

$$\text{mathds}M(A)=x+yi,A\in 2^{\Omega }$$

$$\sqrt{x^2+y^2}\in [0,1].$$

$$\mathbf{m}(A)=\sqrt{x^2+y^2},\text{ and }\theta (A)=\arctan (\frac{y}{x}),$$

$$|\text{mathds}M(A){|}^2=\text{mathds}M(A)\text{mathds}\bar{M}(A)=x^2+y^2,$$

$$\mathbf{m}(A)=|\text{mathds}M(A)|,\text{ and }\theta (A)=\angle \text{mathds}M(A),$$

$$Be{l}_c(A)=\sum _{B\subseteq A}|\text{mathds}M(B)|,$$

Full text

I Introduction

How to measure the uncertainty has been an attracting issue in a variety of areas [1, 2, 3]. The amount of theories had been presented and developed for measuring the uncertainty, including the extended fuzzy sets [4, 5], fuzzy soft sets [6], evidence theory [7, 8], D numbers theory [9, 10], Z numbers [11, 12], R numbers [13, 14], entropy-based [15, 16], and information quality [17]. These theories were broadly applied in various fields, such as the selection [18, 19], recognition [20], prediction [21], medical diagnosis [22], and decision-making [23, 24, 25].

As one of the most effective tools of uncertainty reasoning, Dempster–Shafer (DS) evidence theory [26, 27] can model the uncertainty without prior information in a flexible and effective manner [28, 29, 30]. The fusion results generated by Dempster’s combination rule are fault-tolerant which can be more sufficient and accurate to support the decision-making [31, 32], while the uncertainty can be characterized quantitatively and further be reduced in the process of combination [33, 34, 35]. Besides, the Dempster-Shafer theory satisfies the commutative and associative laws, so that it has been extensively applied in various fields [36, 37]. Nevertheless, through carefully studying the existing methods of evidence theory, it is found that none of these models have the capability to express the fluctuations of data at a given phase of time during their execution. Furthermore, in daily life, uncertainty and imprecision which are inevitably involved in the data occur concurrently with changes to the phase or periodicity of the data. As a result, the existing evidence theories are insufficient to consider these kinds of information, so that some information would loss during the model and process of data.

In this paper, therefore, a generalized Dempster–Shafer (GDS) evidence theory is proposed. To be specific, a mass function in the GDS evidence theory is modeled by a complex number, called as a complex mass function, which has more powerful ability to express uncertain information. On this basis, a generalized Dempster’s combination rule is exploited. Compared with the traditional Dempster’s combination rule, the condition in terms of the conflict coefficient between two evidences $\mathds{K}<1$ is released in the generalized Dempster’s combination rule. Hence, the proposed method is more general and applicable than the traditional Dempster’s combination rule. In particular, when the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the traditional evidence theory under the condition that the conflict coefficient between two evidences $\mathds{K}$ is less than 1. In this context, the GDS evidence theory provides a new framework to be more capable of modeling and handling the uncertainty. Meanwhile, several numerical examples are provided to illustrate the feasibility of the GDS evidence theory. Additionally, an algorithm for decision-making is devised based on the GDS evidence theory. Finally, an application of the new algorithm is implemented to solve the medical diagnosis problem. The results validate the practicability and effectiveness of the proposed algorithm.

The rest of this paper is organised as follows. The preliminaries, including complex number and Dempster–Shafer evidence theory are briefly introduced in Section II. The new GDS evidence theory is proposed in Section III. Section IV provides numerical examples to illustrate the feasibility of the GDS evidence theory. Finally, the conclusion is given in Section V.

II Preliminaries

II-A Complex number [38]

A complex number $z$ is defined as an ordered pair of real numbers

[TABLE]

where $x$ and $y$ are real numbers and $i$ is the imaginary unit, satisfying $i^{2}=-1$. This is called the “rectangular” form or “Cartesian” form.

It can also expressed in polar form, denoted by

[TABLE]

where $r>0$ represents the modulus or magnitude of the complex number $z$ and $\theta$ represents the angle or phase of the complex number $z$.

By using the Euler’s relation,

[TABLE]

the modulus or magnitude and angle or phase of the complex number can be expressed as

[TABLE]

where $x=r\cos(\theta)$ and $y=r\sin(\theta)$.

The square of the absolute value is defined by

[TABLE]

where $\bar{z}$ is the complex conjugate of $z$, i.e., $\bar{z}=x-yi$.

These relationships can be then obtained as

[TABLE]

where if $z$ is a real number (i.e., $y=0$), then $r=|x|$.

The arithmetic of complex numbers is defined as follows:

Give two complex numbers $z_{1}=x_{1}+y_{1}i$ and $z_{2}=x_{2}+y_{2}i$, the addition is defined by

[TABLE]

The subtraction is defined by

[TABLE]

The multiplication is defined by

[TABLE]

The division is defined by

[TABLE]

II-B Dempster–Shafer evidence theory [26, 27]

Uncertain information is inevitable in practical applications [39, 40, 41]. To handle the uncertainty problems in the process of information fusion, many integrated methods have been presented in recent years [42, 43, 44], in which Dempster–Shafer (DS) evidence theory is very common used in the real applications [45, 46, 47]. The basic concepts and definitions are described as below.

Definition 1

(Frame of discernment)

Let $\Omega$ be a set of mutually exclusive and collective non-empty events, defined by

[TABLE]

where $\Omega$ is a frame of discernment [48].

The power set of $\Omega$ is denoted as $2^{\Omega}$,

[TABLE]

where $\emptyset$ represents an empty set.

If $A\in 2^{\Omega}$, $A$ is called a proposition [49].

Definition 2

(Mass function)

A mass function $m$ in the frame of discernment $\Omega$ can be described as a mapping from $2^{\Omega}$ to [0, 1], defined as

[TABLE]

satisfying the following conditions,

[TABLE]

In the DS evidence theory, $m$ can also be called a basic belief assignment (BBA). If $m(A)$ is greater than zero, where $A\in 2^{\Omega}$, $A$ is called a focal element. The value of $m(A)$ represents how strongly the evidence supports the proposition $A$ [50, 51].

Definition 3

(Belief function)

Let $A$ be a proposition in the frame of discernment $\Omega$. The belief function of proposition $A$, denoted as $Bel(A)$ is defined by

[TABLE]

Definition 4

(Plausibility function)

Let $A$ be a proposition in the frame of discernment $\Omega$. The plausibility function of proposition $A$, denoted as $Pl(A)$ is defined by

[TABLE]

The belief function $Bel(A)$ and plausibility function $Pl(A)$ represent the lower and upper bound functions of the proposition $A$, respectively [52, 53, 54]. The value of $m(A)$ represents how strongly the evidence supports the proposition $A$ [55]. Various operations on the BBA are presented, like negation [56, 57], belief interval [58, 59], divergence [60], and entropy function [61, 62, 63].

Definition 5

(Dempster’s rule of combination)

Let $m_{1}$ and $m_{2}$ be two independent basic belief assignments (BBAs) in the frame of discernment $\Omega$. The Dempster’s rule of combination, denoted as $m=m_{1}\oplus m_{2}$ is defined by

[TABLE]

with

[TABLE]

where $A,B\in 2^{\Omega}$ and $K$ is the conflict coefficient between $m_{1}$ and $m_{2}$.

Notice that the Dempster’s combination rule is only feasible under the situation where the conflict coefficient $K<1$ for $m_{1}$ and $m_{2}$ [64, 65]. As an useful uncertainty processing methodology [66, 67, 68], DS evidence theory was widely applied in various areas, like reasoning [69, 70], reliability evaluation [71, 72], fault diagnosis [73], decision-making [74, 75], and classification [76, 77].

III Generalized Dempster–Shafer evidence theory

Let $\Omega$ be a set of mutually exclusive and collective non-empty events, defined by

[TABLE]

where $\Omega$ represents a frame of discernment.

The power set of $\Omega$ is denoted by $2^{\Omega}$, in which

[TABLE]

and $\emptyset$ is an empty set.

Definition 6

(Complex mass function)

A complex mass function $\mathds{M}$ in the frame of discernment $\Omega$ is modeled as a complex number, which is represented as a mapping from $2^{\Omega}$ to $\mathbb{C}$, defined by

[TABLE]

satisfying the following conditions,

[TABLE]

where $i=\sqrt{-1}$; $\mathbf{m}(A)\in[0,1]$ representing the magnitude of the complex mass function $\mathds{M}(A)$; $\theta(A)\in[-\pi,\pi]$ denoting a phase term.

In Eq. (22), $\mathds{M}(A)$ can also expressed in the “rectangular” form or “Cartesian” form, denoted by

[TABLE]

with

[TABLE]

By using the Euler’s relation, the magnitude and phase of the complex mass function $\mathds{M}(A)$ can be expressed as

[TABLE]

where $x=\mathbf{m}(A)\cos(\theta(A))$ and $y=\mathbf{m}(A)\sin(\theta(A))$.

The square of the absolute value for $\mathds{M}(A)$ is defined by

[TABLE]

where $\mathds{\bar{M}}(A)$ is the complex conjugate of $\mathds{M}(A)$, such that $\mathds{\bar{M}}(A)=x-yi$.

These relationships can be then obtained as

[TABLE]

where if $\mathds{M}(A)$ is a real number (i.e., $y=0$), then $\mathbf{m}(A)=|x|$.

The complex mass function $\mathds{M}$ modeled as a complex number in the generalized Dempster–Shafer (GDS) evidence theory can also be called a complex basic belief assignment (CBBA).

If $|\mathds{M}(A)|$ is greater than zero, where $A\in 2^{\Omega}$, $A$ is called a focal element of the complex mass function. The value of $|\mathds{M}(A)|$ represents how strongly the evidence supports the proposition $A$.

Definition 7

(Complex belief function)

Let $A$ be a proposition in the frame of discernment $\Omega$. The complex belief function of proposition $A$, denoted as $Bel_{c}(A)$ is defined by

[TABLE]

where $|\mathds{M}(B)|$ represents the absolute value of $\mathds{M}(B)$.

Definition 8

(Complex plausibility function)

Let $A$ be a proposition in the frame of discernment $\Omega$. The complex plausibility function of proposition $A$, denoted as $Pl_{c}(A)$ is defined by

[TABLE]

where $|\mathds{M}(B)|$ represents the absolute value of $\mathds{M}(B)$.

Obviously, we can notice that $Pl_{c}(A)\geq Bel_{c}(A)$, in which the complex belief function $Bel_{c}(A)$ is the lower bound function of proposition $A$, and the complex plausibility function $Pl_{c}(A)$ is the upper bound function of proposition $A$.

Definition 9

(Generalized Dempster’s rule of combination)

Let $\mathds{M}_{1}$ and $\mathds{M}_{2}$ be two independently complex basic belief assignments (CBBAs) in the frame of discernment $\Omega$. The generalized Dempster’s rule of combination, defined by $\mathds{M}=\mathds{M}_{1}\oplus\mathds{M}_{2}$, which is called the orthogonal sum, is represented as below

[TABLE]

with

[TABLE]

where $A,B\in 2^{\Omega}$ and $\mathds{K}$ is the conflict coefficient between the CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$.

Remark 1

The generalized Dempster’s combination rule is only feasible under the situation where the conflict coefficient $\mathds{K}\neq 1$ for $\mathds{M}_{1}$ and $\mathds{M}_{2}$.

Remark 2

Compared with the traditional Dempster’s combination rule, the condition in terms of the conflict coefficient $\mathds{K}<1$ is released in the generalized Dempster’s combination rule so that it is more general and applicable than the traditional Dempster’s combination rule.

Remark 3

When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the traditional evidence theory under the condition that the conflict coefficient $\mathds{K}<1$.

An example is given to illustrate that the condition $\mathds{K}<1$ is released in the generalized Dempster’s combination rule, where the variation of the magnitude of conflict coefficient $|\mathds{K}|$ between two CBBAs is depicted. Note that $|\mathds{K}|$ can be calculated based on Eq. (27).

Example 1

Supposing that there are two CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$ in the frame of discernment $\Omega=\{A,B\}$, and the two CBBAs are given as follows:

[TABLE]

Since $|\mathds{M}_{1}(A)|=\sqrt{x^{2}+y^{2}}$ and $|\mathds{M}_{1}(B)|=\sqrt{(1-x)^{2}+(-y)^{2}}$, according to Definition 6, the parameters $x$ is set within $[0,1]$ and $y$ is set within $[-1,1]$ satisfying the conditions that $\sqrt{x^{2}+y^{2}}\in[0,1]$ and $\sqrt{(1-x)^{2}+(-y)^{2}}\in[0,1]$ at the same time, where the variations of parameters $x$ and $y$ are shown in Fig. 1.

Fig. 2 and Fig. 3 show the results of the magnitude of conflict coefficient $|\mathds{K}|$ between the two CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$ from different angles.

In particular, as shown in Fig. 2, in the case that $x=1$ and $y=0$, we can obtain that

[TABLE]

The conflict coefficient $\mathds{K}$ is calculated as

[TABLE]

Then, the magnitude of conflict coefficient $|\mathds{K}|$ between the two CBBAs $m_{1}$ and $m_{2}$ is 0.7071.

When $x=0$ and $y=0$, it is obtained that

[TABLE]

The conflict coefficient $\mathds{K}$ is calculated as

[TABLE]

Then, the magnitude of conflict coefficient $|\mathds{K}|$ between the two CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$ is calculated by

[TABLE]

which shows the same result as the case that $x=1$ and $y=0$.

In the case that $x=0.5$ and $y=-0.8660$, we can obtain that

[TABLE]

The conflict coefficient $\mathds{K}$ is calculated as

[TABLE]

Then, the magnitude of conflict coefficient $|\mathds{K}|$ between the two CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$ is calculated by

[TABLE]

In the case that $x=0.5$ and $y=-0.5$, we can obtain that

[TABLE]

The conflict coefficient $\mathds{K}$ is calculated as

[TABLE]

Then, the magnitude of conflict coefficient $|\mathds{K}|$ between the two CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$ is calculated by

[TABLE]

When $x=0.5$ and $y=0.8660$, it is obtained that

[TABLE]

The conflict coefficient $\mathds{K}$ is calculated as

[TABLE]

Then, the magnitude of conflict coefficient $|\mathds{K}|$ between the two CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$ is calculated by

[TABLE]

IV Numerical examples

In this section, several numerical examples are illustrated to show the effectiveness of the generalized Dempster–Shafer evidence theory.

Example 2

Supposing that there are two CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$ in the frame of discernment $\Omega=\{A,B\}$, and the two CBBAs are given as follows:

[TABLE]

Then, the fusing results are calculated by utilising Eq. (30) as follows:

[TABLE]

It is verified that $\mathds{M}(A)$ + $\mathds{M}(B)$ + $\mathds{M}(A,B)$ = 1 in this example.

Example 3

Supposing that there are two CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$ in the frame of discernment $\Omega=\{A,B\}$, and the two CBBAs are given as follows:

[TABLE]

[TABLE]

The fusing results by utilising Eq. (30) are calculated as follows:

[TABLE]

It is obvious that $\mathds{M}(A)$ + $\mathds{M}(B)$ + $\mathds{M}(A,B)$ = 1 in this example.

Through Example 2 and Example 3, it verifies that the generalized Dempster–Shafer evidence theory satisfies the commutative law.

Example 4

Supposing that there are two CBBAs $\mathds{M}_{1}$ and $\mathds{M}_{2}$ in the frame of discernment $\Omega=\{A,B\}$ where they are degenerated to real numbers, and the two CBBAs are given as follows:

[TABLE]

On the one hand, by utilising Eq. (30) of the generalized Dempster’s rule of combination, the fusing results are generated as follows:

[TABLE]

On the other hand, based on Eq. (17) of the classical Dempster’s rule of combination, the fusing results are calculated as follows:

[TABLE]

It is easy to see that the fusing results from the generalized Dempster’s rule of combination is exactly the same as the fusing results from the classical Dempster’s rule of combination. In this example, the conflict coefficient $\mathds{K}$ is 0.2600.

This example verifies that when the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences $\mathds{K}$ is less than 1.

V Conclusions

In this paper, a generalized Dempster–Shafer (GDS) evidence theory is proposed. The main contribution of this study is that a mass function in the GDS evidence theory is modeled as a complex number, called as a complex basic belief assignment. In addition, the definitions of complex belief function and complex plausibility function are also presented in this paper. Based on that, a generalized Dempster’s rule of combination is exploited to fuse the complex basic belief assignments. When the complex mass function is degenerated from complex numbers to real numbers, the GDS evidence theory degenerates to the traditional evidence theory under the condition that the conflict coefficient between the evidences $\mathds{K}$ is less than 1. In summary, this study is the first work to generalize the evidence theory in the framework of complex numbers. It provides a promising way to model and handle more uncertain information in the process of solving the decision-making problems.

Acknowledgment

This research is supported by the Fundamental Research Funds for the Central Universities (No. XDJK2019C085) and Chongqing Overseas Scholars Innovation Program (No. cx2018077).