Generalized belief function in complex evidence theory

Abstract

The complex-value-based generalized Dempster–Shafer evidence theory, also called complex evidence theory is a useful methodology to handle uncertainty problems of decision-making on the framework of complex plane. In this paper, we propose a new concept of belief function in complex evidence theory. Furthermore, we analyze the axioms of the proposed belief function, then define a plausibility function in complex evidence theory. The newly defined belief and plausibility functions are the generalizations of the traditional ones in Dempster–Shafer (DS) evidence theory, respectively. In particular, when the complex basic belief assignments are degenerated from complex numbers to classical basic belief assignments (BBAs), the generalized belief and plausibility functions in complex evidence theory degenerate into the traditional belief and plausibility functions in DS evidence theory, respectively. Some special types of the generalized belief function are further discussed as well as their characteristics. In addition, an interval constructed by the generalized belief and plausibility functions can be utilized for fuzzy measure, which provides a promising way to express and model the uncertainty in decision theory.

Keywords

Complex evidence theory generalized dempster–Shafer evidence theory generalized belief function generalized plausibility function complex basic belief assignment complex mass function uncertainty modelling fuzzy measure decision theory

1 Introduction

How to model and handle the uncertainty in decision theory is still an open issue [1]. To address this problem, many methodologies were exploited, such as the extended intuitionistic fuzzy sets [2], evidence theory [3], evidential reasoning [4], Z numbers [5], D numbers [6], etc. These methodologies have been generally applied in a number of areas, like human reliability analysis [7], medical diagnose [8, 9], and decision making [10 –12]. As one of the most useful methodologies to manage uncertainty, DS evidence theory [13, 14] has many desirable advantages [15]. Specifically, DS evidence theory can not only model the uncertainty quantitatively, but also reduce the uncertainty through using Dempster’s combination rule (DCR) [16]. In addition, the DCR satisfies the commutative and associative laws, so that it can be easy applied [17]. Besides, the result generated by DCR has the characteristic of fault-tolerance, which can better support decision-making [18]. Therefore, the DS evidence theory has been well-studied, including the researches on the negation [19, 20] and information quality [21, 22] of basic belief assignment, belief entropy [23], conflict management [24, 25], evidence reliability evaluation [26], etc.

In the past few years, to make the traditional DS evidence theory more practical, many researchers have extended the belief structure and DCR in different aspects [27]. To be specific, Yang et al. [28] generalized the belief and plausibility functions, and Dempster’s combinational rule to fuzzy sets. Wu et al. [29] generalized the fuzzy belief function in infinite spaces. Yang and Xu [30] relaxed DS evidence theory to deal with evidential reasoning problems. Jiang and Zhan [31] improved the combination rule in the open frame of discernment. Dutta [32] generalized fuzzy focal elements to model the uncertainty in risk assessment. Sabahi [33] devised a novel generalized belief structure by comprising unprecisiated uncertainty. Yager [34] generalized Dempster–Shafer structures that had been extended in satisfying uncertain targets and fuzzy rule bases. Gao and Deng [35] studied the quantum model of mass function. Recently, Xiao [36] presented a complex-value-based generalized Dempster–Shafer evidence theory, also called complex evidence theory, which is a useful methodology to handle uncertainty problems of decision-making on the framework of complex plane.

In this paper, we focus on the generalization of belief function from the view of complex evidence theory. On the basis of complex basic belief assignment in complex evidence theory, the belief function and plausibility function are generalized. Especially, when the complex basic belief assignments are degenerated from complex numbers to classical BBAs, the generalized belief and plausibility functions degenerate into the traditional ones in DS evidence theory, respectively. Some special types of the generalized belief function are further discussed as well as their characteristics. Additionally, an interval which is constructed by the generalized belief function and plausibility function can be utilized for fuzzy measure, which provides a promising way to express and model the uncertainty in decision theory.

The rest part of this paper is organized below. In Section 2, the complex evidence theory is briefly presented based on the framework of complex plane. In Section 3, the new concepts of generalized belief function and plausibility function are defined based on complex evidence theory. In Section 4, some special types of generalized belief function are discussed and analyzed. Section 5 concludes this work.

2 The complex evidence theory

To model and handle the problem of uncertainty, many methods have been presented in the past few years [37 –39]. Among these methods, DS evidence theory [13, 14] that is a useful uncertainty reasoning tool has been proverbially used in a variety of fields, like assessment and selection [40, 41], classification and recognition [42 –44], data fusion [45, 46], etc. As a generalization of the traditional evidence theory, the complex evidence theory [36], which is a complex-value-based Dempster–Shafer evidence theory, is presented based on the framework of complex plane. Some basic concepts of complex evidence theory are introduced below.

Definition 2.1. (Frame of discernment)

The Ω is defined as a frame of discernment which consists of a set of mutually exclusive and collective non-empty events, denoted as $Ω = {ψ_{1}, ψ_{2}, \dots, ψ_{i}, \dots, ψ_{N}} .$ (1)

The power set of Ω is represented by 2^Ω, denoted as $\begin{matrix} 2^{Ω} = & {\emptyset, {ψ_{1}}, {ψ_{2}}, \dots, {ψ_{N}}, {ψ_{1}, ψ_{2}}, \dots, \\ {ψ_{1}, ψ_{2}, \dots, ψ_{i}}, \dots, Ω}, \end{matrix}$ (2) where ∅ is the empty set.

When A ∈ 2^Ω, A is called a proposition or hypothesis.

Definition 2.2. (Complex mass function)

A complex mass function (CMF) $M$ in the frame of discernment Ω is modeled as a complex number, which is represented as a mapping from 2^Ω to $ℂ$ , defined by $IM : 2^{Ω} \to ℂ,$ (3) satisfying: $\begin{matrix} IM (\emptyset) & = 0, \\ IM (A) & = m (A) e^{i θ (A)}, A \neq \emptyset | A \in 2^{Ω} \\ \sum_{A \neq \emptyset | A \in 2^{Ω}} IM (A) & = 1, \end{matrix}$ (4) where $i = \sqrt{- 1}$ ; m (A) ∈ [0, 1] representing the magnitude of IM (A); θ (A) ∈ [- π, π] denoting a phase term.

In Eq. (4), IM (A) can also expressed in the “rectangular” or “Cartesian” forms: $IM (A) = x + yi, A \neq \emptyset | A \in 2^{Ω}$ (5) with $\sqrt{x^{2} + y^{2}} \in [0, 1] .$ (6)

By using the Euler’s relation, the magnitude of IM (A) can be expressed as $m (A) = \sqrt{x^{2} + y^{2}}, and θ (A) = arctan (\frac{y}{x}),$ (7) where x = m (A) cos(θ (A)) and y = m (A) sin(θ (A)).

The square of the absolute value for IM (A) is defined by $| IM (A) |^{2} = IM (A) \hat{IM} (A) = x^{2} + y^{2},$ (8) where $\hat{IM} (A)$ is the complex conjugate of IM (A), such that $\hat{IM} (A) = x - yi$ .

Then, it can obtain the following equations: $m (A) = | IM (A) |, and θ (A) = ∠ IM (A) .$ (9)

Note that for IM (A), if y = 0, it indicates that IM (A) = x is a real number, then m (A) = |x|.

Property 1. The complex mass function IM can also be called a complex basic belief assignment (CBBA), which is a generalization of basic belief assignment (BBA) in DS evidence theory.

Definition 2.3. (Complex Dempster’s combination rule)

Let IM₁ and IM₂ be two independently CBBAs in Ω , where A and B are the propositions of IM₁ and IM₂, respectively. The complex Dempster’s combination rule (CDCR), denoted as IM = IM₁ ⊕ IM₂ is defined by: $IM (C) = {\begin{matrix} \begin{matrix} \frac{1}{1 - IK} \sum_{A \cap B = C} {IM}_{1} (A) {IM}_{2} (B), & C \neq \emptyset, \end{matrix} \\ \begin{matrix} 0, & C = \emptyset, \end{matrix} \end{matrix}$ (10) with $IK = \sum_{A \cap B = \emptyset} {IM}_{1} (A) {IM}_{2} (B),$ (11) in which IK is the conflict coefficient between IM₁ and IM₂.

Note that the CDCR is only feasible when IK ≠ 1.

Property 2. When the CBBAs of IM₁ and IM₂ degenerate from complex numbers to classical BBAs, the CDCR degenerates to the traditional DCR in DS evidence theory with IK < 1. In other word, the CDCR is a generalized DCR.

3 Generalized belief and plausibility functions

In this section, on the basis of complex mass function, the new concepts of belief function and plausibility function are defined in complex evidence theory, where they are the generalization of traditional belief and plausibility functions in DS evidence theory, respectively.

Definition 3.1. (The focal element of complex mass function)

Definition 3.2. (Commitment degree)

For the proposition A of CMF IM in Ω , the commitment degree Com (A) that is committed to A is defined by $\begin{matrix} C om (A) = \frac{| IM (A) |}{\sum_{B \subseteq Ω} | IM (B) |} = \frac{m (A)}{\sum_{B \subseteq Ω} m (B)}, \end{matrix}$ (13) where m (A) and m (B) are the magnitudes of IM (A) and IM (B), respectively, as defined in Eq. (7).

In Eq. (13), ∑_{B⊆
Ω}m (B) is needed to normalize Com (A), which guarantees that 0 ≤ Com (A) ≤1.

Definition 3.3. (Generalized belief function)

Let IM be a CBBA in the frame of discernment Ω . The generalized belief function in complex evidence theory, denoted as GBel for the proposition A ⊆ Ω is defined by a mapping from 2^Ω to [0, 1]: $GBel (A) = \sum_{B \subseteq A} C om (B),$ (14) where $C om (B) = \frac{| IM (B) |}{\sum_{C \subseteq Ω} | IM (C) |} = \frac{m (B)}{\sum_{C \subseteq Ω} m (C)} .$ (15)

When the CBBA IM becomes a classical BBA, for B ⊆ A ⊆ Ω , we have $m (B) = m (B) .$

Under this situation, Eq. (14) can be expressed as $GBel (A) = \sum_{B \subseteq A} \frac{m (B)}{\sum_{C \subseteq Ω} m (C)} = \sum_{B \subseteq A} \frac{m (B)}{\sum_{C \subseteq Ω} m (C)} .$

Since $\sum_{C \subseteq Ω} m (C) = 1,$ in this case, it can be obtained that $GBel (A) = \sum_{B \subseteq A} m (B) = Bel (A) .$

It is obvious that GBel (A) in complex evidence theory is a generalization model of Bel in the traditional DS evidence theory [13, 14].

Property 3. When the CBBAs become classical BBAs, i.e., traditional BBAs, the GBel in complex evidence theory will degrade into Bel in DS evidence theory.

Theorem 1. The GBel satisfies the following axioms:

GBel (∅) =0,

GBel ( Ω ) =1,

For A₁, A₂, …, A_m of the subsets of Ω ,

$\begin{matrix} GBel (A_{1} \cup A_{2} \cup \dots \cup A_{m}) \\ \geq \sum_{i} C om (A_{i}) - \sum_{i < j} C om (A_{i} \cap A_{j}) \pm \dots \\ + (- 1)^{m + 1} C om (A_{1} \cap A_{2} \cap \dots \cap A_{m}) \\ = \sum_{I \neq \emptyset | I \subset {1, \dots, m}} (- 1)^{| I | + 1} C om (⋂_{i \in I} A_{i}) . \end{matrix}$ (16)

The Axiom 1 indicates that the proposition ∅ is supposed to be impossibility. The Axiom 2 indicates that the proposition Ω is supposed to be certainty. The Axiom 3 indicates that beliefs are super-additive, which becomes additive under a certain case where A_i ∈ Ω which means that A_i are singletons.

In Eq. (16), when the CBBAs become classical BBAs, we have $\begin{matrix} GBel (A_{1} \cup A_{2} \cup \dots \cup A_{m}) \\ = Bel (A_{1} \cup A_{2} \cup \dots \cup A_{m}) \\ \geq \sum_{i} m (A_{i}) - \sum_{i < j} m (A_{i} \cap A_{j}) \pm \dots \\ + (- 1)^{m + 1} m (A_{1} \cap A_{2} \cap \dots \cap A_{m}) \\ = \sum_{I \neq \emptyset | I \subset {1, \dots, m}} (- 1)^{| I | + 1} m (⋂_{i \in I} A_{i}) . \end{matrix}$

When A_i∩ A_j = ∅, the above equality holds, such that $\begin{matrix} GBel (A_{1} \cup A_{2} \cup \dots \cup A_{m}) \\ = Bel (A_{1} \cup A_{2} \cup \dots \cup A_{m}) = \sum_{i} m (A_{i}) . \end{matrix}$

Property 4. It can be learned that when the CBBAs degrade into classical BBAs, the GBel can be regarded as a generalization of additive probability function as well as the traditional belief function Bel.

Proof. (1) The proofs for Axiom 1 and Axiom 2 are trivial.

(2) On the basis of [14], we have $\begin{matrix} \sum_{I \neq \emptyset | I \subset {1, \dots, m}} (- 1)^{| I | + 1} GBel (⋂_{i \in I} A_{i}) \\ = \sum_{I \neq \emptyset | I \subset {1, \dots, m}} (- 1)^{| I | + 1} \sum_{B \subset ⋂_{i \in I} A_{i}} C om (B) \\ = \sum_{B \subset Ω | I (B) \neq \emptyset} C om (B) \sum_{I \neq \emptyset | I \subset I (B)} (- 1)^{| I | + 1} \\ = \sum_{B \subset Ω | I (B) \neq \emptyset} C om (B) (1 - \sum_{I \neq \emptyset | I \subset I (B)} (- 1)^{| I |}) \\ = \sum_{B \subset Ω | I (B) \neq \emptyset} C om (B) = \sum_{B \subset A_{i}} C om (B) . \end{matrix}$

Since $\begin{matrix} \sum_{B \subset A_{i}} C om & \leq \sum_{B \subset {A_{1} \cup \dots \cup A_{m}}} C om \\ = GBel (A_{1} \cup \dots \cup A_{m}), \end{matrix}$ it can be calculated that $\begin{matrix} \sum_{I \neq \emptyset | I \subset {1, \dots, m}} (- 1)^{| I | + 1} GBel (⋂_{i \in I} A_{i}) \\ \leq GBel (A_{1} \cup \dots \cup A_{m}), \end{matrix}$ which addresses the proof of Axiom 3.

Theorem 2. For A ⊆ Ω , the Com (A) can be constructed by using a $M \ddot{o} bius$ transformation: $\begin{matrix} C om (A) = \sum_{B \subseteq A} (- 1)^{| A - B |} GBel (B), \end{matrix}$ (17) where |A - B| is the cardinality of A - B, such that |A - B| = |A| - |B|; GBel function is defined in Definition 3.3.

Then, we can get a recursive deduction: $\begin{matrix} C om (A) = GBel (A) - \sum_{B \subset A} C om (B) . \end{matrix}$ (18)

Proof. Since $\begin{matrix} \sum_{B \subseteq A} (- 1)^{| A - B |} GBel (B) \\ = (- 1)^{| A |} \sum_{B \subseteq A} (- 1)^{| B |} GBel (B) \\ = (- 1)^{| A |} \sum_{B \subseteq A} (- 1)^{| B |} \sum_{C \subseteq B} C om (C) \\ = (- 1)^{| A |} \sum_{C \subseteq A} C om (C) \sum_{C \subseteq B \subseteq A} (- 1)^{| B |} \\ = (- 1)^{| A |} C om (A) (- 1)^{| A |} \\ = C om (A), \end{matrix}$ the proof of Eq. (17) in Theorem 2 has been addressed.

Additionally, we have $\begin{matrix} \sum_{B \subseteq A} C om (B) & = \sum_{B \subseteq A} \sum_{C \subseteq B} (- 1)^{| B - C |} GBel (C) \\ = \sum_{C \subseteq A} (- 1)^{| C |} GBel (C) \sum_{C \subseteq B \subseteq A} (- 1)^{| B |} \\ = (- 1)^{| A |} GBel (A) (- 1)^{| A |} \\ = GBel (A) . \end{matrix}$

Because $\begin{matrix} \sum_{B \subseteq A} C om (B) - \sum_{B \subset A} C om (B) = C om (A), \end{matrix}$ it can be calculated that $\begin{matrix} GBel (A) - \sum_{B \subset A} C om (B) = C om (A), \end{matrix}$ which addresses the proof of Eq. (18) in Theorem 2.

Definition 3.4. (Generalized plausibility function)

The plausibility function in complex evidence theory, denoted as GPl (A) for the proposition A ⊆ Ω is defined by a mapping from 2^Ω to [0, 1]: $\begin{matrix} GPl (A) & = 1 - GBel (\bar{A}) = 1 - \sum_{B \subseteq \bar{A}} C om (B) \\ = \sum_{B \cap A \neq \emptyset} C om (B), \end{matrix}$ (19) where $\bar{A}$ is the complement of A, such that $\bar{A} = Ω - A$ .

In Eq. (19), when the CBBA IM becomes a classical BBA, we have $C om (B) = m (B) = m (B) .$

Under this situation, Eq. (19) can be expressed as $GPl (A) = \sum_{B \cap A \neq \emptyset} m (B) = Pl (A) .$

Obviously, GPl in complex evidence theory is a generalization model of Pl in DS evidence theory.

Property 5. When the CBBAs become classical BBAs, i.e., traditional BBAs, the GPl in complex evidence theory will degrade into Pl in DS evidence theory.

Especially, when A ∈ Ω , indicating that all of the propositions are singletons, we have $\begin{matrix} GBel (A) & = C om (A), \\ GPl (A) & = C om (A) . \end{matrix}$

Under this situation, the following equation can be obtained: $\begin{matrix} GBel (A) = GPl (A) . \end{matrix}$ (20)

Property 6. From Definition 3.3 and Definition 3.4, it can be learned that the GBel (A) and GPl (A) represent the lower and upper bound functions of proposition A, respectively. Only when all of the propositions are singletons, GBel (A) is equal to GPl (A).

Based on Property 4, the interval [GBel (A) , GPl (A)] constructed by belief and plausibility functions can be utilized for fuzzy measure, which provides a promising way to express and model the uncertainty in decision theory.

4 Special types of generalized belief function

In this section, some special types of generalized belief function are analyzed. The characteristics of these special types are also discussed.

Definition 4.1. (Generalized vacuous belief function)

A GBel over 2^Ω is called a generalized vacuous belief function when satisfies the condition: $GBel (A) = {\begin{matrix} \begin{matrix} 1, & if A = Ω, \end{matrix} \\ \begin{matrix} 0, & if A \neq Ω . \end{matrix} \end{matrix}$ (21)

It is worth noting that the generalized vacuous belief function expresses complete ignorance. It means that this evidence does not give support to any specific propositions, i.e., any subsets of Ω , and Ω is the only focal element.

Definition 4.2. (Generalized simple support function)

A GBel is called a generalized simple support function focused at the proposition A if $GBel (B) = {\begin{matrix} \begin{matrix} 0, & if A ⊄ B, \end{matrix} \\ \begin{matrix} 1, & if B = Ω, for 0 < c < 1 \end{matrix} \\ \begin{matrix} c, & if A \subset B and B \neq Ω . \end{matrix} \end{matrix}$ (22)

When GBel is a generalized simple support function focussed at the proposition A, the following equations can be deduced: $\begin{matrix} C om (A) & = GBel (A) = c, \\ C om (Ω) & = 1 - GBel (A) = 1 - c, \\ C om (B) & = 0 for others . \end{matrix}$

The generalized simple support function focused at the proposition A indicates the special outcome is in A with belief degree c.

Property 7. When the generalized simple support function focused at the proposition A with Com (A) =1, it is called the certain generalized support function focused at A.

The generalized simple support function expresses the evidence that offers support to one and only one proposition which is a subset of Ω .

Definition 4.3. (Generalized Bayesian belief function)

A GBel over 2^Ω is called a generalized Bayesian belief function if $GPl (A) = GBel (A), for all A \subset Ω .$ (23)

Theorem 3. The generalized Bayesian belief function has the following Axioms:

GBel (∅) =0,

GBel ( Ω ) =1,

GBel (A ∪ B) = GBel (A) + GBel (B), whenever A∩ B = ∅,

$GBel (A) + GBel (\bar{A}) = 1$ .

Proof. The proof is trivial.

Theorem 4. When GBel is a generalized Bayesian belief function, the CBBA IM has non-zero values for only the subsets of Ω that are singletons, denoted as $\sum_{x \in Ω} IM ({x}) = 1 .$ (24)

Proof. The proof is trivial.

When the CBBAs degrade into classical BBAs, i.e., traditional BBAs, the generalized Bayesian belief function expresses probabilistic knowledge which assigns probabilities to all focal elements that are singletons.

Furthermore, on the basis of Theorem 4, we get $GBel (A) = \sum_{B \subseteq A} C om (B) = \sum_{x \in A} C om ({x}) .$

Since for the generalized Bayesian belief function, $GPl (A) = GBel (A),$ it can get $GPl (A) = \sum_{x \in A} C om ({x}) .$

Moreover, because for the generalized Bayesian belief function, $GBel ({x}) = GPl ({x}) = C om ({x}),$ we can obtain $\begin{matrix} GBel (A) & = \sum_{x \in A} GBel ({x}) \\ = GPl (A) = \sum_{x \in A} GPl ({x}) . \end{matrix}$

It is noticed that the generalized Bayesian belief function is fully defined by a point function of Ω equal to Com ({x}).

Definition 4.4. (Generalized consonant belief function)

A GBel over 2^Ω is called a generalized consonant belief function if satisfying the following conditions:

GBel (∅) =0,

GBel ( Ω ) =1,

GBel (A ∩ B) = min {GBel (A) , GBel (B)}, for A, B ⊂ Ω ,

GPl (A ∪ B) = max {GPl (A) , GPl (B)}, for A, B ⊂ Ω ,

$GPl (A) = max_{x \in A} [GPl ({x})]$ , for A ≠ ∅ |A ⊂ Ω .

Proof. (1) The proofs for 1) and 2) are trivial.

(2) Let IM be a CBBA in Ω . Assume GBel is a generalized consonant belief function, where its focal elements are sorted as {A₁, A₂, …, A_i, …, A_j, …, A_m}, such that A_i ⊂ A_j (i < j). Let B and C be arbitrary two subsets of Ω . Let i_α be the largest integer i for A_i ⊂ B, and i_β be the largest integer i for A_i ⊂ C.

Then, $\begin{matrix} A_{i} \subset B, & if and only if i \leq i_{α}, \\ A_{i} \subset C, & if and only if i \leq i_{β}, \\ A_{i} \subset B \cap C, & if and only if i \leq min (i_{α}, i_{β}) . \end{matrix}$

Therefore, for B, C ⊂ Ω , we have $\begin{matrix} GBel (B \cap C) & = \sum_{i = 1}^{min (i_{α}, i_{β})} C om (A_{i}) \\ = min {\sum_{i = 1}^{i_{α}} C om (A_{i}), \sum_{i = 1}^{i_{β}} C om (A_{i})} \\ = min {GBel (B), GBel (C)}, \end{matrix}$ which addresses the proof of 3).

(3) Assume 3) holds. Hence, for B, C ⊂ Ω , we have $\begin{matrix} GPl (B \cup C) & = 1 - GBel (\bar{B \cup C}) = 1 - GBel (\bar{B} \cap \bar{C}) \\ = 1 - min {GBel (\bar{B}), GBel (\bar{C})} \\ = max {1 - GBel (\bar{B}), 1 - GBel (\bar{C})} \\ = max {GPl (B), GPl (C)}, \end{matrix}$ which addresses the proof of 4).

(4) Assume 4) holds. Let A = {x₁, x₂, …, x_m}. Based on 4), we have

$\begin{matrix} GPl (A) & = max [GPl ({x_{1}, x_{2}, \dots, x_{m - 1}}), GPl ({x_{m}})] \\ = max [max_{1 \leq i \leq m - 1} GPl ({x_{i}}), GPl ({x_{m}})] \\ = max_{1 \leq i \leq m} [GPl ({x_{i}})] \\ = max_{x \in A} [GPl ({x})], \end{matrix}$

which addresses the proof of 5).

Property 8. A GBel is consonant if its CBBA IM’s focal elements are nested. In other words, for a family of subsets A_i of Ω (i = 1, 2, …, m), $A_{i} \subset A_{j} for i < j and \sum_{i} IM (A_{i}) = 1 .$ (25)

Next, the relationship between GPl and Com can be derived as follows.

Suppose there is a generalized consonant belief structure. A nested sequence of sets can be built as: ${x_{1}} \subset {x_{1}, x_{2}} \subset \dots \subset {x_{1}, x_{2}, \dots, x_{m}} = Ω,$ which indicates the subsets A₁ ⊂ A₂ ⊂ … ⊂ A_m = Ω satisfying $\sum_{i = 1}^{m} IM (A_{i}) = 1 .$

Because, for B, C ⊂ Ω , $GPl (C) = \sum_{C \cap B \neq \emptyset} C om (B),$ we have $\begin{matrix} GPl ({x}) & = \sum_{{x} \cap B \neq \emptyset} C om (B) \\ = \sum_{i | {x} \cap A_{i} \neq \emptyset} C om (A_{i}) = \sum_{i | x \in A_{i}} C om (A_{i}) . \end{matrix}$

So, $\begin{matrix} GPl ({x_{1}}) = & C om (A_{1}) + C om (A_{2}) + \dots \\ + C om (A_{m - 1}) + C om (Ω), \\ GPl ({x_{2}}) = & C om (A_{2}) + \dots \\ + C om (A_{m - 1}) + C om (Ω), \\ ⋮ \\ GPl ({x_{m}}) & = C om (Ω) . \end{matrix}$

On the contrary, $\begin{matrix} C om (A_{i}) & = GPl ({x}) - GPl ({x_{i + 1}}), \\ C om (Ω) & = GPl ({x_{m}}), \\ C om (B) & = 0, for others . \end{matrix}$

5 Conclusions

In this paper, a new concept of belief function was first proposed in complex evidence theory. Furthermore, the axioms of the proposed belief function were analyzed and proofed. Based on that, a plausibility function was then defined in complex evidence theory. The newly defined belief and plausibility functions were the generalizations of the traditional ones in DS evidence theory, respectively. In particular, when the CBBAs became classical BBAs from complex numbers, the generalized belief and plausibility functions in complex evidence theory degenerated into the traditional belief and plausibility functions in DS evidence theory, respectively. Furthermore, some special types of generalized belief function were discussed as well as their characteristics.

In summary, the main contribution of this study is that this is the first work to consider the belief function and plausibility function in complex evidence theory, which generalizes the traditional belief and plausibility functions on the framework of complex plane. Moreover, the interval constructed by the generalized belief and plausibility functions could be utilized for fuzzy measure. It provides a potential way to express and model the uncertainty in decision theory.

Conflict of interest

The author states that there are no conflicts of interest.

Footnotes

Acknowledgment

The author greatly appreciates the reviewers’ suggestions and the editor’s encouragement. This research is supported by the Research Project of Education and Teaching Reform in Southwest University (No. 2019JY053), Fundamental Research Funds for the Central Universities (No. XDJK2019C085) and Chongqing Overseas Scholars Innovation Program (No. cx2018077).

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