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Abstract

In real applications, sensors may work in complicated environments; thus, how to measure the uncertain degree of sensor reports before applying sensor data fusion is a big challenge. To address this issue, an improved belief entropy–based uncertainty management approach for sensor data fusion is proposed in this article. First, the sensor report is modeled as the body of evidence in Dempster–Shafer framework. Then, the uncertainty measure of each body of evidence is based on the subjective uncertainty represented as the evidence sufficiency and evidence importance, and the objective uncertainty measure is expressed as the improved belief entropy. Evidence modification of conflict sensor data is based on the proposed uncertainty management approach before evidence fusion with Dempster’s rule of combination. Finally, the fusion result can be applied in real applications. A case study on sensor data fusion for fault diagnosis is presented to show the rationality of the proposed method.

Keywords

Dempster–Shafer evidence theory belief entropy Deng entropy improved belief entropy uncertainty management sensor data fusion

Introduction

Sensors play an important role in environment sensing and information acquiring in real applications, such as fault diagnosis,¹ risk analysis,² route optimization,³ image processing,⁴ and so on.^5,6 Driven by real applications, many methods have been proposed for multi-sensor modeling and sensor data fusion,⁷ including neural network model,^8,9 belief function theory,^10,11 Dempster–Shafer evidence theory (D-S theory),^12,13 fuzzy set theory,¹⁴ and so on.^15,16 However, sensors may be affected by the complicated application environment. How to measure the uncertain degree or the reliability of sensor reports with heterogeneous sources is also an open issue. This article will focus on multi-sensor data fusion by designing a new uncertainty measure–based sensor data fusion approach.

Uncertainty management is a key point in uncertain information modeling; many theories and methods have been proposed for intelligent information processing, such as Shannon entropy,¹⁷ probability theory,¹⁸ fuzzy sets,¹⁹ fuzzy inference,^20,21 D-S theory,^22,23 rough sets, ²⁴ support vector machine,^25,26 belief function,^27–29 evidence reasoning,^30–32 and so on.^33,34 Among all these methods, D-S theory can process vague information modeled in fuzzy sets, and it can lead to information convergence in data fusion. Although D-S theory is an effective method for information processing in many fields, such as multiple attribute decision-making,^35–38 risk analysis,^39–44 pattern recognition,^45–50 fault diagnosis,^11,13 controller design,^51,52 and so on,^53–56 the classical Dempster’s rule of combination cannot be used directly for conflict sensor data fusion, especially when there exist highly conflict data, because it may lead to counterintuitive results.^57,58

One way to handle the conflict information in real application like sensor data fusion is to measure the uncertain degree before applying sensor data fusion^11,13,59 But how to measure the uncertain degree of uncertain information in the framework of D-S theory is still an open issue.^60–62 Shannon entropy is a typical way for uncertainty measure in the probabilistic framework.¹⁷ Shannon entropy has been generalized to many fields, for example, as a generalization of Shannon entropy, network entropy is an effective measurement for testing the complexity of networks;^63–66 however, Shannon entropy cannot be used directly among mass functions in the framework of D-S theory; this is because a mass function is the generalized probability assigned on the power set of the frame of discernment (FOD). To address this issue, many uncertainty measures in Dempster–Shafer framework are proposed, such as Höhle’s⁶⁷ confusion measure, Yager’s⁶⁸ dissonance measure, the weighted Hartley entropy,⁶⁹ Klir and Ramer’s⁷⁰ discord measure, Klir and Parviz’s⁷¹ strife measure, George and Pal’s⁷² conflict measure, and so on.^60,61 However, the existed methods may not be that effective in some cases.^73,74

Recently, based on Shannon entropy, Deng entropy is proposed to measure the uncertainty in mass functions.⁷³ Deng entropy can measure uncertain degree more efficiently than the other uncertain measures in some cases. Although Deng entropy had been successfully applied in some real applications,⁷⁵ Deng entropy did not take into consideration the scale of the FOD, which means a loss of available and valuable information in information processing. Thus, the improved belief entropy is proposed by Zhou et al.,^74,76 which can process the uncertain information in the FOD and lead to a more accurate information modeling.

In order to address the uncertain information in multi-sensor, in the work by Fan and Zuo,⁵⁹ a fault diagnosis method based on multi-sensor reports is presented based on D-S theory, and the evidence sufficiency and evidence importance are defined to model the reliability of evidence. Later, the method in the work by Fan and Zuo⁵⁹ is improved with Deng entropy and evidence distance by Yuan et al.,¹³ which leads to a better fusion result for fault diagnosis. However, while modeling the sensor reliability, both Deng entropy and evidence distance are based on basic belief assignments (BBAs) in the work by Yuan et al.,¹³ and a coupling relation may exist in the proposed uncertainty measure. The same problem also exists in the works by Yuan et al.¹¹ and Wang et al.⁷⁵

In this article, based on the improved belief entropy, which can process more uncertain information than Deng entropy, a new uncertainty management approach for sensor data fusion is proposed. In the proposed method, first, the sensor report is modeled as the body of evidence (BOE) in Dempster–Shafer framework. Then, the uncertain degree of each BOE is measured based on a new uncertainty management method. The uncertainty measure of each BOE is based on the subjective uncertainty represented as the evidence sufficiency and evidence importance, and objective uncertainty measure is represented as the improved belief entropy. Third, the evidence is modified with the weight of each BOE. The relative weight of each BOE, which is measured with the proposed uncertainty management method in the previous step, is used to modify the BBA of BOEs. Fourth, evidence fusion of the modified evidence is based on Dempster’s rule of combination. Finally, the fusion result is used for real application.

The rest of this article is organized as follows. In section “Preliminaries,” the preliminaries are briefly introduced, including D-S theory, Deng entropy, and the improved belief entropy. In section “A new uncertainty measure–based sensor data fusion approach,” the new uncertainty measure–based sensor data fusion approach is proposed. In section “Application in sensor data fusion,” a case study on sensor data fusion for fault diagnosis is presented to show the efficiency of the new method. The conclusions are given in section “Conclusion.”

Preliminaries

In this section, some preliminaries are briefly introduced, including D-S theory, Shannon entropy, and some typical uncertainty measures in Dempster–Shafer framework.

Dempster–Shafer evidence theory

Let $Ω = {θ_{1}, θ_{2}, \dots, θ_{i}, \dots, θ_{N}}$ be a finite nonempty set of mutually exclusive and exhaustive events, where $Ω$ is called the FOD. The power set of $Ω$ , denoted as $2^{Ω}$ , is composed of $2^{N}$ elements denoted as follows

2^{Ω} = {\begin{matrix} \emptyset, {θ_{1}}, {θ_{2}}, \dots, {θ_{N}}, {θ_{1}, θ_{2}}, \\ \dots, {θ_{1}, θ_{2}, \dots, θ_{i}}, \dots, Ω \end{matrix}}

(1)

A mass function $m$ is defined as a mapping from the power set $2^{Ω}$ to the interval [0, 1], which satisfies the following conditions^22,23

m (\emptyset) = 0, \sum_{A \in Ω} m (A) = 1

(2)

If $m (A) > 0$ , then $A$ is called a focal element, and the mass function $m (A)$ represents how strongly the evidence supports proposition $A$ .

A BOE, also known as a basic probability assignment (BPA) or BBA, is represented by the focal sets and their associated mass value

(ℜ, m) = {〈 A, m (A) 〉 : A \in 2^{Ω}, m (A) > 0}

(3)

where ℜ is a subset of the power set $2^{Ω}$ , and each $A \in ℜ$ has an associated nonzero mass value $m (A)$ .

A BBA $m$ can also be represented by its associate belief function $Bel$ and plausibility function $Pl$ , respectively, defined as follows

Bel (A) = \sum_{ϕ \neq B \subseteq A} m (B) and Pl (A) = \sum_{B \cap A \neq ϕ} m (B)

(4)

In D-S theory, two independent mass functions, denoted as $m_{1}$ and $m_{2}$ , can be combined with Dempster’s rule of combination defined as^22,23

m (A) = (m_{1} \oplus m_{2}) (A) = \frac{1}{1 - k} \sum_{B \cap C = A} m_{1} (B) m_{2} (C)

(5)

where $k$ is a normalization constant representing the degree of conflict between $m_{1}$ and $m_{2}$ , and $k$ is defined as^22,23

k = \sum_{B \cap C = \emptyset} m_{1} (B) m_{2} (C)

(6)

Deng entropy

Deng entropy is a generalization of Shannon entropy in Dempster–Shafer framework.⁷³ If the information is modeled in the framework of a probability theory, Deng entropy can be degenerated to Shannon entropy. Deng entropy, denoted as $E_{d}$ , is defined as follows⁷³

E_{d} (m) = - \sum_{A \subseteq X} m (A) \log_{2} \frac{m (A)}{2^{| A |} - 1}

(7)

where $| A |$ denotes the cardinality of the proposition $A$ and $X$ is the FOD. If and only if the mass value is assigned to single elements, Deng entropy can be degenerated to Shannon entropy; in this case, the form of Deng entropy is as follows

\begin{array}{l} E_{d} (m) = - \sum_{A \subseteq X} m (A) \log_{2} \frac{m (A)}{2^{| A |} - 1} = \\ - \sum_{A \subseteq X} m (A) \log_{2} m (A) \end{array}

(8)

Some properties of Deng entropy are also discussed in the work by Abelln,⁷⁷ which can contribute to a better understanding of this theory. For more details on Deng entropy, please refer to Deng⁷³ and Abelln.⁷⁷

The improved belief entropy

In Dempster–Shafer framework, the improved belief entropy is defined as follows⁷⁴

E_{Id} (m) = - \sum_{A \subseteq X} m (A) \log_{2} (\frac{m (A)}{2^{| A |} - 1} e^{\frac{| A | - 1}{| X |}})

(9)

where $X$ is the FOD, $| A |$ denotes the cardinality of the focal element $A$ , and $| X |$ is the number of element in the FOD.

The exponential factor $e^{(| A | - 1) / | X |}$ in the improved belief entropy represents the uncertain information in a BOE that has been ignored by Deng entropy and some other uncertainty measures such as the confusion measure, the dissonance measure, the weighted Hartley entropy, the discord measure, and the strife measure. More importantly, by considering the scale of FOD in the improved belief entropy, it can effectively quantify the difference among different BOEs even if the same mass value is assigned on different FODs. The effectiveness of the improved belief entropy is illustrated in Example 1.

Example 1

Consider a target identification problem; assume that two reliable sensors report the detection results independently. The results are represented by BOEs which are shown as follows

m_{1} : m_{1} ({a, b}) = 0.4, m_{1} ({c, d}) = 0.6

m_{2} : m_{2} ({a, c}) = 0.4, m_{2} ({b, c}) = 0.6

Recalling equation (7), the uncertainty measure with Deng entropy is shown as follows

E_{d} (m_{1}) = - \sum_{A \subseteq X} m_{1} (A) \log_{2} \frac{m_{1} (A)}{2^{| A |} - 1} = 2.5559

(10)

E_{d} (m_{2}) = - \sum_{A \subseteq X} m_{2} (A) \log_{2} \frac{m_{2} (A)}{2^{| A |} - 1} = 2.5559

(11)

The results calculated by Deng entropy are counterintuitive. Although the two BOEs have the same mass value, the FOD of the first BOE $m_{1}$ consists of four targets, denoted as $a$ , $b$ , $c$ , and $d$ , while the second BOE $m_{2}$ has only three possible targets, denoted as $a$ , $b$ , and $c$ . Intuitively, it is expected that $m_{2}$ has a less uncertainty than $m_{1}$ . In other words, the belief entropy of $m_{1}$ should be bigger than that of $m_{2}$ . Deng entropy fails to quantify this difference.

With the improved belief entropy in equation (9), the uncertainty measure in Example 1 is calculated as follows

E_{Id} (m_{1}) = - \sum_{A \subseteq X} m_{1} (A) \log_{2} (\frac{m_{1} (A)}{2^{| A |} - 1} e^{\frac{| A | - 1}{| X |}}) = 2.1952

(12)

E_{Id} (m_{2}) = - \sum_{A \subseteq X} m_{2} (A) \log_{2} (\frac{m_{2} (A)}{2^{| A |} - 1} e^{\frac{| A | - 1}{| X |}}) = 2.0750

(13)

It can be concluded that Deng entropy cannot measure the different uncertain degree between these two BOEs in this case, while the improved belief entropy can effectively measure the difference by taking into consideration more available information of the BOE. According to Table 1, it is also safe to say that the first BOE $m_{1}$ has a higher uncertain degree with the new belief entropy; this is reasonable because the FOD of $m_{1}$ includes four possible targets which means a larger information volume than the second BOE $m_{2}$ .

Table 1.

Calculation results in example 3.1.

BOEs⁶⁹	Deng entropy⁷³	Improved belief entropy⁷⁴
$m_{1}$	2.5559	2.1952
$m_{2}$	2.5559	2.0750

BOEs: body of evidences.

Some properties and the merit of the improved belief entropy, such as its compatibility with Shannon entropy, are briefly discussed in the work by Zhou et al.^74,76 In addition, the improved belief entropy function can be used to measure the uncertainty of $D$ numbers⁷⁸ directly or with a modified form in a cautious way. This is because that $D$ numbers is the extension of BPA; thus, it can be represented as the form of BPA in some cases. This open issue deserves further study.

A new uncertainty measure–based sensor data fusion approach

In this section, a new uncertainty measure–based sensor data fusion approach is proposed. The basic feature of the proposed method is the uncertainty management approach for sensor data fusion. The flow chart of the proposed methodology is shown in Figure 1. Five steps are included in the proposed methodology, which are shown as follows.

Figure 1.

The flow chart of the uncertainty measure–based sensor data fusion approach.

Step 1: evidence modeling

Sensor data or other uncertain information is modeled as BOEs in Dempster–Shafer framework. Usually, the information is represented as BOEs; before further study, BBAs in BOE will be used to represent each tiny piece of evidence. Evidence modeling is another important open issue in D-S theory; for more details on this issue, please refer to Deng et al.⁷⁹ and Yang and Liu.⁸⁰

Step 2: uncertainty management

Uncertainty management of the uncertain information is necessary before data fusion or other application types. Uncertainty often comes from several types of uncertain and incomplete information, including ignorance and vagueness. In this article, the uncertainty of information is divided into two types: the objective uncertainty and the subjective uncertainty. The objective uncertainty comes from objective data, such as sensor data modeled in BBAs. In complicated real application environment, many reasons may affect sensor reports; sometimes, sensor data may be in high conflict with each other, thus the uncertain degree of sensor data is modeled as the objective uncertainty in the proposed methodology. The subjective uncertainty means information comes from subjective information, such as the subjective assessment of experts or personal experience of an engineer, which is modeled as evidence sufficiency and evidence importance in the work by Fan and Zuo.⁵⁹

The flow chat of the uncertainty management method, which is based on the improved belief entropy, the sufficiency index, and the importance index, is shown in Figure 2.

Figure 2.

The flow chart of the proposed uncertainty management method.

The subjective uncertainty, denoted as $u_{s}$ , is defined as the product of evidence sufficiency and evidence importance in the work by Fan and Zuo⁵⁹

u_{s} (m) = μ (m) \times v (m)

(14)

where $μ (m)$ is the evidence sufficiency index of the BOE, while $v (m)$ represents the evidence importance index. Both the evidence sufficiency index and the evidence importance index represent subjective judgment information. The evidence sufficiency index is defined as a semi-trapezoidal function and the evidence importance index represents the importance weight of experts and their assessments. For the details of these two indexes, please refer to Fan and Zuo.⁵⁹

The objective uncertainty of each BOE is measured by the improved belief entropy. In the FOD, the objective uncertain degree is a relative value with respect to the highest uncertain degree among all the BOEs; thus, the objective uncertainty, denoted as $u_{o}$ , is defined as follows

u_{o} (m) = \frac{E_{Id} (m)}{max (E_{Id} (m))}

(15)

where $E_{Id} (m)$ is the improved belief entropy of the $i$ th BOE ( $i = 1, 2, 3, \dots, n$ ), while $max (E_{Id} (m))$ represents the maximum value of the improved belief entropy among all these evidence.

With equations (14) and (15), the total uncertainty measure of each evidence (BOE) is defined as follows

u (m) = u_{s} (m) \times u_{o} (m)

(16)

After normalization, the final relative weight of the $i$ th BOE is defined as follows

w (m_{i}) = \frac{u (m_{i})}{\sum_{i = 1}^{n} u (m_{i})}

(17)

where $i = 1, 2, 3, \dots, n$ , $i$ is the order number of all the $n$ BOEs.

Step 3: evidence modification

Since conflict evidence may lead to counterintuitive results,^58,57 evidence modification is a common method for uncertain data preprocessing.^11,13,81,82 In this article, the evidence modification is based on the uncertainty management methodology. Each BOE will be modified with a weight based on the uncertainty management methods. In other words, the BBA of BOE is modified with a comprehensive weight based on the improved belief entropy, evidence sufficiency, and evidence importance.

For a proposition $A$ in the $i$ th BOE, the corresponding mass function (BBA) after evidence modification is defined as follows

m (A) = \sum_{i = 1}^{n} w (m_{i}) m_{i} (A)

(18)

where $u (m_{i})$ represents the uncertainty measure of the $i$ th BOE and $m_{i} (A)$ is the BBA of the proposition $A$ in the $i$ th BOE.

Step 4: evidence fusion

After evidence modification, the modified BBAs now can be combined with Dempster’s rule of combination.

For a proposition $A$ , the fused mass value can be calculated as follows

m (A) = ((m \oplus m) \oplus m \dots \oplus m) (A)

(19)

Step 5: application based on fusion results

The fused results can be applied for specific purpose in real applications, such as decision-making and target identification. Usually, the final result can be got accompany with a belief in terms of percentage; the higher the value, the more helpful it will be for real applications.

Application in sensor data fusion

In this section, the proposed sensor data fusion methodology is applied to the case study on sensor data fusion in the work by Fan and Zuo.⁵⁹

Problem description

Recall the example in the work by Fan and Zuo.⁵⁹ Three fault types are denoted as $F_{1}$ , $F_{2}$ , and $F_{3}$ , the fault hypothesis set is $Θ = {F_{1}, F_{2}, F_{3}}$ , three sensors report the diagnosis results independently, the diagnosis results are modeled as BOEs, denoted as $E_{1}$ , $E_{2}$ , and $E_{3}$ , and the BBAs of the diagnosis results are shown in Table 2. Intuitively, $F_{1}$ is the fault type because both the first BOE $E_{1}$ and the third BOE $E_{3}$ have a big belief (no less than 60%) on $F_{1}$ , while the second BOE $E_{2}$ may come from an abnormal sensor in comparison with the other two BOEs. This is a challenge for combination rule, such as the conventional Dempster’s rule of combination.

Table 2.

BBAs in the fault diagnosis problem.⁵⁹

Sensor report	${F_{1}}$	${F_{2}}$	${F_{2}, F_{3}}$	$Θ$
$E_{1} : m_{1} (\cdot)$	0.60	0.10	0.10	0.20
$E_{2} : m_{2} (\cdot)$	0.05	0.80	0.05	0.10
$E_{3} : m_{3} (\cdot)$	0.70	0.10	0.10	0.10

BBAs: basic belief assignments.

Based on the sensor reports in Table 2, which one is the fault that happens now, $F_{1}$ , $F_{2}$ , or $F_{3}$ . With Dempster’s rule of combination in equation (5), the combination results of sensor reports are shown in Table 3. It is hard to judge which fault has been occurred because the combination results obtained by the conventional Dempster’s rule of combination are very close.

Table 3.

Fused results with only Dempster’s rule of combination.

	${F_{1}}$	${F_{2}}$	${F_{2}, F_{3}}$	$Θ$
Fused results	0.4519	0.5048	0.0336	0.0096

Sensor data fusion with the proposed method

Recall the proposed method in section “A new uncertainty measure–based sensor data fusion approach” to handle the problem in section “Problem description.”

Step 1: evidence modeling

Adopted from Fan and Zuo,⁵⁹ the sensor reports in the problem are modeled as three BOEs, and the BBAs are shown in Table 2.

Step 2: uncertainty management

In the proposed method, the uncertainty management of evidence is modeled as the objective uncertainty and the subjective uncertainty. The subjective uncertainty $u_{s}$ , defined in equation (14), is calculated as the product of the sufficiency index $μ (m)$ and the importance index $v (m)$ . In this article, these two parameters are adopted from Fan and Zuo,⁵⁹ and the corresponding value of each BOE is shown in Table 4. Thus, the subjective uncertainty $u_{s}$ , according to equation (14), is calculated as follows

\begin{matrix} u_{s} (m_{1}) = μ (m_{1}) \times v (m_{1}) = 1.00 \times 1.00 = 1.00 \\ u_{s} (m_{2}) = μ (m_{2}) \times v (m_{2}) = 0.60 \times 0.34 = 0.2040 \\ u_{s} (m_{3}) = μ (m_{3}) \times v (m_{3}) = 1.00 \times 1.00 = 1.00 \end{matrix}

(20)

Table 4.

Parameters in the application adopted from Fan and Zuo.⁵⁹

BOE	$E_{1}$	$E_{2}$	$E_{3}$
Sufficiency index $μ (m)$	1.00	0.60	1.00
Importance index $v (m)$	1.00	0.34	1.00
Subjective uncertainty $u_{s}$	1.00	0.2040	1.00

BOE: body of evidence.

Before calculating the objective uncertain degree of the BOEs, the improved belief entropy of each BOE, according to equation (9), is calculated as follows

\begin{matrix} E_{Id} (m_{1}) = - \sum_{A \subseteq X} m_{1} (A) \log_{2} (\frac{m_{1} (A)}{2^{| A |} - 1} e^{\frac{| A | - 1}{| X |}}) = 2.0505 \\ E_{Id} (m_{2}) = - \sum_{A \subseteq X} m_{2} (A) \log_{2} (\frac{m_{2} (A)}{2^{| A |} - 1} e^{\frac{| A | - 1}{| X |}}) = 1.2617 \\ E_{Id} (m_{3}) = - \sum_{A \subseteq X} m_{3} (A) \log_{2} (\frac{m_{3} (A)}{2^{| A |} - 1} e^{\frac{| A | - 1}{| X |}}) = 1.6517 \end{matrix}

(21)

It is clear that $E_{Id} (m_{1})$ is the maximum of all, thus

max (E_{Id} (m)) = E_{Id} (m_{1}) = 2.0505

(22)

The objective uncertainty, according to equation (15) accompany with equation (22), is calculated as follows

\begin{matrix} u_{o} (m_{1}) = \frac{E_{Id} (m_{1})}{max (E_{Id} (m))} = 1.0000 \\ u_{o} (m_{2}) = \frac{E_{Id} (m_{2})}{max (E_{Id} (m))} = 0.6153 \\ u_{o} (m_{3}) = \frac{E_{Id} (m_{3})}{max (E_{Id} (m))} = 0.8055 \end{matrix}

(23)

where $E_{Id} (m_{i})$ is the improved belief entropy of the $i$ th BOE.

With equation (16), accompanying with equations (20) and (23), the total uncertainty measure of each evidence (BOE) is calculated as follows

\begin{matrix} u (m_{1}) = u_{s} (m_{1}) \times u_{o} (m_{1}) = 1.00 \\ u (m_{2}) = u_{s} (m_{2}) \times u_{o} (m_{2}) = 0.1255 \\ u (m_{3}) = u_{s} (m_{3}) \times u_{o} (m_{3}) = 0.8055 \end{matrix}

(24)

After uncertainty management, with equation (17), the weight for evidence modification of each BOE is calculated as follows

\begin{matrix} w (m_{1}) = \frac{u (m_{1})}{\sum_{i = 1}^{3} u (m_{i})} = 0.5179 \\ w (m_{2}) = \frac{u (m_{2})}{\sum_{i = 1}^{3} u (m_{i})} = 0.0650 \\ w (m_{3}) = \frac{u (m_{3})}{\sum_{i = 1}^{3} u (m_{i})} = 0.4171 \end{matrix}

(25)

Step 3: evidence modification

Evidence modification is based on uncertain management in the previous step; with equations (18) and (25), the evidence in Table 2 is modified as follows

\begin{matrix} \begin{matrix} m ({F_{1}}) = w (m_{1}) \times 0.6 + w (m_{2}) \times 0.05 \\ + w (m_{3}) \times 0.7 = 0.6060 \\ m ({F_{2}}) = w (m_{1}) \times 0.1 + w (m_{2}) \times 0.8 \\ + w (m_{3}) \times 0.1 = 0.1455 \\ m ({F_{2}, F_{3}}) = w (m_{1}) \times 0.1 + w (m_{2}) \times 0.05 \\ + w (m_{3}) \times 0.1 = 0.0967 \\ m (Θ) = w (m_{1}) \times 0.2 + w (m_{2}) \times 0.1 + w (m_{3}) \times 0.1 \\ = 0.1518 \end{matrix} \end{matrix}

(26)

Step 4: evidence fusion

After evidence modification, the modified BBAs now can be combined with equation (19). Since there are three BOEs in total, the modified evidence will be fused with Dempster’s rule of combination by two times, and the combination result is shown as follows

\begin{matrix} m ({F_{1}}) = ((m \oplus m) \oplus m) ({F_{1}}) = 0.8759 \\ m ({F_{2}}) = ((m \oplus m) \oplus m) ({F_{2}}) = 0.0930 \\ m ({F_{2}, F_{3}}) = ((m \oplus m) \oplus m) ({F_{2}, F_{3}}) = 0.0240 \\ m (Θ) = ((m \oplus m) \oplus m) ({Θ}) = 0.0071 \end{matrix}

(27)

The fusion result is presented in Table 5.

Table 5.

Fusion result with different methods.

	${F_{1}}$	${F_{2}}$	${F_{2}, F_{3}}$	$Θ$
Dempster’s rule of combination	0.4519	0.5048	0.0336	0.0096
Tang et al.’s method¹²	0.6522	0.3160	0.0242	0.0076
Fan and Zuo’s method⁵⁹	0.8119	0.1096	0.0526	0.0259
The proposed method	0.8759	0.0930	0.0240	0.0071

Step 5: application based on fusion results

Based on the fused result, it can be concluded that $F_{1}$ is the most possible fault type and the belief on this judgment is 87.59%. Thus, the equipment maintenance action should pay more attention to the components related to fault type $F_{1}$ .

The case study demonstrates the effectiveness of the proposed uncertainty management approach.

Discussion

The fused results with the proposed uncertainty measure are compared with the method in the works by Tang et al.¹² and Fan and Zuo,⁵⁹ and the comparison result is shown in Table 5.

It can be concluded from Table 5 that the proposed method has the most distinguishable fusion result on sensor reports. The highest belief degree on fault type $F_{1}$ is 87.59%, which is higher than the method with Fan et al.’s method with 6.40%. While the fusion results of fault types $F_{1}$ and $F_{2}$ with the conventional Dempster’s rule of combination are close to each other, it is hard to judge which fault has been occurred. Compared with the fusion result based on Tang et al.’s¹² method (the experimental process of this method is shown in Appendix 1), which has a limited belief of 65.22% on the fault type $F_{1}$ , the proposed is far more effective. This is because that the uncertainty management method of the proposed method is based on the improved belief entropy, while the method in the work by Tang et al.¹² is based on the weighted belief entropy, which is a very simple weight factor. Some properties of the improved belief entropy in the work by Zhou et al.⁷⁶ are not available to the weighted belief entropy, such as its compatibility with Shannon entropy. In addition, compared with the method in the work by Tang et al.,¹² the proposed method considers more available information by introducing the importance index and the sufficiency index of evidence into the uncertainty management model.

Three reasons contribute to the effectiveness of the proposed uncertainty management–based sensor data fusion approach. First, the sensor data are preprocessed properly before applying the combination rules based on the subjective uncertainty management method and the proposed objective uncertainty management method. This is very important in sensor data fusion especially when there is conflict evidence. Second, the weights of BOEs are calculated based on a new belief entropy, which can address more uncertain information in BOE in comparison with the old belief entropy. Finally, the final fused rule is based on Dempster’s rule of combination. The merits of Dempster’s rule of combination, such as satisfying the commutativity and associativity, contribute to the rationality of the proposed method.

Conclusion

In this article, an improved belief entropy–based uncertainty management approach is proposed for sensor data fusion. The uncertain degree of sensor report modeled as BOE is measured based on an improved belief entropy. In addition, the importance index and the sufficiency index of evidence are also taken into consideration in the proposed method. After evidence modification with the proposed uncertainty measure method, the weighted BBAs are fused by Dempster’s rule of combination. Finally, decision-making in real applications is based on the conflict data fusion results. The rationality and merit of the proposed method are verified with a case study of fault diagnosis. The on going work of this article is applying the proposed sensor data fusion method to solve more industrial problems in real applications.

Footnotes

Appendix 1 Acknowledgements

The authors greatly appreciate the editor’s encouragement and the three anonymous reviewers’ constructive comments to improve this paper.

Academic Editor: Florentino Fdez-Riverola

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was partially sponsored by the National Natural Science Foundation of China (Grant No. 61671384), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2016JM6018), the Aviation Science Foundation (Grant No. 20165553036), the Fund of Shanghai Aerospace Science and Technology (Grant No. SAST2016083), and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant No. CX201705).

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An improved belief entropy–based uncertainty management approach for sensor data fusion

Abstract

Keywords

Introduction

Preliminaries

Dempster–Shafer evidence theory

Deng entropy

The improved belief entropy

Example 1

A new uncertainty measure–based sensor data fusion approach

Step 1: evidence modeling

Step 2: uncertainty management

Step 3: evidence modification

Step 4: evidence fusion

Step 5: application based on fusion results

Application in sensor data fusion

Problem description

Sensor data fusion with the proposed method

Step 1: evidence modeling

Step 2: uncertainty management

Step 3: evidence modification

Step 4: evidence fusion

Step 5: application based on fusion results

Discussion

Conclusion

Footnotes

Appendix 1

Acknowledgements

Declaration of conflicting interests

Funding

References