A new divergence measure for belief functions in D–S evidence theory for multisensor data fusion
Introduction
Multisensor data fusion is a technique used to combine information from different sources to generate a unified profile [14], [21], [28]. Multisensor data fusion has attracted much attention in the past two decades and has been extensively used in a variety of areas, such as fault diagnosis [9], [16] and reliability evaluation [33], [48]. One of the challenges in multisensor data fusion is how to model and handle information collected from different sensors that may be ambiguous, uncertain or even false due to the influence of the environment and the complexity of the target [2], [10]. Many well-known theories have been presented for addressing uncertain information of this kind, such as extended fuzzy sets [29], [34], evidence theory [18], [23], D numbers [26], evidential reasoning [47], R numbers [30], Z numbers [1], [20], entropy-based approaches [3], and quantum-based approaches [8]. These methods have been applied in various contexts, such as evaluation [49], feature fusion [4], emerging technology commercialization [39], the Cloud of Things paradigm [12], consensus reaching [22], medical diagnosis [44], and decision-making [36], [43].
In this paper, we principally focus on the study of uncertainty in multisensor data fusion on the basis of Dempster–Shafer (D–S) evidence theory [5], [31]. One of the advantages of D–S evidence theory is that it is able to handle uncertainty in a flexible and effective way without prior information [46]. Specifically, in D–S evidence theory, uncertainty is quantitatively expressed in terms of basic belief assignments (BBAs) leveraging singleton sets and/or multielement sets of objectives [25]. Furthermore, D–S evidence theory satisfies the associative law and the commutative law [35]. In addition, Dempster’s combination rule generates fault-tolerant results [37]. Because of the above advantages, D–S evidence theory has been generally applied in many fields [41]. However, an open issue remains in D–S evidence theory, namely, how to reconcile highly conflicting evidence, because Dempster’s combination rule may yield counterintuitive results [45]. So far, a substantial amount of work has been done to address this issue from two different perspectives, namely, the modification of Dempster’s combination rule and the preprocessing of the body of evidence [11], [32]. In this paper, we consider the second perspective, i.e., the preprocessing of the body of evidence to settle the problem of conflicting evidence. One of the major existing related works for this purpose is Murphy’s method [27], which is a simple averaging method for the body of evidence. Later, Deng et al. [7] improved on this approach by utilizing a weighted averaging method. A more recent improvement is Xiao’s method [42], in which the body of evidence is weighted on the basis of divergence. Careful study of the existing methods reveals that the method presented in [42] offers good performance in fusing multisensor data with a high recognition accuracy. However, this method accounts for conflicting evidence only at the level of belief functions, neglecting the relationships between subsets of the sets of belief functions.
In this paper, therefore, a reinforced belief divergence measure, called is proposed to measure the discrepancy between BBAs in D–S evidence theory. The proposed divergence is the first such measure to consider the correlations between both belief functions and subsets of the sets of belief functions, thus allowing it to provide a more effective solution for measuring the discrepancy between BBAs. Furthermore, the proposed divergence has certain benefits in terms of measurement. First, it has the properties of nonnegativeness and nondegeneracy. Second, it is symmetric. Third, it satisfies the triangle inequality. Based on the divergence, an algorithm for multisensor data fusion is newly designed. Through a comparative analysis based on numerical examples and experiments, it is illustrated that the proposed method is more feasible and reasonable than previous methods for measuring the divergence between BBAs. Finally, the proposed algorithm is effectively applied to a real-world classification fusion problem.
The primary contributions of this study are summarized as follows:
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The proposed divergence is the first such measure to consider the correlations between both belief functions and subsets of the sets of belief functions for evaluating the divergence in evidence theory.
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The divergence has benefits in terms of measurement due to its properties of nonnegativeness, nondegeneracy, symmetry and satisfaction of the triangle inequality, which allow it to provide a more convincing and intuitive solution for measuring the discrepancy between BBAs.
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On the basis of the divergence, a new multisensor data fusion algorithm is designed, which can be effectively applied in target recognition based on multiple sensors to achieve a higher recognition accuracy and stronger robustness.
The remaining contents of this work are organized as follows. Section 2 briefly introduces the preliminaries of this study, including D–S evidence theory and the concept of a belief divergence measure. In Section 3, a new divergence measure for use in evidence theory is designed, and a comparative analysis is conducted. In Section 4, a reinforced belief divergence measure is further derived, and subsequently, the performance of this measure and a comparative analysis are discussed. In Section 5, an algorithm for multisensor data fusion is newly proposed on the basis of the reinforced belief divergence measure. In Section 6, experiments on target recognition with multiple sensors are presented to analyze the performance of the proposed method. In Section 7, the proposed method is applied in a real-world study of classification fusion. Section 8 concludes the study.
Section snippets
Dempster–Shafer evidence theory
D–S evidence theory [5], [31], as a generalization of Bayes probability theory, is an effective means of dealing with uncertain information [6], [19]. D–S evidence theory has been applied in many fields, such as pattern classification [24] and decision-making [13], [17]. Definition 2.1 (Frame of discernment). Let be a set of mutually exclusive and collectively exhaustive events,which is defined as a frame of discernment. The power set of denoted by is defined as
A new divergence measure for use in evidence theory
In this section, the correlation between belief functions is first analyzed. Based on this analysis, a new divergence measure for belief functions is then devised. Subsequently, a comparative analysis is conducted.
Reinforced divergence measure for belief functions
In this section, a reinforced divergence measure for belief functions is devised based on the newly defined divergence. Definition 4.1 (Reinforced belief divergence measure). Let be the frame of discernment, which contains h mutually exclusive and collectively exhaustive events. Let m1 and m2 be two belief functions in . The reinforced belief divergence measure, denoted by between the belief functions m1 and m2 is defined aswhere is the belief divergence
A multisensor data fusion algorithm
In this section, a new multisensor data fusion algorithm is proposed based on the reinforced belief divergence measure. A flowchart of the proposed method is shown in Fig. 6.
Step 1: The reinforced belief divergence measure between each two pieces of evidence mi and mj is generated based on Eq. (14). Then, a reinforced belief divergence measure matrix is constructed as follows:
Step 2: The average reinforced belief
Experiments
This section reports experiments involving target recognition with multiple sensors conducted to analyze the performance of the proposed method. Experimental data based on [42] are utilized to compare the proposed method with existing related methods.
Application
In this section, the proposed method is applied in a real-world study of classification fusion.
The Iris data set used in this application was obtained from the UCI machine learning repository (http://archive.ics.uci.edu/ml/datasets/Iris). This data set contains three classes, i.e., Setosa, Versicolor and Virginica, each of which consists of 50 instances. Each instance has four attributes: sepal length, sepal width, petal length and petal width. In this application, to obtain the initial BBAs
Conclusion
In this paper, a new divergence measure for the belief functions considered in Dempster–Shafer evidence theory is proposed, and its effective application in multisensor data fusion is demonstrated. The main contribution of this study is that this is the first work to consider the relationships between multiple sets of hypotheses of belief functions when calculating a divergence measure to be used in evidence theory, rather than considering only the correlations between the belief functions
Declaration of Competing Interest
None.
Acknowledgments
The author greatly appreciates the reviewers’ suggestions and the editor’s encouragement. This research was supported by the Fundamental Research Funds for the Central Universities (No. XDJK2019C085) and the Chongqing Overseas Scholars Innovation Program (No. cx2018077).
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