Elsevier

Information Sciences

Volume 314, 1 September 2015, Pages 184-199
Information Sciences

An information fusion approach by combining multigranulation rough sets and evidence theory

https://doi.org/10.1016/j.ins.2015.03.051Get rights and content

Abstract

Multigranulation rough set (MGRS) theory provides two kinds of qualitative combination rules that are generated by optimistic and pessimistic multigranulation fusion functions. They are used to aggregate multiple granular structures from a set theoretic standpoint. However, the two combination rules seem to lack robustness because one is too relaxed and the other too restrictive to solve some practical problems. Dempster’s combination rule in the evidence theory has been employed to aggregate information coming from multiple sources. However, it fails to deal with conflict evidence. To overcome these limitations, we focus on the combination of granular structures with both reliability and conflict from multiple sources, which has been a challenging task in the field of granular computing. We first address the connection between multigranulation rough set theory and the evidence theory. Then, a two-grade fusion approach involved in the evidence theory and multigranulation rough set theory is proposed, which is based on a well-defined distance function among granulation structures. Finally, an illustrative example is given to show the effectiveness of the proposed fusion method. The results of this study will be useful for pooling the uncertain data from different sources and significant for establishing a new direction of granular computing.

Introduction

In the information age, complex data is often represented by a multi-source information system [7] in which data come from different sources. How to fuse such data has become a challenging task in the community of granular computing (GrC) [60]. Information granulation is one of three basic issues: information granulation, organization, and causation in granular computing. Information granulation involves decomposition of whole data into parts called granules. Then, these granules are organized into a granular structure (or a granular space). In granular computing, the granules induced by an equivalence relation (or a tolerance relation) form a set of equivalence classes (or tolerance classes), in which each equivalence class (or tolerance class) can be regarded as a Pawlak information granule (or a tolerance information granule).

A multi-source information system is used to represent information coming from multiple sources. Single-source information system is a special multi-source information system. According to the granulation approach, the objects in a multi-source information system can be granulated into multiple granular structures induced by a family of binary relations, or a family of attribute sets. In each information subsystem, the objects are organized into a granular structure by an attribute set. It is natural to put a fundamental issue on how to combine multiple granular structures from a multi-source information system. In this paper, we call this kind of information fusion as granulation fusion.

Rough set theory, proposed by Pawlak [17], [18] in 1982, has been proved to be an efficient tool for uncertainty management and uncertainty reasoning. This theory is emerging as a powerful methodology in the field of artificial intelligence such as pattern recognition, machine learning and automated knowledge acquisition. The basic structure of the rough set theory is a known knowledge base (or an approximate space) consisting of a universe of discourse and an indiscernible relation imposed on it. Based on the known knowledge base, the primitive notion of the lower and upper approximate operators can be induced. The lower and upper approximations of a target concept characterize the non-numeric aspect expressed by the known knowledge base. In the view of granular computing (proposed by Zadeh [60]), Pawlak rough set model and its extensions are based on a single relation (such as an equivalence relation, a tolerance relation or a reflexive relation) on the universe, which are called single granulation rough sets [19], [20], [21], [23]. However, if data come from different sources, the data analysis mechanism of the classical rough set theory is not desirable even not efficient. In this circumstance, one often needs to describe concurrently a target concept through multiple binary relations according to a user’s requirements or targets of problem solving, which motivates us to consider how to fuse such data from different sources.

Information fusion is a typical problem that involves the integration of multi-source information in signal processing, image processing, knowledge representation and inference, which has been the objective of many researches over the last few years. Up to now, a variety of qualitative (non-numeric) and quantitative (numeric) information fusion methods [23], [38], [51], [41] have been developed over the years [1], [30], [50], [55], [9]. It is worth pointing out that Qian et al. [22], [23], [24], [25] introduced the multigranulation rough set theory (MGRS) which employed conjunctive/disjunctive operators of multiple binary relations to integrate multiple granular structures induced by a family of binary relations. Furthermore, Khan and Banerjee [7], [8] proposed a weak lower (or strong upper) approximation of a target concept in the framework of a multi-source approximation space. In fact, the essence of the so-called optimistic lower (or upper) approximation defined in [22] is the same as the weak lower (or strong upper) approximation of a target concept proposed in [7]. Similarly, the essence of the so-called pessimistic lower (or upper) approximation defined in [25] is the same as the strong lower (or weak upper) approximation of a target concept proposed in [7]. The former focuses on multiple granulations and the latter focuses on multiple approximation spaces. Since multigranulation rough set model inception, its theoretical framework has been largely enriched, and many extended multigranulation rough set models, as well as relative properties and applications have been also studied extensively [13], [45], [10], [11], [14], [15], [16], [26], [29], [32], [33], [34], [49], [56], [57], [58], [59], [39], [40], [43], [44].

In the view of information fusion, MGRS theory can be regarded as a qualitative fusion strategy through the optimistic and pessimistic fusing paradigms. The optimistic fusion paradigm expresses the idea that in multiple independent granular structures, one needs only at least one granular structure to satisfy with the inclusion condition between an equivalence class and a target concept. Whereas the pessimistic version needs all granular structures to satisfy with the inclusion condition. However, the former seems too relaxed for data analysis and leads to generating a loose uncertain interval and the latter seems too restrictive and generates a tight uncertain interval. Therefore, both of them seem not enough precise to measure the uncertainty in multi-source environment. That is, these two fusing methods are two extreme cases which limit the application scope of MGRS. In addition, previous related work addressing qualitative combination rule deserves to be mentioned here. Yao and Wong [52], [42] proposed qualitative combination rule which requires the definition of a binary relation expressing the preference of one proposition or source, over another. However, the qualitative methods are still not desirable in engineering analysis, so one expects to propose the quantitative approach to deal with quantitative data.

Dempster–Shafer (DS) theory (also known as evidence theory or Dempster–Shafer theory of evidence) [2], [31], a general extension of Bayesian theory, provides a simple method for combining the evidence carried by a number of different sources. This method is called orthogonal sum or Dempster’s combination rule. In Dempster–Shafer theory, inference is made by aggregating independent evidence from different sources via the Dempster’s combination rule. Unfortunately, the unexpected and rather counterintuitive results of Dempster’s combination rule under some situations, as highlighted by Zadeh [61], limit the application of Dempster’s combination rule in intelligent fusion process. To overcome the disadvantage, its alternatives have been developed. Of all the alternatives, there are three distinguished improved combination rules, such as Smets’s unnormalized combination rule (known as the conjunctive combination rule) [36], [37], Yager’s combination rule [54] and Dubois and Prade’s disjunctive combination rule [3]. However, it is pointed out that all combination rules based on DS theory have a common characteristic. That is, they all require prior information to define a basic probability assignment (bpa).

From the above discussions, we find that both multigranulation fusion rules and Dempster’s combination rule have their respective limitations. For this reason, in this paper we focus on a fundamental issue on how to combine multiple granulations from a multi-source information system. To address this issue, we first examine the connection between multigranulation rough set theory and Dempster–Shafer’s theory. One may capture the prior information as the mass function according to the relationship between the single granulation Pawlak rough set and the evidence theory [35], [53], [45], [46], [47], [48]. Then, we propose a two-grade quantitative fusion approach integrating Dempster–Shafer theory and multigranulation rough set theory to deal with uncertainty of a multi-source information system. This new approach is based on a new distance between two granular structures. Finally, an illustrative example is given to show the effectiveness of the proposed fusion method.

This paper is organized as follows. Section 2 reviews some basic concepts of a multi-source information system, Dempster–Shafer theory and MGRS. Section 3 discusses the connection between MGRS and the evidence theory. In Section 4 we propose a new fusion function based on the evidence distance to merge multi-source uncertain information from a multi-source information system. In Section 5, an example is subsequently employed to demonstrate the validity and effectiveness of the integrated information fusion approach. Section 6 concludes the paper with a summary and direction for future.

Section snippets

Preliminaries

In this section, we review some basic concepts of a multi-source information system, the Dempster–Shafer theory and MGRS.

The connection between Dempster–Shafer theory and MGRS theory

In the view of Dempster–Shafer theory, we suppose each granulation structure induced by an attribute set is regarded as a body of evidence in multigranulation rough sets. Therefore, the aim of fusing multiple uncertain information is equivalent to combining granulation structures from different sources. According to the relationship between Dempster–Shafer theory and Pawlak rough set theory, Yao [51] proposed a basic function assignment, i.e., m(G(xi))=|G(xi)||U|. In what follows, we denote the

A two-grade fusion approach by combining Dempster–Shafter theory and MGRS theory

In order to differentiate conflict and reliable evidence, we first introduce the granulation distance for characterizing the difference among granular structures on the multigranulation space and then use the distance to cluster the conflict and reliable evidence.

In what follows, we give a definition of the distance among granular structures based on Liang’s distance [12].

Definition 4.1

Let F=(U,K1,K2,,Kq) be a multigranulation space. Suppose Ki,KjF are two granular structures, where Ki={Gi(xl)|xlU} and Kj=

Example

In what follows, we employ an example to illustrate the effectiveness of the proposed combination rules.

Example 5.1

Let MS={ISi|ISi=(U,ATi,{(Va)aATi},fi,{d})} be a multi-source decision information system, where U={e1,e2,e3,e4,e5,e6} is a universe of six objects which are here regarded as patients. Suppose there are four hospitals (Hi,i={1,2,3,4}) providing us information regarding the attributes A,B,C of the objects. These attributes represent the patients’ physical examination indicators. D is the

Conclusion

In this paper, we proposed the concept of granulation fusion which is one of important issues in the field of granular computing. Multigranulation rough set theory has provided a qualitative fusing method with no demand of prior information. The existing multigranulation fusion functions, optimistic and pessimistic multigranulation fusion functions, are too relaxed or restrictive for data analysis. Dempster’rule of combination and its improved rules have been employed as a major method for

Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their constructive and valuable comments. This work is supported by grants from National Natural Science Foundation of China under Grant (Nos. 71031006, 61322211, 61379021, 11061004, 61303131, 61432011), National Key Basic Research and Development Program of China (973) (No. 2013CB329404), Doctoral Program of Higher Education (No. 20121401110013), Innovative Talents of Higher Learning Institutions of Shanxi, China (No.

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