Approximate concept construction with three-way decisions and attribute reduction in incomplete contexts
Introduction
The theory of formal concept analysis (FCA), proposed by Wille [14], [63], is a mathematical tool for conceptual data analysis and knowledge processing. It mainly concerns the lattice structure, called a concept lattice, constructed by the formal concepts embedded in a concept hierarchy. A formal concept, is a pair consisting of a set of objects (the extension), which includes all objects belonging to the concept, and a set of attributes (the intension), which includes all the attributes common to objects in the extension. In the last two decades, FCA has been successfully applied to information retrieval, knowledge discovery, data mining, machine learning, software engineering, etc. [2], [4], [6], [10], [13], [24], [25], [46], [47], [48], [56], [57].
As is well known, knowledge reduction is a key issue in rough set theory [45], and the main objective is to simplify the representation of knowledge so as to gain optimized decision rules. Recently a large number of reduction methods have been developed for different types of information systems and decision tables [8], [19], [22], [42], [51], [62], [67]. Similarly, knowledge reduction in FCA has also received much attention in the past decades [5], [9], [14], [16], [26]. Specially, much attention has been paid to attribute reduction of formal contexts in FCA. For example, Ganter and Wille [14] proposed to reduce the context by removing both the reducible attributes and objects. The concept lattice obtained from the reduced context is isomorphic to the one induced from the original context. Zhang et al. [77] presented an approach to attribute reduction by keeping all the extensions unchanged. Wang and Zhang [60], Li et al. [34] conducted attribute reduction based on irreducible elements respectively, which are equivalent to Zhang et al.’s method without generating all the concepts. Wu et al. [65] removed redundant attributes to keep information granule, i.e., all extensions of the object concepts, unchanged. Shao and Leung [53] revealed that granular reducts obtained by Wu et al.’s approach [65] are equivalent to dominance reducts. Some researchers focus on the issue of attribute reduction from other perspectives instead of classical FCA. For example, Liu et al. [39] conducted attribute reduction in rough set-based concept lattice [12], [68], Li and Wu [35] reduced attributes from the perspective of covering rough set, Mi et al. [41] gave the approaches to attribute reduction in concept lattices induced by axialities. Besides, some attribute reduction approaches in decision formal context [61] are proposed based on rule acquisition [29], [30], [33], [54].
There is an underlying assumption in classical concept lattice and the extended versions: the binary relation in the formal context is definite, and either the object x has the attribute a, or the opposite holds. In many real-world situations, however, it may happen that the relationship between some objects and some attributes is unavailable or lost, and there may be a third relation between the object and the attribute. In this case, the contexts are called incomplete contexts [3]. (And the classical formal contexts are called complete ones.)
Recently, there is a growing interest in the study of the incomplete contexts [3], [11], [18], [23], [31], [44]. However, most of the research focuses on attribute implication or attribute exploration, and few efforts have been devoted to concept construction in incomplete contexts. Although Li et al. [31] have given an approximate concept construction approach by representing each concept intension as a pair (definite common attributes, possible common attributes), they only take the positive information, i.e., the common (or possible common) attributes shared by the extension, into consideration. Nevertheless, in some cases, some negative information is of equal importance. For example, when a patient chooses some Over-The-Counter (OTC) medicines for a special disease, he must pay attention to the indications (the positive information) and the contraindications (the negative information) of the medicines. In this case, the objects are the OTC medicines, and the attributes are diseases. The indications, the contraindications, and other diseases that are not listed in the medicines’ instruction form a tripartition of the attribute set, which falls into the category of three-way decisions.
The theory of three-way decisions, proposed by Yao [73], is an extension of the commonly used acceptance-rejection binary model with an added third option: non-commitment [70], [71], [72], [73], [74]. The basic ideas in the theory of three-way decisions are based on a tripartition of a universal set. A tripartition consists of three pair-wise disjoint subsets, namely the positive region POS, the negative region NEG and the boundary region BND, of the universe set [73]. Corresponding to the three regions, one can construct rules for acceptance, rejection and non-commitment, respectively [70], [71]. Three-way decisions play a key role in everyday decision-making and have been widely used in many fields and disciplines [1], [17], [20], [21], [28], [32], [36], [37], [38], [40], [58], [66], [74], [75], [76], [78].
In fact, some researchers have introduced the theory of three-way decisions into FCA in complete contexts. For example, Ganter and Kuznetsov [15] pointed out that the version space is by definition the set of classifiers that match all positive examples and do not match any negative examples. Rodriguez-Jimenez et al. proposed the mixed formal concept lattice, in which both positive and negative attributes are contained, to extract mix rules [52]. Qi et al. proposed two three-way formal concept lattices, namely the OE-concept lattice and the AE-concept lattice [49].
Motivated by the works mentioned above, this paper focuses on approximate concept construction with three-way decisions and attribute reduction in incomplete contexts. Compared with other methods, for example [31], that deal with missing information, the advantage of using three-way decisions is that it deals with the missing information from two aspects simultaneously, the positive aspect and the negative aspect.
The rest of the paper is organized as follows. Section 2 briefly reviews some basic notions related to complete contexts and incomplete contexts. Section 3 proposes two models to construct three-way approximate concepts in incomplete contexts, and then the equivalence of the two models is proved. After that, Section 4 discusses attribute reduction and attribute characteristics in the three-way approximate concept lattices. Finally, Section 5 concludes this paper.
Section snippets
Preliminaries
In this section, we briefly review some basic notions related to complete contexts, and then introduce the notion of incomplete context.
Definition 1 A formal context consists of two sets U and A and a binary relation R between U and A. The elements of U are called objects, and the elements of A are called attributes. For an object x and an attribute a, (x, a) ∈ R indicates the object x has the attribute a, and indicates the opposite.[14]
A formal context can be represented by a two-dimensional
Approximate concept construction with three-way decisions in incomplete contexts
Just as mentioned in the introduction, the positive information (obtained by the positive operators) is as important as the negative information (obtained by the negative operators) to describe a concept. To achieve this goal, in this section, we propose two models based on three-way decisions to construct the approximate concept lattice of an incomplete context, one is based on three-way approximate operators, and the other is based on the tripartition of the ternary relation.
Attribute reduction and attribute characteristics in three-way approximate concept lattices
In this section, we mainly focus on the issues of attribute reduction and attribute characteristics in three-way approximate concept lattices. First, we recall some basic notions and existing approaches to the two issues related to the classical concept lattice.
Conclusions
In this paper, we proposed two models to construct approximate concept with three-way decisions in incomplete contexts. One is from the perspective of the three-way approximate operators, and the other is from perspective of the tripartition of the ternary relation of the incomplete context. In each model, the approximate concepts contains both the positive and the negative information simultaneously. It has been proved that the two models are equivalent to each other. After the approximate
Acknowledgment
The authors would like to thank the Editor and the anonymous reviewers for their valuable comments and constructive suggestions. This work is supported by the National Natural Science Foundation of China (Nos. 61272060 and61379114), and Key Natural Science Foundation of Chongqing (No. CSTC2013jjB40003), and the Social Science Foundation of the Chinese Education Commission (No. 15XJA630003).
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