A constructing approach to multi-granularity object-induced three-way concept lattices

Abstract

Three-way concept analysis is a research hotspot. Inspired by multi-scale information system, some scholars put forward and researched multi-granularity formal context. Most of the existing algorithms for constructing three-way concept lattice take care of the static formal contexts and can not deal with the multi-granularity formal context. To address this problem, this paper primarily focuses on a constructing approach to object-induced three-way concept (OE-concept) lattices for multi-granularity formal contexts. Firstly, transformation relationships between OE-concepts based on different granularities are researched. Secondly, the mutual transformation between OE-concept lattices under different granularities is proposed via these relationships, and then corresponding algorithms are proposed. Finally, several groups of datasets are selected from UCI for comparative experiments. Experimental results exhibit that our algorithms are more effective and advantageous than the latest construction algorithms.

Introduction

Formal concept analysis (FCA) was put forward by mathematician Wille in 1982 [1]. This theory provides a concept hierarchy structure based on the binary relation between objects and attributes of datasets. The point of departure for FCA is Boolean data [2], which is named as a formal context in FCA, indicating each object possesses what attributes. The formal concept consists of a set of objects and a set of attributes established by two set-theoretic operators [1]. Concept lattice model has been widely concerned since it was put forward, many scholars [3], [4], [5], [6], [7], [8], [9] extend the original model proposed by Wille. As an effective tool for knowledge characterization and processing, FCA has been widely used in knowledge engineering, machine learning, expert system, information retrieval, data mining, ect [10], [11], [12], [13], [14].

Three-way decision (3WD) is an effective decision-making method with its main idea of “dividing and acting” proposed by Yao in 2012 [15]. It adds a hesitant decision, i.e. delayed decision, on account of the accept-reject two-way decision. The basic opinion of 3WD derives from a three partitions of the universe, including three disjoint regions, i.e. positive, boundary, and negative regions. Rules of accept (reject, delay) decision are extracted from the positive (negative, boundary) region. 3WD has become a theory and method for the fields of granular computing and knowledge discovery together with rough set, fuzzy set, cloud computing and formal concept analysis [16], [17], [18], [19], [20], [21]. In fact, when information is not enough, the non-commitment is usually more in line with the needs of practical problems, therefore the 3WD theory has gradually become an important theory and method to deal with uncertain problems, which has been widely concerned, and has been successfully applied to information, engineering, medical treatment, management and other fields [22], [23], [24].

Previous studies on formal concept analysis were mainly based on positive operators. Missaoui et al. [25] considered both the positive and negative attributes of the formal context, and then performed computing implications. Ganter et al. [26] put forward partial formal contexts, and then researched implicational knowledge. 3WD not only considers positive and negative, but also introduces hesitation. Thus the combination of 3WD and formal concept analysis can lead to richer knowledge based on the formal context. Qi et al. [27] presented three-way concept analysis, and proposed two types of three-way concept lattice. After that, Qi and Mao et al. further studied connections among three-way concept and classical concept lattices [28], [29]. Lots of scholars conducted extended research on the three-way concept lattices [30], [31], [32]. Zhao et al. [33] investigated the connections among the current several three-way concept lattice models. Wei et al. [34] defined decision rules of formal decision contexts and proposed the rules acquisition method based on three-way concept lattices. Qian et al. [35] proposed a construction algorithm by two new types of formal contexts whose concept lattices and three-way concept lattice are isomorphic, and showed that their algorithm outperforms the algorithm of constructing the OE-concept lattice by its definition. Yang et al. [36] proposed a new construction algorithm via the concepts of a formal context and its complement context, and expressed that their method outperforms existing methods. Yao [37] systematically analyzed and summarized FCA based on 3WD perspectives. Qi et al. [38] proposed the notion of 3-valued formal contexts, defined 3-valued operators and constructed 3-valued concept lattices. The relationships between 3-valued lattices and existing approximation concept lattices are examined.

Lin [39] proposed granular computing theory in 1998, this idea was derived from fuzzy information granulation put forward by Zadeh [40]. Its early results mainly focused on granulation principles, granulation methods, and granulation normal form research [41], [42], [43], [44]. Pawlak [45] presented rough set theory in 1982, which is an effective mathematical tool for dealing with incomplete and imprecise information. The classic rough set theory is based on an equivalence relation (granularity). However, single-granular rough sets cannot be used to describe the multi-perspective idea in granular computing theory. Therefore Wu et al. [46] proposed a multi-scale information system from the perspective of granular computation which has different levels of granulations. In such a system, each object under each attribute is represented by different scales at different levels of granulations having a granular information transformation from a finer to a coarser labelled value. In recent years, inspired by multi-scale information system, some scholars have introduced multi-scale ideas into FCA and proposed multi-granularity (labelled) formal context. In fact, the idea of multi-scale in FCA has been reflected in the literature [47]. Constructing concept lattices is one of the most important problem of FCA. Therefore, the construction of concept lattices based on the multi-granularity formal context is a research hotspot. From the perspective of algorithm, the zoom algorithms for classical concept lattice and object (property) oriented concept lattice are given through a granularity tree proposed by Belohlavek and Shao et al. [47], [48], [49], respectively. Based on the perspective of the transformation relationship among concepts, Hu and Qin [50] proposed constructing methods for classical concept lattice under a multi-granularity formal context.

In real life, if the attribute granularity level is too fine, lots of redundant concepts may be generated, thus it takes a lot of time to extract useful information. On the other hand, if the attribute granularity level is too coarse, some useful knowledge may be ignored. Therefore, different application situations may require different granularity level. When the granularity level changes, it takes a lot of time to reconstruct the three-way concept lattice by the algorithms proposed in [36], because the concept lattice of the original formal context is ignored. Compared with the classical concept lattice construction, there are not researches about constructing object-induced three-way concept lattices based on a multi-granularity formal context. Therefore, this paper mainly researches a constructing approach to object-induced three-way multi-granularity concept lattices.

The structure of this article is as follows: FCA and three-way concept lattices are concisely reviewed in section 2. Section 3 investigates connections among object-induced three-way concepts under a multi-granularity formal context and the construction algorithms for OE-concept lattice based on a multi-granularity formal context. In section 4, we compare these algorithms with the recent construction algorithms [36] for the three-way concept lattice generation to show the superiority of these proposed algorithms. In section 5, the summary of this article and the future outlook are drawn.