Concept Granular Computing Systems

The study of concept lattices, property oriented concept lattices and object oriented concept lattices provides complementary conceptual structures, which can be used to search, analyze and extract information from data sets. This paper is devoted to the study of rule acquisition and attribute reduction of formal decision context. Based on object oriented concepts and property oriented concepts, the notions of object oriented decision rules and property oriented decision rules are proposed. By using some equivalence relations on the set of extents of the related conditional concept lattices and decision concept lattices, the rule acquisition methods are presented. The attribute reduction approaches for formal decision context to preserve the object oriented decision rules and property oriented decision rules are put forward by using discernibility attributes.

In traditional formal concept analysis, the attributes in the formal context are considered fixed. However, in the real world data set, attributes always have different levels of granularity, correspondingly, the derived concept lattice may reveal different information and patterns. Therefore, the capability to change the level of granularity of an attribute in formal concept analysis to capture relevant patterns in data is a natural requirement. In this paper, a theoretical study has been undertaken in multi-scale formal contexts, where attributes with different levels of granularity possess different attribute values. Two types of formal concepts, i.e., object-oriented and property-oriented multi-scale concepts, are introduced and studied in detail. The collection of object-oriented concept lattices and property-oriented concept lattices can be obtained at different granularity levels of attributes. It has been shown that the set of extents in the derived concept lattices increases when we choose to use a finer level of granularity. Moreover, a corresponding bidirectional approach to concept construction(i.e., from coarser to finer and from finer to coarser, respectively) is exhibited, and some characterization theorems have been obtained.

Attribute reduction is one of the main issues in the study of concept lattice. This paper mainly deals with attribute reductions of an object-oriented concept lattice constructed on the basis of rough set. Attribute rank of object-oriented concept lattice is first defined, and relationships between attribute rank and object-oriented concepts are then discussed. Based on attribute rank, generating algorithm of object-oriented concepts is investigated. The object-oriented consistent set and object-oriented reduction of an object-oriented concept lattice are defined. Adjustment theorems of the object-oriented consistent set, and the necessary and sufficient conditions for a attribute subset to be an object-oriented consistent set of an object-oriented concept lattice are discussed. Then the object-oriented discernibility matrix of an object-oriented concept lattice is defined and its properties are also studied. Based on the object-oriented discernibility matrix, an approach to object-oriented reductions of an object-oriented concept lattice is proposed, and the attribute characteristics are also analyzed.

As an important tool for data analysis and knowledge processing, formal concept analysis has been applied to many fields. Currently, granular computing (GrC) provides a novel approach for data analysis. In this paper, pictorial diagrams of a formal context are first defined based on equivalence relations. And then, from the viewpoint of GrC, the transformation between equivalence classes and three kinds of concepts (formal concept, property oriented concept, and object oriented concept) in formal contexts are studied. Further, using the relationships between arrow relations proposed by Wille and Ganter and irreducible elements of a concept lattice, the new judgment theorems for join(meet)-irreducible elements of a concept lattice, join-irreducible elements of a property oriented concept lattice and an object oriented concept lattice are achieved.

This paper studies a new definition and an approach to attribute reduction in an object oriented concept lattice based on congruence relations. Firstly, dependence space based on the object oriented concept lattice is researched to obtain the relationship among object oriented concept lattices and the corresponding congruence relations. Then the notion of attribute reduct in this paper, resembling that in rough set theory, is defined to find minimal attribute subsets which can preserve all congruence classes determined by the attribute set. Finally, an approach of discernibility matrix is presented to calculate all attribute reducts. It is shown that attribute reducts can also keep all object oriented extents and their original hierarchy in the object oriented concept lattice.KeywordsObject oriented concept latticerough setattribute reductioncongruence relationformal context

Dual multi-adjoint concept lattices

Four kinds of basic decision rule sets are respectively produced from the formal concept lattice, the dual formal concept lattice, the object-oriented concept lattice and the attribute-oriented concept lattice in a formal context. Based on these sets, two types of decision inferences are established via an inclusion degree. The corresponding decisions by inferences are proved to be the lower and the upper approximated decisions. Thus total decision rules described as the lower and the upper approximated decision rules are obtained. And they are accordant and consistent with the basic decision rule sets.

Relations of attribute reduction between object and property oriented concept lattices

Attribute reduction is one basic issue in the analysis of information tables. In this paper, the approaches to attribute reduction in formal context based on the object oriented concept lattice are investigated. We first introduce the notions of context matrix and the operations of corresponding column vectors. Then present some judgment theorems for attribute reduction in formal contexts. Based on the judgment theorems, we propose an attribute reduction approach and show concrete reduction algorithm.

Rough set theory and formal concept analysis are two complementary mathematical tools for data analysis. In this paper, based on object oriented concept lattice, we first propose one kind of consistency of conditional concepts and decision concepts. Subsequently, notions of consistent set and reduct set of attributes are introduced, and some judgment theorems for consistent set are presented. Finally, discernibility matrix is employed to determine consistent sets and calculate reduct sets in consistent formal decision contexts.

Formal concept analysis of binary relations has proved to be useful in resolve of many problems of theoretical or interest. This paper deals with approach to knowledge reduction in formal context based on object oriented concept lattice and attribute oriented concept lattice. The concept of consistent set in formal context is first introduced and its equivalent definitions are given. The judgment theorem and discernibility matrix associated with formal context reducts are established, from which we can provide the approach to knowledge reduction in formal context.

Basic relationships between Rough Set Theory (RST) and Formal Concept Analysis (FCA) are discussed. Differences between the “partition oriented” RST and the “order oriented” FCA concerning the possibility of knowledge representation are investigated. The fundamental connection between RST and FCA is that the knowledge bases of RST and the scaled many-valued contexts of FCA are shown to be nearly equivalent.

Complex and noisy financial eco-system requires reliable models and proven techniques to predict the market movements and investor decisions. This study uses competent soft computing techniques: rough set theory (RST) and formal concept analysis (FCA) to study the investors' preferences, behavioural drivers and their actual behaviour in Gold-ETF (G-ETF) market. G-ETF, though a safe-haven and an alternate for reducing portfolio risks, inherits all complexities of financial markets. The employed RST helps in generating decision rules; and FCA to identify key factors affecting investment decision. This study is first of its kind, as integration of the foresaid techniques was not employed to study financial behaviour, earlier. The study has analysed 250 responses of G-ETF investors, in 12 listed G-ETFs, to conclude with a rich insight on the investment decisions discretised by different decision rules, strongly recommending the combined use of RST and FCA for data driven decisions.

The paper examines Formal Concept Analysis (FCA) and Rough Set Theory (RST) against the background of the theory of finite approximations of continuous topological spaces. We define the operators of FCA and RST by means of the specialisation order on elements of a topological space X which induces a finite approximation of X. On this basis we prove that FCA and RST together provide a semantics for tense logic S4.t. Moreover, the paper demonstrates that a topological space X cannot be distinguished from its finite approximation by means of the basic temporal language. It means that from the perspective of topology S4.t is a better account of approximate reasoning then unimodal logics, which have been typically employed.

Motivated by Rough Set Theory we describe an interval arithmetic on complete lattices. Lattice elements get approximated by approximations which are pairs consisting of a lower and an upper approximation. The approximations form a complete lattice again. We describe these lattices of approximations by formal contexts. Furthermore, we interpret the result for concept lattices as restricting the scope to a subcontext of interesting objects and attributes.