Formal context reduction in deriving concept hierarchies from corpora using adaptive evolutionary clustering algorithm star

Formal concept analysis (FCA) is an important theory on the study of hierarchical structures caused by a binary relation between objects and attributes. And it is well known that a soft set always induces a binary relationship between objects and attributes for a formal context. Based on these facts, we study a new system called a soft context, which is induced using a soft set instead of a binary relation in the formal context, and investigate its theoretical foundations. First, we introduce the notions of soft context and soft concept for objects and study some basic properties. In particular, for the purpose of studying soft concepts, we introduce and study the notions of independent and dependent soft concepts. And we obtain the important things: (1) the set of all soft concepts can be decomposed into the set of independent soft concepts and the set of dependent soft concepts; (2) every dependent soft concept is generated by some independent soft concepts. Second, we obtain important properties about the relationship between formal context and soft contexts as the following: (1) each formal context always induces an associated soft context; (2) for a given formal context, there exists a bijection between the set of all formal concepts and the set of all soft concepts of the associated soft context induced by the given formal context. And we apply the above-mentioned properties to get the set of all formal concepts of a given formal context and explain that this method is more effective than the traditional way of computing all formal concepts in formal contexts using the example. Finally, we introduce the notion of soft concept lattice. Also, for a given formal context, we show that the formal concept lattice is order-isomorphic to the soft concept lattice of the associated soft context.

Constructing three-way concept lattices based on apposition and subposition of formal contexts

The theories of concept lattice expansion and recovery were presented in this paper. Firstly, a formal context can be reduced from the viewpoint of objects or attributes by inclusion reduction, and its corresponding concept lattice can be constructed. And then, the concept lattice can be expanded dynamically according to the need, thus, the concept lattice of original formal context can be recovered eventually. Finally, the paper shows that the concept lattice of the original formal context and that of the reduced formal context are monomorphic.

Adjunctive three-way concepts from positive and negative concepts in lattice-valued formal contexts

We investigate whether semantic information related to object categories can be obtained from human fMRI BOLD responses with Formal Concept Analysis (FCA). While the BOLD response provides only an indirect measure of neural activity on a relatively coarse spatio-temporal scale, it has the advantage that it can be recorded from humans, who can be questioned about their perceptions during the experiment, thereby obviating the need of interpreting animal behavioral responses. Furthermore, the BOLD signal can be recorded from the whole brain simultaneously. In our experiment, a single human subject was scanned while viewing 72 gray-scale pictures of animate and inanimate objects in a target detection task. These pictures comprise the formal objects for FCA. We computed formal attributes by learning a hierarchical Bayesian classifier, which maps BOLD responses onto binary features, and these features onto object labels. The connectivity matrix between the binary features and the object labels can then serve as the formal context. In line with previous reports, FCA revealed a clear dissociation between animate and inanimate objects with the inanimate category also including plants. Furthermore, we found that the inanimate category was subdivided between plants and non-plants when we increased the number of attributes extracted from the BOLD response. FCA also allows for the display of organizational differences between high-level and low-level visual processing areas. We show that subjective familiarity and similarity ratings are strongly correlated with the attribute structure computed from the BOLD signal.

In recent times Formal Concept Analysis (FCA), in which the data is represented as a formal context, has gained popularity for Association Rules Mining (ARM). Application of ARM in health care datasets is challenging and a highly rewarding problem. However, datasets in the medical domain are of high dimension. As the dimensionality of dataset increases, size of the formal context as well as complexity of FCA based ARM also increases. To handle the problem of high dimensionality and mine the associations, we propose to apply Singular Value Decomposition (SVD) on the dataset to reduce the dimensionality and apply FCA on the reduced dataset for ARM. To demonstrate the proposed method, experiments are conducted on Tuberculosis (TB) and Hypertension (HP) datasets. Results indicate that with fewer concepts, SVD based FCA has achieved the performance of FCA on TB data and performed better than FCA on HP data.

Formal contexts is a common framework for rough set theory and formal concept analysis, and some rough set models in formal contexts have been proposed. In this paper, based on the theory of abstract approximation spaces presented by Cattaneo [1], a Brouwer orthocomplementation on the set of objects of a formal context is presented, as a result, a pair of new lower and upper rough approximation operators is introduced. Comparison between the new approximation operators and the existing approximation operators is made, and two necessary and sufficient conditions about equivalence of the operators are obtained. Relationships and algebraic structures among the definable subsets of these approximation operators are investigated.

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.

With the expansion of the research field, the research object of some original seemingly unrelated properties have been studied together. At this time, the number of attribute in formal context has changed. For the increased attributes, we need to construct a new concept lattice. The existing incremental building algorithms of concept lattice need the original formal context as the basis, with single attribute or a set of attribute of the object to rebuild the concept lattice. They can't effectively utilize these existing concept lattice that have not relation in attributes. Here, the paper presents one new algorithm for incorporating concept lattice based on the existed concept lattices. We can directly build the together lattice from bottom to top by direct product operation on the existed concept lattices and the mapping relation between the direct product lattice of two existed concept lattices and the together lattice. Formal contexts that attribute sets have no intersection are fit for this algorithm.

The method of concept lattice building is one of the most important contents in the formal concept analysis. Concept lattice is corresponding with the formal context, so the distributed treatment of concept lattice certainly relates to some operations such as the decomposition and combination of context. This paper defines the horizontal and vertical combination operations in formal contexts and discusses three relations between same-field formal contexts, then especially present a novel similarity method to convert the inconsistent contexts into independent or consistent contexts. This paper also defines some basic operations among concepts and proves that the concept lattice of sub-contexts vertically combined is isomorphic to the union of sub-lattices of these sub-contexts in same-field and consistent contexts, which establishes the theoretical foundation for the distributed construction of concept lattices.

Formal Concept Analysis (FCA) can be used to analyze data given in the form of a formal context. In particular, FCA provides efficient algorithms for computing a minimal basis of the implications holding in the context. In this paper, we extend classical FCA by considering data that are represented by relational structures rather than formal contexts, and by replacing atomic attributes by complex formulae defined in some logic. After generalizing some of the FCA theory to this more general form of contexts, we instantiate the general framework with attributes defined in the Description Logic (DL) EL, and with relational structures over a signature of unary and binary predicates, i.e., models for EL. In this setting, an implication corresponds to a so-called general concept inclusion axiom (GCI) in EL. The main technical result of this report is that, in EL, for any finite model there is a finite set of implications (GCIs) holding in this model from which all implications (GCIs) holding in the model follow.

Knowledge discovery process from databases has gained importance recently. Finding and using the valuable and meaningful data which is hidden in large databases can have strategic importance for the organizations to gain competitive advantage. In this context, data mining methods are used to analyze customer purchasing data which is called “market basket analysis”. This analysis provides insight into the combination of products within a customer’s ‘basket’. In this study, a market basket analysis is conducted to identify a customer purchasing behaviour with Formal Concept Analysis (FCA). The FCA technique is one of the data mining methods using formal contexts and concept lattices. So, this method provides to create association rules based on lattices reflecting the relationships among the attributes in a database.

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.

As an important tool for data analysis and knowledge processing, formal concept analysis (FCA) has been applied to many fields. In this paper, we introduce a new method to find all formal concepts based on formal contexts. The amount of intents calculation is reduced by the method. And the corresponding algorithm of our approach is proposed. The main theorems and the corresponding algorithm are examined by examples, respectively. At last, several real-life databases are analyzed to demonstrate the application of the proposed approach. Experimental results show that the proposed approach is simple and effective.

Knowledge reduction in formal contexts using non-negative matrix factorization