Interval-Valued Fuzzy Formal Concept Analysis

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.

Semantic Web provides a knowledge-based environment that enables information to be shared and retrieved effectively. In this research, we propose the Scholarly Semantic Web for the sharing, reuse and management of scholarly information. To support the Scholarly Semantic Web, we need to construct ontology from data which is a tedious and difficult task. To generate ontology automatically, Formal Concept Analysis (FCA) is an effective technique that can formally abstract data as conceptual structures. To enable FCA to deal with uncertainty in data and interpret the concept hierarchy reasonably, we propose to incorporate fuzzy logic into FCA for automatic generation of ontology. The proposed new framework is known as Fuzzy Formal Concept Analysis (FFCA). In this paper, we will discuss the Scholarly Semantic Web, and the ontology generation process from the FFCA framework. In addition, the performance of the FFCA framework for ontology generation will also be evaluated and presented.

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fuzzy relational formal concept analysis (FRCA)</i> mines collections of fuzzy concept lattices from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fuzzy relational context families</i> , which are special datasets made of fuzzy formal contexts and fuzzy relations between objects of different types. Mainly, FRCA consists of the following procedures: first, an initial fuzzy relational context family is transformed into a collection of fuzzy formal contexts; second, a fuzzy concept lattice is generated from each fuzzy formal context by using one of the techniques existing in the literature. The principal tools to transform a fuzzy context family into a set of fuzzy formal contexts are the so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fuzzy scaling quantifiers</i> , which are particular fuzzy quantifiers based on the concept of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evaluative linguistic expression</i> . FRCA can be applied whenever information needs to be extracted from multirelational datasets including vagueness, and it can be viewed as an extension of both <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">relational concept analysis</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fuzzy formal concept analysis</i> . This article contributes to the development of fuzzy relational concept analysis by achieving the following goals. First of all, we present and study a new class of fuzzy quantifiers, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-scaling quantifiers</i> , to extract fuzzy concepts from fuzzy relational context families. Subsequently, we provide an algorithm to generate, given a t-scaling quantifier, a collection of fuzzy concept lattices from a special fuzzy relational context family, which is composed of a pair of fuzzy formal contexts and a fuzzy relation between their objects. After that, we introduce an ordered relation on the set of all t-scaling quantifiers, which allows us to discover a correspondence among fuzzy concept lattices deriving from different t-scaling quantifiers. Finally, we discuss how the results obtained for t-scaling quantifiers can be extended to the class of fuzzy scaling quantifies. Therefore, this analysis highlights the main differences between t-scaling and fuzzy quantifiers.

Similarity Reasoning in the presence of vague information is becoming fundamental in several research areas and, in particular, in the Semantic Web. Fuzzy Formal Concept Analysis (FFCA) is a generalization of Formal Concept Analysis (FCA) for modeling uncertainty information. Although FFCA has become very interesting for supporting different activities for the development of the Semantic Web, in the literature it is usually addressed at a technical level and intended for a restricted audience. This paper proposes a similarity measure for FFCA concepts. The key notions underlying the proposed approach are presented informally, in order to reach a broad audience of readers.

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.

Effective identification of interdependence information between decision variables is crucial for variable grouping in large-scale global optimization (LSGO). This paper introduces a novel approach called FCA-G (Formal Concept Analysis-Driven Grouping) to solve LSGO problems. FCA, an effective tool for data analysis, is employed in this approach. The primary contribution involves transforming decision variables into the formal context within FCA and utilizing the FCA methodology to solve LSGO problems based on a cooperative coevolution framework. Based on the formal context, a formal concept lattice is constructed, from which equivalent concepts are extracted. All variables within these concepts exhibit explicit interactions. This approach ensures a high degree of correlation among variables within subgroups and a low degree of correlation between subgroups, thereby enhancing cooperative coevolution. Experimental results indicate the significant potential of FCA-G in LSGO, as it outperforms state-of-the-art LSGO algorithms across the majority of LSGO test problems, including those with up to 1000 decision variables and a large-scale cloud task scheduling problem.

Rough-set-driven approach for attribute reduction in fuzzy formal concept analysis

It is beneficial to automate the process of deriving concept hierarchies from corpora since a manual construction of concept hierarchies is typically a time-consuming and resource-intensive process. As such, the overall process of learning concept hierarchies from corpora encompasses a set of steps: parsing the text into sentences, splitting the sentences and then tokenising it. After the lemmatisation step, the pairs are extracted using formal context analysis (FCA). However, there might be some uninteresting and erroneous pairs in the formal context. Generating formal context may lead to a time-consuming process, so formal context size reduction is require to remove uninterested and erroneous pairs, taking less time to extract the concept lattice and concept hierarchies accordingly. In this premise, this study aims to propose two frameworks: (1) A framework to review the current process of deriving concept hierarchies from corpus utilising formal concept analysis (FCA); (2) A framework to decrease the formal context’s ambiguity of the first framework using an adaptive version of evolutionary clustering algorithm (ECA*). Experiments are conducted by applying 385 sample corpora from Wikipedia on the two frameworks to examine the reducing size of formal context, which leads to yield concept lattice and concept hierarchy. The resulting lattice of formal context is evaluated to the standard one using concept lattice-invariants. Accordingly, the homomorphic between the two lattices preserves the quality of resulting concept hierarchies by 89% in contrast to the basic ones, and the reduced concept lattice inherits the structural relation of the standard one. The adaptive ECA* is examined against its four counterpart baseline algorithms (Fuzzy K-means, JBOS approach, AddIntent algorithm, and FastAddExtent) to measure the execution time on random datasets with different densities (fill ratios). The results show that adaptive ECA* performs concept lattice faster than other mentioned competitive techniques in different fill ratios.

Graded polygons of opposition in fuzzy formal concept analysis

Fuzzy Formal Concept Analysis (FFCA) is a robust mathematical tool for analyzing data, particularly where uncertainty or fuzziness is inherent. FFCA is utilized across various domains, including data mining, information retrieval, and knowledge representation. However, fuzzy concepts extraction is a crucial yet computationally intensive task. This paper addresses the challenge of time efficiency in extracting single-sided fuzzy concepts from large datasets. A parallel algorithm is proposed to reduce computational time and optimize resource utilization, thus enabling the scalable analysis of expanding datasets. By computing fuzzy concepts across multiple threads in parallel, each thread processes an attribute independently to extract fuzzy concepts, which are then merged in the final step. The proposed algorithm extracts fuzzy-crisp concepts, which are more concise than other types of fuzzy concepts. Experiments were conducted to evaluate the performance of the proposed parallel algorithm against existing sequential methods. Experimental results demonstrate significant gains in computational efficiency, with the algorithm achieving an average time reduction of 68% compared to the attribute-based algorithm and up to 83%-time reduction compared to the fuzzy CbO algorithm across various types of datasets, including binary, quantitative, and fuzzy.

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.

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.

Creative knowledge discovery—finding useful, previously unknown links between concepts—is a vital tool in unlocking the economic and social value of the vast range of networked data and services that is now available. We define "standard" knowledge discovery as the search for explanatory and predictive patterns in a specific domain, usually with a large volume of data. In contrast, creative knowledge discovery is concerned with the creation of new (and effective) patterns—either by generalization of existing patterns or by analogy to patterns in other domains. An important precondition for creative knowledge discovery is that we understand the relations within the data. Fuzzy formal concept analysis is a powerful approach that enables us to find embedded structure in data and to extract novel concepts that can be used in subsequent processing such as creative knowledge discovery. This paper outlines a fast algorithm for computing fuzzy formal concepts and provides a brief illustration of the use of fuzzy formal concept analysis in creative knowledge discovery.

Text knowledge analysis plays a very important role in knowledge processing. In order to find interesting patters, we introduce text analysis into the Formal Concept Analysis(FCA) field. FCA is a valid tool for data ming and knowledge processing. In the paper, we present a method for exacting the conceptual patterns and concept lattice from multiple knowledge sources. The method consists of two steps: the first is to insert some related many-valued contexts(MVCs)(which represent text knowledge) into some corresponding column or cell of other MVC in accordance with certain rules, and then extract formal concepts and concept lattice. Our insertion method not only is semanticpreserving, but adds the dimension of attributes for describing objects. Hence, the discovered concepts have the more richer intents and are more denser in the concept lattice. The paper mainly applied the method to column insertion operation and cell insertion operation, and obtained some propositions, which can be used by knowledge engineers to represent and analyze text knowledge(especially,text objects) and to extract new semantics information or associate rules from concept lattices.

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.