Fuzzy Formal Context Research Articles

Inner product reduction for fuzzy formal contexts

Formal concept analysis finds application across multiple domains, including knowledge representation, data mining, and decision analysis. Within this framework, the exploration of attribute reduction for fuzzy formal contexts represents a substantial area of research. We introduce a novel form of attribute reduction for fuzzy formal contexts named inner product reduction, and an algorithm for finding all inner product reducts is given by using the indiscernibility matrix, and a calculation example is given. Furthermore, for consistent fuzzy decision formal contexts, the definition and algorithm of inner product reduction are given. Finally, the concept and algorithm of inner product reduction are extended to general fuzzy decision formal contexts. Through experimental verification, the viability and efficacy of the inner product reduction algorithm for fuzzy formal contexts are verified.

Two-way concept-cognitive learning method: a perspective from progressive learning of fuzzy skills

ABSTRACT Concept-cognitive learning and two-way concept-cognitive learning have been widely used to learn concepts from given cues, and fruitful results have been obtained. However, there are still some limitations: (i) The existing two-way concept-cognitive learning methods based on competences focused on dichotomous skills, but they did not consider the proficiency of skills in the concept-cognitive learning process and ignored the dynamic learning of skills. (ii) The existing necessary and sufficient information granules based on competences cannot be learned directly from any information granule. (iii) Most of the other existing two-way concept-cognitive learning models in fuzzy formal context have been studied through an antitone Galois connection, but they cannot directly deal with an isotone Galois connection between items and fuzzy skills. To overcome these issues, we propose a novel two-way concept-cognitive learning model based on fuzzy skills to learn fuzzy skill-based cognitive concepts (i.e. necessary and sufficient fuzzy skill-based information granules) from a perspective of skill proficiency. And we optimize the two-way concept-cognitive learning model to learn directly fuzzy skill-based cognitive concepts from any information granule and save time for learning concepts. Furthermore, a progressive learning mechanism is explored to deal with dynamic data. The experimental results of 12 UCI data sets show that the proposed concept-cognitive learning models in this paper are feasible and effective.

$$\delta $$-granular reduction in formal fuzzy contexts: Boolean reasoning, graph represent and their algorithms

$$\delta $$-granular reduction in formal fuzzy contexts: Boolean reasoning, graph represent and their algorithms

Decision Implication-Based Knowledge Representation and Reasoning Within Incomplete Fuzzy Formal Context

Decision Implication-Based Knowledge Representation and Reasoning Within Incomplete Fuzzy Formal Context

Three-way concepts in the interval-valued formal contexts

The theory of interval-valued formal contexts was originally derived from fuzzy formal contexts. While the fuzzy formal context can extract information from fuzzy formal contexts more precisely, it lacks theoretical analysis of formal contexts with interval-valued data types. This paper incorporates the three-way concept into interval-valued formal contexts, and partitions the interval value range of objects and attributes into three regions utilizing the notion of three-way decisions. On the basis of interval-valued information granules, the concepts of negative operators and interval-valued three-way concepts are proposed. They can conduct profounder knowledge discovery in interval-valued formal contexts, and a generation algorithm of interval-valued three-way concepts is devised. Finally, the effectiveness of the algorithm is substantiated through experimentation

Mining positive and negative rules via one-sided fuzzy three-way concept lattices

Mining association rules are a major area of research in data mining, as there are typically a large number of rules in each data set. And extracting necessary rules from each data set requires efficiency and careful consideration. In this paper, we propose a novel mining algorithm for efficiently acquiring positive and negative quantitative association rules (positive and negative QA-rules) in a fuzzy or multi-valued ordered context based on a three-way concept lattice (3CL). To this end, a new 3CL, called a one-sided fuzzy three-way concept lattice (OF3WC-lattice), is constructed based on a fuzzy formal context (FFcontext). The designed OF3WC-lattice describes the dependencies between objects and attributes, whether in the original FFcontext or its complement context. And the OF3WC-lattice is based on two kinds of two-way operators to make it consistent with practical linguistics. In addition, we propose an algorithm for constructing an OF3WC-lattice through analyzing relationships between the novel concept lattice and traditional one-sided fuzzy concept lattices. Experimental results demonstrate that our method effectively mines positive and negative QA-rules.

Dynamic updating variable precision three-way concept method based on two-way concept-cognitive learning in fuzzy formal contexts

Cognitive learning and two-way learning are effective knowledge representations, which can simulate the human brain to learn concepts. The combination of both topics has achieved some results and improved conceptual evolution ability in fuzzy formal concept analysis (FFCA), but there are still some shortcomings: 1) FFCA does not consider the flexibility of concepts, which makes it difficult to select suitable concepts; 2) the existing necessary and sufficient concepts fail to learn directly from any information granules; 3) the cognitive learning mechanism ignores to integrate previously acquired knowledge into the present state in the process of dynamic concept learning. To tackle these issues, in this paper, we put forward a novel two-way concept-cognitive learning (TCCL) model based on three-way decision in fuzzy formal contexts. Firstly, we introduce the object and attribute operators to learn variable precision object induced three-way concept, where such concept has flexibility by adjusting thresholds. Then, to learn directly necessary and sufficient three-way concept from the given clues, we investigate a new TCCL model, which has low computation cost for concept learning. Furthermore, updating mechanism of three-way concept is discussed in dynamic learning environment. Finally, the conducted experiments explicate the effectiveness and feasibility of our proposed approach in the large-scale datasets.

Fuzzy Decision Rule-Based Online Classification Algorithm in Fuzzy Formal Decision Contexts

Formal concept analysis (FCA) extracts interpretable rules by using implication relationships between concepts, which is an effective method for acquiring knowledge. In this paper, a fuzzy decision rule-based online classification algorithm called OFRCA is designed by combining online learning theory and FCA in a fuzzy formal decision context. First, the original weight vector of all attributes is obtained by fusing the original deterministic decision rules. Second, incremental fuzzy decision rules are obtained according to the added objects. Then for the incremental fuzzy decision rules, OFRCA updates the weight vector based on minimizing the loss function. At the end of the process, all the rules are fused into an attributeweighted classifier with the final weight vector. In this paper, we employ the regret to obtain a learning guarantee for OFRCA. The growth rate of OFRCA's regret converges to 0 as the number of iterations increases in the ideal state, which provides an effective learning guarantee for OFRCA. To verify the effectiveness of the proposed algorithm, this paper conducted numerical experiments and systematically analyzed the experimental results, which demonstrated that OFRCA achieved the highest advanced classification performance among the compared algorithms.

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

Three-way concept analysis is an integration of formal concept analysis and three-way decision, which has been found wide applications in knowledge representation, cognitive computing and decision sciences. There have been already a huge number of literature on theory and applications on such topics. Most researches on formal concept analysis focus, independently, on specific scenarios such as Boolean formal contexts and fuzzy formal contexts. In this paper, a unified framework for three-way concept analysis is built in lattice-valued formal contexts. The notions of positive concept and negative concept are given in a lattice-valued formal context based on an overlapping function, one of significant operators for information fusion. Two pairs of object induced adjunctive three-way concept operators and attribute induced adjunctive three-way concept operators are introduced. Their properties have been fully investigated. With those operators, the algebraic structures of adjunctive three-way concepts, their extensions and intensions have been explored. Specifically, the local structure within pairs of sub-concept and sup-concept has been finely characterized. The derived results provide ones a way to construct the complete lattice of a lattice-valued formal context or its sublattice by determining all pairs of sub-concept and sup-concept provided that an object fuzzy set or an attribute fuzzy set is given. A case analysis on a real world dataset from UCI repository is presented to verify the rationality and significance of obtained results.

Granular reduction in formal fuzzy contexts: graph representation, graph approach and its algorithm

Granular reduction in formal fuzzy contexts: graph representation, graph approach and its algorithm

Attribute granulation in fuzzy formal contexts based on L-fuzzy concepts

Formal concept analysis is an effective tool for granular computing. In applications, the selection of attributes greatly determines the structure and size of the obtained concept lattice. The existing studies have dealt with this problem on the basis of ordinary formal concepts. And thus, these studies are not capable of dealing with uncertainties caused by fuzziness. As a solution, we investigate attribute granulation in the settings of fuzzy contexts based on L-fuzzy concepts. Firstly, we present the mathematical descriptions of this open problem as well as the detailed meanings of this study with an illustrative example. And then, we present zoom-in algorithm to get more specific fuzzy concepts, and zoom-out algorithm to derive more abstract fuzzy concepts. Finally, systematic experiments are performed to show the influence of granulating of attributes on the sizes of fuzzy concept lattices.

Formal concept analysis approach to understand digital evidence relationships

The number of cyber attacks is constantly increasing daily, which demands organizations to respond quickly and adequately to security incidents. Digital forensics plays an essential role in these activities. In the digital investigation process, it is necessary to identify and separate relevant digital evidence from that which is not. In this paper, we describe the construction of ordinary and fuzzy formal contexts based on digital evidence collected from event logs and the filesystem (New Technology File System). We generated four concept lattices for various subsets of attributes regarding timestamps, types of files, or event logs. The association rules and their connections with Formal Concept Analysis are explored, and several algorithms, including GUHA methods, are applied to our data. We compare, evaluate and interpret the various methods for association rules mining. Moreover, we describe the state-of-the-art of fuzzy attribute implications in Formal Concept Analysis and provide the interpretation of implications in our epoch-time attributes. Our solution provides warnings for the security analyst to manually check and inspect the suspicious records in data. Hence, the analyst can quickly find relevant records for the case and perform further analysis.

Extracting Concepts From Fuzzy Relational Context Families

<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.

Attribute reduction and information granulation in Pythagorean fuzzy formal contexts

Pythagorean fuzzy set theory is one of the significant tools to deal with real-life problems which are prone to imprecision, partial truth or uncertainty. A Pythagorean fuzzy set reports the degree of truthness as well as falsity of a statement in order to illustrate the imprecision of that statement. Motivated by the resourcefulness of this theory, this paper is aimed to explore formal concept analysis in Pythagorean fuzzy environment. It defines Pythagorean fuzzy formal concept and Pythagorean fuzzy concept lattice associated with a Pythagorean fuzzy formal context. The proposed Pythagorean fuzzy formal concept is a duple whose first argument is (crisp) extension and second argument is Pythagorean fuzzy intension. Moreover, the join of different combinations of object concepts of a Pythagorean fuzzy formal context returns different Pythagorean fuzzy formal concepts. We also studied the knowledge reduction in Pythagorean fuzzy formal contexts in association with Pythagorean fuzzy concept lattice. The key ideas of attribute reduct, core and discernibility matrix for a Pythagorean fuzzy formal context are illustrated. Further, we merged the notions of information granulation and attribute reduction with Pythagorean fuzzy formal context. Various results of formal concepts and concept lattices are extended in Pythagorean fuzzy set theory. At last, an algorithm is developed which discovers non-redundant Pythagorean fuzzy formal concepts for a given Pythagorean fuzzy formal context. Eventually, an example is presented which elaborates the procedure of this algorithm.

Sharing profits in formal fuzzy contexts

Cooperative game theory is concerned with situations where a group of agents coordinate their actions to get a common benefit. An allocation rule for these situations is a way to share the common benefit among the agents. The search for a fair allocation rule may depend on the information one has about these agents. A formal context represents information about certain attributes of a set of objects in a table, and they have been used in the literature to describe information about the agents in a game. More recently, formal contexts are extended to the fuzzy setting. Now in this paper we establish a methodology to share the profits of a group of agents that have some information about them collected in a fuzzy formal context when those benefits depend on a set of attributes.

Multilevel Conflict Analysis Based on Fuzzy Formal Contexts

Uncertainties are widespread in real-world situations. In the existing studies, the uncertainties expressed by the agents engaged in a conflict situation have been extensively investigated. However, to what extent these agents can be trusted is often ignored and seldom taken into account. In this article, to manage these two sides of uncertainties, we propose a mathematical framework for multilevel conflict analysis from an outsider's perspective. Concretely, in order to indicate the analysis levels and manage the conflict situation with an extent-intent view, we resort to fuzzy formal concept analysis. At first, we explain what is multilevel data analysis. After that, based on fuzzy concept lattices, we present a general framework for multilevel conflict analysis and a method for computing the maximal coalitions and minimum conflict sets. And then, we describe how to update conflict analysis results when increasing analysis levels. Experimental analysis demonstrates that managing two sides of uncertainties can bring much more conflict resolution, and increasing analysis levels is helpful for postponing decision making and bringing more economical conflict resolutions. Finally, case studies are made to show the application of multilevel conflict analysis model and some interesting findings are obtained. Specifically, once the cost of increasing analysis levels is taken into consideration, it is interesting to choose the optimal analysis level for making the most economic decision. The main contribution of this study is the model of multilevel conflict analysis based on fuzzy concept lattices, which can incorporate the uncertainties of two sides.

Incremental concept-cognitive learning approach for concept classification oriented to weighted fuzzy concepts

Fuzzy formal concept analysis (FFCA) is generally used to describe the processes of concept-cognitive learning (CCL). However, for fuzzy formal contexts, each attribute has the same weight (i.e, the same degree of importance) before constructing fuzzy concepts, which limits mining interesting knowledge and affects its application promotion. On the other hand, this model is hard to resist the influence of noise hidden in data, which results in poor classification learning. Moreover, the existing incremental CCL algorithms still face some challenges that the previously acquired knowledge is not fully utilized to improve the classification accuracies for dynamic data. To address these issues, we introduce different weights into fuzzy formal contexts and propose a novel incremental CCL mechanism in dynamic environment. Firstly, weight values of attributes from different decisions based on fuzzy entropy are established to measure the significant degree of attributes. Then, to comprehensively explicate the hierarchical relationships of fuzzy concepts, we construct the weighted fuzzy concept lattice and the weighted fuzzy concept space. Secondly, we design an algorithm to update the weighted fuzzy concepts for facilitating concept classification. To overcome the individual cognitive limitation, we put forward the progressive weighted fuzzy concept to remove repeated information. Furthermore, the classification prediction label and dynamic updating mechanism after adding objects are systematically discussed. Finally, we perform an experimental evaluation on ten data sets which explicate the feasibility and efficiency of our proposed approach.

Attribute reduction in intuitionistic fuzzy formal concepts

Knowledge acquisition in intuitionistic fuzzy information systems is of importance because those fuzzy information systems are often encountered in many real-life problems. Formal concept analysis is a simple and effective tool for knowledge acquisition. However, there is still little work on introducing knowledge acquisition methods based on formal concept analysis into intuitionistic fuzzy information systems. This paper mainly extends the formal concept theory into intuitionistic fuzzy information systems. Firstly, two pairs of adjoint mappings are defined in intuitionistic fuzzy formal contexts. It is verified that both pairs of adjoint mappings form Galois connections. Secondly, two types of intuitionistic fuzzy concept lattices are constructed. After that, we also present the main theorems and propositions of the intuitionistic fuzzy concept lattices. Thirdly, we deeply discuss the attribute characteristics for type-1 generalized one-sided intuitionistic fuzzy concept lattice. Furthermore, a discernibility matrix-based algorithm is proposed for attribute reduction and the effectiveness of this algorithm is demonstrated by a practical example. The construction of intuitionistic fuzzy conceptS is meaningful for the complex and fuzzy information in real life.

Selection of appropriate bonds between [formula omitted]-fuzzy formal contexts for recommendation tasks

The bond between L-fuzzy formal contexts can be defined as a (Galois) connection between L-fuzzy concept lattices of L-fuzzy formal contexts. The selection of appropriate bond from the set of all bonds between L-fuzzy formal contexts is an important challenge to apply it in recommendation tasks. We propose the general method for the selection of bonds regarding external information given by L-fuzzy relation. The alternative versions of direct products of L-fuzzy formal contexts are formulated and explored since we demonstrate that the extent of direct product is a bond between input L-fuzzy formal contexts. We present examples of the benevolent and rigorous recommendations in several application domains including the real dataset about music genres. Finally, the connections with factorization of L-fuzzy formal contexts and Sugeno integral are thoroughly studied in our paper.

A proposal to extend Relational Concept Analysis with fuzzy scaling quantifiers

In this article, we integrate Relational Concept Analysis with fuzzy logic to explore multi-relational datasets including also vague information. Mainly, we aim to extract a family of fuzzy concept lattices from data organized as a collection of fuzzy formal contexts and fuzzy relations between objects of different types. To achieve this goal, we use existing fuzzy FCA techniques and fuzzy scaling quantifiers. Our principal contribution here consists of introducing and studying fuzzy scaling quantifiers, which are fuzzy quantifiers based on the concept of evaluative linguistic expression.