Formal Decision Contexts 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.

A novel method for generating a canonical basis for decision implications based on object-induced three-way operators

Three-way concept analysis can express not only the information that the objects of study have in common but also the information that they do not have in common, which is an extension of classical formal concept analysis. The decision implication is a special representation of the implication in the decision formal context, which reveals the dependency between conditional attributes and decision attributes. This paper is devoted to the study of a novel method for generating a canonical basis for decision implications based on object-induced operators (OE operators). First, the decision implication logic based on OE operators is described from the two aspects of semantics and syntax, which is applied to the decision formal context. Then, a method of obtaining the canonical basis for decision implications based on decision the premise of the OE operators is given, and the completeness, the nonredundancy and the optimality of the canonical basis are proven. Finally, a method of generating true premises based on OE operators is proposed by means of an incremental method, and the canonical basis for decision implications is then generated by the true premises based on OE operators.

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.

Rule acquisition of three‐way semi‐concept lattices in formal decision context

AbstractThree‐way concept analysis is an important tool for information processing, and rule acquisition is one of the research hotspots of three‐way concept analysis. However, compared with three‐way concept lattices, three‐way semi‐concept lattices have three‐way operators with weaker constraints, which can generate more concepts. In this article, the problem of rule acquisition for three‐way semi‐concept lattices is discussed in general. The authors construct the finer relation of three‐way semi‐concept lattices, and propose a method of rule acquisition for three‐way semi‐concept lattices. The authors also discuss the set of decision rules and the relationships of decision rules among object‐induced three‐way semi‐concept lattices, object‐induced three‐way concept lattices, classical concept lattices and semi‐concept lattices. Finally, examples are provided to illustrate the validity of our conclusions.

Rule reductions of decision formal context based on mixed information

Formal concept analysis is an effective tool in knowledge discovery. Nonetheless, it ignores the negative information, especially in rule extraction. And some applications require to mix positive and negative information for management and representation explicitly. This paper discusses an emerging method for rule reduction in formal contexts with mixed information by presenting a mixed decision rule based on a mixed concept lattice. Based on the mixed concept model, we discuss the relationships between mixed concept, formal concept and three-way concept, respectively. In this paper, to fully consider the mixed information, we construct a mixed decision rule. Moreover, to reduce the redundance of rule, a novel approach for weak-basis from the viewpoint of granular computing is designed to selection the core mixed decision rules and necessary mixed decision rules. Finally, the comparison of mixed decision rules and three-way decision rules is perfectly discussed, and case study is presented for the difference.

A novel approach to attribute reduction and rule acquisition of formal decision context

A novel approach to attribute reduction and rule acquisition of formal decision context

Three-way reduction for formal decision contexts

Attribute reduction is an important component in rough set theory and formal concept analysis. The three-way concept lattice is a combination of concept lattices and three-way decision theory. We investigate granular reduction, three-way granular reduction, and three-way distribution reduction for general formal decision contexts. We furthermore obtain discernibility matrix-based reduction algorithms for each type of reduction. In particular, we demonstrate that in decision contexts, three-way granular reduction coincides with positive region reduction for decision tables. Furthermore, we evaluate the effectiveness of the three-way granular reduction algorithm using 17 UCI datasets.

A dynamic rule-based classification model via granular computing

As an effective tool for data representation and processing, granular computing has been incorporated into formal decision contexts for finding granular reducts to achieve the task of mining granular rules. However, the classification performance of granular rules has not been evaluated, and this type of method is not suitable for dynamic data. To solve this problem, the current study updates granular reducts and evaluates the obtained granular rules in terms of classification performance. Concretely, we first give a theoretical analysis of updating granular reducts and granular rules and then present a novel dynamic rule-based classification model (DRCM) based on the updating mechanism. Finally, we discuss the feasibility of the proposed model and compare it with several popular classification algorithms. The conducted experiments demonstrate that the granular reducts can improve the classification ability to a certain extent and that DRCM can achieve better classification performance on some consistent datasets.

Rule extraction based on linguistic-valued intuitionistic fuzzy layered concept lattice

As one of the research tools for data processing and knowledge discovery, concept lattice can effectively extract information. In daily life, due to the ambiguity and uncertainty of the decision environment, different experts may provide different evaluation information according to individual needs which may be expressed with linguistic values. In order to handle these problems, we study the rule extraction method of linguistic-valued intuitionistic fuzzy layered concept lattice. First, we propose a linguistic-valued intuitionistic fuzzy formal decision context based on the intuitionistic fuzzy lattice implication algebra, which can simultaneously process the obtained comparable and incomparable linguistic information from both positive and negative aspects. Furthermore, by setting different linguistic-valued trust degrees, we put forward the linguistic-valued hierarchical concept construction operator. On this basis, a linguistic-valued intuitionistic fuzzy layered concept lattice can be constructed to meet the requirements of different experts at different levels. And then through the relationship between the conditional concept and decision concept in the obtained concept set, we get a rule set of the formal decision context. Finally, we present a rule extraction method to help people make more reasonable decisions, combining the confidence and support degree of linguistic-valued intuitionistic fuzzy decision rules. And a practical example involving individual financial investment decision-making is used to verify the efficiency and applicability of the proposed approach.

Concept-Cognitive Learning Model for Incremental Concept Learning

Concept-cognitive learning (CCL) is an emerging field of concerning incremental concept learning and dynamic knowledge processing in the context of dynamic environments. Although CCL has been widely researched in theory, the existing studies of CCL have one problem: the concepts obtained by CCL systems do not have generalization ability. In the meantime, the existing incremental algorithms still face some challenges that: 1) classifiers have to adapt gradually and 2) the previously acquired knowledge should be efficiently utilized. To address these problems, based on the advantage that CCL can naturally integrate new data into itself for enhancing flexibility of concept learning, we first propose a new CCL model (CCLM) to extend the classical methods of CCL, which is not only a new classifier but also good at incremental learning. Unlike the existing CCL systems, the theory of CCLM is mainly based on a formal decision context rather than a formal context. In learning concepts from dynamic environments, we show that CCLM can naturally incorporate new data into itself with a sufficient theoretical guarantee for incremental learning. For classification task and knowledge storage, our results on various data sets demonstrate that CCLM can simultaneously: 1) achieve the state-of-the-art static and dynamic classification task and 2) directly accomplish preservation of previously acquired knowledge (or concepts) under dynamic environments.

Multiple multidimensional linguistic reasoning algorithm based on property-oriented linguistic concept lattice

Aiming at the difficult problems of dealing with mass linguistic information in uncertain environment, this paper mainly focuses on a linguistic reasoning algorithm based on property-oriented linguistic concept lattice by combining concept lattice and neural network. Specifically, we present a property-oriented linguistic concept lattice to express linguistic information between concepts based on linguistic term set. Furthermore, we are devoted to the study of rule extraction of fuzzy linguistic formal decision context by constructing property-oriented linguistic decision concepts. In order to obtain more decision knowledge from known linguistic rules, an extension method of linguistic decision rules is developed, which takes advantage of the dominance relation between similar rules on the premise of consistent property-oriented linguistic decision rules. In addition, we construct a multiple multidimensional linguistic reasoning model for predicting uncertain decision information. Moreover, we input a multiple multidimensional linguistic reasoning model with property-oriented linguistic decision information into neural network to obtain unknown decision results, which can not only improve the accuracy of inference results but also reduce the loss of linguistic information. Finally, some experiments are conducted to demonstrate the efficiency of the proposed method.

Reduction of fuzzy-crisp concept lattice based on order-class matrix

Knowledge reduction is one of critical problems in data mining and information processing. It can simplify the structure of the lattice during the construction of fuzzy-crisp concept lattice. In terms of fuzzy-crisp concept, we develop an order-class matrix to represent extents and intents of concepts, respectively. In order to improve the computing efficiency, it is necessary to reduce the size of lattices as much as possible. Therefore the judgement theorem of meet-irreducible elements is proposed. To deal with attribute reductions, we develop a discernibility Boolean matrix in formal fuzzy contexts by preserving extents of meet-irreducible elements via order-class matrix. A heuristic attribute-reduction algorithm is proposed. Then we extend the proposed model to consistent formal fuzzy decision contexts. Our methods present a new framework for knowledge reduction in formal fuzzy contexts.

Semi-Supervised Concept Learning by Concept-Cognitive Learning and Concept Space

In human concept learning, people can naturally combine a handful of labeled data with abundant unlabeled data when they make classification decisions, which is also known as semi-supervised learning (SSL) in machine learning. Especially, human concept learning not only is a static process in human cognition but also can vary gradually with dynamic environments. Nevertheless, the classical SSL algorithms must be redesigned to accommodate newly input data. In this sense, concept-cognitive learning may be a good choice, as it can implement dynamic processes by imitating human cognitive processes. Meanwhile, numerous SSL methods were designed based on the feature vector information of instances, while ignoring concept structural information that is a very important process in human knowledge organization. Based on this idea, a novel SSL method, named semi-supervised concept learning method (S2CL), is proposed for dynamic SSL by employing concept spaces, in which knowledge is represented by hierarchical concept structures. Moreover, to make full use of the global and local conceptual information, we further propose an extended version of S2CL (namely, <inline-formula><tex-math notation="LaTeX">$\text{S2CL}^{\alpha }$</tex-math></inline-formula> ) for concept learning. More specifically, to effectively exploit the unlabeled data, this paper first shows some new related theories for S2CL (or <inline-formula><tex-math notation="LaTeX">$\text{S2CL}^{\alpha }$</tex-math></inline-formula> ) based on a regular formal decision context; then a novel SSL framework is designed, and its corresponding algorithm is developed. Finally, we conduct some experiments on various datasets to demonstrate the effectiveness of our methods, which include concept classification and incremental learning under a large quantity of unlabeled data.

Attribute reduction in formal decision contexts and its application to finite topological spaces

Attribute reduction in formal decision contexts has become one of the key issues in the research and development of formal concept analysis (FCA) and its applications. As far as we know, however, most of the existing reduction methods for formal decision contexts are time-consuming especially for the large-scale data. This paper investigates the attribute reduction method for large-scale formal decision contexts. The computation of a discernibility matrix is an important step in the development of the corresponding reduction method. A simple and powerful method to efficiently calculate the discernibility matrix of formal decision contexts is first presented. In addition, a heuristic algorithm for searching the optimal reduct is then proposed. Thirdly, as an application of the new results, we discuss the problem of finding the minimal subbases of finite topological spaces. It has shown that the method of attribute reduction in formal decision contexts can be used to obtain all the minimal subbases of a finite topological space. Furthermore, we present an algorithm for computing the minimal subbase of a topological space, based on the attribute reduction method proposed in this paper. Finally, two groups of experiments are carried out on some large-scale data sets to verify the effectiveness of the proposed algorithms.

Fuzzy-Based Concept Learning Method: Exploiting Data With Fuzzy Conceptual Clustering.

Concepts have been adopted in concept-cognitive learning (CCL) and conceptual clustering for concept classification and concept discovery. However, the standard CCL algorithms are incapable of tackling continuous data directly, and some standard conceptual clustering methods mainly focus on the attribute information, ignoring the object information that is also important to improve clustering analysis and concept classification ability. Therefore, in this article, we present a novel concept learning method, called the fuzzy-based concept learning model (FCLM), to address these two issues by exploiting concept hierarchical relations in concept lattices. More specifically, we first show some new related notions for FCLM based on a regular fuzzy formal decision context; among these notions, the object-oriented and attribute-oriented fuzzy concept similarities are used to achieve the concept similarity measure in concept lattices. Moreover, a novel fuzzy concept learning framework is designed, and its corresponding learning algorithms are developed. Finally, we conduct some experiments on various real-world datasets to demonstrate that the proposed method can achieve the state-of-the-art classification performance among similarity-based learning methods. In addition, we further verify the effectiveness of our method in concept discovery on the MNIST dataset.

Formal concept analysis of multi-scale formal context

The case that attributes have a single value is considered in the classical formal concept. However, real-life data involve multi-level attribute values, which means that attributes have different values on different scales. In order to deal with multi-level attributes, we introduce the concept of multi-scale formal context in this paper. In this context, with the change of scale, objects owned by each attribute change monotonously. Thus, under the multi-scale formal context, the change of formal concept is discussed. Moreover, this paper defines the consistent multi-scale formal decision context and discusses the optimal scale selection problem of the consistent multi-scale decision context.

Finding the Key Structure of Mechanical Parts with Formal Concept Analysis

Aiming at the problem that the assembly body model is difficult to classify and retrieve (large information redundancy and poor data consistency), an assembly body retrieval method oriented to key structures was presented. In this paper, a decision formal context is transformed from the 3D structure model. The 3D assembly structure model of parts is defined by the adjacency graph of function surface and qualitative geometric constraint graph. The assembly structure is coded by the linear symbol representation of compounds in chemical database. An importance or cohesion as the weight to a decision-making objective on the context is defined by a rough set method. A weighted concept lattice is introduced on it. An important formal concept means a key structure, since the concept represents the relations between parts’ function surfaces. It can greatly improve the query efficiency.

Rules acquisition of formal decision contexts based on three-way concept lattices

Three-way concept lattices can supply much more information than classical concept lattices since they contain the positive information and negative information between objects and attributes simultaneously. Taking advantage of this, the rules acquisition for formal decision contexts from the perspective of three-way concept lattices are discussed, and the results are compared with the common decision rules based on classical concept lattices. Firstly, the definition of object-induced three-way consistence of a formal decision context is presented. Then, positive decision rules and negative decision rules are proposed for an object-induced three-way consistent formal decision context. Furthermore, we give semantic explanation for these rules, and compare them with common rules obtained from a strongly consistent lattice. In parallel, the similar issues are investigated from the perspective of attributes, including the definition of attribute-induced three-way consistence of a formal decision context, rules acquisition based on an attribute-induced three-way consistent formal decision context, and rules comparison. Finally, the relationships among the three types of consistence, namely, object-induced three-way consistence, attribute-induced three-way consistence, and existing strongly consistence are discussed to make the contents compact. Some theoretical examples are given to illustrate the main results of this paper.

Neighbourhood systems based attribute reduction in formal decision contexts

Attribute reduction of formal decision context mainly uses the relationship between two concept lattices generated by the condition and decision attributes to remove redundant condition attributes. By using decision attributes to observe the covering of objects, this study defines two types of consistent sets and reducts in a consistent formal decision context based on neighbourhood systems. Four types of reductions in inconsistent formal decision contexts are also studied. The methods to calculate all types of reductions are formulated by discernibility matrix. Finally, an approach to obtain the decision rules in consistent formal decision context is proposed.

Granular matrix-based knowledge reductions of formal fuzzy contexts

Knowledge reduction is an important issue in formal fuzzy contexts, which can simplify the structure of concept lattices. In this paper, a novel granular matrix-based for knowledge reduction of crisp-fuzzy concept is investigated. Firstly, matrix representations of extents and intents of concepts are defined, respectively, which are used to characterize the join-irreducible elements and propose the corresponding algorithm. In this framework, granular consistent set and granular reduct are developed. Then the judgement theorem of reduction and its corresponding algorithm in formal fuzzy context are proposed. Furthermore, we generalize the matrix approach to formal fuzzy decision contexts. Finally, numerical experiments are conducted to evaluate the effectiveness of the proposed approaches. Our methods present a new framework for knowledge reduction in formal fuzzy contexts.