Data mining using L-fuzzy concept analysis.

Association rules in data mining are implications between attributes of objects that hold in all instances of the given data. These rules are very useful to determine the properties of the data such as essential features of products that determine the purchase decisions of customers. Normally the data is given as binary (or crisp) tables relating objects with their attributes by yes-no entries. We propose a relational theory for generating attribute implications from many-valued contexts, i.e, where the relationship between objects and attributes is given by a range of degrees from no to yes. This degree is usually taken from a suitable lattice where the smallest element corresponds to the classical no and the greatest element corresponds to the classical yes. Previous related work handled many-valued contexts by transforming the context by scaling or by choosing a minimal degree of membership to a crisp (yes-no) context. Then the standard methods of formal concept analysis were applied to this crisp context. In our proposal, we will handle a many-valued context as is, i.e., without transforming it into a crisp one. The advantage of this approach is that we work with the original data without performing a transformation step which modifies the data in advance

A Relation-Algebraic Approach to L - Fuzzy Topology

Any science deals with the study of certain models of the real world. However, a model is always an abstraction resulting in some uncertainty, which must be considered. The theory of fuzzy sets is one way of formalizing one of the types of uncertainty that occurs when modeling real objects. Fuzzy sets have been applied in various real-world problems such as control system engineering, image processing, and weather forecasting systems. This research focuses on applying the categorical framework of abstract L - fuzzy relations to L-fuzzy topology with ideas, concepts and methods of the theory of L-fuzzy sets. Since L-fuzzy sets were introduced to deal with the problem of approximate reasoning, t − norm based operations are essential in the definition of L - fuzzy topologies. We use the abstract theory of arrow categories with additional t − norm based connectives to define L - fuzzy topologies abstractly. In particular, this thesis will provide an abstract relational definition of an L - fuzzy topology, consider bases of topological spaces, continuous maps, and the first two separation axioms T0 and T1. The resulting theory of L - fuzzy topological spaces provides the foundation for applications and algorithms in areas such as digital topology, i.e., analyzing images using topological features

Relational Approach to the L-Fuzzy Concept Analysis

Modern industrial production systems benefit from the classification and processing of objects and their attributes. In general, the object classification procedure can coincide with vagueness. Vagueness is a common problem in object analysis that exists at various stages of classification, including ambiguity in input data, overlapping boundaries between classes or regions, and uncertainty in defining or extracting the properties and relationships of objects. To manage the ambiguity mentioned in the classification of objects, using a framework for L-fuzzy relations, and displaying such uncertainties by it can be a solution. Obtaining the least unreliable and uncertain output associated with the original data is the main concern of this thesis. Therefore, my general approach to this research can be categorized as follows: We developed an L-Fuzzy Concept Analysis as a generalization of a regular Concept Analysis. We start our work by providing the input data. Data is stored in a table (database). The next step is the creation of the contexts and concepts from the given original data using some structures. In the next stage, rules, or patterns (Attribute Implications) from the data will be generated. This includes all rules and a minimal base of rules. All of them are using L-fuzziness due to uncertainty. This requires L-fuzzy relations that will be implemented as L -valued matrices. In the end, everything is nicely packed in a convenient application and implemented in Java programming language. Generally, our approach is done in an algebraic framework that covers both regular and L -Fuzzy FCA, simultaneously. The tables we started with are already L-valued (not crisp) in our implementation. In other words, we work with the L-Fuzzy data directly. This is the idea here. We start with vague data. In simple terms, the data is shown using L -valued tables (vague data) trying to relate objects with their attributes at the start of the implementation. Generating attribute implications from many-valued contexts by a relational theory is the purpose of this thesis, i.e, a range of degrees is used to indicate the relationship between objects and their properties. The smallest degree corresponds to the classical no and the greatest degree corresponds to the classical yes in the table