Konceptuální svazy, jejich redukce a aplikace

Abstract

Formal Concept Analysis (FCA) is well known method for object-attribute data analysis. It is based on understanding the world in terms of concepts which can be simply described as sets of objects and their attributes. One of the main FCA outputs is the concept lattice, which organizes all concepts present in the analyzed data and allows the visualization of the dataset, navigation within particular subsets of the dataset and generally supports the exploration of the data. However the application of FCA in various fields is not always straightforward and the methodology of the usage differs. In this thesis the Formal Concept Analysis is applied on three different types of data - social networks, images and web pages. All of these types have become hot topics in the last decade. And all of them may represent large complex structures which can be difficult to inspect. The usage of FCA may be clearly helpful in this situation. But when it comes to practical applications of FCA, we often find that the obtained results are too complicated to interpret or the input data may be too large to even perform the analysis. As a secondary output of this thesis we may see the range of input data which is processable by the FCA. However the main topic of this thesis is the reduction which will allow the analysis of larger data. Although there are more approaches discussed, the thesis is focused on using matrix factorization methods to simplify the input data. These methods allow the input data to be broken down to parts and provide information about their importance. Therefore we are able to select the most important parts for further processing only. This thesis shows that this specific type of simplification leads to the reduction of concept lattice. Additionally, the properties of the reduction process are studied using both general and specific measures. The behaviour of the reduction is partially explained. A novel formalization is introduced allowing the separation of formal concepts into two groups, while each of them behaves differently during the reduction. This point of view has the potential for future analysis. This thesis contains a brief overview of object-attribute data and their scaling, introduction to the Formal Concept Analysis and basics of three matrix factorization methods - Singular Value Decomposition, Nonnegative Matrix Factorization and Semidiscrete Matrix Decomposition. My thesis is accompanied by an overview of the historical development of mentioned methods and related approaches. For the validation of the results various quantitative measures from different fields are used. Namely -- the general measures, such as entropy, normalized correlation dimension and Lorenz curves were used in conjunction with specific Formal Concept Analysis measures such as relatedness, closeness and concept stability. Attribute implications -- as different type of FCA output -- are also discussed.