Scaling, Granulation, and Fuzzy Attributes in Formal Concept Analysis

The present paper deals with scaling within the framework of formal concept analysis (FCA) of data with fuzzy attributes. In ordinary FCA, the input is a data table with yes/no attributes. Scaling is a process of transformation of data tables with general attributes, e.g. nominal, ordinal, etc., to data tables with yes/no attributes. This way, data tables with general attributes can be analyzed by means of FCA. We propose a new way of scaling, namely, scaling of general attributes to fuzzy attributes. After such a scaling, the data can be analyzed by means of FCA developed for data with fuzzy attributes. Compared to ordinary scaling to yes/no attributes, our scaling procedure is less sensitive to how a user defines a scale which eliminates the arbitrariness of user's definition of a scale. This is the main advantage of our approach. In addition, scaling to fuzzy attributes is appealing from the point of view of knowledge representation and is connected to Zadeh's concept of linguistic variable. We present a general definition of scaling, examples comparing our approach to ordinary scaling, and theorems which answer some naturally arising questions regarding sensitivity of FCA to the definition of a scale.