Perspective-oriented data analysis through the development of information granules of order 2

The problem of interest in this study is to describe and quantify a structure of data sets X₁,X₂,…,X_p with the use of a certain referential structure (composed of a collection of referential fuzzy sets) constructed on the basis of some previously available data X. The essence of the proposed approach is to carry out clustering in any X_i completed in a new granular feature space constructed with the aid of referential fuzzy sets. As a result, the clusters formed in X_i in this way emerge in the form of fuzzy sets of order-2. The lack of precision (variability) being associated with this description is quantified with the aid of entropy measure and directly relates the new structure with the notion of surprise (unexpectedness, interestingness) of the concepts and anomalies occurring in the data. Experimental studies are reported for synthetic data and real-world multivariable time series.