The design of granular classifiers: A study in the synergy of interval calculus and fuzzy sets in pattern recognition

In this study, we are concerned with a development of a certain category of granular classifiers referred here to as hyperbox-driven classifiers (HDC). The approach fully capitalizes on the two key technologies of granular computing, namely set (interval) calculus and fuzzy sets in dealing with a description of geometry of patterns (data) belonging to a certain category. We take advantage of the capabilities of sets (intervals and their Cartesian products) when describing a core structure of classes of patterns in the form of some hyperboxes. Their combinations are referred to as a core structure of the feature space. Next, we refine the geometry of the classifier by bringing forward the concepts of regions of the feature space characterized by fuzzy sets. They are sought as a secondary structure. The construction of the core structure is realized by means of the supervised version of DBSCAN—one of the popular clustering algorithms of data mining. The secondary structure is described by fuzzy sets whose membership functions are a solution to some optimization problem of allocation of degrees of belongingness. In the formation of the secondary structure we exploit a concept of the Hausdorff distance that determines a distance between some information granule (of a well defined geometry) and a numeric pattern (a point in the highly dimensional feature space). The two-stage description of the granular classifier is also beneficial in the characterization of the classification error. By virtue of the design of the core structure, the classification error occurring there assumes very small, almost zero values. On the other hand, the secondary structure coming with a substantial level of overlap between classes is characterized by relatively higher values of the classification error. Being cognizant of that, we suggest a way of forming a synthetic measure built on the membership values and named a separability index that helps quantify the classification error in this setting. A series of numeric examples are used to demonstrate the effectiveness of the proposed granular classifiers.