A model of granular data: a design problem with the Tchebyschev-based clustering

We introduce a model of granular data emerging through a summarization and processing of numeric data. It supports data analysis and casts it in the setting of data mining. The structure of data is revealed through the FCM equipped with the Tchebyschev (l_∞ ) metric. The study offers a novel contribution to a gradient-based learning of the prototypes developed in the l_∞ -based data space. The l_∞ metric promotes a development of easily interpretable information granules, namely hyperboxes. A detailed discussion of their geometry is provided. In particular, we discuss a deformation effect of the hyperbox-shape of granules due to an interaction between the granules. We also show how the clustering gives rise to a two-level topology of information granules. A core part of the topology comes in the form of hyperbox information granules. A residual structure is expressed through detailed, yet difficult to interpret, membership grades. Illustrative examples including synthetic data are studied