Granular mappings

We are concerned with the granular representation of mappings (or experimental data) coming in the form R:R→[0,1] (for one-dimensional cases) and R:Rⁿ→[0,1] (for multivariable cases) with R being a set of real numbers. As the name implies, a granular mapping is defined over information granules and maps them into a collection of granules expressed in some output space. The design of the granular mapping is discussed in the case of set and fuzzy set-based granulation. The proposed development is regarded as a two-phase process that comprises: 1) a definition of an interaction between information granules and experimental evidence or existing numeric mapping and 2) the use of these measures of interaction in building an explicit expression for the granular mapping. We show how to develop information granules in case of multidimensional numeric data by resorting to fuzzy clustering (fuzzy C-means). Experimental results serve as an illustration of the proposed approach.