On the Role of Numerical Preciseness for Generalization, Classification, Type-1, and Type-2 Fuzziness

When performing data analysis on a computing device no mathematically idealized real number set IR is available. A basic resolution is given, so that a fuzzy model is in fact always a discrete model and not a continuous one. Due to the limited preciseness the computing device offers only a limited number of decimals in a limited discrete number space OR. This contribution considers effects on the generalization and fuzziness of data when replacing IR by IIR The effects are studied for data, that are numerically rounded or when intervals are considered. Often part of the data is missing or is of limited quality, so that it is of practical interest to consider the exact underlying space IIR and not the hypothetical space IR. We calculate precisely type-1 and type-2 fuzzy membership functions under preciseness assumptions of the elements in IIR.