Empirical Distribution of Non-Precise Data: Roughness and Fuzziness

In many fields, especially in environmetrics and social sciences, it is impossible to obtain exact quantitative data about a variable of interest. Many researchers have suggested that vague, non-precise observations should be described by fuzzy sets. Fuzzy set theory originated by Zadeh (1965) relies on ordering relations that express intensity (degree) of membership of an object in a set. Applications of statistical methods to fuzzy data can be found in several monographs, for example, has proposed rough set methodology as a new approach in handling classificatory analysis of vague concepts. In this methodology any vague concept is characterized by a pair of precise concepts called the lower and the upper approximations. Rough set theory is based on equivalence relations describing partitions made of classes of indiscernible objects. This new approach proved to be useful in many applications. For reference see, e.g., Ziarko (1994) or Polkowski and Skowron (1998). Several studies have already been conducted about combinations of rough and fuzzy sets. The notions of rough fuzzy sets and fuzzy rough sets were introduced by several researchers, among them, e.g., Dubois and Prade (1990), or Kuchneva (1992). Suppose that a finite number of non-precise measurements of quantity X is given. These measurements create a non-precise sample S. Suppose that a family C of crisp or fuzzy concepts is defined on the domain of X. The following questions arise: • How to classify non-precise observations into non-precise classes? • How exact (rough) is the result of classification? • How to reduce fuzziness of non-precise observations? The first question can be answered by using some techniques based on degrees of inclusion of fuzzy sets. A method for calculating a fuzzy frequency function from sampling data will be suggested. This function gives the fuzzy proportions (or fuzzy counts) of fuzzy numbers in fuzzy categories (intervals). Any measure of fuzziness of a fuzzy frequency function provides a quantitative evaluation of fuzziness of empirical distribution of non-precise data. The fuzzy frequency function can be graphically represented by a generalized histogram (Bodjanova, 1999). The classical histogram is a special case of the generalized histogram. The general form of the fuzzy frequency function allows customization by choosing the measurement of the degree of inclusion of an observation into a group, which is appropriate for the data set under consideration. Exactness (roughness) of classification of objects from a non-precise sample S into fuzzy categories from C depends on the …