This research monograph presents basic foundational aspects for a theory of statistics with fuzzy data, together with a set of practical applications. Fuzzy data are modeled as observations from random fuzzy sets. Theories of fuzzy logic and of random closed sets are used as basic ingredients in building statistical concepts and procedures in the context of imprecise data, including coarse data analysis. The monograph also aims at motivating statisticians to look at fuzzy statistics to enlarge the domain of applicability of statistics in general. Hung T. Nguyen is a professor of Mathematical Sciences at New Mexico State University, USA. Berlin Wu is a professor of Mathematical Sciences at National Chengchi University, Taipei, Taiwan.
This monograph aims at laying down a rigorous framework for statistical analysis of fuzzy data. By fuzzy data we mean imprecise data which are recorded linguistically, i.e. expressed in some natural language as opposed to precise numerical measurements. Clearly, this type of data is more complex and general than set-valued observations (in, say, coarse data) which generalize data in statistical multivariate analysis.
Fuzzy data need to be modeled mathematically before they can be subject to analysis. In this monograph, we will model fuzzy data as fuzzy sets in the sense of Zadeh. Postulating that those data are generated by random mechanisms, we will proceed to formulate the basic concept of random fuzzy sets as bona fide random elements in appropriate metric spaces. With this mathematical model for populations, we are entirely in the framework of standard statistical analysis. With this goal, this monograph is in fact more oriented towards probabilistic foundations than statistical procedures. We formulate, however, basic problems in statistics such as estimation, testing and prediction. We will also illustrate statistical methods for analyzing fuzzy data through some studies.