Fuzzy Specification of Uncertain Knowledge and Vague or Imprecise Information

Abstract

Impreciseness and vagueness are facets of uncertainty and mean that an object or some of its features can not be recorded or described precisely. Hence both must be considered in contrast with randomness, which describes variability, being another kind of uncertainty. Whereas probability theory and mathematical statistics deal with the behaviour of (perhaps hypothetical) populations, impreciseness and vagueness are concerned with each single piece of information, called a datum. For handling such data they must be described as mathematical items. The “classical” model for impreciseness is given by set-value-description, e.g. by intervals in the simplest case. However, as is known, the main problem in application of interval mathematics is fixing precise ends of the intervals. Moreover, data can be given by verbal descriptions. Then they are called vague data, usually coded by numbers or letters. However, in such a form they lose much of their semantic meaning, which would be very important for processing them and for interpreting the conclusions drawn from this processing. In the paper some examples are provided how such imprecise or vague data had been specified by fuzzy sets in real-world applications, e.g. measuring from blurred pictures in the micro area and comparing imprecise sample-spectrograms with standard spectrograms. Methods included are, e.g., using grey-tone levels in pictures on the screen and using structure elements from mathematical morphology for specifying fuzzy functions. Finally a numerical example is considered to show how impreciseness influences the results of simple statistical procedures.