Information and measurement

Abstract

Measurements are basic for all quantitative science and for many human activities. By the limited accuracy of every measurement equipment the result of one measurement of a continuous quantity is not a precise number but more or less non-precise containing different kinds of uncertainty. Besides systematic errors and random errors individual measurement results are also subject to another type of uncertainty, so-called fuzziness. This unavoidable imprecision has to be analyzed in order to obtain realistic results. It turns out that special fuzzy subsets x* of the set of real numbers R, called fuzzy numbers, are useful to model fuzziness of measurement results. Using the concept of fuzzy numbers a more realistic description of measurement results is possible. In the thesis fuzzy numbers, fuzzy vectors and vectors of fuzzy numbers are defined. I have proposed and proved a characterization of membership functions of fuzzy sets, and especially a characterization of characterizing functions of fuzzy numbers. I have proposed and proved a generalization of the extension principle for fuzzy vectors, and given a graphical visualization. I have proposed different methods of how to construct the characterizing function of a fuzzy number and the vector-characterizing function of a fuzzy vector, explained all the methods by examples. I have explained a method for computing with fuzzy numbers, and compared the resulting characterizing functions depending on the used t-norm. Fuzzy valued functions are introduced and I have proved a statement about its integral. At the end of the thesis, I have demonstrated how to apply fuzzy concepts to measurement theory. The dissertation thesis is written in a formally mathematical way using definitions, theorems, proofs and examples. I have explained in detail various approaches how to use fuzzy numbers through their representation via characterizing functions. I have shown the presented methods on examples including detailed figures to enable better understanding. A significant part of the work was published as a requested paper in the impact journal "Iranian Journal of Fuzzy Systems". The dissertation thesis can serve as theoretical background for experts engaging with measurement in practice.