Various Representations and Algebraic Structure of Linear Imprecision Indices

The paper is devoted to the investigation of imprecision indices, introduced in [7]. They are used for evaluation of uncertainty (or more exactly imprecision), which is contained in information given by fuzzy (nonadditive) measures, in particular, by lower or upper probabilities. We argue that there exist various types of uncertainty, for example, randomness, investigated in probability theory, imprecision, described by interval calculi, inconsistency, incompleteness, fuzziness and so on. In general these types of uncertainty have very complex behavior, caused by their interaction. Therefore, the choice of uncertainty measures is not unique, and is defined by the problems addressed. The classical uncertainty measures are Shannon’s entropy and Hartley’s measure. In the paper imprecision indices and their linear representatives are introduced axiomatically. The system of axioms enables to define various imprecision indices. So we investigate the algebraic structure of all imprecision indices and describe their families with best properties.