Uncertainty measures for evidental reasoning. II: A new measure of total uncertainty
In Part I [\textit{N. Pal}, \textit{J. Bezdek} and \textit{R. Hemasinha}, Int. J. Approx. Reasoning 7, No. 3/4, 165-183 (1992; Zbl 0769.68112)] we discussed limitations of two measures of global (non-fuzzy) uncertainty of \textit{M. Lamata} and \textit{S. Moral} [Measures of entropy in the theory of evidence, Trab. Estad. 3, No. 1, 15-31 (1988; Zbl 0731.62015)] and a measure of total (non-fuzzy) uncertainty due to \textit{G. J. Klir} and \textit{A. Ramer} [Int. J. Gen. Syst. 18, No. 2, 155-166 (1990; Zbl 0732.60004)] and established the need for a new measure. We propose a set of intuitively desirable axioms for a measure of total uncertainty (TU) associated with a basic assignment \(m(A)\), and then derive an expression for a (unique) function that satisfies these requirements. Several theorems are proved about the new measure. Our measure is additive, and unlike other TU measures, has a unique maximum. The new measure reduces to Shannon's probabilistic entropy when the basic probability assignment focuses only on singletons. On the other hand, complete ignorance - basic assignment focusing only on the entire set, as a whole - reduces it to Hartley's measure of information. We show that the computational complexity of the new measure is \(O(N)\), whereas previous measures of \(TU\) are \(O(N^ 2)\). Finally, we compare the new measure to its predecessors by extending the numerical example of Part I so that it includes values of the new measure.
scientific article; zbMATH DE number 168019
scientific article