Analysis of “mixing” combination rules and Smet’s combination rule

Abstract

The process of solution of different practical and ecological problems, using hyperspectral satellite images usually includes a procedure of classification. Classification is one of the most difficult and important procedures. Some image classification methods were considered and analyzed in this work. These methods are based on the theory of evidence. Evidence theory can simulate uncertainty and process imprecise and incomplete information. It were considered such combination rules in this paper: “mixing” combination rule (or averaging), convolutive x-averaging (or c-averaging) and Smet’s combination rule. It was shown, that these methods can process the data from multiple sources or spectral bands, that provide different assessments for the same hypotheses. It was noted, that the purpose of aggregation of information is to simplify data, whether the data is coming from multiple sources or different spectral bands. It was shown, that Smet’s rule is unnormalized version of Dempster rule, that applied in Smet’s Transferable Belief Model. It also processes imprecise and incomplete data. Smet’s combination rule entails a slightly different formulation of Dempster-Shafer theory. Mixing (or averaging) rule was considered in this paper too. It is the averaging operation that is used for probability distributions. This rule uses basic probability assignments from different sources (spectral bands) and weighs assigned according to the reliability of the sources. Convolutive x-averaging (or c-averaging) rule was considered in this paper too. This combination rule is a generalization of the average for scalar numbers. This rule is commutative and not associative. It also was noted, that convolutive x-averaging (c-averaging) rule can include any number of basic probability assignments. It were also considered examples, where these proposed combination rules were used. Mixing, convolutive x-averaging (c-averaging) rule and Smet’s combination rule can be applied for analysis of hyperspectral satellite images, in remote searching for minerals and oil, solving different environmental and thematic problems.