The integrated sigma-max system and its application in target recognition

Abstract

Randomness and fuzziness are two fundamental kinds of uncertain phenomena, which can be handled respectively by probability theory (sigma system) and possibility theory (max system). For many practical problems of information processing, random uncertainty may often co-exist with fuzzy uncertainty, which reminds us to look for a novel mechanism called sigma-max hybrid uncertainty inference. This mechanism can cope with randomness and fuzziness jointly and achieve a direct fusion of heterogeneous information modeled by probability or possibility. Specifically, we are interested in two typical forms of uncertainty inference, i.e., uncertainty update equation combining heterogeneous information as well as composition rule of heterogeneous relations. Such an objective is achieved by the adoption of joint description of a random variable and a fuzzy variable, i.e., hybrid distribution of probability and possibility, which lays the foundation for connecting the two uncertainty systems of probability and possibility, and results in the integrated sigma-max system. The hybrid distribution, other than joint probability (or possibility) distribution, has a couple of special normalization requirements that are order-dependent with respective to “addition” operation over random variable and “max’ operation over fuzzy variable. The derived sigma-max inference is applied to target recognition, an example of which is simulated to demonstrate the merit of the proposed method over the classic Bayesian classifier when randomness and fuzziness are both involved.