Fuzzy hypothesis testing with hybrid data

The formulation of binary hypothesis testing, where the available data under one hypothesis is a superposition of a random and a fuzzy component, is addressed. In this formulation the likelihood ratio, which is normally utilized in such types of decision making problems, is fuzzified and compared to a threshold. This involves the application of principles of ordering fuzzy sets which have been developed elsewhere by the authors. Decision rules are described which are independent of the shape of the membership function representing the fuzzy data. These decision rules are illustrated for the case where the random component of the data is normally distributed. Probability of error performance curves, using different ordering criteria, are obtained and compared to the performance of non-fuzzy hypothesis testing.