A convolution-based distance measure for fuzzy singletons and its application to a pattern recognition problem

Abstract

A new method to measure the distance between fuzzy singletons (FSNs) is presented. It first fuzzifies a crisp number to a generalized trapezoidal fuzzy number (GTFN) using the Mamdani fuzzification method. It then treats an FSN as an impulse signal and transforms the FSN into a new GTFN by convoluting it with the original GTFN. In so doing, an existing distance measure for GTFNs can be used to measure distance between FSNs. It is shown that the new measure offers a desirable behavior over the Euclidean and weighted distance measures in the following sense: Under the new measure, the distance between two FSNs is larger when they are in different GTFNs, and smaller when they are in the same GTFN. The advantage of the new measure is demonstrated on a fuzzy forecasting trading system over two different real stock markets, which provides better predictions with larger profits than those obtained using the Euclidean distance measure for the same system.