Centroids of fuzzy sets when membership functions have spikes

Centroids are a common defuzzification method for fuzzy sets defined on the real line, and thus are practically important in type-1 and type-2 fuzzy logic systems. We propose a modified definition of the centroid that takes account of singleton spikes in membership. Such spikes are encountered in aggregates that combine fuzzy sets whose support is discrete with fuzzy sets whose support is defined on an interval. Our modified definition avoids a computational problem with the choice of discretization in the standard definition when spikes are present, and reduces to the usual one in the absence of spikes.