L/sub 1/-Norm based Fuzzy Clustering for Data with Tolerance

In this paper, the clustering algorithms for data with tolerance are constructed based on L
₁
-norm and the effectiveness is verified through numerical examples. First, two objective functions, which are based on SFCM-T and EFCM-T respectively, is defined. It is more complex to calculate exact solutions of these functions theoretically in the L
₁
-norm space than the L
₁
-norm space (Euclidean space) so that two methods to obtain the solutions are proposed. Next, two kinds of clustering algorithms based on L
₁
-norm are proposed using the two methods to obtain the exact solutions. Last, the effectiveness of the proposed algorithms is verified through the numerical examples of an artificial data set and the Iris data set.