Fuzzy clustering of data with uncertainties using minimum and maximum distances based on L1 metric

Fuzzy c-means is well-known among the various methods of fuzzy cluster analysis. L₁-based fuzzy c-means has also been studied in recent years. This paper discusses the L₁-based fuzzy c-means of data with fuzzy uncertainties. The data unit is supposed to be the Cartesian product of fuzzy numbers. The metric between a data unit with uncertainty and a cluster center is defined using minimum and maximum distances. The fuzzy c-means algorithm is an alternative procedure for the optimization of the cluster center and the fuzzy set membership, while the solution of the cluster center for uncertain data cannot be obtained directly. An algorithm for the solution of cluster centers based on the L₁ metric for uncertain data is developed in this paper. Using this algorithm, an exact alternate optimization procedure is obtained. Numerical examples show that the results for uncertain data are different from the results for data without uncertainties