Generating an Interpretable Family of Fuzzy Partitions From Data

Abstract

In this paper, we propose a new method to construct fuzzy partitions from data. The procedure generates a hierarchy including best partitions of all sizes from n to two fuzzy sets. The maximum size n is determined according to the data distribution and corresponds to the finest resolution level. We use an ascending method for which a merging criterion is needed. This criterion is based on the definition of a special metric distance suitable for fuzzy partitioning, and the merging is done under semantic constraints. The distance we define does not handle the point coordinates, but directly their membership degrees to the fuzzy sets of the partition. This leads to the introduction of the notions of internal and external distances. The hierarchical fuzzy partitioning is carried independently over each dimension, and, to demonstrate the partition potential, they are used to build fuzzy inference system using a simple selection mechanism. Due to the merging technique, all the fuzzy sets in the various partitions are interpretable as linguistic labels. The tradeoff between accuracy and interpretability constitutes the most promising aspect in our approach. Well known data sets are investigated and the results are compared with those obtained by other authors using different techniques. The method is also applied to real world agricultural data, the results are analyzed and weighed against those achieved by other methods, such as fuzzy clustering or discriminant analysis.