Visualization of Fuzzy Clustering Result in Metric Space

Abstract

This paper presents a visualization of a result of fuzzy clustering. The feature of fuzzy clustering is to obtain the degree of belongingness of objects to fuzzy clusters so the result will be more commensurate with reality. In addition, the number of clusters requires less and the solution of the result will be more robust when compared with conventional hard clustering. In contrast, the fuzzy clustering result interpretation tends to be more complicated. Therefore, measuring the similarity (or dissimilarity) between a pair of fuzzy classification status of objects is important. In order to measure the similarity (or dissimilarity) mathematically, it is necessary to introduce a scale to the fuzzy clustering result. That is, the obtained solutions as a fuzzy clustering result must be in a metric space. In order to implement this, we have proposed multidimensional joint scale and cluster analysis. In this analysis, we exploit a scale obtained by multidimensional scaling. This paper clarifies that the multidimensional joint scale and cluster analysis introduces scale to the fuzzy clustering result and then the visualization of the fuzzy clustering result in the metric vector space has a theoretical mathematical meaning through the Euclidean distance structure. In this paper, this is shown by using several numerical comparisons with ordinary visualizations of the fuzzy clustering result.