Robust Diagnostics of Fuzzy Clustering Results Using the Compositional Approach

Abstract

Fuzzy clustering, like the known fuzzy k-means method, allows to incorporate imprecision when classifying multivariate observations into clusters. In contrast to hard clustering, when the data are divided into distinct clusters and each data point belongs to exactly one cluster, in fuzzy clustering the observations can belong to more than one cluster. The strength of the association to each cluster is measured by a vector of membership coefficients. Usually, an observation is assigned to a cluster with the highest membership coefficient. On the other hand, the refinement of the hard membership coefficients enables to consider also the possibility of assigning to another cluster according to prior knowledge or specific data structure of the membership coefficients. The aim of the paper is to introduce a methodology to reveal the real data structure of multivariate membership coefficient vectors, based on the logratio approach to compositional data, and show how to display them in presence of outlying observations using loadings and scores of robust principal component analysis.