Volume and Surface Area-Based Cluster Validity Index

Abstract

Cluster validity index plays an important role in assessing the quality of clustering results. However, most of the existing validity indices take a trial-and-error strategy, and their correctness depend on not only the measurements of intra-and inter-cluster distances but also the specific clustering algorithms and data structures. Consequently, the applications of these indices are limited in practice. In this paper, we firstly define the total surface area and volume of all clusters in a 2-dimensinal data space, thereby recovering their natural interrelation among various numbers of clusters. On this basis, a novel validity index is proposed to directly assess the clustering results of any dataset, which does not require any trail-and-error process, clustering algorithms, data structures, or the measurements of intra-and inter-cluster distances. In the case of a high-dimensional data space, all clusters are transformed into spherical clusters of normalized size in a 2-dimensinal data space through a multidimensional scaling transformation. Two groups of typical synthetic datasets and real datasets with various characteristics are used to validate the novel validity index.INDEX TERMS Cluster validity index, multidimensional scaling transformation, volume and surface area.

Toolbox for Distance Estimation and Cluster Validation on Data With Missing Values

“…However, the assumption that a data set contains clusters with spherical shapes can be unrealistic, making the clustering and cluster validation problems more challenging. In [32], a new approach for transforming and normalizing an arbitrarily shaped subset of data to an approximately spherical shape with a specified radius was introduced. The method is based on the notation of chains around high-density key points.…”

“…Density-based connections are created until the key points are visited. In [32], the number of key points M was suggested to be selected as…”

“…Key point selection (presented in Section III-D1) was used in the cluster validation. The number of key points can be fixed to || √ N ||, as recommended in [32]. However, we provided two modified versions of the original algorithm based on key point pruning, i.e., the algorithms started from the given maximum for the key points and then removed irrelevant points one by one.…”

“…The second part compares clustering methods and cluster validation indices on two-dimensional (2D) data sets with missing values. In addition, the validation results, which are based on a key point selection function [32], are given. The results are validated against the reference results given in previously published research papers.…”