With the aim to propose a non parametric hypothesis test, this paper carries out a study on the Matching Error (ME), a comparison index of two partitions obtained from the same data set, using for example two clustering methods. This index is related to the misclassifica-tion error in supervised learning. Some properties of the ME and, especially, its distribution function for the case of two independent partitions are analyzed. Extensive simulations show the efficiency of the ME and we propose a hypothesis test based on it.
A cluster operator takes a set of data points and partitions the points into clusters (subsets). As with any scientific model, the scientific content of a cluster operator lies in its ability to predict results. This ability is measured by its error rate relative to cluster formation. To estimate the error of a cluster operator, a sample of point sets is generated, the algorithm is applied to each point set and the clusters evaluated relative to the known partition according to the distributions, and then the errors are averaged over the point sets composing the sample. Many validity measures have been proposed for evaluating clustering results based on a single realization of the random-point-set process. In this paper we consider a number of proposed validity measures and we examine how well they correlate with error rates across a number of clustering algorithms and random-point-set models. Validity measures fall broadly into three classes: internal validation is based on calculating properties of the resulting clusters; relative validation is based on comparisons of partitions generated by the same algorithm with different parameters or different subsets of the data; and external validation compares the partition generated by the clustering algorithm and a given partition of the data. To quantify the degree of similarity between the validation indices and the clustering errors, we use Kendall's rank correlation between their values. Our results indicate that, overall, the performance of validity indices is highly variable. For complex models or when a clustering algorithm yields complex clusters, both the internal and relative indices fail to predict the error of the algorithm. Some external indices appear to perform well, whereas others do not. We conclude that one should not put much faith in a validity score unless there is evidence, either in terms of sufficient data for model estimation or prior model knowledge, that a validity measure is well-correlated to the error rate of the clustering algorithm. Five external criteria were used to evaluate the extent of recovery of the true structure in a hierarchical clustering solution. This was accomplished by comparing the partitions produced by the clustering algorithm with the partition that indicates the true cluster structure known to exist in the data. The five criteria examined were the Rand, the Morey and Agresti adjusted Rand, the Hubert and Arabie adjusted Rand, the Jaccard, and the Fowlkes and Mallows measures. The results of the study indicated that the Hubert and Arabie adjusted Rank index was best suited to the task of comparison across hierarchy levels. Deficiencies with the other measures are noted. Cluster validation is an important part of any cluster analysis. External measures such as entropy, purity and mutual information are often used to evaluate K-means clustering. However, whether these measures are indeed suitable for K-means clustering remains unknown. Along this line, in this paper, we show that a data distribution view is of great use to selecting the right measures for K-means clustering. Specifically, we first introduce the data distribution view of K-means, and the resultant uniform effect on highly imbalanced data sets. Eight external measures widely used in recent data mining tasks are also collected as candidates for K-means evaluation. Then, we demonstrate that only three measures, namely the variation of information (VI), the van Dongen criterion (VD) and the Mirkin metric (M), can detect the negative uniform effect of K-means in the clustering results. We also provide new normalization schemes for these three measures, i.e., VI"n"o"r"m^', VD"n"o"r"m^' and M"n"o"r"m^', which enables the cross-data comparisons of clustering qualities. Finally, we explore some properties such as the consistency and sensitivity of the three measures, and give some advice on how to use them in K-means practice.