Pablo Andretta Jaskowiak

The area under the receiver operating characteristics (ROC) Curve, referred to as AUC, is a well-known performance measure in the supervised learning domain. Due to its compelling features, it has been employed in a number of studies to evaluate and compare the performance of different classifiers. In this work, we explore AUC as a performance measure in the unsupervised learning domain, more specifically, in the context of cluster analysis. In particular, we elaborate on the use of AUC as an internal/relative measure of clustering quality, which we refer to as Area Under the Curve for Clustering (AUCC). We show that the AUCC of a given candidate clustering solution has an expected value under a null model of random clustering solutions, regardless of the size of the dataset and, more importantly, regardless of the number or the (im)balance of clusters under evaluation. In addition, we elaborate on the fact that, in the context of internal/relative clustering validation as we consider, AUCC is actually a linear transformation of the Gamma criterion from Baker and Hubert (1975), for which we also formally derive a theoretical expected value for chance clusterings. We also discuss the computational complexity of these criteria and show that, while an ordinary implementation of Gamma can be computationally prohibitive and impractical for most real applications of cluster analysis, its equivalence with AUCC actually unveils a much more efficient algorithmic procedure. Our theoretical findings are supported by experimental results. These results show that, in addition to an effective and robust quantitative evaluation provided by AUCC, visual inspection of the ROC curves themselves can be useful to further assess a candidate clustering solution from a broader, qualitative perspective as well.

Background: Clustering is crucial for gene expression data analysis. As an unsupervised exploratory procedure its results can help researchers to gain insights and formulate new hypothesis about biological data from microarrays. Given different settings of microarray experiments, clustering proves itself as a versatile exploratory tool. It can help to unveil new cancer subtypes or to identify groups of genes that respond similarly to a specific experimental condition. In order to obtain useful clustering results, however, different parameters of the clustering procedure must be properly tuned. Besides the selection of the clustering method itself, determining which distance is going to be employed between data objects is probably one of the most difficult decisions. Results and conclusions: We analyze how different distances and clustering methods interact regarding their ability to cluster gene expression, i.e., microarray data. We study 15 distances along with four common clustering methods from the literature on a total of 52 gene expression microarray datasets. Distances are evaluated on a number of different scenarios including clustering of cancer tissues and genes from short time-series expression data, the two main clustering applications in gene expression. Our results support that the selection of an appropriate distance depends on the scenario in hand. Moreover, in each scenario, given the very same clustering method, significant differences in quality may arise from the selection of distinct distance measures. In fact, the selection of an appropriate distance measure can make the difference between meaningful and poor clustering outcomes, even for a suitable clustering method.

Abstract One of the most challenging aspects of clustering is validation, which is the objective and quantitative assessment of clustering results. A number of different relative validity criteria have been proposed for the validation of globular, clusters. Not all data, however, are composed of globular clusters. Density-based clustering algorithms seek partitions with high density areas of points (clusters, not necessarily globular) separated by low density areas, possibly containing noise objects. In these cases relative validity indices proposed for globular cluster validation may fail. In this paper we propose a relative validation index for density-based, arbitrarily shaped clusters. The index assesses clustering quality based on the relative density connection between pairs of objects. Our index is formulated on the basis of a new kernel density function, which is used to compute the density of objects and to evaluate the within- and between-cluster density connectedness of clustering results. Experiments on synthetic and real world data show the effectiveness of our approach for the evaluation and selection of clustering algorithms and their respective appropriate parameters.