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. *

BackgroundClustering 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 conclusionsWe 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.

BackgroundSeveral missing value imputation methods for gene expression data have been proposed in the literature. In the past few years, researchers have been putting a great deal of effort into presenting systematic evaluations of the different imputation algorithms. Initially, most algorithms were assessed with an emphasis on the accuracy of the imputation, using metrics such as the root mean squared error. However, it has become clear that the success of the estimation of the expression value should be evaluated in more practical terms as well. One can consider, for example, the ability of the method to preserve the significant genes in the dataset, or its discriminative/predictive power for classification/clustering purposes.Results and conclusionsWe performed a broad analysis of the impact of five well-known missing value imputation methods on three clustering and four classification methods, in the context of 12 cancer gene expression datasets. We employed a statistical framework, for the first time in this field, to assess whether different imputation methods improve the performance of the clustering/classification methods. Our results suggest that the imputation methods evaluated have a minor impact on the classification and downstream clustering analyses. Simple methods such as replacing the missing values by mean or the median values performed as well as more complex strategies. The datasets analyzed in this study are available at http://costalab.org/Imputation/.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-015-0494-3) contains supplementary material, which is available to authorized users.

Cluster analysis is usually the first step adopted to unveil information from gene expression microarray data. Besides selecting a clustering algorithm, choosing an appropriate proximity measure (similarity or distance) is of great importance to achieve satisfactory clustering results. Nevertheless, up to date, there are no comprehensive guidelines concerning how to choose proximity measures for clustering microarray data. Pearson is the most used proximity measure, whereas characteristics of other ones remain unexplored. In this paper, we investigate the choice of proximity measures for the clustering of microarray data by evaluating the performance of 16 proximity measures in 52 data sets from time course and cancer experiments. Our results support that measures rarely employed in the gene expression literature can provide better results than commonly employed ones, such as Pearson, Spearman, and euclidean distance. Given that different measures stood out for time course and cancer data evaluations, their choice should be specific to each scenario. To evaluate measures on time-course data, we preprocessed and compiled 17 data sets from the microarray literature in a benchmark along with a new methodology, called Intrinsic Biological Separation Ability (IBSA). Both can be employed in future research to assess the effectiveness of new measures for gene time-course data.

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Decision tree induction algorithms are widely used in knowledge discovery and data mining, specially in scenarios where model comprehensibility is desired. A variation of the traditional univariate approach is the so-called oblique decision tree, which allows multivariate tests in its non-terminal nodes. Oblique decision trees can model decision boundaries that are oblique to the attribute axes, whereas univariate trees can only perform axis-parallel splits. The majority of the oblique and univariate decision tree induction algorithms perform a top-down strategy for growing the tree, relying on an impurity-based measure for splitting nodes. In this paper, we propose a novel bottom-up algorithm for inducing oblique trees named BUTIA. It does not require an impurity-measure for dividing nodes, since we know a priori the data resulting from each split. For generating the splitting hyperplanes, our algorithm implements a support vector machine solution, and a clustering algorithm is used for generating the initial leaves. We compare BUTIA to traditional univariate and oblique decision tree algorithms, C4.5, CART, OC1 and FT, as well as to a standard SVM implementation, using real gene expression benchmark data. Experimental results show the effectiveness of the proposed approach in several cases.

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