Numerous entropy-type characteristics (functionals) generalizing Rényi entropy are widely used in mathematical statistics, physics, information theory, and signal processing for characterizing uncertainty in probability distributions and distribution identification problems. We consider estimators of some entropy (integral) functionals for discrete and continuous distributions based on the number of epsilon-close vector records in the corresponding independent and identically distributed samples from two distributions. The estimators form a triangular scheme of generalized U -statistics. We show the asymptotic properties of these estimators (e.g., consistency and asymptotic normality). The results can be applied in various problems in computer science and mathematical statistics (e.g., approximate matching for random databases, record linkage, image matching).

BackgroundClustering of gene expression data is widely used to identify novel subtypes of cancer. Plenty of clustering approaches have been proposed, but there is a lack of knowledge regarding their relative merits and how data characteristics influence the performance. We evaluate how cluster analysis choices affect the performance by studying four publicly available human cancer data sets: breast, brain, kidney and stomach cancer. In particular, we focus on how the sample size, distribution of subtypes and sample heterogeneity affect the performance.ResultsIn general, increasing the sample size had limited effect on the clustering performance, e.g. for the breast cancer data similar performance was obtained for n = 40 as for n = 330. The relative distribution of the subtypes had a noticeable effect on the ability to identify the disease subtypes and data with disproportionate cluster sizes turned out to be difficult to cluster. Both the choice of clustering method and selection method affected the ability to identify the subtypes, but the relative performance varied between data sets, making it difficult to rank the approaches. For some data sets, the performance was substantially higher when the clustering was based on data from only one sex compared to data from a mixed population. This suggests that homogeneous data are easier to cluster than heterogeneous data and that clustering males and females individually may be beneficial and increase the chance to detect novel subtypes. It was also observed that the performance often differed substantially between females and males.ConclusionsThe number of samples seems to have a limited effect on the performance while the heterogeneity, at least with respect to sex, is important for the performance. Hence, by analyzing the genders separately, the possible loss caused by having fewer samples could be outweighed by the benefit of a more homogeneous data.

Cancer subtype identification is important to facilitate cancer diagnosis and select effective treatments. Clustering of cancer patients based on high-dimensional RNA-sequencing data can be used to detect novel subtypes, but only a subset of the features (e.g., genes) contains information related to the cancer subtype. Therefore, it is reasonable to assume that the clustering should be based on a set of carefully selected features rather than all features. Several feature selection methods have been proposed, but how and when to use these methods are still poorly understood. Thirteen feature selection methods were evaluated on four human cancer data sets, all with known subtypes (gold standards), which were only used for evaluation. The methods were characterized by considering mean expression and standard deviation (SD) of the selected genes, the overlap with other methods and their clustering performance, obtained comparing the clustering result with the gold standard using the adjusted Rand index (ARI). The results were compared to a supervised approach as a positive control and two negative controls in which either a random selection of genes or all genes were included. For all data sets, the best feature selection approach outperformed the negative control and for two data sets the gain was substantial with ARI increasing from (−0.01, 0.39) to (0.66, 0.72), respectively. No feature selection method completely outperformed the others but using the dip-rest statistic to select 1000 genes was overall a good choice. The commonly used approach, where genes with the highest SDs are selected, did not perform well in our study.

The Rényi entropy is a generalization of the Shannon entropy and is widely used in mathematical statistics and applied sciences for quantifying the uncertainty in a probability distribution. We consider estimation of the quadratic Rényi entropy and related functionals for the marginal distribution of a stationary m-dependent sequence. The U -statistic estimators under study are based on the number of ǫ-close vector observations in the corresponding sample. A variety of asymptotic properties for these estimators are obtained (e.g., consistency, asymptotic normality, Poisson convergence). The results can be used in diverse statistical and computer science problems whenever the conventional independence assumption is too strong (e.g., ǫ-keys in time series databases, distribution identification problems for dependent samples).

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