Zeev Volkovich

We discuss a new approach for the proof of the Levy-Khintchine formula for the V -infinitely divisible laws. Our proof is based on a description of the conditionally positive definite functions as positive functionals on semi-normed algebras of suitable test functions. In the framework of this approach we obtain integral representations of the common continuous positive definite functions and the logarithms of characteristic functions of the ordinary infinitely divisible and V -infinitely divisible distribution.

The estimation of the suggested number of clusters in dataset is an ill posed problem of essential relevance in cluster analysis. A group (cluster) is characterized by a relatively high similarity among its elements in addition to a relatively low similarity to elements of other groups. High stability in partitions, obtained from the same data source, is logically classified as a high consistency of the clustering process. Thus, the number of clusters that maximizes cluster stability can serve as an estimator for the "true" number of clusters. In the current paper we consider a probabilistic approach to this problem resting upon the Gaussian clusters model. We claim that sequences of clustered samples can be interpreted as Gaussian distributed i.i.d. samples drawn from the same source, if the number of clusters is chosen correctly. The samples closeness, within the clusters, can be measured by means of the p-values, calculated for the appropriate Hotelling's T-square s...

In this paper we introduce a new heuristic approach for local clustering of the protein-protein interaction networks (PPIN), which can be applied to very large graphs. The method is based on idea of repeated bisections (rbr) proposed earlier for global clustering of PPIN. Each round of bisection is carried out by multilevel graph clusterization method realized by "Graculus" tool.

By this paper we contribute to an area of inverse problems with applications in medicine and further areas of science, engineering and social sciences. Most expressive gene selection is a form of the inverse problem where we start from given gene expressions level for DNA microarrays and intend to turn out the causes which can lead to such occurrences. In this paper, an effective gene selection algorithm is proposed for evaluating the gene expression differences in the compared datasets based on a two-sample distribution-free test statistics. We focus on the analysis of the genes activeness on a certain disease using intrinsic information about corresponding dataset structure. Further the inverse problem of parameter defining for 'active' gene set selection is formulated as follows: given set of 'active' genes, quality of sub diagnoses differentiation determines parameters for gene selection. The algorithm was evaluated on the Acute Lymphoblastic Leukemia (ALL) Dataset.

In this paper, an efficient gene selection algorithm is proposed, which employs a two-sample distribution-free test statistics for evaluating the gene expression di¤erences in the compared datasets. The experimen- tal results obtained for the Acute Lymphoblastic Leukemia (ALL) Dataset con…rm the e¢ ciency of the algorithm.

In this paper, a method for the study of cluster st ability is purposed. We draw pairs of samples from the data, according to two sampling distributions. The first distribution corresponds to the high density zones of data-elements distribution. Thus it is associated with the cluste rs cores. The second one, associated with the clust er margins, is related to the low density zones. The samples are clustered and the two obtained partitions are compared. The partitions are considered to be consistent if the obtained cluster s are similar. The resemblance is measured by the t otal number of edges, in the clusters minimal spanning trees, conn ecting points from different samples. We use the Fr iedman and Rafsky two sample test statistic. Under the homogen eity hypothesis, this statistic is normally distrib uted. Thus, it can expected that the true number of clusters correspon ds to the statistic empirical distribution which is closest to normal. Numerical experiments demonstrate the abi...

The exon-intron structures of fungi genes are quite different from each other, and the evolution of such struc-tures raises many questions. We tried to address some of these questions with an accent on methods of revealing evolu-tionary factors based on the analysis of gene exon-intron structures using statistical analysis. Taking whole genomes of fungi, we went through all the protein-coding genes in each chromosome separately and calculated the portion of intron-containing genes and average values of the net length of all the exons in a gene, the number of the exons, and the average length of an exon. We found striking similarities between all of these average properties of chromosomes of the same spe-cies and significant differences between properties of the chromosomes belonging to species of different divisions (Phyla) of the kingdom of Fungi. Comparing those chromosomal and genomic averages, we have developed a technique of clus-tering based on characteristics of the exon-intr...

A new representation for the characteristic function of the multivariate strictly geo-stable distribution is presented. The representation is appealing from a parametric viewpoint: its parameters have an intuitive probabilistic interpretation; and it is particularly useful for estimating the parameters of the geo-stable distribution.

Massive determination of complete genomes sequences has led to development of different tools for genome comparisons. Our approach is to compare genomes according to typical genomic distributions of a mathematical function that reflects a certain biological function. In this study we used comprehensive genome analysis of DNA curvature distributions before starts and after ends of prokaryotic genes to evaluate the