### Article

## Comparative Analysis of Two Similarity Measures for the Market Graph Construction

Market graph is built on the basis of some similarity measure for financial asset returns. The paper considers two similarity measures: classic Pearson correlation and sign correlation. We study the associated market graphs and compare the conditional risk of the market graph construction for these two measures of similarity. Our main finding is that the conditional risk for the sign correlation is much better than for the Pearson correlation for larger values of threshold for several probabilistic models. In addition, we show that for some model the conditional risk for sign correlation dominates over the conditional risk for Pearson correlation for all values of threshold. These properties make sign correlation a more appropriate measure for the maximum clique analysis.

This article is about the comparative verse research with the new probability models and statistical methods in the comparative vese studies.

A common network representation of the stock market is based on correlations of time series of return fluctuations. It is well known that financial time series have a stochastic nature. Therefore there is uncertainty in inference about filtered structures in market network. Thus market network analysis need to be complemented by estimation of uncertainty of the obtained results. However as far as we know there are no relevant research in the literature. In the present paper we maake the first step in this direction. We propose the approach to measure statistical uncertainty of different market network structures. This approach is based on conditional risk for corresponding multiple decision statistical procedures. The proposed appoach is illustrated by numerical evaluation of statistical ucertainty for popular network structures. Our experimental study validates the possibility of application of the approach for comparison of uncerttainty of different network structures.

Problem of construction of the market graph as a multiple decision statistical problem is considered. Detailed description of a optimal unbiased multiple decision statistical procedure is given. This procedure is constructed using the Lehmann’s theory of multiple decision statistical procedures and the conditional tests of the Neyman structures. The equations for thresholds calculation for the tests of the Neyman structure are presented and analyzed.

The paper presents an analysis of the stocks traded on MICEX from 2007 to 2011. In order to analyze the data, we construct a market graph model. The vertices of the graph represent stocks; the edges represent strong similarity between considered stocks returns. We suggest using the following way to calculate the similarity measure: we calculate the number of the periods when two considered stocks have the positive return simultaneously. Our results show that the market graph model with the suggested similarity measure can be used to describe the stock market dynamics in an effi- cient and concise manner.

Full texts of third international conference on data analytics are presented.

The paper presents the analysis of the network model referred to as market graph of the BRIC countries stock markets. We construct the stock market graph as follows: each vertex represents a stock, and the vertices are adjacent if the price correlation coefficient between them over a certain period of time is greater than or equal to specified threshold. The market graphs are constructed for different time periods to understand the dynamics of their characteristics such as correlation distribution histogram, mean value and standard deviation, size and structure of the maximum cliques. Our results show that we can split the BRIC countries into two groups. Brazil, Russia and India constitute the first group, China constitutes the second group.

We consider a market graph model of the Russian stock market. To study the peculiarity of the Russian market we construct the market graphs for different time periods from 2007 to 2011. As characteristics of constructed market graphs we use the distribution of correlations, size and structure of maximum cliques, and relationship between return and volume of stocks. Our main finding is that for the Russian market there is a strong connection between the volume of stocks and the structure of maximum cliques for all periods of observations. Namely, the most attractive Russian stocks have a strongest correlation between their returns. At the same time as far as we are aware this phenomenon is not related to the well developed USA stock market.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.