### Article

## Optimal decision for the market graph identification problem in a sign similarity network

Research into the market graph is attracting increasing attention in stock market analysis. One of the important problems connected with the market graph is its identification from observations. The standard way of identifying the market graph is to use a simple procedure based on statistical estimations of Pearson correlations between pairs of stocks. Recently a new class of statistical procedures for market graph identification was introduced and the optimality of these procedures in the Pearson correlation Gaussian network was proved. However, the procedures obtained have a high reliability only for Gaussian multivariate distributions of stock attributes. One of the ways to correct this problem is to consider different networks generated by different measures of pairwise similarity of stocks. A new and promising model in this context is the sign similarity network. In this paper the market graph identification problem in the sign similarity network is reviewed. A new class of statistical procedures for the market graph identification is introduced and the optimality of these procedures is proved. Numerical experiments reveal an essential difference in the quality between optimal procedures in sign similarity and Pearson correlation networks. In particular, it is observed that the quality of the optimal identification procedure in the sign similarity network is not sensitive to the assumptions on the distribution of stock attributes.

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.

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 multiple comparisons of several populations on small samples and specificity of the method of it solution are analyzed. It is proposed to extend a classical method for constructing statistical tests by the use of information preprocessing. Examples of the application of the proposed method are given.

In this paper we address the problem of forecasting the target events of a time series given the distribution ξξ of time gaps between target events. Strong earthquakes and stock market crashes are the two types of such events that we are focusing on. In the series of earthquakes, as McCann et al. show [W.R. Mc Cann, S.P. Nishenko, L.R. Sykes, J. Krause, Seismic gaps and plate tectonics: seismic potential for major boundaries, Pure and Applied Geophysics 117 (1979) 1082–1147], there are well-defined gaps (called seismic gaps) between strong earthquakes. On the other hand, usually there are no regular gaps in the series of stock market crashes [M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314 (2002) 749–755]. For the case of seismic gaps, we analytically derive an upper bound of prediction efficiency given the coefficient of variation of the distribution ξξ. For the case of stock market crashes, we develop an algorithm that predicts the next crash within a certain time interval after the previous one. We show that this algorithm outperforms random prediction. The efficiency of our algorithm sets up a lower bound of efficiency for effective prediction of stock market crashes.

Market graph is known to be a useful tool for market network analysis. Cliques and independent sets of the market graph give an information about con- centrated dependent sets of stocks and distributed independent sets of stocks on the market. In the present paper the connections between market graph and classical Markowitz portfolio theory are studied. In particular, efﬁcient frontiers of cliques and independent sets of the market graph are compared with the efﬁcient frontier of the market. The main result is: efﬁcient frontier of the market can be well ap- proximated by the efﬁcient frontier of the maximum independent set of the market graph constructed on the sets of stocks with the highest Sharp ratio. This allows to reduce the number of stocks for portfolio optimization without the loss of quality of obtained portfolios. In addition it is shown that cliques of the market graphs are not suitable for portfolio optimization.

The paper deal with uncertainty in market network analysis. The main problem addressed is to investigate statistical uncertainty of Kruskal algorithm for the minimum spanning tree in market network. Uncertainty of Kruskal algorithm is measured by the probability of q incorrectly included edges. Numerical experiments are conducted with the returns of a set of 100 financial instruments traded in the US stock market over a period of 250 days in 2014. Obtained results help to estimate the reliability of minimum spanning tree in market network analysis.

The geographic information system (GIS) is based on the first and only Russian Imperial Census of 1897 and the First All-Union Census of the Soviet Union of 1926. The GIS features vector data (shapefiles) of allprovinces of the two states. For the 1897 census, there is information about linguistic, religious, and social estate groups. The part based on the 1926 census features nationality. Both shapefiles include information on gender, rural and urban population. The GIS allows for producing any necessary maps for individual studies of the period which require the administrative boundaries and demographic information.

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.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in EXPNP, or even in EXP that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are *selfreducible*? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that EXPNP does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of EXP is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for NEXP.