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## On the Preservation of Comparison of Distorted Histograms

The necessity of comparison of histograms with the help of relationship of type “more or less” arises in many problems of decision-making. There are many approaches to solve this problem. But the histograms can be distorted. Then, we have to find the conditions on the distortions under which the comparison of the two histograms does not change. The solution of this problem is researched in the paper with respect to three popular probabilistic methods of comparison.

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This paper is devoted to study of stability of comparison of histograms with help of different probability methods. Background: The comparison of histograms is necessary in many applied problems of data processing. The comparison of type ”more-less” is considered in this paper. But the histograms may be distorted. The nature of these distortions can be different. Then we have a problem to find the conditions on distortions under which the comparison of the two histograms is not changed. Methods: There are many approaches to comparison of histograms. The three popular proba- bilistic methods of comparison of histograms are considered in this paper: comparison of math- ematical expectations, comparison with help of principle of stochastic dominance, comparison with help of stochastic precedence. We consider the interval distortions of histograms in this paper. Results:The necessary and sufficient conditions of preservation for comparison of distorted his- tograms found with respect to different probability indices of comparison. The description of set of admissible distortions preserving the comparison of two histograms found. The characteristics of stability of histograms to distortion are introduced. These characteristics are calculated for histograms of USE (Unified State Exam) of applicants admitted in 2012 in Russian universities. It is shown that the stability of comparison of histograms to distortion can does not correspond to the values of difference index of comparison (margin). Conclusions: The found conditions invariability of comparing histograms can be used to es- timate the reliability of results of different rankings, data processing, etc. in terms of different types of uncertainty: stochastic uncertainty, the uncertainty associated with the distortion of the data in filling data gaps, etc.

We study the problem of the stability of probabilistic methods of comparing histograms in case of their distortions. The comparing of histograms is given by preorder relation on the set of histograms. This relation should to be agreed with the condition of ordering of arguments histograms by ascendingly their importance. The pointwise interval uncorrelated changes are meant by the distortions of histograms. Necessary and sufficient conditions on the level of distortion of histograms are found under which the comparison of the two histograms is not changed. The study was conducted for three popular probabilistic methods of comparison: a) using the mathematical expectation; b) using the distribution function (stochastic dominance); c) using probability of inequalities. Proved approval illustrated by studies of stability of comparing histograms of USE applicants admitted to universities.

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

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.

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.