## No 4 (2020)

ESTIMATION OF THE NUMBER OF SUMMANDS OF THE CENTRAL LIMIT THEOREM

#### Abstract

The problem of estimating the number of summands of random variables for a total normal distribution law or a sample average with a normal distribution is investigated. The Central limit theorem allows us to solve many complex applied problems using the developed mathematical apparatus of the normal probability distribution. Otherwise, we would have to operate with convolutions of distributions that are explicitly calculated in rare cases. The purpose of this paper is to theoretically estimate the number of terms of the Central limit theorem necessary for the sum or sample average to have a normal probability distribution law. The article proves two theorems and two consequences of them. The method of characteristic functions is used to prove theorems. The first theorem States the conditions under which the average sample of independent terms will have a normal distribution law with a given accuracy. The corollary of the first theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 1. The second theorem defines the normal distribution conditions for the average sample of independent random variables whose mathematical expectations fall in the same interval, and whose variances also fall in the same interval. The corollary of the second theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 2. According to the formula relations proved in theorem 1, a table of the required number of terms in the Central limit theorem is calculated to ensure the specified accuracy of approximation of the distribution of the values of the sample average to the normal distribution law. A graph of this dependence is constructed. The dependence is well approximated by a polynomial of the sixth degree. The relations and proved theorems obtained in the article are simple, from the point of view of calculations, and allow controlling the testing process for evaluating students ' knowledge. They make it possible to determine the number of experts when making collective decisions in the economy and organizational management systems, to conduct optimal selective quality control of products, to carry out the necessary number of observations and reasonable diagnostics in medicine.

**Applied Mathematics and Control Sciences**. 2020;(4):7-19

RESTRICTIONS OF THE TYPE OF POLYHEDRAL DISPLACED CONE IN LINEAR PROGRAMMING PROBLEMS

#### Abstract

The paper considers the linear programming problem. A method is proposed to simplify its solution by identifying a class of constraints of a special type, due to which the desired plan will belong to a polyhedral convex cone located in a non-negative orthant. In this case, you must perform the following algorithm of actions. First, the original coordinate system is parallel transferred to the top of the selected cone. Then a transition to another space is made, which will lead to significant changes: a decrease in the number of restrictions. Next is the solution to the problem in any convenient way, for example, by the simplex-method - the most frequently used algorithm for finding solutions to linear extremal problems. One of its features is that with a large number of restrictions, its effectiveness decreases. This is a significant drawback in solving a number of problems, in particular, those of an economic nature, which, as a rule, striving to most accurately reflect the real state of affairs, impose a large number of restrictions on the desired plan. Therefore, if possible, it is better to reduce their number, even by increasing the variables, as this can happen in the proposed method for solving the selected class of problems. After finding the optimal plan, you need to return to the original space, and then to the old coordinate system. An important condition of this algorithm is the non-negativity of the cone elements. Thanks to this assumption, when a task is modified, new constraints are excluded. To track the implementation of this requirement, a condition is given in the work that guarantees its fulfillment. At the end of the work, the technique of searching for the operator (transition matrix) is described, with the help of which the task is transferred to another space. It is based on the search for vectors aligned with the generators of the cone.

**Applied Mathematics and Control Sciences**. 2020;(4):20-31

A MATHEMATICAL MODEL FOR SUMMING ROTARY MOTIONS

#### Abstract

In this work, using the means of applied mathematics, problems are solved related to the field of automation and control of technological processes, namely, the analytical description of superpositions of rotations that occur during the operation of numerous mechanisms. The practical aspect of the topic is determined by the fact that in mechanisms such as planetary gears, cutter drives in machines for cleaning pipes of large diameters, etc. summation of rotational motions is realized, and the shape of the hodograph is useful information in the design of such devices. The prerequisite for consideration is the principle of summation of rectilinear uniform movements. The aim of the work is to determine how things are in a similar situation when adding rotational synchronous movements. It was found that just as the result of the addition of two uniform rectilinear mechanical movements is also a uniform rectilinear movement, the result of the addition of two uniform unidirectional circular movements is also a uniform circular movement. The hodograph when two uniform oppositely directed circular motions are added is an ellipse. In a particular case, the ellipse can degenerate into a straight line segment. When two asynchronous rotations are added, hodographs in the form of a cochlea are possible, which is similar to Pascal's cochlea.

**Applied Mathematics and Control Sciences**. 2020;(4):32-45

COINTEGRATION ANALYSIS METHOD FOR FAULT DETECTION BASED ON SENSOR DATA

#### Abstract

Sensors are a popular source of information about the operation of complex dynamic technical systems. Considering data from sensors as a multidimensional time series is also used to describe cyber-physical systems. The article proposes a method for detecting system malfunctions based on the method of analyzing cointegration dependencies. It is determined that in the data for analysis it is possible to reveal cointegration dependences as facts of interdependence of data from different sensors. Calculations are given on the example of a system with 52 parameters. Out of 1,326 data pairs, 75 are cointegrated. The conducted analysis shows that the proposed method enables one to clearly illustrate situations with changes in behavior. Having identified cointegrated pairs, we can follow them, and if cointegration has ‘disappeared’, that is, at some new time interval we can no longer talk about the presence of a cointegration ratio, then something has changed in the process itself. In practice, this means either a change in technology (which the operator knows about), or a breakdown/accident/failure, due to equipment errors, changes in some parameters of the resources used. In the latter case, such information (that the process has changed) can be used to attract attention in general, which may ultimately lead to the need for equipment repair or maintenance or readjustment, etc. The analysis shows that the proposed method enables one to clearly illustrate situations with changes in behavior. As an example of using the method, we used the ready-made Tennessee Eastman Process (TEP) data set. Different pairs of data may have the ability to identify different errors. All errors cause a change in the behavior of one or several pairs of data, thus tracking the behavior of the value of random component enables identifying cases of deviation of the process from long-term equilibrium (in terms of cointegration), that is, cases of failure from the normal system operation. The results obtained are clear and objective and can be used by process operators or by a source for automatic process control.

**Applied Mathematics and Control Sciences**. 2020;(4):49-64

APPLICATION OF MATRIX APPROACH OF FUZZY LOGIC FOR DECISION SUPPORT IN OIL MINING EQUIPMENT SERVICE

#### Abstract

The work solves the problem of building an intelligent decision support system for servicing oil production equipment. At the first stage - the choice of an intelligent model - it is shown that in the existing conditions it is difficult to obtain a training sample in digital form. On the other hand, there is an opportunity to gain knowledge of subject matter experts - masters and technologists - in the form of a set of linguistic rules. Based on this, a conclusion about the effectiveness of the use of fuzzy logic to solve this problem is made. At the stage of constructing an intelligent model, the use of the matrix approach of fuzzy logic is proposed. To elaborate this approach an algorithm of fuzzy inference based on vector fuzzy predicates is developed. Capabilities and advantages of new algorithm are demonstrated. In particular, it is shown that the matrix representation makes possible reducing computations to solving a system of linear equations. Matrix inference also allows to explicitly determine the range of values of the analyzed parameters at which the knowledge base does not allow making a clear conclusion. A model of a fuzzy logic machine in the form of a fuzzy combinational circuit that analyzes an external memory block is proposed for the analysis of retrospective information on the change in the values of the parameters of technological equipment over time. Specific cases allowing the transition from a state machine to a combinational circuit are shown. Article also shows how this can be done. The main advantage of this approach is the absence of the need to use the difficult to formalize concept of a fuzzy state, which leads to a simplified construction of fuzzy logical devices with memory. At the end work contains brief conclusions about the application of the proposed methods and algorithms for building, testing, implementing a decision support system and about its effectiveness.

**Applied Mathematics and Control Sciences**. 2020;(4):65-88

THE DEVELOPMENT OF CONCEPTUAL ARCHITECTURE OF SELF-DRIVING PORTABLE VEHICLE’S UNIVERSAL MULTI-MODULAR PLATFORM

#### Abstract

This article is the logical sequel of my previous one, which deals with literature review in terms of self-driving portable vehicles with functionality of movement inside undetermined dynamic environment. That research paper describes the process and results of the best movement algorithm selection. Also it contains review of available virtual platforms, frameworks and existing non-commercial projects. In this article the research was continued and made more specific. I decided to concentrate on universal multi-module platforms, which can be used to standardize the production of self-driving portable vehicles. This study covers the results of conceptual architecture development. Unified Modeling Language (UML) was selected as the notation for it. Also there are three diagrams, that were defined to illustrate this architecture: Use Case diagram, sequence diagram and deployment diagram.

**Applied Mathematics and Control Sciences**. 2020;(4):89-102

IMPROVING THE ALGORITHM FOR MANAGING THE PROVISION OF MEDICAL SERVICES BASED ON COMMITTEE METHODS

#### Abstract

In matters of management in social and economic systems, medical issues are becoming increasingly important, and the processes of diagnosis of diseases are becoming the most important task of various health care institutions. Incorrect diagnosis of diseases leads not only to the fact that the patient's treatment becomes ineffective, and sometimes even harmful, but also leads to a significant increase in the costs associated with expensive procedures for further analysis and treatment of the patient. It is for this reason that it is necessary to improve the algorithm of the diagnostic procedure, including two qualitatively new blocks based on the developed methods of latent analysis procedures based on Committee and discriminant analysis. Methods of Committee and discriminant analysis as an auxiliary device are also effective and effective in solving the problem of diagnosing any diseases, including neurological and vascular diseases, as particularly dangerous and significantly impairing the standard of living of the population and leading to increased mortality, as well as worsens the statistical results of health care institutions. The improved algorithm of the diagnostic procedure proposed in the article is supplemented by the authors with two stages, which are added if the picture of symptoms is blurred and the preliminary diagnosis is not presented, or when several diagnoses fit the existing symptoms. This usually occurs due to conflicting, redundant information that corresponds to a variety of diseases. In this case, the method of differential diagnosis is used, which consists in re-interviewing, examining the patient, studying his medical history, prescribing additional clinical, functional and laboratory tests, which in some cases are excessive and even harmful to the patient, as well as increase the total cost of treating the disease, and usually increase the costs of the health care institution as a whole.. To resolve this situation, you can use discriminant analysis methods, which will allow you to find out the necessary diagnosis by assigning factors to a particular disease. However, often there is a situation related to the inconsistency of data that does not allow us to solve the problem of discriminant analysis using standard methods. In this case, you can use current mathematical methods, such as decision trees, the support vector method, and neural networks, but they are poorly interpreted and difficult to understand. therefore, we propose a method of analysis, which is improved by the authors in order to reduce the size of the problem, which helps reduce the number of factors necessary for the diagnosis of diseases. This method allows you to eliminate the inconsistency of data and unambiguously make a diagnosis, some features of which can later be identified by other standard clinical or instrumental methods. Thus, we can conclude that the use of methods of Committee and discriminant analysis in the diagnosis of diseases is not only an auxiliary, but also a mandatory attribute, in order to improve the quality of medical services. Moreover, the main methodological position of the application of these methods is to use the search for latent risk factors for diseases, which allows increasing the proportion of timely diagnosed patients. Also, do not forget about the provision of correct information collection and processing.

**Applied Mathematics and Control Sciences**. 2020;(4):105-120

PREDICTION OF THE RISK GROUP (BY ACADEMIC PERFORMANCE) AMONG FIRST COURSE STUDENTS BY USING THE DECISION TREE METHOD

#### Abstract

Mass education in Russian universities in specialties (direction of study) related to the exact and technical sciences is characterized by a high dropout rate, starting from the first year of study. The current level of school education, the system for selecting applicants through the USE procedure, in many cases does not guarantee that future students will be able to successfully master science-intensive specialties. An emphasis on student-centered, individual learning is possible only after students have proven themselves in the early stages of their studies. Therefore, the anticipatory identification of the ability of yesterday's applicants to study effectively is a very urgent task. In this paper, we consider methods for constructing decision trees designed to classify students, highlighting from them a lot of those (risk group) who, with a high degree of probability, will be expelled after the first academic cycle (trimester). At the same time, the minimum information about the freshmen, recorded in their personal file, is used as input data. The construction of the model was carried out according to the data on students of the applied mathematics and computer science direction of the Perm State National Research University for a five-year period of sets of 2014-2018. At the same time, the information from 2014-2017 was used for training, and the flow of 2018 was used as a test one. At the stage of machine learning, several models of decision trees were considered, which were optimized using balancing, restrictions on the maximum tree depth and the minimum number of elements in a leaf. The effectiveness of the binary classification was assessed using a matrix of inaccuracies and a number of numerical criteria obtained on its basis. As a result of machine learning, a decision tree was built, which predicted 16 out of 17 people expelled from the first trimester into the risk group. That is, for a number of reasons, they turned out to be incapable of learning in the direction of applied mathematics and computer science. In addition, it was possible to determine the level of significance of various types of initial data, showing that the results of the USE largely determine the success of students at this stage of training. The definition of the risk group provides certain guidelines for the purposeful activity of teachers and university psychologists, which ultimately can serve as a basis for improving the quality of education and reducing dropout rates. The work performed demonstrates the capabilities of data mining methods in solving poorly formalized tasks characteristic of this type of human activity.

**Applied Mathematics and Control Sciences**. 2020;(4):121-136

APPLICATION OF NEURAL NETWORKS IN SIMULATION OF CLUSTER-NETWORK RELATIONS IN OIL AND GAS INDUSTRY

#### Abstract

Despite the fact that the cluster approach is quite common in scientific works, the issues of the formation, development and evaluation of the effectiveness of cluster-network interactions remain unresolved. The research of the scientific community is based mainly on qualitative methods of cluster analysis (expert analysis, retrospective analysis, comparison method, etc.), however, the need to transform regional development and the transition to neo-economics require the use of economic and mathematical methods of analysis, and their arsenal is relatively small. which necessitates the search for new solutions. An attempt is made in the work to simulate the cluster-network mechanism in the oil and gas industry using neural networks, since the oil sector is one of the key sectors of the Russian economy, which influences the determining rates and paths of the country's socio-economic development, and is subject to the greatest regulation by the government of the country than most other sectors. The most important specific feature of the oil sector is that it is not only able to generate huge monetary resources, but also to accumulate them to solve a large number of socio-economic problems. Based on the results of the trained neural network, using the example of the indicators of the Perm Territory, predicted values of the gross regional product were made and, as a possible core of the oil industry cluster, the profit forecast of the company of the LUKOIL group.

**Applied Mathematics and Control Sciences**. 2020;(4):137-152

METHODOLOGY OF DISCRETE SPATIAL-PARAMETRIC REAL ESTATE MARKET MODELLING

#### Abstract

This paper gives a formalized description of the procedure for constructing widely used discrete spatial-parametric models of the real estate market in terms of set theory - an apparatus specially created for describing discrete spaces. The presentation is carried out in comparison with the approaches and concepts of a related methodology - regression models of mass appraisal of real estate objects. The methodology of discrete spatial-parametric modeling of the real estate market is used for market monitoring, for building dynamic market indices and for mass appraisal of real estate objects. The methodology is based on statistical cluster analysis and also allows for static interpolation spatial-parametric forecasting of the values of market indicators in small clusters with insufficient sample size and in narrow markets with little or no supply. The application of the methodology of discrete spatial-parametric modeling of the real estate market is demonstrated on the example of the residential real estate market in Moscow.

**Applied Mathematics and Control Sciences**. 2020;(4):155-185