Distributions Of Random Variables Research Articles

Information system of remote sensing data analysis

The information system is designed to solve the problems of analyzing remote sensing data and making decisions using non-parametric pattern recognition algorithms, the synthesis of which is based on kernel estimates of probability density. Initial information is compiled by training samples on the states of the investigated natural objects and their corresponding spectral data, which are presented in Microsoft Excel files. The application of the software provides the following functions: primary processing of remote sensing data to evaluate the main quantitative characteristics of the laws of distribution of random variables; testing of the hypothesis on distributions of multivariate random variables based on non-parametric pattern recognition algorithms; estimating the state of natural objects by spectral data; mapping of territories by state of natural objects based on spectral data of earth surface elements and their corresponding spatial coordinates. Results of the presented functional tasks are saved in the form of reports in graphic files and spreadsheets of Microsoft Excel. Software tools can be used in the study of various nature objects from remote sensing data.

On the subject of the effectiveness of the ATC and pilot joint simulation training (retraining) program

The purpose of the article is to conduct a statistical data analysis obtained after the experimental study. The experimental research deals with rating the trainees’ level of proficiency before and after the proposed method of the ATC and pilot joint training (retraining). It’s common knowledge that progress in aviation has moved far forward. The fourth, fifth and sixth-generation aircraft have come into existence. Aircraft and automated systems have become digital. In order to develop and implement ATM systems, ICAO has identified three fundamental concepts that will allow us to enhance human reliability in the future. One of the concepts is human-oriented automation. Automation involves the transition from the analog onboard and ground systems to digital ones. As a result, the onboard and ground systems, after modernization, will receive the fifth and sixth-generation abbreviation. Subsequently, a new approach will be required to train (retrain) air traffic controllers and pilots to operate these systems. To solve this problem, it is necessary to change the concept of training flight and traffic control personnel. The article provides an evaluation of the proposed new technique for the joint training (retraining) of ATC and pilots by conducting a statistical analysis of experimental data. The proposed training method will enable us to improve the quality of training, reduce a number of pilot and ATC errors, which are the underlying principles of the ICAO concept. The article presents the statistical analysis of data based on the Student t-criterion and the law of random variables distribution, which will make it possible to draw a conclusion about the effectiveness of the proposed method for training specialists.

SIMULATION MODELING OF INVESTMENT PROJECTS IN THE SERVICE SECTOR

Scientific article is devoted to researching the possibility of developing a simulation model of an investment project for a travel agency. In recent years, the tourism industry in the world is experiencing rapid development. The development of the tourism industry requires new investments. The investor must assess the degree of risks, the likelihood of profit or loss. It is noted that the most important advantage of simulation modeling is that it makes it possible to study economic systems at the design stage. Due to this, simulation models can be used as a universal tool in making appropriate decisions under conditions of uncertainty and taking into account those factors that are difficult to predict and take into account, which is why simulation is so often used in the development of investment projects. When assessing the risks of investment projects, the collection of information requires significant financial expenses, is quite time-consuming, and sometimes impossible. The subject of the study is a set of the theoretical, methodological and organizational problems related to the management decisions regarding the investment of the projects in the field of tourism. The methodology is based on the stochastic simulation modeling of economic processes. The aim of the study is to develop a simulation model for an investment project in the tourism sector, which will allow the investor to appreciate the degree of the risk and the likelihood of return on the investment. The article reveals the features of computer simulation in the MathCAD system. It is proved that it is convenient to develop simulation models of investment projects in the MathCAD system. It has a powerful mathematical support, remains one of the systems in which the description of the solution of mathematical problems is given with the use of conventional mathematical formulas and symbols and does not require special training in programming. The simulation model of the investment project of the travel agency was developed in the MathCAD system. Statistical processing of the results of the experiments with the model has been carried out. The necessity of the detailed study of critical intervals of the histogram at the transition from loss-making to profitable NPV values has been substantiated. Based on the results of simulation experiments with the model, the errors of the model were estimated, the law of NPV random variables distribution was established. The developed simulation model allows the investor to estimate the risk factor, the probability of profit or loss from the investment, to assess the possible uncertainty of the results of their own decision to invest in the project. Conclusion: the simulation model of the investment project built and investigated in MathCAD system allows investor to estimate the risk factor, probability of profit or loss for choosing the optimal pricing policy and optimization of economic strategy of tourist agency.

Ten Ways to Specify a Gini Coefficient Using Entropy

Our friend and frequent collaborator Prof. Michael McAleer loved to enumerate lists and to give practical advice. Here, we present a review of 10 ways to derive the well-known Gini coefficient based on entropy measures. In fact, Mike was a collaborator on some of this work, as will be discussed in the paper. All are useful ways to combine two powerful tools, entropy measures and Gini coefficients to examine inequality in income distribution functions (IDFs) and can be applied to distributions of other random variables. Others have shown that a Gini coefficient can be derived from the first moment of an observed share function. Ryu and Slottje demonstrated that by projecting the observed share function with other moments and functions different approximated share functions can be derived. By doing so, more information can be conveyed about the underlying IDF that generated the observed Gini value. In the spirit of Prof. Michael McAleer, we present 10 different ways to utilize entropy functions to generate Gini coefficient measures.

Structural Materials Durability Statistical Assessment Taking into Account Threshold Sensitivity

This work presents an experimental—analytical study of the possibility of applying the method of the greatest probability to evaluate the sensitive thresholds of the bottom (N0) and top (Nk) of the statistical distribution of the mechanical structural characteristics. For the structural materials alloyed steel 15Cr2MoVA, steel C45 and aluminium alloy D16T1, the statistical distribution of proportional limit, yield strength, ultimate tensile strength, reduction in area, cyclic stress was estimated, as well as the following statistical parameters: mathematical mean, average square deviation, dispersion, asymmetry, variation coefficient, and excess. Purpose: to determine whether the limits of the sensitivity of the statistical distribution of the mechanical characteristics have been computed using the maximum likelihood method. Value: there is a certain upward and downward flattening of the probability curves in the statistical distribution curves of the fatigue test results. This implies that the chosen law of the distribution of random variables has an effect on the appearance of errors. These errors are unacceptable given the importance of accurately determining the reliability and durability of transport means, shipbuilding, machinery, and other important structures. Our results could potentially explain why sensitive limits cannot be applied to the statistical distribution of the mechanical characteristics of structural materials.

QUANTITATIVE ASSESSMENT OF CREDIT RISK

Currently, various mathematical methods and models are used for quantitative assessment for risks, most of which are linear, as in case of nonlinear models normal distribution of random variables is assumed. This approach, in our opinion, significantly devalues the model, since other distributions of random variables that change over time are more common. It is known that the financial and credit policy pursued by banks in connection with large investments is carried out in conditions of certain risks. As a result, we come to a situation where there is some degree of uncertainty, when the expected return on bank investment in a loan portfolio is a random variable or random function. In such conditions, the development of new methods of quantitative analysis and assessment of the risk contained in the financial operations of a bank is becoming a very urgent problem, the solution of which will allow financial institutions to estimate in advance the likelihood of obtaining the expected profit, or calculate estimates of other equivalent indicators. In this article tools of probability theory were used as well as modern risk classifying systems and management strategies. Thus, the presented approach can be used in decision making in different conditions and fields for quantitative risk assessment.

УЧЕТ РИСКОВ ПРИНЯТИЯ РЕШЕНИЙ ПРИ МОДЕЛИРОВАНИИ ВОЗДЕЙСТВИЙ ОКРУЖАЮЩЕЙ СРЕДЫ НА ЭКОНОМИЧЕСКИЕ ОБЪЕКТЫ

Risk analysis of decision-making serves as the basis for ensuring industrial and environmental safety and achieving economic efficiency in the process of economic activity. It is shown that modeling of environmental impacts on the complex systems under study, involving risk analysis in the form of probabilities of decision-making errors, is one of the conditions for achieving economic efficiency and safety. The environmental impact model, which takes into account the probabilities of errors of the first and second kind, is defined by an infinite-dimensional matrix stochastic operator. It is shown that under the influence of the matrix operator, the observed probability distributions of discrete random variables are transformed to a bimodal form. The conformity of the proposed model to the principles of structural harmony is investigated. An example of effective pricing taking into account the environmental impact model is given.

Stochastic Optimal Power Flow in Hybrid Power System Using Reduced-Discrete Point Estimation Method and Latin Hypercube Sampling

Stochastic nature of some input variables dictates the requisite of probabilistic analysis in power systems operation and planning. Wind generation is considered as a main source of intermittency in power systems due to the uncertain nature of wind speed. The proposed probabilistic optimal power-flow (POPF) method investigates spatial correlation among sources to attain more practical output distributions. The method established reduced-discrete point estimate method (RDPEM) along with the Latin hypercube sampling (LHS) in order to attain the stochastic characteristic of optimization’s outputs. Despite needing less computational effort, highly accurate results can be obtained, while there is no prerequisite for probability distribution of the input random variables. In order to more validate the efficiency of the proposed method, the Gram–Charlier (GC) expansion is used to compare the outputs’ cumulative distribution functions (CDFs) that are obtained from Monte Carlo (MC) with RDPEM methods. The performance and precision of the proposed solution are ascertained by comparison with those of Monte Carlo with discrete LHS (MCDLHS) in a hybrid IEEE 14-bus test system.

Методика определения оптимальных маршрутов в условиях оперативного планирования автомобильных грузовых перевозок

The article presents a methodology for determining optimal routes in dynamically changing conditions of operational planning of road freight transport. The methodology consists of two main elements: theoretical software (router) and implementing software (routing protocol). It has unique properties, namely, the analytical apparatus of the router provides the possibility of choosing an effective action in the presence of stochastic uncertainty (when there is no information, which would guarantee the justified application of the well known laws of distribution of random variables). It works with an unlimited number of entries into the system, therefore, not only effective performance indicators of cars can and should be included in the databases of the studied data, but effective indicators of cargo handling facilities as well.

Вплив конструкторсько-технологічних чинників на довговічність вставного твердосплавного породоруйнівного оснащення шарошок тришарошкових бурових доліт

Aim. In the study learning trial was made to make a mathematical model of the assembling process in connections " tungsten carbide insert cutter - cone", while the methods of mathematical statistics were applied. This makes it possible to determine the parameters of the distribution of the closing link. Accordingly, the tension is determined by samples of constituent links. For this purpose, the amount of tension was investigated. Tension affects the change in stress, which characterizes the accuracy of the assembly on the strength of the connection " tungsten carbide insert cutter - cone ". Depending on the magnitude of the insert pressing force, it is customary to estimate the tension in the connection. Method. The analysis of tungsten carbide insert cutter and holes in the layers of bit established that the size errors of the component links of dimensional chains are distributed according to laws that they are similar to the normal law of the distribution of random variables. The test was carried out according to the criteria of Pearson and Kolmogorov Results. The increase in pressing force with an increase in tension is due to an increase in elastic deformation both in the cemented layer and in the core layers of the rolling-cutter row. The extreme nature of the dependence of the pressing force on the amount of tension for the connection " tungsten carbide insert cutter - cone " has been established. This dependence, as a characteristic of the strength in the connection, can be used to design technological pressing operations for roller cone bits steels with different strength indicators. Scientific novelty. a mathematical model of the assembling process of connections " tungsten carbide insert cutter - cone " was made. It makes it possible to determine the parameters of the distribution of the closing link - tension, based on the samples of the component links. The established dependencies make it possible to reasonably and most accurately approach the formation of selective groups of teeth and the marking of holes for them. Practical significance. The appointment of tensions, the value of which exceeds the value of the extreme point, causes the occurrence of unfavorable plastic deformation or destruction of the layers of the shell (which are conjugated along the height of the shank of the insert cutter). They also result to the occurrence of irrational stresses in the body of the tungsten carbide insert cutter. It was established that the tension values have a greater influence on the force of pressing carbide teeth into the holes of roller cone bits steel 9310H (USA), when its hardness value is Rockwell hardness 59-60 compared to Rockwell hardness 48-50. Within the limits of the studied tension values, this dependence is linear.

ЭЛЕКТИВНЫЙ КУРС «ЗНАКОМСТВО С МИРОМ СЛУЧАЙНОСТЕЙ И СТАТИСТИЧЕСКИХ ЗАКОНОМЕРНОСТЕЙ» В ЦИФРОЙ СРЕДЕ КАК СРЕДСТВО РАЗВИТИЯ СТОХАСТИЧЕСКОГО МИРОВОЗЗРЕНИЯ СТАРШЕКЛАССНИКОВ

The article considers the main provisions describing the features of the introduction and implementation of the elective course on stochasticity for high school students of secondary schools. The stochastic line of the school mathematics course for 10-11 grades is aimed at identifying general patterns, assessing the probabilistic nature of real-world dependences, defining phenomena of random nature, processing statistical data, constructing distributions of random variables, using the law of large numbers and probabilistic sampling method of measurement. The expediency of developing this elective course is due to the importance of knowledge of probability theory for both completing the Unified State Exam in Mathematics and preparing for university studies. The article argues that students who have mastered the methods of measuring and evaluating the phenomena associated with the concept of chance achieve high individual results. As part of the elective course and working in the digital environment, students learn how to use probabilistic and statistical methods to solve problems effectively, the focus of which is conditioned by the establishment of interdisciplinary links with biology, physics, chemistry, astronomy, etc. (for example, the study of Mendel's laws of heredity, Brownian motion (random motion of physical particles), Gibbs' rule, etc.). Skills and abilities obtained help students better understand variability and uncertainty inherent in social and natural processes, transfer their knowledge to new conditions and to new tasks, accumulate personal worldview experience. Thanks to this, high school students have a positive attitude towards a chance and its manifestations, perceive the surrounding reality more objectively. According to the results of the elective course implementation pupils have stable cognitive motives for the cognition of the stochastic component in the environment, deep subject knowledge, which conditions the development of motivational, value, cognitive and activity constructs of stochastic world-view.

GRay: Ray Casting for Visualization and Interactive Data Exploration of Gaussian Mixture Models.

The Gaussian mixture model (GMM) describes the distribution of random variables from several different populations. GMMs have widespread applications in probability theory, statistics, machine learning for unsupervised cluster analysis and topic modeling, as well as in deep learning pipelines. So far, few efforts have been made to explore the underlying point distribution in combination with the GMMs, in particular when the data becomes high-dimensional and when the GMMs are composed of many Gaussians. We present an analysis tool comprising various GPU-based visualization techniques to explore such complex GMMs. To facilitate the exploration of high-dimensional data, we provide a novel navigation system to analyze the underlying data. Instead of projecting the data to 2D, we utilize interactive 3D views to better support users in understanding the spatial arrangements of the Gaussian distributions. The interactive system is composed of two parts: (1) raycasting-based views that visualize cluster memberships, spatial arrangements, and support the discovery of new modes. (2) overview visualizations that enable the comparison of Gaussians with each other, as well as small multiples of different choices of basis vectors. Users are supported in their exploration with customization tools and smooth camera navigations. Our tool was developed and assessed by five domain experts, and its usefulness was evaluated with 23 participants. To demonstrate the effectiveness, we identify interesting features in several data sets.

Research on Dynamic and Static Test Methods for Evaluating the Poisson’s Ratio of Oriented Strand Board

In this article, dynamic method and static method of testing Poisson’s ratio of OSB (Oriented Strand Board) were proposed. Through modal and static numerical analyses, the position where the transverse stress is equal to zero was determined. The binary linear regression method was applied to express the gluing position of the strain gauge as a relational expression that depended on the length-width ratio and width-thickness ratio of the cantilever plate. Then the longitudinal and transverse Poisson’s ratios of OSB were measured by the given dynamic and static methods. In addition, the test results of OSB Poisson’s ratio were analyzed with the probability distribution of random variables. The results showed that using the proposed dynamic method and static method, the test results for longitudinal and transverse Poisson’s ratios of OSB were quite consistent, despite the gluing position of the strain gauges being different. And these OSB Poisson’s ratios were accorded with that obtained by the axial tensile method and the four-point bending method. OSB longitudinal and transverse Poisson’s ratios followed Weibull distribution.

STOCHASTIC MODELING OF OPTIMAL LOGISTICS IN THE FUNCTION OF MAXIMIZING INTERNATIONAL BUSINESS EFFICIENCY OF SMEs

The concept of business activity conditions specific procedures and activities in order to maximize the difference between output and input variables while taking into account the uncertainty of the business outcome. The business outcome is determined by a number of factors that are under the control of the decision maker. However, a number of factors are conditioned by stochastic quantities, which obey the laws of probability of a random variable whose value the decision maker cannot influence but must anticipate and respect in the business decision making process. Conditionality of business results with the market component refers to user expectations, and it requires a model approach by which the uncertain business future is recomposed into business expectations, with a high level of reliability. Modeling of the system by the process of mathematical simulation enables the calculation of variants of business future in the present time without realizing business strategies before their selection and classification. The modeling process includes business system analysis, factor selection, qualitative and quantitative expression, classification of variables, functional linking, formulation of probability distribution of random variables, and the choice of time frame for simulation. The process of mathematical simulation indicates the management consequences of business alternatives, thus the decision maker is guided by business expectations and recruits business logistics in accordance with the decision. The simulation model is adapted to the specific management problem, specific conditions and circumstances of decision making. It does not have a universal character and must be constructed specifically for each management situation.

Improving Sample Average Approximation Using Distributional Robustness

Sample average approximation is a popular approach to solving stochastic optimization problems. It has been widely observed that some form of robustification of these problems often improves the out-of-sample performance of the solution estimators. In estimation problems, this improvement boils down to a trade-off between the opposing effects of bias and shrinkage. This paper aims to characterize the features of more general optimization problems that exhibit this behaviour when a distributionally robust version of the sample average approximation problem is used. The paper restricts attention to quadratic problems for which sample average approximation solutions are unbiased and shows that expected out-of-sample performance can be calculated for small amounts of robustification and depends on the type of distributionally robust model used and properties of the underlying ground-truth probability distribution of random variables. The paper was written as part of a New Zealand funded research project that aimed to improve stochastic optimization methods in the electric power industry. The authors of the paper have worked together in this domain for the past 25 years.

Reliability Analysis of Stone Columns Bearing Capacity

Abstract Reliability analysis and stone columns techniques are widely used, with an increasing trend, in different geotechnical engineering problems. Design methods of stone columns is still based on deterministic approach, in spite some degree of uncertainties is associated with soil and stone column parameters. In the present work, reliability analysis of bearing capacity of stone columns is examined; using three methods of reliability analysis, First Order Second Moment FOSM, Point Estimate Method PEM, and Monte Carlo simulations MCs. Parametric analysis using all variables involved in stone columns design has been conducted. Distributions of random variables involved in stone columns design are considered either as normal or lognormal. The random variables correlation effect is examined. The study has allowed to evaluate parameter uncertainty effect on reliability index of stone columns, and has permitted to figure out reduced set of parameters that must be considered as random in reliability analysis of stone columns beside sensitivity analysis.

Influence of parameters inhomogeneity on the existence of chimera states in a ring of nonlocally coupled maps

The aim of the research is to study the influence of inhomogeneity in a control parameter of all partial elements in a ring of nonlocally coupled chaotic maps on the possibility of observing chimera states in the system and to compare the changes in regions of chimera realization using different methods of introducing the inhomogeneity. Methods. In this paper, snapshots of the system dynamics are constructed for various values of the parameters, as well as spatial distributions of cross-correlation coefficient values, which enable us to determine the regime observed in the system for these parameters. To improve the accuracy of the obtained results, the numerical studies are carried out for fifty different realizations of initial conditions of the ring elements. Results. It is shown that a fixed inhomogeneous distribution of the control parameters with increasing noise intensity leads to an increase in the range of the coupling strength where chimera states are observed. With this, the boundary lying in the region of strong coupling changes more significantly as compared with the case of weak coupling strength. The opposite effect is provided when the control parameters are permanently affected by noise. In this case increasing the noise intensity leads to a decrease in the interval of existence of chimera states. Additionally, the nature of the random variable distribution (normal or uniform one) does not strongly influence the observed changes in the ring dynamics. The regions of existence of chimera states are constructed in the plane of «coupling strength – noise intensity» parameters. Conclusion. We have studied how the region of existence of chimeras changes when the coupling strength between the ring elements is varied and when different characteristics of the inhomogeneous distribution of the control parameters are used. It has been shown that in order to increase the region of observing chimera states, the control parameters of the elements must be distributed inhomogeneously over the entire ensemble. To reduce this region, a constant noise effect on the control parameters should be used.

Reliability assessment of renewable energy integrated power systems with an extendable Latin hypercube importance sampling method

The increasing integration of renewable energy sources has brought about a great computational burden for the traditional methods that assess the power system reliability. To reduce the computational costs, an extendable Latin hypercube importance sampling (ELHIS) method that combines importance sampling (IS) and Latin hypercube sampling (LHS) is proposed in this paper. First, the challenge of combining IS and LHS is analysed, and a customized sampling process of LHS is designed accordingly. Then, in the IS part of ELHIS, the cross entropy theory and Gaussian mixture model are adopted in the stage that constructs the quasi-optimal probability distribution of random variables. In addition, the samples in this stage are also utilized in the estimation of reliability indices to reduce the waste of computational efforts. In the LHS part, an extendable LHS approach is used to make ELHIS adaptive and flexible in determining the sample size to reach the required accuracy. Finally, numerical tests are performed on the modified IEEE-RTS 79 test system, where the real historical data from three wind farms and a photovoltaic station in Northwest China are employed. The results show that ELHIS is obviously faster than recent IS methods when used to assess the power system reliability.

Inventory Management at the Enterprise in the Field of Probability Models

The inventory management system is designed to continuously ensure the production activities of the enterprise with all necessary resources. The purpose of this study is to build a probabilistic model that can be proposed as a new inventory model, which establishes the relationship of period factors between the purchase of parts and the duration of their suitability, which affect inventory management. The research methods are based on a probabilistic approach using continuous distributions. Using the statistical method, point estimates were found for the studied parameters: mean and standard deviation. The histograms of relative frequencies between dates of two next purchases, volume of purchases of details and days of replacement of the fulfilled details are constructed. The critical areas for the studied parameters are illustrated. The values of the difference in days between the purchases of parts and the values of purchases of parts that meet the normal distribution of random variables with the appropriate parameters, as well as the critical values of the need for parts in the production process. The size of the part reserve, which corresponds to Erlang distribution, was found, depending on the established risk factor. For different values of this factor, the value of the difference in days between the purchases of parts, the size of purchases and the reserve of parts that correspond to the distributions of random variables, as well as the critical value of the need for parts in the production process to avoid downtime. Using the central limit theorem, it is shown that the purchase volume of parts and the volume of used parts are distributed according to the normal law. The study concludes that the probabilistic approach is the basis for forecasting inventory management in the enterprise, taking into account the risks associated with determining the optimal demand for raw materials in the enterprise

Stability of Dependencies of Contingent Subgroups with Merged Groups: Vaccination Case Study

The answers to extreme phenomena both in nature and in business sectors are the constructions of the distribution of random variables with extreme values. Another area in which appropriate theoretical research is conducted regarding the influence of suppressor (third) variables in categorical data. When examining dependencies in PivotTables, we often find it necessary to merge data into larger sets (e.g., due to a greater number of theoretical frequencies lower than their critical value). A phenomenon many exist wherein the partial relation is stronger than the zero relation. For example, in such a combination, instability may occur, which indicates contingent subgroups with the merged group. The dependence of dependencies is practically manifested because the data of contingent subgroups indicate inconsistent (inverted) conclusions compared to the associated group. For this reason, this paper aimed to find the critical ratios of partial probabilities in the contingency table of subgroups of the original variables, and to determine the conditions of result consistency and contingency stability, including the proof. For practical use and for the ease of repeating the proposed procedure, the solution is based on a case study that compares the effectiveness of vaccination.