A Data-Driven Expectation Prediction Framework Based on Social Exchange Theory

Along with the rapid application of new information technologies, the data-driven era is coming, and online consumption platforms are booming. However, massive user data have not been fully developed for design value, and the application of data-driven methods of requirement engineering needs to be further expanded. This study proposes a data-driven expectation prediction framework based on social exchange theory, which analyzes user expectations in the consumption process, and predicts improvement plans to assist designers make better design improvement. According to the classification and concept definition of social exchange resources, consumption exchange elements were divided into seven categories: money, commodity, services, information, value, emotion, and status, and based on these categories, two data-driven methods, namely, word frequency statistics and scale surveys, were combined to analyze user-generated data. Then, a mathematical expectation formula was used to expand user expectation prediction. Moreover, by calculating mathematical expectation, explicit and implicit expectations are distinguished to derive a reliable design improvement plan. To validate its feasibility and advantages, an illustrative example of CoCo Fresh Tea & Juice service system improvement design is further adopted. As an exploratory study, it is hoped that this study provides useful insights into the data mining process of consumption comment.

Evolution of teaching the probability theory based on textbook by V. P. Ermakov

The paper is devoted to the study of what changes the course of the probability theory has undergone from the end of the 19th century to our time based on the analysis of The Theory of Probabilities textbook by Vasyl P. Ermakov published in 1878. In order to show the competence of the author of this textbook, his biography and creative development of V. P. Ermakov, a famous mathematician, Corresponding Member of the St. Petersburg Academy of Sciences, have been briefly reviewed. He worked at the Department of Pure Mathematics at Kyiv University, where he received the title of Honored Professor, headed the Department of Higher Mathematics at the Kyiv Polytechnic Institute, published the Journal of Elementary Mathematics, and he was one of the founders of the Kyiv Physics and Mathematics Society. The paper contains a comparative analysis of The Probability Theory textbook and modern educational literature. V. P. Ermakov's textbook uses only the classical definition of probability. It does not contain such concepts as a random variable, distribution function, however, it uses mathematical expectation. V. P. Ermakov insists on excluding the concept of moral expectation accepted in the science of that time from the probability theory. The textbook consists of a preface, five chapters, a synopsis containing the statements of the main results, and a collection of tasks with solutions and instructions. The first chapter deals with combinatorics, the presentation of which does not differ much from its modern one. The second chapter introduces the concepts of event and probability. Although operations on events have been not considered at all; the probabilities of intersecting and combining events have been discussed. However, the above rule for calculating the probability of combining events is generally incorrect for compatible events. The third chapter is devoted to events during repeated tests, mathematical expectation and contains Bernoulli's theorem, from which the law of large numbers follows. The next chapter discusses conditional probabilities, the simplest version of the conditional mathematical expectation, the total probability formula and the Bayesian formula (in modern terminology). The last chapter is devoted to the Jordan method and its applications. This method is not found in modern educational literature. From the above, we can conclude that the probability theory has made significant progress since the end of the 19th century. Basic concepts are formulated more rigorously; research methods have developed significantly; new sections have appeared.

Reliability of the belt conveyor bed when restoring failed roller supports

The reliability of modern belt conveyors, whose length reaches tens of kilometers, is primarily determined by the reliability of the roller supports that support the belt and ensure its movement. As they wear out, some roller bearings fail and need to be repaired or replaced. The dynamics of the number of working roller supports is determined by the system of Kolmogorov equations. Their solution allows us to find the probabilities of finding the system in states with a different number of working elements. The article finds probabilities for two cases. In the first case, when restoring, only one roller support is repaired each time. In the second case, all roller supports are repaired or replaced. In the case of sequential recovery, the mathematical expectation of the number of properly working roller supports may be less than the total number by several units. There are always elements that need to be repaired. If the recovery rate of the elements is many times higher than the failure rate, the mathematical expectation of the number of properly operating roller supports is less than the total number of roller supports by less than one, during most of the time all roller supports are serviceable. In the case of simultaneous recovery of elements, an equally high level of reliability is achieved even with comparable failure and recovery rates. The results obtained can be used to determine the necessary reserve of spare structural elements and to plan the maintenance of conveyors.

Weapon-Target Assignment Problem Using Modified Water Wave Optimization Algorithm

The weapon-target assignment (WTA) is a classic problem. The WTA mathematical model is that warship formations are reasonably equipped with weapons resources for each weapon system to attack the air threaten targets. The purpose of targets optimization is to maximize combat effectiveness, that is to say, the mathematical expectation is maximum. We adopt the greedy strategy and improved propagation operation is to strengthen the water wave optimization (WWO) search performance. This article elaborates a modified water wave optimization (MWWO) to solve the WTA problem, which can detect optimized allocation decision matrix and search for the maximum mathematical expectation. Based on parameter optimization, the overall performance of the MWWO is more stable, the search speed is accelerated and the accuracy is improved. The experiment results indicate that the MWWO are verified and avoids local optimum, and can be more convenient for solving the WTA and obtain better performance.

DISCREPANCY PARAMETERS OF APPROXIMATIONS OF DISCRETELY SPECIFIED DEPENDENCIES BY ANALYTICAL FUNCTIONS AND SEARCH CRITERIA FOR OPTIMAL VALUES OF THEIR COEFFICIENTS

Universal discrepancy parameters of approximations of discretely specified dependencies by analytical functions and search criteria for optimal values of their coefficients, as well as analysis of features of their application are described. Discrepancy parameters of approximations, which do not depend on the ranges of variation of the values of functions and the number of points of a discretely specified dependence, are proposed. They can be effective for objectively comparing the quality of approximations of any dependencies by any functions. Approximations of a discretely specified dependence of the mathematical expectation of the equivalent electrical resistance of a layer of aluminum granules during spark-erosion dispersion in water on the instantaneous values of the discharge current are carried out. As approximating functions, we chose a power function with an exponent factor –1 and a function based on exponential. Using the criteria of the least approximation error, the optimal values of the coefficients of both approximating functions are founded. It is shown in which cases it is advisable to use the combined search criteria for the optimal values of the coefficients of the approximating functions, and in which are enough simple one-component ones. Ref. 27, fig. 2, tables 2.

Study of the temporal distribution density of the shot sheaf span to create a simulation model of the optical sensor signal

When studying the space path of a cloud of pellets from a shotgun and evaluating its parameters, it is advisable to have a simulation model of the optical sensor signal when a shot sheaf crosses the light screen. To create such a model, you need to get the relationship between the scale parameter in the Rayleigh distribution and the time span of the shot sheaf. Studies of the temporal distribution density of the shot sheaf span are performed, and graphs of the distribution density of the span for near-real situations are constructed. A linear dependence of the mathematical expectation of the span distribution density on the scale parameter in the Rayleigh distribution is established. A simplified expression is obtained for calculating the mathematical expectation of the time span of the shot sheaf. Statistical modeling confirmed the possibility of the practical use of the proposed formulas, including with a large number of pellets (up to 1000). The dependence of the mathematical expectation of the span on the number of pellets is investigated, its approximation by various functions is carried out, and the approximation errors are estimated. The research conducted allows us to create a simulation model of the optical sensor signal when the shot sheaf crosses the light screen on the basis of empirical data on the real-signal duration (with averaged measurement data for several shots being helpful).

ANALYTICAL NONLINEAR-PROBABILISTIC MODEL OF THE EQUIVALENT ELECTRICAL RESISTANCE OF A LAYER OF METAL GRANULES

As a result of processing the experimental data, an analytical continuous nonlinear-probabilistic model of the equivalent electrical resistance of a layer of metal granules in the working liquid was created. It is described by four equations: the modified Gaussian distribution and the dependences on the instantaneous values of the discharge current in the layer of metal granules of the mathematical expectation, dispersion and correction coefficient of the range of its equivalent electrical resistance. Based on the form of the dependences obtained during the experiments and the physics of the processes that occur in this case, two main groups of analytical functions are considered that approximate the obtained dependences. Criteria and methods for finding the optimal values of their coefficients are described. The adequacy of the approximation of each of the three obtained dependences by several analytical functions was investigated, the optimal values of the coefficients of which were found by the described method. Analytical functions was compared, which approximate the dependence of the mathematical expectation of the equivalent electrical resistance of a layer of metal granules on the instantaneous values of the discharge current in it with the known nonlinear models of the resistance of such a medium. References 33, figures 3, tables 3.

Mathematical Modelling of the Fruit-Stone Culture Seeds Calibration Process Using Flat Sieves

The paper provided describes a mathematical model of calibration process of fruit-stone culture seeds of cherry, sweet cherry, cherry-plum, apricot and almond using flat sieves with impact shock ball cleaners oscillating in the horizontal plane. It has been defined that the mathematical expectation of time of knocking out the fruit-stone from the sieve opening T ⌢ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T} is the minimum value of ratio of average time of complete ball motion cycle in space under sieve to the probability of knocking out the stone by a ball with the kinetic energy level of 2 Mj. The dependences of energy distribution density of ball on impact on the sieve have been obtained, based on which the intervals of ball cleaner parameters have been determined, i.e. the ball diameter D belongs to the interval 25–35 mm; the space height H under sorting sieve belongs to the interval 1.2D–1.4D mm; the value range for distance between rods t belongs to the interval 0.5D–0.7D mm. Using the method of golden section, the following parameters of ball cleaner were obtained: D = 33 mm, t = 23 mm, H = 40 mm. The parameters obtained provide mathematical expectation of time of knocking out the fruit-stone from the sieve opening T ⌢ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T} = 0.03 s. Consequently, the average ball velocity v ⌢ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over v} is = 0.4 m∙s-1, and the average ball path is L ⌢ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over L} = 0.006 m.

Approximate formulas for the evaluation of the mathematical expectation of functionals from the solution to the linear Skorohod equation

This paper is devoted to the construction of approximate formulas for calculating the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod stochastic differential equation with a random initial condition. To calculate the mathematical expectations of nonlinear functionals from random processes, functional analogs of quadrature formulas have been developed, based on the requirement of their accuracy for functional polynomials of a given degree. Most often, formulas are constructed that are exact for polynomials of the third degree [1–9], which are used to obtain an initial approximation and in combination with approximations of the original random process. In the latter case, they are usually also exact for polynomials of a given degree and are called compound formulas. However, in the case of processes specified in the form of compound functions from other random processes the constructed functional quadrature formulas, as a rule, have great computational complexity and cannot be used for computer implementation. This is exactly what happens in the case of functionals from the solutions of stochastic equations. In [1, 2], the approaches to solving this problem were considered for some types of Ito equations in martingales. The solution of the problem is simplified in the cases when the solution of the stochastic equation is found in explicit form: the corresponding approximations were obtained in the cases of the linear equations of Ito, Ito – Levy and Skorohod in [3–11]. In [7, 8, 11], functional quadrature formulas were constructed that are exact for the approximations of the expansions of the solutions in terms of orthonormal functional polynomials and in terms of multiple stochastic integrals. This work is devoted to the approximate calculation of the mathematical expectations of nonlinear functionals from the solution of the linear Skorokhod equation with a leading Wiener process and a random initial condition. A new approach to the construction of quadrature formulas, exact for functional polynomials of the third degree, based on the use of multiple Stieltjes integrals over functions of bounded variation in the sense of Hardy – Krause, is proposed. A composite approximate formula is also constructed, which is exact for second-order functional polynomials, converging to the exact expectation value, based on a combination of the obtained quadrature formula and an approximation of the leading Wiener process. The test examples illustrating the application of the obtained formulas are considered.