Distributions Of Random Variables Research Articles

Robust optimization bidding strategy for user-side resource-side participation in the market distribution of electrical energy and peaking ancillary services considering risk expectations and opportunity constraints

Compared to traditional resources, user-side resources are of various types and have more significant uncertainty about their regulatory capacity, leading to difficulties in coordinating decisions about their simultaneous participation in the electric energy and peaking ancillary services markets. This paper proposes a joint bidding decision-making method for the day-ahead electricity energy and peak shaving auxiliary service market based on distributed robust opportunity constraints, which addresses the problem of difficulty in using an accurate probability density distribution to represent the uncertainty process of user-side resources. Firstly, a data-driven method for characterizing the uncertainty of load regulation capacity is investigated, and fuzzy sets are constructed without assuming specific probability distributions of random variables. Then, to minimize the risk expectation of the joint bidding cost on the customer side, a bidding strategy that considers the uncertainty is proposed. Finally, an example simulation verifies the reasonableness and effectiveness of the proposed joint bidding method, and the results show that the constructed model overcomes the problem of over-conservatism of the robust model, and the computational adaptability is better than that of the stochastic model, which achieves a better balance between robustness and economy.

Reformulation and Enhancement of Distributed Robust Optimization Framework Incorporating Decision-Adaptive Uncertainty Sets

Distributionally robust optimization (DRO) is an advanced framework within the realm of optimization theory that addresses scenarios where the underlying probability distribution governing the data is uncertain or ambiguous. In this paper, we introduce a novel class of DRO challenges where the probability distribution of random variables is contingent upon the decision variables, and the ambiguity set is defined through parameterization involving the mean and a covariance matrix, which also depend on the decision variables. This dependency makes DRO difficult to solve directly; therefore, first, we demonstrate that under the condition of a full-space support set, the original problem can be reduced to a second-order cone programming (SOCP) problem. Subsequently, we solve this second-order cone programming problem using a projection differential equation approach. Compared with the traditional methods, the differential equation method offers advantages in providing continuous and smooth solutions, offering inherent stability analysis, and possessing a rich mathematical toolbox, which make the differential equation a powerful and versatile tool for addressing complex optimization challenges.

Quantitative method for the probability of structural damage based on moment theory

In the context of uncertainty in the inverse problem of damage identification, this study proposes a moment theory based probabilistic damage identification (MbPDI) method. Noise errors in the identification process are treated as uncertain parameters, while structural stiffness parameters are treated as functions of noise errors. A limit state function for damage identification is defined, and its probability is computed. Utilizing the concept of moment theory, the expansion series is performed for the limit state function around the most probable point of failure, with gradient information derived using sensitivity matrices. Based on the gradient information and uncertainty inputs, damage probabilities for various elements can be obtained. Additionally, this paper addresses the calculation of structural damage probabilities under the non-normal distribution of random variables by transforming non-normal variables into equivalent normal variables. The proposed probability index and expected index of structural damage in the probabilistic damage identification method has a clear physical meaning, and the calculation has high accuracy. The proposed method is validated through a mathematical example and a numerical simulation. Meanwhile, the influence of damage extent, noise level, and modal order on identification results are analyzed. Experimental validation of the method is also presented.

The distribution of power-related random variables (and their use in clinical trials)

AbstractIn the hybrid Bayesian-frequentist approach to hypotheses tests, the power function, i.e. the probability of rejecting the null hypothesis, is a random variable and a pre-experimental evaluation of the study is commonly carried out through the so-called probability of success (PoS). PoS is usually defined as the expected value of the random power that is not necessarily a well-representative summary of the entire distribution. Here, we consider the main definitions of PoS and investigate the power related random variables that induce them. We provide general expressions for their cumulative distribution and probability density functions, as well as closed-form expressions when the test statistic is, at least asymptotically, normal. The analysis of such distributions highlights discrepancies in the main definitions of PoS, leading us to prefer the one based on the utility function of the test. We illustrate our idea through an example and an application to clinical trials, which is a framework where PoS is commonly employed.

Modelling Risk Dependencies in Insurance Using Survival Clayton Copula

Our aim in this paper is to show the use of survival Clayton copula as a suitable tool for modelling risk dependencies in insurance. A purpose-built simulation of an adequate upper tail dependence can be an important part of the aggregation of risks in an insurer’s internal models. The occurrence of extreme values of the aggregate random variable might have a very negative impact on the insurer when securing coverage of unexpected losses. The upper conditional quantile exceedance probability of the copula is a suitable indicator. In addition an analysis of its effect on the level of modelling of the risk scenario is available. This effect is measured using the Tail Value at Risk of the aggregate random variable. To simplify our description of the given principle for aggregating risks we will in this paper only consider the two-dimensional case. The programming language R was used to simulate the values of the joint distribution of the marginal random variables.

The reliability function of the case-hardened teeth of wheels of cylindrical gears

BACKGROUND: Methods for calculating the load capacity and reliability of gears, accepted as standardized both at the national and international levels, are based on the laws of distribution of random variables, considered as the only possible ones, which is not entirely true. As a result, the developed gears have either overestimated or underestimated reliability, which leads to their low competitiveness. Therefore, the development of methods for assessing the reliability of the case-hardened gears, taking into account the actual laws of distribution of random variables, remains relevant, as it will make it possible to design competitive transmissions. AIM: Improvement of the current approach to calculating the reliability function of case-hardened gears according to their strength performance criteria. METHODS: The improved approach to calculating the reliability function is based on improved classical methods for testing the load capacity of cylindrical gears for contact and bending stresses (GOST 21354-87 and ISO 6336). The methodology for calculating the reliability function according to the criterion of deep contact durability is based on the Lebedev-Pisarenko criterion, the formulae of which have been modified for application to gears by V.I. Korotkin. The implementation of the proposed methods was carried out in MathCAD software. RESULTS: The methods for calculating the reliability function of the case-hardened gears according to the criteria of contact and bending durability, which take into account the variable value of the misalignment in the gearing caused by the deformation of the shafts, the bearing rings and the housing (force misalignment), are proposed. In addition, the dependence of the calculation results of the reliability function of cylindrical gears on the method of setting the force misalignment in the gearing of teeth (constant or variable value) is shown. Validation of the improved approach was carried out using the data available in the scientific and technical literature on failures of the case-hardened gears. The scientific novelty of the research lies in the proposed method for calculating the reliability function of the case-hardened gears according to the criterion of deep contact durability, which performs the calculation under an unknown law of distribution of effective stresses using the Parzen-Rosenblatt method (the method is also used in methods for contact and bending durability criteria), as well as in taking into account the variable value of the force misalignment in the meshing. CONCLUSION: The practical significance of the research lies in the ability of probabilistic determination of the cause of gear failure according to six criteria (pitting, tooth breakage, tooth interior flank fracture of both the pinion and the wheel), which makes it possible to adjust the design, the manufacturing technology, and the operating requirements in order to ensure the required gear performance during its design.

Fisher information bounds and applications to SDEs with small noise

In this paper, we first establish general bounds on the Fisher information distance to the class of normal distributions of Malliavin differentiable random variables. We then study the rate of Fisher information convergence in the central limit theorem for the solution of small noise stochastic differential equations and its additive functionals. We also show that the convergence rate is of optimal order.

Limit distribution of partial transposition of block random matrices

It is well known that, under some assumptions, the limit distribution of block random matrices and their partial transposition converges to the distributions of random variables in some non-commutative probability spaces. Using free probability theory, we obtain the relation between the free cumulants of the corresponding random variables. As an application, we can derive a new family of co-completely positive and [Formula: see text]-positive maps by using the Wishart ensemble.

Determination of the probabilistic content of organic fertilizer fractions after exposure to arch-destroying and crushing agrotechnological devices

Abstract Achieving stable and uniform dosing of organic fertilizers during local application necessitates the crushing of large lumps to a specific size range. The process of breaking down these lumps follows a random distribution. By leveraging the integral function of the normal distribution of random variables intertwined with principles of probability theory, equations are formulated to ascertain the distribution of lump sizes within organic fertilizers. Furthermore, these equations enable the calculation of the likelihood of obtaining lumps of specified sizes post the interaction of the vault-breaker-shredder’s fingers with counter-cutting ribs on the fertilizer. This analytical approach provides a systematic method to optimize the sizing and distribution of organic fertilizer particles for enhanced application efficiency and effectiveness.

Stochastic modeling and optimization of discrete-time cold standby repairable systems with unreliable repair facility and retrial mechanism

This study develops novel reliability models for discrete-time cold standby repairable systems with unreliable repair facility and retrial mechanism. In view of the situation that the state of some engineering systems cannot be continuously monitored, discrete distribution is used to describe the distribution of random variables involved in this paper. For the case that multiple events can occur simultaneously in the discrete-time system model, the priority order of multiple events occurring simultaneously is defined. Two different models are proposed based on two types of priority rules. The retrial mechanism of the components is considered in the case that the failure information of the system components cannot be successfully transmitted to the repair facility. The preventive maintenance (PM) and repair after the breakdown of the repair facility are considered. The key reliability indices are derived by using the iterative algorithm of the difference equation and the generative function method. The algorithms for calculating system state probability, key reliability indices and cost–benefit ratio (CBR) are designed. To demonstrate the impact of each parameter on the system reliability indices, numerical results are provided. The optimal repair rate configuration scheme is chosen using the Particle Swarm Optimization (PSO) algorithm to achieve the lowest possible CBR. The impact of the system model with or without PM on stationary availability and CBR is shown. In addition, the impact of the number of system components on a series of system performance indices is analyzed, and the optimal number of components of the system corresponding to the minimum value of CBR is obtained.

Determination of the minimum number of periods for assessing the sustainable development indices of the EU countries using the methods of ordinal statistics

The subject matter of the article is the process of assessing the sustainable development indices of the European Union countries. The goal of the article is to develop a methodology for determining the number of periods for which it is necessary and sufficient to assess the sustainable development indices of states. The article results the following task: to develop a methodology for determining the law of distribution of random variables of sustainable development indices. Determination of the minimum number of periods for assessing the sustainable development indices of the European Union countries. Methods used: parametric and ordinal statistics. The following results are obtained: parametric and non-parametric methods of statistics, their advantages and disadvantages are considered. Various methods of estimating the distribution function for small samples, in particular the method of rectangular contributions and the method of uncertainty reduction are analysed. Particular attention is paid to the problem of changing the law of scattering of quality indicators when changing the conditions of technology. A graph-analytical method for identifying the law of distribution of random variables based on a small amount of statistical information is proposed. For this purpose, the theory of ordinal statistics was used. A step-by-step methodology for identifying the law of distribution of random variables using 10 ordered values has been developed. The mathematical expectations of ordinal statistics for three distribution laws are proposed. A methodology for determining the number of periods for assessing the indices of sustainable development of countries using ordinal statistics is developed. The study is based on the analysis of statistical data for the last ten years and their ordering in ascending order. The mathematical expectations of ordinal statistics are used to select appropriate distribution laws. Given the limited information available when working with small samples, the article proposes a methodology that allows obtaining the maximum amount of information from the available data. The developed approach makes it possible to take into account the uncertainty of the phenomenon under study and make informed decisions based on statistical analysis. Conclusions: based on the knowledge of the law of distribution, a methodology for determining the minimum number of periods for assessing the sustainable development indices of the European Union countries is proposed. Testing of the methodology on real numerical data has confirmed that the minimum number of periods is seven, provided that the distribution law follows the normal law.

Assessment of quality indicators in construction

Introduction. Ensuring proper quality of construction is one of the pressing challenges of our time. Consumers expect cheap products from developers in the form of commercial or residential real estate that meet all the required quality indicators. Most accidents in monolithic construction occur due to non-compliance with technological procedures at low temperatures. The purpose of the study is to develop a framework for assessing the quality of concrete work at subzero temperatures (during winter work) based on the results of studies of technological parameters of winter concreting, their impact on the quality and safety of erected monolithic structures of industrial and civil buildings (structures).Materials and methods. The study used the method of approximating distributions of random variables, expert estimates, the theory of multiple principles, and mathematical statistics. The experiments were carried out in the laboratories of the Don State Technical University, as well as in the conditions of actual concrete work on a construction site in winter. The work uses a correlation-regression analysis of the dependence of the organizational and technological causes of inconsistencies (factors) and quality indicators of concrete work production. Statistical information on structural damage and collapse of buildings during their operation was processed. To make organizational and technological decisions to improve the quality of concrete work, it is proposed to take the influence of defects on the load-bearing capacity of buildings as an indicator of the quality level.Results. Using practical and expert methods, the most significant reasons (factors) for the occurrence of inconsistencies in the level of quality of concrete work production have been established. Factors have been identified that have the greatest influence on the change in the load-bearing capacity and reliability of buildings and structures being erected when performing the main types of construction and installation work. The nomenclature of technological quality indicators during winter production of concrete work, their influence on quality substantiated, and methods for assessing the quality of winter concreting technology are developed. Quantitative indicators of the quality of construction of monolithic reinforced concrete structures were obtained.Discussion and conclusions. The practical value of the work lies in the development of a methodological tool for creating a regulatory framework for operational construction control of the production of monolithic and prefabricated monolithic construction and installation works under conditions of exposure to low temperatures, as well as for acceptance control of compliance of completed construction projects with quality requirements. Proposals are given for establishing regulatory tolerances, ways to improve the manufacturability of the equipment used, a new method for designing winter concreting technology, and a quality control method focused on ensuring structural safety have been developed. By developing the research, it is further possible to develop methods for calculating technology parameters taking into account technological variability, as well as establish new indicators to regulate the required level of quality and safety of buildings being built.

Quantum model regression for generating fuzzy numbers in adiabatic quantum computing

This paper addresses the problem of inherent fuzziness in real-world data, arising from uncertainties, complexities, and limitations of traditional statistical methods. We introduce a pioneering method leveraging Adiabatic Quantum Computing (AQC), based on an adiabatic quantum regression model, to generate fuzzy numbers—ideal tools owing to their extension of real numbers and robust arithmetic properties. This innovative approach overcomes challenges in Quantum Machine Learning (QML) and limitations of Noisy Intermediate-Scale Quantum (NISQ) computers, providing a solution superior to conventional statistical methods and addressing the crucial issue of exponential power increase. Unique to this work, we offer rare closed-form expressions to produce fuzzy numbers through AQC and emphasise the integration of random variables and fuzzy numbers to encapsulate uncertainty fully. We propose a novel transformation of the Cumulative Distribution Function (CDF) into triangular and trapezoidal fuzzy numbers using AQC, enabling a comprehensive description of probability distributions of random variables. Experimental results are detailed, demonstrating the significant applicability and breakthroughs of this research in addressing data fuzziness.

Weak approximation of Schrödinger–Föllmer diffusion

We consider the weak convergence of the Euler–Maruyama approximation for Schrödinger–Föllmer diffusions, which are solutions of Schrödinger bridge problems and can be used for sampling from given distributions. We show that the distribution of the terminal random variable of the time-discretized process weakly converges to the target one under mild regularity conditions.

Estimation of traditional numerical characteristics of lognormal distribution laws of a one-dimensional random variable in conditions of large volume statistical data

The efficiency of estimating the numerical characteristics of a family of the lognormal distribution law of a onedimensional random variable under conditions of large volumes of statistical data is considered. To circumvent the problem of large samples, methods of discretization the range of values of a random variable based on the formulas of Sturges, Brooks-Carruthers, Heinhold-Gaede and the formula proposed by the authors of this article are used. Data arrays have been generated that make it possible to evaluate the numerical characteristics of the laws of distribution of random variables, taking into account their discrete values. Based on the transformed data arrays, estimates of the mathematical expectation, standard deviation, skewness and kurtosis coefficients were calculated. Estimates of the numerical characteristics of the considered distribution laws under the conditions of a continuous and discrete random variable are compared for different volumes of initial statistical data. The effectiveness of methods for estimating the numerical characteristics of the family of the lognormal distribution law based on the initial statistical data and on the results of transformations of these data using known discretization formulas has been established. The reliability of the comparison of the effectiveness indicators of the studied methods was confirmed by using the Kolmogorov-Smirnov criterion. It is shown that the discretization formula proposed by the authors of this article is better and more effective compared to traditional methods.

Application of the interval approach to determine the exploitation time of pipelines

The fundamental problem from the point of view of pipeline exploitation in KGHM Polska Miedz S.A. is the very high overwearing of the pipes used for the transport of tailings, as well as determining the time of trouble-free operation of pipe system components. Failures involve significant financial outlays, severe restrictions on operation and in some cases even stopping operation. For this reason, it is vital to monitor the condition of the transport systems, as well as to determine the permissible service life of the pipe sections, after which segments at risk should be replaced or turned over in order to extend their further operation. This paper focuses on the application of interval numbers to assess the durability of piping systems. The calculations were made using classical interval numbers by using code written in INTLAB libraries. The correctness of the solutions obtained was verified using the Monte Carlo method, assuming a uniform distribution of random variables.

Consignment contract in a supply chain with price, demand and yield randomness: A distributionally robust approach

We investigate a supply chain consisting of a supplier and an e-commence platform who sign a consignment contract with uncertainty in price, demand, and yield. Due to the challenges of actually determining the joint distribution of random variables, we assume that only the partial distributional information is available for these three uncertainties. To address the issue of distributional ambiguity and to more accurately account for arbitrary correlations among these uncertainties, we introduce a moment-based distributionally robust optimization (DRO) approach that assumes only the first and second moments of these three uncertainties are known. Given an exogenous consignment rate, we derive the closed-form expression for the supplier’s optimal production decision and her worst-case expected profit. We study the impact of random price and random yield within the ambiguity set on the supplier’s production decision and the worst-case expected profit. From the e-commerce platform’s perspective, we further examine the contract design problem when the consignment rate is endogenous and analyze equilibrium solutions for both contracting parties. Our results show that the additional consideration of price or yield uncertainty within the ambiguity set does not always mitigate or enhance the degree of conservatism of the contracting parties, which is closely related to the correlation between any two types of uncertainty, as well as the degree of uncertainty itself.

An adaptive data-driven subspace polynomial dimensional decomposition for high-dimensional uncertainty quantification based on maximum entropy method and sparse Bayesian learning

Polynomial dimensional decomposition (PDD) is a surrogate method originated from the ANOVA (analysis of variance) decomposition, and has shown powerful performance in uncertainty quantification (UQ) accuracy and convergence recently. However, complex high-dimensional problems result in a large number of polynomial basis functions, leading to heavy computational burden, and the probability distributions of the input random variables are indispensable for PDD modeling and UQ, which may be unavailable in practical engineering. This study establishes an adaptive data-driven subspace PDD (ADDSPDD) for high-dimensional UQ, which employs two types of data for modeling the PDD basis function and the low-dimensional subspace directly, namely, the data of input random variables and the input-response samples. Firstly, we propose a data-driven zero-entropy criterion-based maximum entropy method for reconstructing the probability density functions (PDF) of input variables. Then, with the aid of the established PDFs, a data-driven subspace PDD (DDSPDD) is proposed based on the whitening transformation. To recover the subspace of the function of interest accurately and efficiently, we put forward an approximate active subspace method (AAS) based on the Taylor expansion under some mild premises. Finally, we integrate an adaptive learning algorithm into the DDSPDD framework based on the sparse Bayesian learning theory, obtaining our ADDSPDD; thus, the real subspace and the significant PDD basis functions can be identified with limited computational budget. We validate the proposed method by using four examples, and systematically compare four existing dimension-reduction methods with the AAS. Results show that the proposed framework is effective and the AAS is a good choice when the corresponding assumptions are satisfied.

Mueller matrix polarization interferometry of optically anisotropic architectonics of biological tissue object fields: the fundamental and applied aspects

The presented results concern the diagnostic application of vector-parameter polarization mapping and polarization-interference phase scanning applied to layered azimuths and ellipticity polarization maps obtained with the help of digital holographic reproduction for phase-inhomogeneous tissue layers in the female reproductive system. The differential diagnosis of the pathological changes in optically anisotropic polycrystalline tissue components, specifically “benign (fibroids) and precancerous (endometriosis),” was investigated. All studies within the representative groups of the histological samples were conducted using circularly polarized laser radiation, ensuring azimuthal invariance and polarization measurement reliability. Integral and layered maps and polarization azimuths and ellipticity random variable distribution histograms were obtained from the microscopic images of histological sections from the female reproductive system tumors. Systematized tables present statistical moments which characterize azimuths and ellipticity polarization maps for fibroids and endometriosis tissues. It is shown that the statistical parameters (diagnostic markers) are the most sensitive to benign and precancerous changes in the female reproductive system tissues. The results of information analysis, including the accuracy determination of the diagnostic vector-parameter polarization and polarization-interference methods for detecting and differentiating the samples of fibroids and precancerous endometriosis tissues in the female reproductive system, are presented.

New results on convergence in distribution of fuzzy random variables

We study some properties of convergence in distribution of fuzzy random variables in the metric dp, more precisely, if this type of convergence is preserved when we apply some functions between the space of fuzzy sets and other metric spaces. In particular, we show that this convergence is preserved when we take the closed convex hull, the maximum, the minimum, the product and the quotient. Moreover, it is preserved when we apply some functionals, such as the value or the ambiguity. Finally, we show some results regarding convergence in distribution with respect to the metric d∞.